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Glossary

Back

A 1 B vibration

Vertical vibration of an aircraft represented by the sum of the vibrations measured at the two sides of the

cockpit.

A 2 B vibration

Roll vibration of an aircraft represented by the difference in the vibrations measured at the two sides of

the cockpit.

A-weighted sound level

Sound pressure level that has been weighted according to A-weighting networks, commonly used in

measurements that relate to human response to noise (dBA). See Sound level.

Abnormality scaling

The process of scaling the level of abnormality in each vibration feature between 0 and 1 to signify the

degree of deviation from its normal value.

Absolute acceleration shock response spectrum

This is an SRS for acceleration. See Response spectrum (RS) and Shock response spectrum (SRS).

Absolute error

The absolute value of the difference between the exact solution and the computed solution, or between

the true value and the measured value.

Absolute limit

Conditions that could result in catastrophic failure.

Absorber

See Vibration absorber.

Absorption coefficient

Fraction of the incident energy absorbed by a sound surface.

Alternating current.

AC bridge

A bridge circuit that uses an AC excitation. It is more stable than a DC bridge.

G-1

AC motor

A motor that uses an AC excitation. Single-phase and multiphase, induction, and synchronous varieties

are available.

Acceleration

Rate of change of velocity with respect to time along a specified axis (m/sec2 or g) or about an axis

(rad/sec2). Second derivative of displacement. Gives both magnitude and direction (vector).

Accelerogram

A recording of the acceleration of the ground during an earthquake.

Accelerograph

An instrument (accelerometer) that records the acceleration of the ground during an earthquake.

Accelerometer

Sensor or transducer measuring acceleration; output (typically voltage) is proportional to acceleration.

A piezoelectric accelerometer contains one or more piezo-electric crystal elements, which produce

electric charges or voltages when stressed in tension, compression, or shear. Can be two or three axial.

The output can be in digital in some cases, with proper transducers.

Acceptability index

The percentage of times an adjustment set will result in the vibration satisfying the specification.

Accuracy

How well a measurement (reading) or a computed result represents the true (correct) value. Accuracy of

a value and accuracy of an instrument are related because the latter can determine the former.

Acoustic absorption coefficient

A ratio (fraction) expressing the ability of a material to absorb acoustic power.

Acoustic emission (AE)

Sound or ultrasound pulses generated during crack initiation or propagation in materials and coatings

from being subjected to stress.

Acoustic enclosure

A rigid box surrounding noise sources to provide noise isolation for machinery or equipment.

Acoustic holography

A method using phase and amplitude information over a closed surface to reconstruct a sound field in

the region outside the surface.

Acoustic impedance

Acoustic pressure acting on a surface divided by the volume (or particle) velocity at the surface. Assumes

a plane wave is falling perpendicular to the surface. May assume acoustic material of infinite depth on

which the wave is applied.

Acoustic power

The rate of acoustic energy flow emitted or radiated by a specified sound or noise source (W).

Acoustic resonance

Condition of peak (maximum) response of a fluid-borne system to a periodic driving pressure

oscillation. Typically, the frequency at which the system tends to oscillate naturally.

G-2 Vibration and Shock Handbook

Across wind response

Response perpendicular to the direction of wind.

Action plan

When measured values of noise or vibration violate the local or national regulatory limits, “action plans”

shall be developed for correcting of the situation.

Active bridge

A bridge circuit (with four bridges; e.g., Wheatstone bridge) that uses an operational amplifier (an active

device, which uses an external power source).

Active control

See Active vibration control.

Active damping

Controlled damping where the damping force is generated by an actuator (with an external power

source) in response to sensing (e.g., active suspensions and magnetorheology).

Active device

A device (hardware) that needs an external power source to operate.

Active filter

Uses an operational amplifier (an active device), as opposed to a passive filter, which uses only passive

devices.

Active gauge

The strain gauge mounted on the deforming member and forms a part of the bridge circuit. See Dummy

gauge. The bridge output is proportional to the change in the active gauge (automatically compensated

for ambient changes by the dummy gauges).

Active mount

An engine mount that uses actuators (with external power) for providing proper suspension and

isolation (force and motion).

Active resonator absorber

See Vibration absorber and Resonator absorber. A resonator absorber that utilizes an active device to

generate resonance (see Resonance) within the absorber subsection.

Active suspension

A suspension system (of a vehicle) that uses actuators (with external power) with feedback control for

providing proper suspension.

Active vibration control

A vibration control method in which an external force (utilizing external power) is applied to control

vibrations.

ADC

An analog-to-digital converter, which samples an analog signal at some frequency (sampling rate) and

converts the sampled data into a digital form.

Adiabatic process

A process that takes place with no heat transfer with the environment.

Glossary G-3

Adjustable suspension element

Typically includes variable rate damper and stiffness.

Aeroelastic model

The model that simulates the dynamic properties of buildings to capture the motion-dependent loads.

Air-borne sound

A sound that is transmitted from one structure to another.

Aliasing distortion

The error that results due to sampling (discretization) of an analog signal.

Along-wind response

Response in the direction of wind.

Amplitude

Peak magnitude of variation (displacement, velocity, acceleration, etc.) of each oscillation of vibration.

The maximum value of a sinusoidal quantity. A measure of floor vibration.

Amplitude-dependent damping

Nonlinear form of damping where its value depends on the amplitude but not frequency of motion,

e.g., fluid friction or aerodynamic damping or whose magnitude is proportional to the square of the

velocity; nonhysteretic internal damping.

Amplitude ratio

The ratio of forced vibration amplitude to the static amplitude.

Analog device

A device with analog circuitry (hardware) but no digital components. It deals with analog or timecontinuous

signals but not digital signals.

Analog-to-digital conversion

See ADC.

Anelasticity

Property of real solids responsible for internal friction (prefix “an” means “other than”). Anisotropic

damping where there are differences depending on directions relative to crystalline axes.

Angular frequency

Frequency of a periodic quantity, measured in radians per unit time. Cyclic frequency multiplied by 2p:

The usual symbol is v:

Angular velocity

The rate of change of angular position (i.e., orientation) of an object (radians per second [rad/sec],

revolutions per minute [rpm], etc.).

Annular clearance

The uniform gap between the rotor and stator of a turbine or compressor motor or some rotating

machine.

Antialiasing filter

A low-pass filter with its cut-off set near half the sampling rate of the filtered signal in order to minimize

the aliasing error.

G-4 Vibration and Shock Handbook

Appendages

Additional secondary devices. Additional nonstructural elements to the main structure to increase the

building’s function to the occupants.

Asymptotic stability

Stability representation where the response naturally decays to zero.

Asymptotic stability borderline

Maximum depth-of-cut below which chatter will not occur for all spindle speeds.

Attenuate

Decrease. To make small.

Attenuation of muffler

Decrease in sound power between two points in a muffler system (noise suppressor).

Autocorrelation

Correlation of a signal with itself, with a phase shift (reference time shift) expressed as a function of the

shift. Inverse Fourier transform of the power spectrum.

Autonomous system

Self-contained system with internal control (no external control). A system whose parameters are timeinvariant

(i.e., do not vary with time).

Autotuning

Automatic tuning (see tuning) of controller parameters to achieve desired performance (e.g., in vibration

control).

Average tuning iteration number (ATIN)

The average number of iterations taken for tuning an aircraft.

Average value

A representative value of a data set. The centriod of the data. The approximate point at which the sum of

moments of all the data values is zero. A first-order statistic of random data. Mean value. Expected value.

Mathematical expectation.

Axi-symmetric wave

A sound wave generated by a point dipole. When propagation of sound wave is described by polar

coordinates, its magnitude is independent of the angle about the given axis.

Axial

Along the shaft centerline. Along the axis.

Back e.m.f.

An electromotive force (e.m.f.) is a voltage generated in a conductor located in a magnetic field when

there is a relative motion (in an orthogonal direction). The back e.m.f. is the voltage generated in a

moving coil that is connected to a voltage source (like motor rotor windings) and located in a magnetic

field, where the generated e.m.f. opposes the applied voltage.

Backlash

Mechanical looseness in between meshed gear wheels due to unavoidable clearances between meshing

teeth.

Glossary G-5

Baffled space

The idealized boundary condition to model the sound field generated by a compact source embedded in

a plane large in size compared to wavelength, e.g., a radiation field generated by a circular vibrating

piston mounted flush with the surface.

Balance

A condition where the rotating centerline between bearings coincides with that which defines the center

of mass distribution. See Static balancing and Dynamic balancing.

Balance weight adjustments

Weights added to or removed from the rotor hub to tune vibrations through changes in blade mass.

Balance weights are measured in grams or ounces, with positive adjustments representing the addition of

weight.

Balancing

A procedure for adjusting the mass distribution of a rotor by adding or removing weight, with the

goal of achieving less vibration amplitude at rotational speed. See Static balancing and Dynamic

balancing.

Band-pass filter

A filter that allows through the signal components within a given range of frequencies, stopping

(filtering) those above and below that band.

Bandstop filter

A filter that suppresses a given range of frequencies, transmitting only those above and below that band.

Also called band-elimination filter. Also called a notch filter when the band is very small.

Bandwidth

Operating frequency range of a device. The difference between the upper- and lower-cut-off frequencies

at which the signal is attenuated by 3 dB. The difference between the limiting frequencies within which

performance of a device falls within specified limits with respect to some characteristic. A measure of

speed of response of a device.

Barkhausen effect

Jerkiness in the magnetization curve of a ferrous specimen.

Base isolation

See Isolation. An engineering technology that uncouples the motion of a structural system from the

ground motion by inserting an isolation layer under the structure.

Basis

Countable set of linearly independent functions or vectors that can represent a general function or a

vector space.

Bathtub curve

Curve (based on anecdotal and statistical data) describing the frequency of failures. Applicable to an

individual machine or population of machines of the same type.

Beam

A structural (vibrating) component that is long and slender and has some flexibility (bending or flexure).

Commonly modeled by Euler– Bernoulli equation or Timoshenko equation.

G-6 Vibration and Shock Handbook

Bearing

A support for the ends of a shaft, a rotating member, etc. May be dry or lubricated. Will introduce some

damping, but it should be minimized for proper operation. Linear bearing is one that provides a

guideway for rectilinear motion (rather than rotary motion).

Beats

The alternating and significant rise and fall of vibration amplitude caused by two sources vibrating at

nearly equal frequencies (or engines or rotating machines rotating at nearly equal speeds).

Bernoulli – Euler beam

A common analytical model (partial differential equation with respect to time and length coordinate)

used to analyze slender beams. Rotatory inertia and deformation due to shear are neglected in this model

(holds for slender beams), unlike the Timoshenko beam model.

Bicoherence

Second-order correlation measure between signals.

Biorthogonal wavelets

Symmetric wavelets with compact support created by two sets of wavelets; one set for decomposition and

one set for reconstruction.

Bisection method

A nonlinear root-finding method which repeatedly bisects the interval to find a root. For this method to

be successful, the initial interval has to be such that the function is positive at one endpoint and negative

at the other.

Blast mitigation

Physical defensive measures taken to reduce the damaging effects of explosions from devices

introduced by hostile parties. Includes both exterior hardening (wall, etc.) and internal measures

taken to improve the survival of the protected resources (e.g., using more stable chemicals in a

protected process).

Bode plot

The pair of plots of amplitude versus frequency and phase versus frequency. Usually, the amplitude is

given in dB (or in log scale) as a nondimensional magnification factor of a transfer-function magnitude.

The frequency is given in log scale.

Boom

A long, overhanging pole or beam like structure. As in a crane or a robot, a boom can be used to position

and manipulate objects at a remote location.

Boundary conditions (BCs)

The conditions (force, moment, motion, deformation, etc.) at the boundary locations of an object

(e.g., beam, plate, structure) used for solving the governing equations of the object. These are

conditions expressed with respect to spatial coordinates (space) rather than time. Also require initial

conditions (ICs) which are conditions at the initial time, for a complete solution of the system

equations.

Boundary layer

A thin stationary layer of liquid with a hypothetical boundary next to a solid surface where much of the

velocity changes (gradients) occur.

Glossary G-7

Bracing

Temporary supports of a structure using struts in vertical and lateral directions during seismic

rehabilitation of the structure.

Bridge circuit

An electric circuit with four arms having resistances or impedances, and input (source) and an output.

Where pure resistances are used, we have a Wheatstone bridge. Hydraulic circuits also may be represented

as bridge circuits, and analyzed using similar techniques.

Bridge balance

The condition (the relation between bridge resistances or impedances) such that the output of the bridge

circuit is zero (even when the source provides an excitation).

Bridge constant

The constant parameter relating a variation in a bridge arm (resistance or impedance) to the

corresponding bridge output, starting with a balanced bridge.

Broadband (wideband) process

A random process where most of the energy is in a wideband of frequency.

Broadband excitation

An excitation with a significant range of frequency (almost a white noise spectrum).

Buffer

A device provided at an isolation bearing which will be hit by the moving plate of an isolation bearing to

limit the bearing displacement during extreme earthquakes.

Bulk modulus

Change in volume divided by initial volume gives the bulk strain. The bulk stress (or pressure) divided by

the bulk strain is the bulk modulus.

Bump

Simple shock that is generally repeated many times when testing, e.g., half-sine, 10g; 16 msec, 3000

bumps (shock) per axis, 3 bumps per second.

Bump test

See Bump. An impact test. A test carried out on an object (equipment, structure, etc.) using a bump

impact.

Butterworth filter

A low-pass filter with many poles (i.e., high order) to provide good roll-off (i.e., sharp cut-off).

Calibration

Adjusting some parameters or conditions (using knobs, keys, computer interface, etc.) of a device so that

its performance (output or reading) is accurate with known engineering units.

Cantilever

A flexible beam fixed at one end (zero displacement and zero slope) and free at the other (zero bending

moment and zero shear force).

Carrier

A sufficiently high-frequency signal that can carry another (transient) signal of lower frequency for

maintaining the accuracy of the carried signal during signal transmission and analysis.

G-8 Vibration and Shock Handbook

Carrier frequency

Frequency of the carrier signal.

Carrier signal

See Carrier.

Cartesian coordinates

Orthogonal (perpendicular) coordinates in a rectangular coordinate frame with a common origin.

Cascade connection

Connection in series of two or more devices.

Cascade plots

Successive spectra plotted with respect to time and displayed in a three-dimensional manner. Also known

as Waterfall plots.

Catastrophic failure

Sudden and complete failure of a machine. Cannot be analyzed by conventional methods of nonlinear

system analysis.

Causality

Cause – effect behavior of a dynamic system. The numerator order (or the input order) cannot be greater

than the denominator order (or the output order) of the system transfer function. This determines the

physical realizability of the system.

Center of gravity

The point representing the average position of weight in a body. The point of intersection of orthogonal

axes, relative to which the first moment of weight is zero. Typically the same as the center of mass.

Denoted by COG or CG.

Center of mass

The point representing the average position of matter (mass) in a body. The point of intersection of

orthogonal axes, relative to which the first moment of mass is zero. Typically the same as the center of

gravity, denoted by COG or CG.

Center of stiffness

The point of intersection of orthogonal axes relative to which the first moment of stiffness is zero.

Centered difference

Finite difference approximation that uses the Taylor series expansions on either side of the point in

question.

Centrifugal acceleration

Acceleration of a rotating mass (e.g., attached to a string or arm) directed towards the center rotation.

Equal to v2r where v is the angular speed of rotation and r is the radius of rotation.

Centrifugal compressor

A compressor (a machine in which air or gas is compressed) by the mechanical action of rotating vanes or

impellers. The flow is in a radial direction through the impeller, which imparts velocity (kinetic energy)

to the fluid. Passing through a diffuser section at exit, the kinetic energy is converted into pressure

(potential energy).

Glossary G-9

Centrifugal pump

See Centrifugal compressor. As the impeller of the pump (enclosed in a casing or volute with an inlet and

a discharge connection), kinetic energy is imparted to the fluid by the rotating impeller and is converted

into pressure at the outlet.

Centripetal acceleration

See Centrifugal acceleration (synonymous).

Centroid

Same as the center of mass or center of gravity for practical purposes.

Cepstra

Plural of cepstrum. Equivalent to spectra, but in the “quefrency domain” rather than the frequency

domain.

Cepstrum

The inverse Fourier spectrum of the logarithm of the Fourier spectrum of a signal.

Change limit

Warning levels that provide early warning well in advance of the absolute limit.

Channel

A signal line in a piece of equipment or hardware. Both input channels and output channels are

important. The capability and complexity increase as the number of channels increases.

Chaos

Deterministic form involving the “butterfly effect” (work of E. Lorenz).

Characteristic equation

See Characteristic polynomial.

Characteristic polynomial

The polynomial, when set to zero (characteristic equation) and solved, gives the eigenvalues

(or poles or characteristic roots) of a dynamic system. The denominator polynomial of

the transfer function of a dynamic system. The roots are complex in general. The imaginary

parts give the natural frequencies. The real parts give the reciprocal of time constants or damping

level.

Charge amplifier

An electronic amplifier used to convert the high-output impedance of a device (e.g., piezoelectric

accelerometer) into low-output impedance so that it is compatible with common signal acquisition and

readout devices (processors, controllers, displays, recorders, etc.).

Chatter

An instability in a machine tool, which occurs at high frequencies, causing poor

machining quality, increasing tool wear, generating noise, and causing other problems to the

machine.

Chatter frequency

Dominant frequency of vibration during the occurrence of chatter (unstable vibration in a machine

tool).

G-10 Vibration and Shock Handbook

Chatter marks

Distinctive undulating patterns left on a part after machining that are indicative of chatter.

Cholesky decomposition

A symmetric positive definite matrix may be written as the product of a lower triangular matrix and its

transpose (an upper triangular matrix): A ¼ U tU:

Circle fit

Fitting vibration test data to a transfer function that approximates to a circle on the complex plane

(horizontal real axis and vertical imaginary axis). True for a single degree-of-freedom (single-DoF) in

relation to some specific transfer functions (e.g., mobility) with some types of damping. Parameters of

the system (damping, natural frequency) can be extracted from the fitted circle. Used in experimental

modal analysis or modal testing.

Cofferdam

A temporary watertight enclosure that is pumped dry to expose the bottom of a body of water so that

construction (e.g., of piers) may be undertaken.

COG

Centre of gravity (or center of mass). See their definitions.

Coherence

First-order correlation measure (normalized) between different signals, expressed in the frequency

domain (as a function of frequency). It has a magnitude and a phase.

Coincidence effect

A form of acoustic resonance phenomena where the transmission loss decreases rapidly when the

acoustically forced bending wave speed is equal to the free bending wave speed (flexural natural

frequency) in a structure (plate).

Cold welding

Bonding between two parts of seismic isolation bearings (which are supposed to move relatively to each

other during earthquake attacks) because of long durations of contact.

Color centers

Defects introduced into a crystal such as lithium fluoride by ionizing radiation.

Combined adjustments

The net adjustments to opposite blade-pairs in a four-bladed aircraft.

Common-mode rejection

Improvements in signal-to-noise ratio (SNR) by the reduction of noise signals that are common to both

inputs of a differential amplifier.

Commutator

A device that provides power to a set of coil segments (circuits) in a rotating electric machine (motor).

Analogous methods may be used in linear actuators (which perform rectilinear motion rather than rotary

motion). A commutator with a slip-ring with split segments connected to the circuit segments, and

brushes (carbon blocks) in contact to provide the power, is called a “slip-ring brush commutator.”

Alternatively, electronic commutation may be used where the power is switched into to different circuit

segments by electronic means.

Glossary G-11

Compact source condition

When the scale of the source region is small (compared with the wavelength of generated sound wave),

the compact source condition is mathematically valid for modeling the sound field. See Monopole and

Dipole.

Compensating element

A resistor or a coil in a circuit (e.g., eddy-current sensor, bridge circuit, potentiometer circuit), which is

used to compensate for environmental changes (e.g., change in temperature).

Complex modes

In some vibrating systems with some complex types of damping, the mode shapes (eigenvectors)

obtained analytically (see mode shapes) can be complex (as opposed to real). Strictly then, these mode

shapes are not realizable in practice. In fact, some distinct shapes will appear (similar to real modes) but

with some changes (e.g., movements of nodes, small changes in phase of the moving elements) at the

corresponding natural frequencies.

Complex propagation constant

This constant g has two parts; the real part, d; called the attenuation constant in decibels per meter

(dB/m), and the imaginary part, b; called the phase constant in radians per meter (rad/m).

Compound pendulum

See Pendulum. A pendulum having more than one degree of freedom (e.g., two swinging degrees and one

twisting degree).

Compressor circuit

In controlling a vibration shaker, this circuit ensures that the drive signal to the shaker satisfies the

required spectrum of the test signal that needs to be applied to the test object. It compresses the frequency

components that exceed the specification.

Condition-based maintenance

A strategy where maintenance is performed after assessing the actual condition of the

machinery (rather than scheduling it at fixed intervals) in order to optimally schedule

maintenance, thereby achieving maximum production and still avoiding unexpected catastrophic

failures.

Condition monitoring

Monitoring the operation of a machine or other dynamic system, for the purpose of checking proper

operation and possible faults or malfunction. Vibration signal analysis is one approach to condition

monitoring.

Confined concrete

Concrete confined by transverse reinforcement, commonly using closely spaced steel spirals or hoops to

increase the ductility.

Consistent mass

Representation of the mass of a system based on the virtual work principle and that results in a

nondiagonal mass matrix.

Constant percentage bandwidth

Frequency bandwidths over the entire frequency range that remain a constant percentage of the

frequency being monitored.

G-12 Vibration and Shock Handbook

Consumer product noise

Noise perceived by a user or bystander of a noise generating product over which the individual has some

control. Excludes occupational noise and environmental noise.

Contrast enhancement

Adaptation of the weights associated with features classified as normal or abnormal to enhance the

contrast.

Continuous monitoring

Constant or very frequent data collection and analysis, usually with a permanently installed monitoring

system.

Continuous wavelet transform

Wavelet transform on continuous ranges of scale-shift parameters.

Controllability

The ability of a controller to drive the response of the dynamic system, which it controls, to any desired

value in a finite time, using a finite input (excitation).

Convolution integral

Integral of the product of two functions (usually of time), where one function is the shifted complement

of the other (i.e., one is a function of t and the other is a function of t 2 t; and the integration is done

over t). If one function is the system input and the other is the response of the system to a unit impulse,

then the convolution integral gives the system response to the given input. The Fourier transform of a

convolution integral is the product of the Fourier spectra of the individual time functions. In general,

a product in the frequency (or Laplace) domain is a convolution integral in the time domain, and

vice versa.

Correlation function

The correlation function is the mathematical expectation of a random function with another random

function in the time domain. It is a measure of how two random functions correlate with each other over

various time differences.

Coulomb damping

Friction force that is constant in magnitude, and acts on a body in the opposite direction to the velocity

of the body relative to the surface in contact.

Coupling loss factor

The fraction of the original vibrating energy between two subsystems that is lost when interconnected.

Coverage index

The reach of each sensor (accelerometer) location in reflecting faulty components. It is defined in terms

of the structural influences that represent the proximity of each sensor (accelerometer) to various system

(gearbox) components.

Creep

Slow dimensional changes of a specimen due to structural defects. They progress through three

distinctive phases called primary, secondary, and tertiary.

Crest factor

The ratio

􀀄

Peak Level

RMS Level

􀀅

of a waveform. It can indicate early stages of rolling element (bearing) failure.

Glossary G-13

Critical damping

The minimum value of damping that produces a nonoscillatory decaying motion of a system under free

vibration. Value of damping that provides most rapid response to a step function without overshoot.

When this occurs, damping ratio ¼ 1.

Critical damping coefficient

Damping coefficient corresponding to critical damping. See Damping coefficient and Critical damping.

This occurs when ðc=2mÞ2 ¼ k =m; where k ¼ spring constant, m ¼ mass, and c ¼ damping coefficient or

damping constant. At this value, damping ratio ¼ 1.

Critical frequency

The minimum frequency at which the coincidence effect occurs when the propagation speed of a bending

wave in a plate equals the speed of sound in air.

Critical speed

The rotational speed that is equal to a natural frequency of bending of a rotor– shaft – bearing system

(rotor dynamic resonant frequency). It can cause large amplitude vibration or whirling with only slight

excitation.

Cross sensitivity

The sensitivity of a device (e.g., sensor) in a direction orthogonal (perpendicular) to the direction of

direct sensitivity. Normally, the cross sensitivity of a sensor has to be minimized and the direct sensitivity

maximized.

Crust

The outermost major layer of the Earth, ranging from about 10 to 65 km in thickness worldwide.

The uppermost 15 to 35 km of crust is brittle enough to produce earthquakes.

Cumulative modification set

The total number of tuning iterations performed in the field on an aircraft.

Curve-fitting

Fitting a set of data to an analytical curve, as accurately as possible. See Circle fit, which is a single-DoF fit.

Multi-DoF curve-fitting is used for complex systems with closely-spaced modes. Used in experimental

modal analysis and modal testing.

Cut-off frequency

The frequency of a filter beyond which the signal entering it is cut-off (filtered out).

Cycle

The full range of states or values through which a periodic (cyclic) phenomenon or function passes

before completely repeating itself. The complete sequence of values of a periodic quantity that occur

during one period. For a periodic waveform (signal), this corresponds to one period (or 1/frequency).

Cyclic frequency

Frequency of a periodic variation in cycles per unit time (Units: Hertz or Hz, or cycles or cycles). Typical

symbol is f. See also Angular frequency.

D’Alembert’s principle

An equivalent version of Newton’s Second Law where the acceleration is multiplied by mass, and the

resulting “inertia force” is applied in the direction opposite to that of the acceleration. The dynamic

systems become an equivalent static system in this manner.

G-14 Vibration and Shock Handbook

DAC

Digital-to-analog conversion. Digital data from a digital device (e.g., computer) are interpolated and

converted by a DAC into an analog signal for use with analog instruments.

Damage boundary curve

For a given acceleration, curve giving the critical velocity change and, for a given velocity change, the

critical maximum shock acceleration that leads to a damage on the material (deformation, fracture,

faulty operation after the shock, etc.).

Damped natural frequency

The natural frequency of free vibration of a damped linear system.

Damper

A device that dissipates mechanical energy and progressively diminishes the response of a mechanical

system (in free motion). In the case of linear viscous damping, this device is sometimes called a dashpot.

The resulting damping force is the product of the damping coefficient and the relative velocity of the two

components of the damper.

Damping

The dissipation of energy of a mechanical system, which progressively diminishes (dampens) the

magnitude of response (or the amplitude of vibration) of the system in free motion. The damping

process converts and dissipates mechanical energy into other forms of energy such as thermal energy

(heat transfer) and noise.

Damping capacity

Energy loss per cycle of the motion.

Damping coefficient

The constant by which the relative velocity of a body is multiplied to yield the viscous damping force

acting on the body.

Damping constant

The same as damping coefficient.

Damping ratio

Ratio of actual damping coefficient to the critical damping coefficient in a system with viscous damping.

Expressed mathematically, the damping ratio z ¼ c=cc: If less than 1, the system will be underdamped and

will exhibit oscillations when disturbed or excited. If larger than 1, the system will be overdamped and the

system response will diminish and die out with no oscillations.

Damping redshift

Change in frequency of a harmonic oscillator due to damping. Predicted by the viscous damping model,

it is virtually immeasurable because of the small size of the shift.

Dashpot

Same as “damper” but assumes a linear viscous damping model and generally has three orthogonal

damping forces.

Daubechies wavelets

A family of orthogonal wavelets with compact support and a specified number of vanishing

moments.

Glossary G-15

Direct current.

DC bridge

A bridge circuit using a DC excitation. Compare with an AC bridge.

DC motor

An electric motor that uses a DC excitation. Requires a commutator for operation. Commonly used as a

servomotor, where motion sensing and feedback control are employed for accurate generation of motion

trajectories.

DC tachometer

A speed sensor that uses the principle of an electric generator (electromagnetic induction). The excitation

is DC. The output DC voltage is proportional to the measured speed.

Detuning

Deviation of a passive vibration control system (see passive vibration control) from its original design

due to structural parameters changes, change in the excitation frequency, or change in nature of

disturbance.

Deflection

Movement (or deformation) of a structure or parts of a structure under applied loads. In the case of an

elastic body or spring, the distance it moves when subjected to a static or dynamic force.

Degrees of freedom

Denoted by DoF or dof or df. Minimum number of independent coordinates necessary to determine

completely the state (position and orientation) of the considered system. This definition assumes a

holonomic system. For a general (nonholonomic) system, the definition must use “incremental”

coordinates and corresponding “incremental” displacements. Often used to classify mechanical

structures, such as robotic arms. In the finite element analysis, DoF may be associated with nodes or

elements. Three-dimensional shell and beam elements have six DoF per node (three translations and

three rotations).

Delayed resonator (DR) absorber

A form of a resonator absorber (see resonator absorber) to optimally utilize the effect of time-delay in the

processed signal.

Delta function

Dirac delta function. A unit impulse function. A function whose value is infinity at one point and zero

elsewhere, with the integral of the function equal to 1.

Demodulation

Recovering the original signal from a modulated signal by removing the carrier signal. A common

method is to multiply (by digital or analog means) the modulated signal by the carrier signal and then

low-pass filter the result.

Depth-of-cut selection

Technique where the depth-of-cut is chosen to avoid or suppress chatter in a machine tool. See Chatter.

Design (basis) earthquake

Denoted by DBE. The earthquake (as defined by various parameters, such as peak ground acceleration or

PGA, response spectra, etc.) for which the structure will be, or was, designed.

G-16 Vibration and Shock Handbook

Desorption

Release of gas molecules from a solid undergoing evacuation.

Determinacy of machine structure

A degree at which a physical structure complies with the law of cause and effect (see causality). This

allows the system dynamic behavior to be modeled and analyzed mathematically, and the modeled

system to be physically realizable.

Detonation

An explosive reaction that consists of the propagation of a shock wave through the explosive,

accompanied by a chemical reaction that furnishes energy to sustain the shock propagation in a stable

manner, with gaseous formation and pressure expansion following shortly thereafter.

DFT

Discrete Fourier transform or digital Fourier transform. A discrete version of the Fourier integral for use

in numerical computation. The fast Fourier transform (FFT) uses a much more efficient algorithm for

this computation.

Dielectric constant

A parameter of a capacitive material indicating the degree of its ability to store electric charges. Also

known as permittivity.

Differential transformer

See LVDT and RVDT. A transformer with a movable ferromagnetic element in its core. Serves as a

noncontact displacement sensor.

Difference equation

A discrete version of a differential equation. Used in computer simulation of dynamic systems. In the

nonhomogeneous case (i.e., when there is an input function) there is an equivalent z-transform transfer

function (just like a nonhomogeneous differential equation having an equivalent Laplace transfer

function or a frequency response function).

Diffraction parameter

This indicates the importance of the diffraction force. A larger parameter value means that the diffraction

force is more important than the drag or inertia forces.

Diffuser

A component in a centrifugal pump adjacent to the impeller exit, which has multiple passages of

increasing area for converting velocity (kinetic energy) to pressure (potential energy).

Digital filter

In contrast to an analog filter, it uses a digital processor and a filter algorithm to perform filtering on

digital data.

Digital Fourier analysis

See DFT and FFT. Fourier analysis using digital data. Converts time-domain information into frequencydomain

(spectral) information. Can determine Fourier spectra, transfer functions, power spectra

densities, coherence functions, etc. through this analysis.

Digital oscilloscope

An instrument that has the usual oscilloscope capabilities (acquisition and display of one or more

signals) together with a variety of other capabilities such as extraction of signal parameters (mean, time

constants, periods) and even FFT and spectral analysis.

Glossary G-17

Digital signal analyzer

Signal analyzer. Digital spectrum analyzer. A hardware digital device that carries out Fourier analysis and

related operations on signals at very high speed (almost real time). Faster than a software-based analyzer.

Digital-to-analog conversion

See DAC.

Digital transducer

A transducer that can sense a signal and present it as a pulse sequence (pulse count or pulse rate) or

digital data. It is generally very accurate, but necessarily introduces a quantization error.

Dilation

The squeezing or stretching of the mother wavelet in the time domain in order to analyze the signal in

different scales.

Dipole

See Monopole. The second basic sound source where a pair of monopoles, close together, opposite in

sign, and equal in magnitude are located along one axis. The sound sources like a compact vibrating

body, with a characteristic axis or a point force acting in the medium that may be modeled as a dipole.

Dirac delta function

See Delta function.

Direct sensitivity

See Cross sensitivity.

Directivity

The dependency of a sound field on the angular position. Although the monopole sound field has

no preferred direction, the dipole and the multipole sound fields have a particular dependency on

the angular position. The dependency varies with the condition of the compactness of the sound

source.

Discrete Fourier transform

See DFT. A mathematical transformation used to transform a discrete time signal (sampled data) into a

discrete set of spectral values equally spaced in the discrete frequency domain. It can be used to

numerically (computationally) determine the spectrum of any general (e.g., nonperiodic) time signal.

Discrete wavelet transform

Wavelet transform on a discrete set of scale-shift parameters.

Dislocation

Disruption to the order of an ideal crystal, e.g., by an extra plane of atoms inserted partway into the

crystal to form an edge dislocation. Edge and screw dislocations are described by means of the Burgers

vector.

Displacement

Change of position of a body, usually measured from the mean position or position of rest. Usually

applies to the uniaxial case. In the case of angular motions, it may be termed a “rotation.” In a general

mutidimensional case, this is a vector quantity that specifies the change of a body or particle, and is

usually measured from the mean position or position of rest. In general, it can be represented as a

translation vector, a rotation vector, or both.

G-18 Vibration and Shock Handbook

Dissipative muffler

A muffler where the acoustic performance is determined mainly by the presence of sound absorbing

(e.g., flow-resistive) material. See Muffler and Reactive muffler.

Distinct eigenvalues

The case where there are no repeated eigenvalues. All the eigenvalues are unequal.

Distributed-parameter system

A system where at least one member is continuous (e.g., beam, plate), needing one or more spatial

coordinates to analytically represent (model) it. A partial differential equation in time and space is

needed. Compare with lumped-parameter system.

Doppler effect

When the wave source moves towards the receiver, the received frequency increases. When it moves away

from the receiver, the received frequency decreases. True for both sound and light.

Drag force

Force on a body in the direction of wind or any other fluid (liquid or gas).

Drift

The undesirable variation of the output of an instrument when the input conditions are kept constant.

Ductile detailing

Special requirements for civil engineering structures, such as close spacing of lateral reinforcement to

attain confinement of a concrete core in reinforced concrete and masonry, appropriate relative

dimensioning of beams and columns, hooks on main beam reinforcement within the column, etc.

Ductile structure

A structure that is able to undergo a repeated large displacement after yielding, without significant

degradation of yield strength.

Ductility factor

The ratio of the total displacement (elastic plus inelastic) to the elastic (i.e., yield) displacement.

Duffing oscillator

Classic nonlinear elastic (mechanical) oscillator characterized by a double-well potential.

Dummy gauge

A strain gage in a bridge that is not active. Used for bridge completion and compensation (for

temperature changes, etc.). Active gauge is the strained gauge, which is mounted on the deforming

member.

Dunkerley’s formula

A formula that provides a lower bound for the fundamental (first or lowest) natural frequency of a

mechanical (vibrating) system.

Duration of shock pulse

The time required for the acceleration of a shock pulse to rise from some specified fraction of the

maximum amplitude (peak) of the pulse and to decay to this value (see Shock pulse).

Dwell frequency

The frequency at which a “sine-dwell” test is carried out.

Glossary G-19

Dyadic

Follows a geometric sequence of ratio 2.

Dynamic absorber

See Absorber or Vibration absorber.

Dynamic balancing

See Balancing. Balancing achieved on two parallel planes of vibration perpendicular to the axis of

rotation.

Dynamic magnification factor

The ratio of the maximum displacement of a single-DoF system excited by a harmonic force to the

deflection that would result if a force of that magnitude was applied statically.

Eccentric mass

An equivalent point mass in a rotary device, located at a radial offset (eccentricity) from the axis of

rotation such that the corresponding centrifugal force is equal to the imbalance force of the original

device. The mass that needs to be placed on a rotor (eccentrically) in order to balance it.

Eccentricity

The distance between the geometric center and the center of mass of a rotor.

Eddy current damping

Damping due to eddy currents generated in a conducting surface and associated dissipation of the energy

through electrical resistance to eddy currents.

Eddy current sensor

Eddy currents are generated on the surface of a conducting medium when a high frequency (e.g., radio

frequency in the MHz level) magnetic field is applied to it. An eddy current sensor is a proximity sensor

where the probe emits a high-frequency magnetic field towards the monitored object, which acts as a thin

conducting layer. The inductance change caused by the eddy currents in the probe coil is a measure of the

intensity of eddy currents and the distance (proximity) of the object. The inductance change can be

measured using an inductance bridge circuit.

Effective isolation period

A linearized equivalent isolation period of a nonlinear isolation system.

Eigenfrequencies

Natural frequencies determined by an eigenvalue analysis.

Eigenfunctions

Mode-shape functions, in a general sense, including complex modes.

Eigenvalue analysis

Analysis carried out to obtain the roots of the characteristic equation of the matrix formulation of an

engineering problem. In system dynamics and vibration problems, the imaginary parts of the eigenvalues

are the natural frequencies of the system. The reciprocal of the imaginary parts are the time constants of

the system. Imaginary parts divided by the corresponding undamped natural frequencies provide the

modal damping ratios. The analysis also provides eigenvectors, which give the mode shape information.

The analysis that provides both natural frequencies and mode shapes is called modal analysis. See

Eigenvalue problem. In stress analysis, the eigenvalues and eigenvectors give the values and the directions

G-20 Vibration and Shock Handbook

of the principal stresses or principal strains. In other engineering problems, there are eigenvalue problems

with corresponding interpretations.

Eigenvalues

The equation of motion for undamped free vibrations of a system with n DoF can be represented in

matrix notation as the product of the characteristic matrix and the vector representing the

corresponding displacement coordinates. For nontrivial solutions, the determinant of the characteristic

matrix is set equal to zero, yielding a characteristic equation. The roots of the characteristic equation of

the system are called eigenvalues. See Eigenvalue analysis for the interpretation of eigenvalues.

Eigenvalue problem

The mathematical formulation of the dynamics of a system as a problem of determining eigenvalues and

eigenvectors of a matrix. This matrix is M21K; where M is the mass matrix and K is the stiffness matrix in

the undamped case, giving natural frequencies, time constants and modal damping ratios. See Eigenvalue

analysis. In stress analysis, the eigenvalue analysis gives the values and directions of the principal stresses

or principal strains. In other engineering problems, there are eigenvalue problems with corresponding

interpretations.

Eigenvectors

Many engineering problems can be cast as the solution of eigenvalues and eigenvectors of a characteristic

matrix. In vibrations problems, this matrix is M21K; where M is the mass matrix and K is the stiffness

matrix, in the undamped case. The corresponding eigenvectors give mode shape vectors (see Eigenvalue

analysis). In stress analysis, the eigenvectors give the direction of the principal stresses or strains.

Elastomer

A generic term that encompasses all types of rubber, natural or synthetic.

Elastomeric bearing

One type of base isolators, which are usually made of rubber or soft materials.

Elastomeric coupling

A flexible shaft coupling, which employs nonmetallic elastomeric elements for transmitting power, and

accommodates shaft misalignment from one rotating shaft to another.

Electricite de France (EDF)

One type of seismic isolators, which consists of an elastomeric component and a sliding bearings in

series.

Electrical impedance

The transfer function (voltage or current), in the frequency domain.

Electro-rheological (ER) fluid

Materials that undergo significant instantaneous reversible changes in characteristics when subjected to

electric field.

Element

In finite element analysis, this is a small portion or a sub domain of a system having small but finite

dimensions and bounded by straight or curved boundaries. Elements are connected to each other via a

set of connection points called nodes. The minimum number of nodes for an element would be the

corner nodes defining its shape.

Glossary G-21

EMA

See Experimental modal analysis.

e.m.f.

Electromotive force. See Back e.m.f.

Emission

Emitting or radiation of sound energy or power from a noise source. Noise emission is controlled

through noise government legislation.

Energy equivalence

A simplified analysis method of a dynamic system where a simplified (e.g., lumped-parameter) model (of

reduced number of degrees of freedom) is established such that its energy (kinetic energy and potential

energy) is equal to that of the original system. Information about the original system can be derived

(approximately) by analyzing the equivalent system.

Ensemble average

The average of ensembles or samples (records) of possible outcomes. This is different from averaging a

single record or signal (temporal average).

Enveloped spectrum

The envelope of a set of spectra obtained by different records of a vibration signal.

Environmental noise

Noise received by individuals indoors and outdoors, caused by noise sources that are controlled by

others.

Environmental quality standard

World Health Organization (WHO) is the world body that recommends this regulatory standard of noise

exposure. E.g., see Guideline for Community Noise published by WHO in 2000.

Epicenter

The projection on the surface of the earth directly above the hypocenter of an earthquake.

Equalization

In vibration testing, the spectrum of the actual excitation signal applied to the test object has to be

(approximately) equal to the specified test spectrum. The process of achieving this by examining small

frequency band of the actual spectrum and changing the spectral values (by adjusting the gains) is called

equalization.

Equalizer

A device that achieves equalization in a dynamic test system.

Equipment interaction

Feedback effect between the motions of the structure and attached equipment.

Equipment tuning

When the natural frequency of a piece of equipment is such that undesirable resonances occur in

vibratory response and equipment tuning may be necessary to avoid this condition. One option is to tune

a vibration absorber (dynamic absorber) to this frequency.

G-22 Vibration and Shock Handbook

Equivalent static acceleration

Steadily applied acceleration producing the same deformation of the structure as that resulting from the

action of a shock. Also termed pseudoacceleration.

Equivalent mass

See Energy equivalence. The mass of a single-DoF model, in the “energy equivalence” method.

Equivalent stiffness

See Energy equivalence. The stiffness of a single-DoF model, in the “energy equivalence” method.

Equivalent viscous damping

The value of viscous damping assumed to analyze a vibratory motion, such that the dissipation of energy

per cycle is the same for the viscous and the actual cases.

Ergodic hypothesis

For a stationary process, the assumption that the ensemble average can be approximated by the time

(temporal) average.

Ergodic process

A stationary process where the ensemble average can be approximated by the time (temporal) average.

This assumption is known as the ergodic hypothesis.

Euclidean norm

Square root of sum of squares (SRSS) of a set of independent values.

Euclidean space

See Cartesian coordinates. Space represented by a general Cartesian coordinate system with n coordinate

axes. Cannot be represented in a physical space when n . 3:

Euler angles

A set of orthogonal rotation angles normally used to describe orientation of bodies in three-dimensional

(3-D) space.

Euler – Bernoulli beam

See Bernoulli – Euler beam.

Euler’s method

First-order Runge – Kutta method of numerical integration.

Excel

Microsoftw spreadsheet software.

Excitation

Force (or even a motion such as support motion) applied to a mechanical system. This is the input to the

system. The resulting response is the output.

Exciter

Shaker. The device that generates and applies an excitation to the test object in dynamic (vibration)

testing.

Glossary G-23

Expansion joint

A connection between two or more seismically-isolated structures for the passage of occupants from one

building to another, and also to let the adjacent buildings move relatively to each other during a seismic

excitation.

Expected value

Also known as “mathematical expectation” of a random function. It is the integral of that function

weighted by its probability density function. If the function is the random variable X; then the expected

value gives the mean value ðX􀀊 Þ; which is the first moment statistic. If it is X2; then we get the second

moment statistic. If it is ðX 2 X􀀊 Þ2; then we get the variance, the square root of which is the standard

deviation of a random parameter. Other higher-order statistics of X can be derived similarly.

Experimental modal analysis (EMA)

A method of determining a model of a dynamic (vibrating system) through testing and modal parameter

extraction. A signal is applied to the system using an exciter. Various responses are acquired and analyzed

(usually by Fourier analysis methods in the frequency domain) to determine the natural frequencies,

damping ratios, and mode shapes of the system, and using these an analytical model (mass matrix,

stiffness matrix, and a damping matrix) is computed for the system.

Explosion interaction

Two structures collide during an explosion.

Extrinsic

Opposite of intrinsic. A method where the device is used as the medium (or vehicle) in transmitting the

objective (e.g., signal), rather than generating the objective. Some types of fiber-optic and ultrasound

sensors use the extrinsic approach as well as the intrinsic approach.

Fast Fourier Transform (FFT)

A fast and efficient computer algorithm to calculate the discrete Fourier transform (DFT). The input data

are samples of a time waveform from (e.g., digitized voltage measurements). The result is a display of

discrete spectral points giving amplitude versus frequency, and phase versus frequency. The frequency

interval is so small that the spectrum appears continuous in the display or printout.

Fatigue

Failure of a structural member exposed to a repeated load whose amplitude is well under the member’s

strength.

Fault

A zone of the Earth’s crust within which the two sides have moved. Faults may be hundreds of miles long,

from 1 to over 100 miles deep, and not readily apparent on the ground surface.

Fault detection

The process of determining if the overall machinery is healthy or a fault has occurred. Fault detection may

performed by such methods as oil analysis and vibration monitoring. Data gathering, data analysis (e.g.,

FFT, cepstrum, wavelet analysis), comparison to standards, comparison to limits set in-plant for specific

equipment, etc., may be involved when detecting whether machinery is developing a faulty condition.

Fault diagnosis

The process of identifying the type, source, and severity of a failure (e.g., a faulty gearbox) for a detected

fault. Vibration-based fault diagnosis may be performed by studying the abnormality of vibration

features combined with the locations of the accelerometers generating those features.

G-24 Vibration and Shock Handbook

Fault prognosis

Estimating the expected time to failure, and trends in the condition of the equipment being monitored,

and planning and scheduling appropriate maintenance procedures.

FEA

Finite element analysis. See also Finite element method (FEM). A computational technique that starts by

discretizing the continuum equations of the problem.

Feasible region

All sets of blade adjustment in jet engines that will reduce the aircraft vibration to within

specifications.

Featural influence

The coefficients representing the weight of each vibration feature in the structure-based connectionist

network.

Feed selection

Technique where the feed is chosen to avoid or suppress chatter.

Feedback control

A common control procedure where a sensor measures or monitors the system response and feeds the

information into the controller. The controller compares the measurement with the desired value. If there

is an error, then a correcting signal is generated by the controller, which is then applied to the system by

the drive unit (or actuator).

Feedback resistor

The resistance element in the feedback path of an operational amplifier. It makes the device stable.

Feedforward action

A control signal generated without monitoring the response. Using a model of the system or

measurement of unknown inputs may be used to generate it.

Feedforward control

See Feedforward action.

FEM

Finite element method. See FEA.

Ferromagnetic core

A member (e.g., rod or disk made of laminations) made of ferromagnetic material for use in a magnetic

circuit (e.g., windings) to concentrate the field and reduce losses.

FFT

See Fast Fourier transform.

Fiber-optic

Devices made using optical fibers (a bundle of thin and flexible fibers that can carry light or laser signals).

Fictitious spring method

An analysis method for sliding isolated structures, in which a fictitious spring is introduced between the

base mat and ground to simulate nonlinear behavior of friction.

Glossary G-25

Filter

An analog-hardware or software (programmed) device used to filter out some frequency bands of a signal

(while allowing through the other bands).

Final peak sawtooth shock (FPS)

See Terminal peak sawtooth shock (TPS).

Finite difference

Computational methods involving a difference approximation for the derivatives.

Finite element method

A method for solving differential equations by approximating the solution by a linear combination of

trial functions, which are typically polynomials.

Finite element method (FEM)

A numerical technique to solve the equations of an engineering (e.g., vibration) problem. A

continuum is divided into a number of relatively small regions called elements that are interconnected

at selected nodes. The kinetic energy or potential energy of the entire continuum is

expressed as the sum of the element kinetic or potential energy. The solution is approximated by a

linear combination of trial functions, which are typically polynomials. This leads to the assembled

mass matrix and the assembled stiffness matrix, and finally to the finite element equations of the

problem (vibration).

Fixed end

A boundary condition where the displacement and slope are zero (bending moment and shear force are

not zero). Cantilever is obtained by fixing one end of a beam.

Flexible mount

An engine mount made of flexible material (e.g., elastomer or spring) to provide vibration or shock

isolation.

Flexibility matrix

The aij element of this matrix is the displacement at i due to a unit force at j with all other forces equal to

zero. Inverse of the stiffness matrix.

Flexural vibration

Bending vibration. The oscillatory motion of a flexible shaft (possibly with a rotor) or a beam or plate in

any transverse plane of bending.

Flow resistance

Resistance due to friction between a fluid and some constriction (e.g., a pipe or duct) or between a gas

and fibers in acoustic problems. The energy loss may take place in a very narrow layer of air adjacent to

each fiber of acoustic material.

Flow resistivity

Unit area flow resistance per unit thickness of a porous material. Given by the applied air pressure

difference across the two sides of the material divided by particle velocity through the material.

Fluid damping

Force of external friction whose magnitude is proportional to the square of the velocity, and whose

direction is opposite the velocity.

G-26 Vibration and Shock Handbook

Foil-type strain gauge

A strain gauge consisting of a thin metallic foil. As it is stretched (strained), its resistance increases, which

can be measured using a bridge circuit to provide a strain reading.

Force isolation

See Isolation. Vibration isolation of a system from a force source.

Force transmissibility

See Transmissibility. Transmitted vibration force divided by an applied vibration force (or actuator force).

Forced vibration

The steady-state vibration caused by an external excitation (force or input). Typically (in linear problems

in particular), forced vibration occurs at the frequency of the exciting force.

Forcing frequency

The frequency of the excitation function (force).

Fourier analysis

Analysis carried out using the Fourier transform. This provides frequency domain information

(magnitude and phase with respect to frequency) of a time signal.

Fourier integral

See Fourier transform.

Fourier series expansion

Expressing a general periodic signal into a series (sum) of harmonic (sinusoidal, real or complex)

components of integer multiples of a fundamental frequency.

Fourier spectrum

Frequency spectrum of a signal. The spectrum of a signal, as obtained by Fourier transform.

Fourier transform

An integral transform (weighted by a complex exponential function). This converts timedomain

information (time signals) into frequency-domain information (frequency spectra) and vice

versa.

Frahm absorber

See Absorber. A dynamic (vibration) absorber.

Frame

Countable set of linearly dependent functions.

Free-body diagram

Diagram showing the basic components of an object and all the forces acting on them.

Free vibration

Vibration that occurs in a system after the removal of excitation. Typically, free vibrations occur at one or

more natural frequencies.

Free vibration

The oscillatory motion (vibration) of a system that occurs in the absence of an external forcing function

(excitation).

Glossary G-27

Frequency

The repetition rate or periodicity of a periodic event, usually expressed in cycles per second (hertz or Hz),

or revolutions per minute (rpm). This is the cyclic frequency, which is the reciprocal of the period. The

angular frequency (radians per second or rad/sec) is obtained by multiplying cyclic frequency by 2p:

Frequency domain

The representation of the system variables (including inputs and outputs) as functions of frequency.

Related to the time domain through the Fourier transform. Vibration data or results are displayed or

usually analyzed as a function of frequency. The independent variable is frequency ðvÞ:

Frequency-domain analysis

Response analysis of a dynamic system with respect to frequency. Typically, uses Fourier analysis,

frequency spectra, and transfer functions.

Frequency-domain index

A single number calculated from the frequency spectrum and used for trend analysis and comparison.

Frequency modulation (FM)

A signal modulation method, where a carrier signal is used to transmit another signal (modulating signal,

carried signal) over a long distance without distortion. In this method, the amplitude of the carrier is

fixed, but its frequency is varied according the amplitude of the carried (modulating) signal. Typically,

radio signals are transmitted by this FM method.

Frequency response

The amplitude and phase response characteristics of a system as functions of frequency under external

excitation.

Frequency response function (FRF)

Output or input in the frequency domain, assuming zero initial conditions. Laplace transfer function can

be converted into FRF by simply changing the Laplace variable into complex frequency variable jv: The

FRF has amplification and phase change information of a response with respect to an input at various

frequency values. It may be plotted as a Bode diagram or a Nyquist diagram.

Frequency transfer function (FTF)

Same as Frequency response function (FRF).

Friction pendulum system (FPS)

One type of sliding isolators, which has a spherical concave sliding surface to produce a recentering force

for the isolated structure under excitations.

Fringes

A pattern of dark and light lines obtained when two light waves of nearly equal frequency are mixed. This

method is used in some sensors, for example, for measuring small velocity changes (by Doppler effect).

Full bridge

A bridge circuit with four arms. It can be used to sense small changes in one of the arms (consisting of a

resistor, inductor, capacitor, etc.). A half bridge contains two arms.

Full-scale drift

The drift (undesirable variation of the output when the input conditions are kept constant) of an

instrument, when the input conditions correspond to its full-scale (maximum) value.

G-28 Vibration and Shock Handbook

Fully differential capacitive sensor

A bridge in which all four components change in symmetric manner.

Fully operational

The highest performance level at which a structure is expected to operate with all functions, after an

earthquake attack, if there is no (or at least very little) structural and nonstructural damage.

Fundamental frequency

Fundamental natural frequency. The lowest (first) natural frequency of a (vibrating) system.

Fundamental natural mode of vibration

The mode of vibration of a system having the lowest natural frequency.

Fuzzy logic

An approach to computing (soft computing) and knowledge-based decision making, based on the

“degrees of membership” in a set, rather than the conventional binary membership (member or

nonmember). Fuzzy data are matched with a fuzzy knowledge base to generate fuzzy inferences

(decisions). A more general form of “true or false” boolean logic on which the modern computer is

based, and incorporates such logic states as slightly true or very true.

FVC

Frequency to voltage converter.

g units

Units of acceleration due to the Earth’s gravity (by international agreement, g ¼ 9:80665 m=sec2 or

386.087 in/sec2 or 32.1739 ft/sec2). Divide an acceleration value by the value of g to obtain (as a ratio) the

acceleration in g units.

Gabor transform

A sampled short-time Fourier transform.

Gaussian elimination

The process by which a matrix is converted by elementary row operations to an upper triangular matrix.

Generalized coordinates

A coordinate system, which does not necessarily use physical coordinates to represent the dynamics of the

system. An example is the modal coordinates.

Generalized forces

See Generalized coordinates. When multiplied by the increment of the corresponding generalized

coordinate, the product is equal to the actual incremental work done by that force in moving through

that incremental motion.

Geometric modeling

The process of transcribing the geometry of a system into a finite element or CAD data base.

Geometric nonlinearity

Nonlinearity caused by geometrical constraints, such as rubber mount casing, or by trigonometric

functions in the kinematic equations of the system (e.g., robot).

Gradient wind

Wind velocity above the boundary layer.

Glossary G-29

Green’s function

The solution that represents the sound field generated by an impulsive source in space and in time. Useful

in describing the sound field generated by a continuously distributed source, with convolution

integration, both in free space and bounded space.

Ground vibration

A wave transmitted through the ground that causes the surface particles of ground to oscillate as it

passes.

Guyan reduction

A procedure to reduce the model scale for vibration analysis by removing those DoF (called slave DoF)

that can be approximately expressed by the rest of the DoF (called master DoF) through a static relation.

Gyroscopic action

A torque (or moment) is necessary to change the direction of angular momentum vector (the product of

inertia and angular velocity vector) of a spinning member. Without such a torque, the member will

remain oriented in a fixed direction.

Gyroscopic motion

Motion of bodies in 3-D space subjected to multiple angular rotations, causing gyroscopic effects.

Gyroscopic sensor

A sensor based on the gyroscopic principle, commonly used to measure angles from a fixed reference,

when a physical surface representing such a reference is not available.

h-conversion

The process of achieving convergence or improving the accuracy of a finite element solution through

increasing the number of elements, while keeping the element order the same.

Haar wavelet

The first known wavelet derived from a step function.

Half bridge

See Full bridge.

Half-power bandwidth

The frequency interval in a spectrum where the magnitude drops to 1=

ffiffi

2 p the peak, resonant value

(i.e., half the power).

Half-power points

See Half-power bandwidth. The two frequency points representing the half-power bandwidth.

Half-sine shock

Simple shock for which the acceleration-time curve has the form of a half-period (positive or negative

part) of a sinusoid.

Hamilton’s principle

Variation of the difference of the system kinetic energy and the system potential energy plus the

line integral of the virtual work done by the nonconservative forces during any time interval must be

equal to zero.

G-30 Vibration and Shock Handbook

Hammer test

An impact test where the test object is impacted with an instrumented hammer and the corresponding

excitation and response signals of the object are measured. It can be used to determine natural

frequencies and damping ratios.

Harmonic

A sinusoidal quantity having a frequency that is an integral multiple of the frequency of a periodic

quantity to which it is related (normally the fundamental frequency).

Harmonic oscillator

This oscillator produces periodic oscillation described mathematically by either of the harmonic

trigonometric functions (sine or cosine), which has only one frequency component (undamped natural

frequency of the oscillator). Similar to an undamped simple oscillator.

Harmonic response analysis

Harmonic response analysis is used to determine the response of a vibration system to a harmonic

(i.e., sinusoidal) excitation force. A typical output is a graph showing response (usually displacement

of a certain degree of freedom) vs. frequency. The response has an amplitude (amplification)

and a phase change with respect to the harmonic excitation, which are characteristics of the system

itself.

Hearing loss

Permanent hearing loss is caused by excessive and prolonged noise exposure, which destroys the auditory

sensor cells.

High-tech facility

Production lines for printed circuit boards, high precision devices, semiconductor devices, electronic

appliances, etc., employing robotics, automation, and other modern and advanced technologies.

Holonomic system

A mechanical dynamic system where all of the constraints can be represented by algebraic equations (not

differential equations). In this case, the number of DoF is equal to the number of independent

coordinates needed to represent an arbitrary position and orientation of the system.

Homogeneous equation

An unforced ordinary differential equation. For a dynamic system with zero input, the right-hand side of

the system differential equation will be zero, giving a homogeneous equation.

Hooke’s Law

Linear spring equation. Deflection is proportional to the applied force. Strain is proportional to stress in

one dimension.

Hydraulic actuator

A piston-cylinder (i.e., ram) device to which a controlled and pressurized fluid is sent, thereby moving

the piston. This motion can be used to move another object. This described a linear (rectilinear) actuator,

although rotary hydraulic actuators are also available.

Hyperpatch or Volume

A portion of tri-cubic solid completely bounded by four areas (for tetrahedron volumes) or six areas (for

brick volumes).

Glossary G-31

Hypocenter

The location of initial radiation of seismic waves (i.e., the first location of dynamic rupture).

Hysteresis

Greek word meaning “arrive late,” used to describe a specific nonlinear processes where the response

curve when the input is increasing is different from that when the input is decreasing, resulting in

a hysteresis loop. Examples include magnetic systems (due to residual magnetism) and plastic systems

(due to residual stresses and strains).

Hysteresis loop

A diagram that describes the force – deformation relation of a nonlinear device of the hysteresis type. See

Hysteresis. Vibration isolator with hysteretic damping produces a hysteresis loop, the area of which gives

the energy dissipation in one cycle of vibration.

Hysteretic damping

Internal friction type damping in material where the energy dissipation per cycle is independent of

frequency.

Imbalance

The eccentricity that exists between the center of mass of a rotor and the axis of rotation that generates

centrifugal forces, resulting in vibrations and undesirable forces at the bearings.

IC engine

Internal combustion engine. A mixture of fuel and air is ignited in a cylinder, causing the piston of the

cylinder to move.

Imission

Receiving, perceiving or observing radiated noise, causing noise exposure at a location near the noise

source. Noise imission is legislated using an Environmental Quality Standard.

Impedance

Resistance exhibited by a “flow” variable (e.g., current, velocity) to an applied “effort” variable (e.g.,

voltage, force), expressed as a transfer function in the frequency domain. See Mechanical impedance and

Electrical impedance.

Impedance bridge

A bridge circuit with inductor or capacitor elements in addition to resistors in at least one of its four

arms.

Impedance transformer

A device with a very high input impedance and very low output impedance. Typically, an operational

amplifier with feedback.

Impeller

A bladed member of the rotor of a pump or compressor that imparts principal energy into the fluid that

is being pumped or compressed.

Impulse

The product of the force on a body and the time during which that force acts, equal to the change in

momentum.

G-32 Vibration and Shock Handbook

Impulse response function

The response of a system to a unit impulse input. Inverse Fourier transform of the frequency response

function (or inverse Laplace transform of the transfer function) of the system.

Incipient failures

Partial and usually gradual failure of a machine.

Inelastic response

A structure subjected to large excitations, resulting in yielding of longitudinal rebars.

Inertia

A property of matter where it resists a change in motion, including resistance to change from a position

at rest.

Inertia asymmetry

The moments of inertia in two orthogonal planes are not equal.

Inertia block

A block or a heavy plate attached to the base of a machine in order to achieve good vibration isolation.

Inertia matrix

A matrix consisting of direct and cross moment of inertia terms in three dimension of a rigid body.

Mass matrix. The matrix multiplied by the acceleration vector of a multi-DoF system in its Newtonian

equations of motion expressed in the vector-matrix form.

Inertial actuator

A class of actuators in which the actuation force is generated through the addition of inertial masses and

applying the resulting inertial forces on the system at the actuator point of attachment.

Influence coefficient

A coefficient representing the effect at a particular point of an action some other point.

Initial conditions (ICs)

The conditions of the response of a dynamic system at time t ¼ 0: These are temporal conditions,

different from boundary conditions, which are special conditions. The required number of ICs is equal to

the “order” of the system, or the number of state variables needed for the system, or the order of the

highest derivative of the input – output differential equation.

Inner ear

Cochlea is the main component of inner ear, where the actual reception of sound takes place. The

mechanical motion caused by the sound wave is transmitted to basilar membrane and its associated

sensing cells, where this motion is sensed as sound.

Inner modulation

Current structural vibration.

Input

The force or energy (or a motion such as support motion) applied to a mechanical system.

Input – output equation

A single differential equation with the response as the dependent variable and the input as the

independent variable.

Glossary G-33

Insertion loss

The insertion loss of a muffler is defined as the difference between the sound pressure or acoustic power

level at the same point in space before and after the muffler is connected to the noise source.

Instability

A self-induced excitation that can occur in rotating machinery, which does not require a sustained

forcing phenomenon to initiate or maintain the motion. At the inception of instability, the rotor

deflection will continue to build up with speed, and if further increased the large amplitudes of motion

will normally result, causing damage to the machine.

Intensity

A metric of the effect, or the strength, of an earthquake hazard at a specific location, commonly measured

on qualitative scales such as MMI, MSK, and JMA.

Internal friction

Damping that derives from structural defects internal to structural components of a vibrating system.

Interplate

Processes between the Earth’s crustal plates.

Interval model

A linear regression model with interval coefficients to represent the range of aircraft vibrations caused by

blade adjustments. This model can account for the stochastic and nonlinear nature of aircraft vibration.

Intraplate

Processes within the plates.

Intrinsic

Opposite of extrinsic. A method where the device is used to directly generate the objective (e.g., signal)

rather than simply to transmit it. Some types of fiber-optic and ultrasound sensors use the intrinsic

approach (as well as extrinsic approach).

Intrinsic loss factor

Factor in a subsystem to represent the loss of vibration power with respect to the total power.

ISO

International Organization for Standardization. Technical Committee 108, Subcommittee 5 is

responsible for standards for Condition Monitoring and Diagnostics of Machines.

Isolation

Diminishing the transfer of vibration amplitude from a machine to another machine or a structure by the

judicious selection of barrier materials and barrier configuration. Vibration isolation of a system from a

vibration source. Inverse of “transmissibility.”

Isolation period

Vibration period of the isolation system.

Isotropic

Being the same in all directions. See Orthotropic.

Jerk

The rate of change of acceleration with time (third derivative of displacement with respect to time).

G-34 Vibration and Shock Handbook

Johnson noise

White noise (approx.), whose value depends on both bandwidth and resistance.

Journal bearings

Journal bearings consist of a circular section of shaft, called the journal, rotating inside a cylindrical bush,

called the bearing. The space between the two is partially filled by the lubricating fluid, which is drawn up

around the journal due to the effect of elasto-hydrodynamic lubrication.

Jump phenomenon

A nonlinearity where the frequency response of a system sharply shifts up or down (jumps) at a particular

frequency, as the frequency of excitation to the system increases or decreases through this frequency.

K-normal

A modal vector z that is normalized with respect to the stiffness matrix K, such that zT Kz ¼ 0

Keulegan – Carpenter number

The Keulegan – Carpenter number indicates the relative importance of the drag and inertia forces for

small (small compared to the wavelength) diameter structures. A small Keulegan – Carpenter number

means that the inertia force is more important than the drag force.

Key point

A coordinate location in space.

Kinematic design

Design of a structure by its motion, independent of mass and force.

Kinematics

Motion geometry of bodies.

Kinetic energy

Energy associated with motion (velocity).

Kronecker delta

A function of two indices, which is equal to one when the two indices are identical, and zero otherwise. It

may be used to represent a unit matrix (or identity matrix), a diagonal matrix with unity diagonal elements.

Kurtosis

Fourth probability density moments sensitive to the impulsiveness in the vibration signal and, therefore,

sensitive to the type of vibration signal generated in the early stages bearing fault of a rolling element.

LabVIEW

Industry standard equipment, software, and hardware marketed by National Instruments.

LaCoste zero-length spring

Spring of a vertical seismometer, which would collapse to zero length if wire of the coil was of

infinitesimal diameter.

Lagrange’s equations

Differential equations of motion of a dynamic system, obtained using the Lagrangian energy approach, in

terms of a set of generalized coordinates. There are as many Lagrange’s equations as there are generalized

coordinates. The forcing terms are generalized forces.

Glossary G-35

Lagrangian

Equal to total kinetic energy minus the total potential energy of the system.

Laminated rubber bearing (LRB)

A type of elastomeric isolators that is usually composed of alternating layers of steel and hard rubber

ensuring it is stiff enough to sustain the vertical loads, yet flexible under the lateral forces.

Laplace transform

An integral transform weighted by an exponential function of the Laplace variable s. It converts a

derivative into a power of s, of the same order. Hence, a differential equation is converted into an

algebraic equation. Related through the Fourier transform through s ¼ jv; where v ¼ angular frequency

and j ¼

ffiffiffiffi

21 p : An input – output equation is converted into a transfer function.

Lateral force resisting system

A structural system for resisting horizontal forces due, for example, to earthquake or wind (as opposed to

the vertical force resisting system, which provides support against gravity).

Learning

Adaptation of the interval coefficients to represent the range of vibration caused by blade adjustments

more effectively.

Life safety

The third performance level in which a structure is expected to survive an earthquake. The life safety of

occupants is not endangered either by structural collapse or the presence of heavy falling debris, although

substantial damage may occur to both structural and nonstructural building components.

LIGO

Laser Interferometer Gravitational Wave Observatories of the U.S.

Limit cycle

An instability in a nonlinear system that is exhibited as a steady oscillation of amplitude that is

independent of the input amplitude.

Limiting stable depth-of-cut

Depth-of-cut below which chatter will not occur for a given spindle speed of a machine tool.

Line noise

Noise at the frequency of the voltage signal in a power line (60 or 50 Hz) caused by the power lines in the

vicinity of a system (e.g., through ground loops).

Line or line segment

A portion of a cubic spline curve bounded on both ends by a key point. A line may be straight or curved.

Linear quadratic regulator (LQR)

A linear feedback controller with constant gains, which minimizes a cost function consisting of the time

integral of the sum quadratic functions (positive definite and semidefinite) in the state variables and

input variables.

Linear system

A system is linear if it can be represented by a set of linear differential equations. The principle of

superposition holds. Its magnitude of response is directly proportional to its magnitude of excitation for

every part of the system.

G-36 Vibration and Shock Handbook

Linear-variable differential transformer

See LVDT.

Load – displacement relationship

The relationship between the load on a structure and the corresponding displacement.

Logarithmic Decrement

Measure of damping. Equal to the natural logarithm of the ratio of successive amplitudes of free decay.

It is inversely proportional to the magnification factor (quality factor), Q.

Long-term statistics

Statistics of the short-term parameters such as the significant wave height.

Lorentizian

Resonant response of a driven harmonic oscillator with damping. Also, the shape of spectral lines in

atomic physics.

Lorenz force

Force on a current-carrying wire located orthogonally in a magnetic field.

Loudness level

The loudness of sound is determined by subjective comparison with the loudness of a pure tone of

1000 Hz. The level, measured in “phons,” is numerically equal to the sound pressure level, in decibels

(dB), of the 1000 Hz tone.

LPA

Last point algorithm for performing numerical integration. Will not become unstable like Euler’s first

point approximation does.

Lumped mass

A point presentation of the mass of a system. Results in a diagonal matrix.

Lumped mass matrix

The diagonal mass matrix obtained by lumping the inertia of a system by element masses at the nodes

finite element mesh.

Lumped-parameter model

A model where the system components are represented by parameters (e.g., mass, stiffness, damping

coefficient) concentrated at a finite number of points. An approximation of a distributed-parameter

system.

LVDT

Linear variable differential transformer. A popular noncontact sensor for measuring linear (rectilinear)

displacements. Operates by the transformer effect, with a movable ferromagnetic core element (attached

to the sensed object).

M-normal

A modal vector z that is normalized with respect to the mass matrix M, such that zT Mz ¼ 0:

Machinery failure

The inability of a machine to perform its required function due to some defect in the machine components.

Glossary G-37

Machining

Removal of excess material from a piece of material (workpiece) by moving a working tool over the

surface of the workpiece to achieve a desired shape.

Magneto-rheological (MR) fluid

A magnetic fluid analogous to electro-rheological (ER) fluid. Typically consists of micron-sized,

magnetically polarizable particles dispersed in a carrier medium such as mineral or silicon oil. When a

magnetic field is applied to the fluid, particle chains are formed, and the fluid becomes a semisolid and

exhibits viscoplastic characteristics.

Magnetorheological damping

Damping associated with a magnetic fluid and a magnetic field.

Magnitude

A measure of an earthquake’s release of strain energy, measured on a variety of scales, of which the

moment magnitude MW (derived from seismic moment) is preferred. The absolute value of a complex

quantity (e.g., FRF).

Masks

Baseline spectra that are used with an allowable tolerance limit to “filter out” or block specific

frequencies. Optical plates with windows for concentrating light from a source to generate better contrast

in an optical pulse-generating transducer (e.g., optical encoder).

Masonry structure

A structure built from bricks and cement mortar that has relatively small lateral and tensile strength,

which is vulnerable to an earthquake attack.

Mass law

For a single plate, the transmission loss depends on frequency and the surface density of the plate in the

frequency range from the fundamental (lowest) natural frequency to the critical frequency of the plate.

Mass matrix

The matrix multiplied by the acceleration vector of a multi-DoF system, which, in its Newtonian

equations of motion, is expressed in the vector-matrix form. Typically, a diagonal matrix with its

diagonal elements giving the masses corresponding to various DoF of the system.

Mathematical expectation

Also known as “expected value.” The mathematical expectation of a random function is the integral of

that function weighted by its probability density function. If the function is the random variable X, the

expected value gives the mean value ðX􀀊 Þ; which is the first moment statistic. If it is X 2, we get the second

moment statistic. If it is ðX 2 X􀀊 Þ2 we get the variance, the square root of which is the standard deviation

of a random parameter. Other higher order statistics of X can be derived similarly.

Matlab

MATLABw is a commercial software package (tool) for numerical computation, visualization and

symbolic manipulation. Various toolboxes are available for specialized computation on important

subjects such as control and signal analysis.

Maximax shock response spectrum

Envelope of the absolute values of the positive and negative shock response spectra. See Response

spectrum.

G-38 Vibration and Shock Handbook

Mechanical design

Design of a structure or building utilizing the properties of electronic and mechanical subassemblies and

parts.

Mechanical impedance

Force or velocity in the frequency domain. This transfer function is one of many used in the analysis and

design of vibrating systems.

Mechanical looseness

A condition where two or more mechanical components, which are designed to be rigidly connected or in

firm contact, can move relative to one another. See Backlash (in gears).

Mechatronics

Synergistic integration of electronic and electrical engineering, computer technology and control

engineering with mechanical engineering, using concurrent and integrated design techniques. This area

forms a crucial part in the design, manufacture, and maintenance of a wide range of engineering products

and processes.

Mean free path

Distance a molecule travels on average in a gas between molecular collisions.

Measure of noise evaluation

Energy based indices like LAeq (equivalent A-weighted sound pressure level) for noise evaluation. All of

these measures are legislated by the government, and are dependent on legislative regulations or standards.

Member

Any individual component of a structural frame. An element is within a set (membership ¼ 1).

MEMS

Microelectromechanical systems. A relatively new and important field concerned with modeling,

analysis, design, and development of very small mechanical devices with electrical or electronic features.

Mesodynamics

A new term to describe complex behavior of otherwise harmonic mechanical oscillators with energies

approximating 10211 J.

Mesomechanics

Focused on the complex behavior of granular materials.

Mexican hat wavelets

Obtained by calculating the second derivative of a Gaussian.

Meyer wavelets

Wavelets associated with enveloped trigonometric functions.

Microdynamics

Dynamics describing complex features of contact friction, largely advanced by NASA.

Microphone

A device for producing an electrical signal from an acoustic pressure (sound or voice). Standard

equipment in acoustic or sound acquiring and generating systems.

Glossary G-39

Microphone array

A directional measuring device that consists of an array of miniature microphones distributed spatially in

an acoustic field.

Middle ear

Consists of three small ear bones, the hammer, anvil and stirrup. Serves as an impedance

transformer, which matches the low impedance of the air in ear canal with the high impedance of

inner ear.

MIMO

Multi-input – multioutput system. A system with many inputs and many outputs. A multivariable system.

Mirror microphone system

A directional measuring device that consists of a reflector of elliptic or parabolic shape and an

omnidirective microphone located at its focus.

Misalignment

A condition when two rotating machines (or components) are not coupled together along a single axis

(straight line) of rotation.

Mobility

The transfer function (velocity or force) in the frequency domain. The inverse of mechanical

impedance.

Modal analysis

The dynamics of a system may be formulated as a problem of determining eigenvalues and

eigenvectors of a matrix. This matrix is M21K; where M is the mass matrix and K is the stiffness

matrix in the undamped case giving natural frequencies, time constants and modal damping ratios.

See Eigenvalue analysis and Eigenvalue problem. The imaginary parts of the eigenvalues are the

natural frequencies of the system. The reciprocal of the imaginary parts are the time constants of

the system. Imaginary parts divided by the corresponding undamped natural frequencies provide the

modal damping ratios. The eigenvectors provide the mode shape information. The analysis

that provides both natural frequencies and mode shapes is called modal analysis. This is the process

of decomposing the vibrations (or response) of a complex system into its component modes of

vibration (motion). One goal may be to find the locations and frequencies of maximum

(undesirable) response for application of vibration control or suppression methods (e.g., stiffener

and dampers).

Modal density

The number of mode shapes present in a unit spatial area or the number of natural frequencies in a unit

frequency interval.

Modal matrix

The square matrix formed by placing all the independent mode shape vectors as column vectors.

Modal vector

A mode shape vector expressed as a column vector.

Mode

The condition where a mechanical system moves at one natural frequency while maintaining a

characteristic shape (mode shape). Can also apply to nonmechanical systems.

G-40 Vibration and Shock Handbook

Mode mixing

Transfer of energy among modes of different frequencies. This assumes nonlinearity and is not the same

as beating.

Mode shape

The deflected characteristic shape, usually normalized, that a system would take when it vibrates at one

of its natural frequencies. All elements move in step at this frequency while maintaining a specific

shape. Some points will remain stationary throughout, which and they are called nodes. Only the shape

of the curve would be known from a modal analysis. If the mode shapes are real, they correspond to

natural modes. Otherwise, we have complex modes, which is the case in the presence of some types of

damping.

Model identification

Experimental modeling. Determining an analytical model of a dynamic system using input – output test

data. See Experimental modal analysis.

Model reduction

Reduction of the order or the number of DoF (of a vibrating system) by using some accuracy limits (error

tolerance). Modal truncation is used, where modes above a specific frequency are neglected. Means of

simplification and idealization based on the nature of the vibration problem.

Modulation

Embedding a signal within a carrier signal so that the original signal (carried signal or modulating signal)

does not lose accuracy during transmission. Examples are amplitude modulation (AM), frequency

modulation (FM), and pulse-width modulation (PWM).

Modulus of elasticity

Young’s modulus. Direct stress divided by direct strain in one dimension.

Mohr’s circle

A circle for representing stresses, strains, or moments of inertia at a particular point of system in different

directions. The horizontal axis indicates the direction of the principal values and the vertical axis the

shear (or cross) values. The angle around the circle is double the physical angle in the system.

Moment of inertia

A mass parameter that is measure of the resistance (inertia) of a body to angular acceleration about a

given axis. Equal to the sum of (or integral to) the products of each element of mass in the body and the

square of the element’s distance from the axis.

Momentum

A property of a moving body that exists by virtue of its mass and motion, to resist any change to that

motion. Equal to the product of the body’s mass and velocity. It is a vector (with a direction as well as a

magnitude) in the same direction as the velocity.

Monopole

An acoustic source with energy density concentrated at a single point in space. The most basic sound

source, where the source region is compact and the generating motion has no preferred direction.

Monopole produces a wave that spreads spherically outwards.

Mother wavelet

The function used to generate all the basis functions of the space by dilation and translation.

Glossary G-41

Motion isolation

See Isolation. Vibration isolation of a system from a motion source.

Motion transmissibiliy

See Transmissibility. Transmitted vibration motion divided by an applied vibration motion (or support

motion).

Mounts

Usually, vibration mounts used for vibration isolation.

Muffler

A structural device that suppresses noise when fluid (e.g., hot and high-speed gas) flows through it.

See Dissipative muffler and Reactive muffler.

MTBF

Mean time between failures.

MTTF

Mean time to failure.

Multibody-spring-dashpot

Collection of rigid bodies joined together by springs and dashpots.

Multiresolution analysis

The process of studying a signal at different scales by use of different time windows to capture

characteristics varying from coarse to fine.

Narrowband

Corresponding to a narrow (small) frequency interval (band).

Narrowband process

A random process where most of the energy is concentrated in a narrow band (interval) of frequency.

Natural frequency

The frequency of free vibrations of a system without the influence of an external forcing, except for an

initial excitation to start the motion. Also a self-induced phenomenon. For a single-DoF system, vn ¼

ðk=mÞ1=2; where vn is the undamped natural frequency, k is the spring stiffness, and m is the mass. When

damping is present, the natural frequency decreases to vd; which is the damped natural frequency given

by vd ¼

ffiffiffiffiffiffiffiffi

1 2 z2

vn; where z ¼ damping ratio. For a multi-DoF system, the natural frequencies are the

frequencies of the normal modes of vibration.

Natural mode

See Mode shape.

Natural period

The time period of one complete cycle of free vibration. The inverse of the natural frequency (cyclic).

Natural vibration analysis

See Modal analysis. The natural vibration analysis is to determine the natural vibration frequencies and

the associated mode shapes of a vibration problem when the excitation forces are zero. They are intrinsic

characteristics of the vibration problem and independent of the excitation forces.

G-42 Vibration and Shock Handbook

Near-fault ground motions

Ground motions, typically assumed to be within a distance of about 20 to 60 km from a ruptured fault,

which usually have long-period velocity pulses that impart considerable momentum to a long-period

structure.

Negative averaging

A baseline signal that is recorded and then subtracted from subsequent signals before analysis to reveal

changes and transients.

Negative definite

A symmetric matrix is negative definite if xt Ax , 0 for every nonzero vector x.

Negative shock response spectrum

See Shock response spectrum (SRS). SRS plotted, considering the highest negative response due to the

shock, without reference to the duration of the shock. It is the envelope of the negative primary and

residual spectra.

Neural Network (NN)

A nonanalytical (computational) modeling technique, inspired by the human nervous system.

Considered as a “learning” network. A network of massively parallel network is trained (by adjusting

the network weights, etc.) to yield a known (correct) output for a given input. Once trained, it may be

used as a computational model of the system. Further training and adaptations may be done

subsequently. Several variations of this feed-forward network architecture are available (e.g., feedback,

recurrent, dynamic, reinforcement learning).

Neutral axis

The axis along a bending beam where the stress and strain are zero. Undeformed axis when a beam is

bent.

Nodal point

In finite elements: a point of connection between elements. A node will have a specific number of degrees

of freedom, indicating its ability to move or rotate. In vibrations: a point or a line in the mode shape of a

structure that does not move (see Node).

Node

See Mode shape. Points that are stationary in a modal vibration. The mode shape curve passes through

the relaxed configuration of the structure at these points.

Noise (1/f )

Also known as pink or flicker noise. Common in complex systems of both electronic and mechanical

type.

Noise Criteria (NC)

NC curves are utilized for rating the background noise level in a room. Each curve specifies the maximum

octave-band sound pressure level for a given NC rating. See Preferred noise criteria (PNC).

Noise regulation law

This regulatory standard defines the national minimum for noise suppression, for example, factory noise,

construction work noise, and road traffic noise. The central government is the regulating agency.

See Emission.

Glossary G-43

Nomograph

A logarithmic frequency – velocity plotting graph sheet. The sheet gives constant deplacement, constant

velocity, and constant acceleration lines, properly graduated. A period axis (period ¼ 1/cyclic frequency)

is given for convenience of plotting.

Nonclassical damping

Damping that cannot lead to modal decoupling of a structural system. For this case, the mode shapes of a

structure can only be expressed in terms of complex (not real-valued) functions.

Nonholonomic system

A mechanical dynamic system where one or more of its constraints need differential relations (not

algebraic equations) for mathematical representation. In this case, the number of DoF is equal to the

number of independent “incremental” coordinates needed to represent an arbitrary increment (change)

in position and orientation of the system, and is less than the number of independent coordinates needed

to represent an arbitrary position and orientation of the system.

Nonlinearity

A characteristic that is not linear (e.g., hysteresis, saturation, deadband, jump phenomenon, nonlinear or

nonproportional force– motion relationship). Represented by nonlinear equations. The principle of

superposition does not hold. See Linear system.

Normal modes

See Natural mode. In a multiple DoF system, the normal modes are the free natural configurations in

which the system vibrates. A mode is characterized by its natural frequency and mode shape. The mode

shape depends only on the mass and stiffness of the system and how they are distributed, and not on

damping. Normal modes exist for all undamped systems and only for a selected class of damped systems

(e.g., proportional damping).

Normal wear period

A relatively long period of operation during which the frequency of failure occurrence is relatively low.

Notch filter

A bandstop filter when the band is very small.

Nyquist frequency

Half the sampling rate. The frequency above which the spectrum of a digital signal has no significance

and should be ignored.

Nyquist plot

Plot of frequency transfer function (FTF) on a complex plane where the horizontal axis gives the real part,

and the vertical axis gives the imaginary part of the FTF.

OBE

Operating basis earthquake. An earthquake representation (e.g., response spectrum) under which a

nuclear power plant can operate safely.

Occupational Noise

Noise received at the workplace, indoors and outdoors, caused by all noise sources in the vicinity of the

workplace.

ODS

Operational deflection shape of a structure.

G-44 Vibration and Shock Handbook

Offset

Steady-state error in response of a device.

Oil whip

A condition that occurs when a subsynchronous instability (oil whirl) excites a critical speed (resonance),

which then remains at a constant frequency regardless of speed changes.

Oil whirl

A condition that occurs when the lubrication in a lightly loaded journal bearing does not exert a constant

force on the supported shaft, and a stable operating position is not maintained.

Oil whirl/whip

See Whirling. The whirling motions that occur due to oil action in journal bearings when operated at or

above double the critical speed of the rotor. The whirling motion is in the forward direction and its speed

will match the critical speed of the rotor. The oil film is believed to rotate at half the velocity of the shaft

due to friction drag. At rotational speeds near twice the critical speed, the oil film provides the stimulus as

its speed matches the critical speed value resulting in large displacements and whipping.

Omnidirective microphone

A microphone whose response does not depend on the direction of the incident sound.

Operating basis earthquake

See OBE.

Operational

The second performance level in which a structure is expected to function in a limited manner after an

earthquake because of some damage to structural or nonstructural members.

Optical encoder

A digital transducer that measures displacement. The encoder plate (disk) has light-passing areas

(windows) on one or more track, with a light source on one side and a light sensor on the other side. As

the plate moves, a pulse signal is generated by the light sensor. The pulse count gives displacement and a

pulse rate gives speed.

Orbit

A plot of the displacement Y-direction (ordinate) versus the displacement in X-direction (abscissa).

Order

The order of the highest derivative of a differential equation (DE) and the highest difference in a

difference equation. The order of a dynamic system corresponds to the number of first-order DEs needed

to represent the system. Factor of rotational speed of a machine used to represent nondimensional

frequency in tracking (monitoring) its spectral response (in the frequency domain). See also order

tracking.

Order tracking

Cascade plots of machine response (vibration) spectra that are synchronously linked to the machine

rotational speed via a tachometer.

Orthogonality

Generalized perpendicularity. If the vectors x and y are orthogonal, then xT y ¼ yT x ¼ 0: If the vectors x

and y are orthogonal with respect to matrix P, then xT Py ¼ yT Px ¼ 0:

Glossary G-45

Orthotropic

Not uniform in all directions (not isotropic).

Oscillation

A periodically increasing and decreasing variation of a response. In forced oscillation, the force itself

oscillates causing an oscillatory response. In free oscillations, the cause is the continuous interchange of

energy between two forms (e.g., kinetic energy and potential energy), where the variation can be either

constant amplitude (undamped) or decaying amplitude (damped). If the amplitude progressively

increases, it is an unstable oscillation. No oscillations occur in overdamped systems because energy is

more rapidly dissipated than it is interchanged, leaving no energy for a returning (reversed) response.

Outer modulation

Structural vibration delayed by one tooth passing period.

Outer ear

Composed of pinna, ear canal, and eardrum. Pinna produces only a small enhancement of the sounds

arriving from the front of the listener. Ear canal works as a resonant cavity allowing for a nearly

reflection-free termination, and thus a good impedance match of eardrum. Eardrum transforms sound

wave to vibration in the membrane.

Overdamped response

No oscillations occur in overdamped systems because energy is more rapidly dissipated than it is

interchanged between two forms (e.g., kinetic and potential), leaving no energy for a returning (reversed)

response. For a simple DoF mass (m) – spring (k) – damper (c) system, the condition for overdamped

behavior is 4k=m , ðc=mÞ2:

Overall sound pressure level

The pressure is evaluated as the root mean square (RMS) of all components of pure tones. Can be

converted into decibels (dB) as for any sound pressure.

Overlap index

Denotes the overlap in coverage of more than one gearbox component by one accelerometer location.

p-conversion

The process of achieving convergence or improving the accuracy of a finite element solution through

increasing the order of the elements, while keeping the number of elements the same.

Parametric instability

A self-excited motion, representing an instability that is induced by the periodic variation of the system

parameters such as inertia, mass, and stiffness. The system cannot be represented by linear differential

equations with constant coefficients.

Particle velocity

The fluctuating velocity of a material particle associated with a sound field (units: m/s). Using Newton’s

Second Law, we can solve for the particle velocity u in terms of sound pressure p.

Particular integral

A particular solution (just one solution) that is satisfied by a nonhomogeneous differential equation.

Particular solution

Same as particular integral.

G-46 Vibration and Shock Handbook

Passive vibration control

A vibration control method in which a resilient member (stiffness) and an energy dissipater (damper) are

utilized to either absorb the vibratory energy or dissipate it inside the damper. By definition, no external

power source is needed in a “passive” system.

Patch or area

A portion of a bicubic surface completely bounded by three line segments (for triangular patches) or four

line segments (for quadrilateral ones).

Peak

Maximum value of a varying quantity (or a spectrum) measured from the zero or mean value.

Peak ground acceleration (PGA)

The maximum amplitude of a recorded acceleration (similar to the ZPA, or zero period acceleration,

which is the acceleration at infinite frequency).

Peak level

The highest absolute value in a vibration signal.

Peak-to-peak value

For an oscillating quantity, this is the algebraic difference between the maximum and minimum extremes

of the quantity.

Pendulum

A simple pendulum. A point mass suspended from a fixed point by a string. A mechanical-swing

oscillator whose motion is harmonic at small amplitudes, but nonlinear otherwise — an archetype of

deterministic chaos. These are assumed to have just one DoF (the angle of swing). More complex pendula

such as compound pendulum and double pendulum, which have more than one DoF, are available.

Perimeter

The edge or boundary of an area.

Period

Duration of one complete cycle in a cyclic waveform (e.g., vibration signal, pressure or particle local

velocity in an acoustic system). The inverse of the cyclic frequency (Hz).

Period shift effect

Reducing the structural response by increasing the structural period.

Periodic monitoring

Regular and intermittent data gathering and analysis with portable and removable, monitoring

equipment.

Periodic quantity

An oscillating quantity whose value repeats itself in cyclic (periodic) manner.

Phase

The fractional part of a period through which a variable (signal) has advanced, measured from an

arbitrary reference. A time difference, expressed in degrees, between two signals gives their phase

difference. A shift of one period of the signal is considered a phase shift of 2p (or 3608). Phase lag

corresponds to a time delay and phase lead corresponds to a time advance of the signal.

Glossary G-47

Phase angle

The fractional part of a period through which the quantity has advanced as a measure from a reference

independent variable.

Phase difference

See Phase. The differences between the phase angles of two periodic quantities. In theory, only those

components with the same frequency can be compared for phase.

Phase noise

See Phasor. In phasor representation, the noise vector that is perpendicular, rather than parallel, to the

oscillator (primary signal) vector.

Phasor

A rotating vector of length (magnitude) a and angular speed v: Its projection on the vertical axis gives a

sinusoidal wave of amplitude a and angular frequency v: The phase angle may be represented by the

starting angle of the phasor from a reference line (e.g., vertical axis).

Phons

Unit of loudness level. See Loudness level.

Photogate

Infrared beam device used for time interval measurement.

Physical pendulum

See Pendulum. Contrasted with a simple pendulum, its motion at very small amplitudes is not harmonic

— an archetype of mesodynamic complexity.

Physical realizability

Whether an analytical model is physically realizable (whether the cause – effect condition or causality is

satisfied).

Pierson – Moskowitz spectrum

A parameter ocean-wave spectrum first derived for the North Atlantic, which is applicable for deep water,

fully developed, and unidirectional seas.

Piezoelectric

A phenomenon exhibited by some materials in which application of a strain (or stress) causes the

establishment of an electric field (or charge) and vice versa. The forward principle is used in piezoelectric

accelerometers and the reverse principle is used in piezoelectric actuators.

Pinned beam

See Simply supported beam.

Pitch control rod adjustments

Extension or contraction of pitch control rods by a number of notches to alter the pitch of the rotor

blades. Positive pitch rod adjustments indicate extension.

Plane wave

A sound wave that is uniform over any plane normal to a specified axis. This is applicable when

propagation of sound wave is one-dimensional and the amplitude of the sound pressure variation is

constant. This is approximately valid locally for a sound field very far from the source region.

G-48 Vibration and Shock Handbook

Poisson’s ratio

Lateral strain divided by the direct strain (absolute value, disregarding the sign) in a member in tension

in one dimension.

Poles

Eigenvalues of a system. The roots of the characteristic equation. The characteristic polynomial is the

denominator of the transfer function.

Portevin – LeChatelier Effect

“Jerky” strain of a loaded specimen as a result of internal friction arising from mesoscale structural

changes.

Post mortem

The investigation of root cause failure mechanisms.

Postprocessing

The process of calculating and graphically representing additional quantities and results (e.g., stresses,

reactions) from the primary finite element solution results (e.g., displacements).

Positive definite

A symmetric matrix is positive definite if xt Ax . 0 for every nonzero vector x.

Positive shock response spectrum

SRS plotted, considering the highest positive response due to the shock, without reference to the duration

of the shock. It is about the envelope of the positive primary and residual spectra.

Postshock

Shock pulses added after the specified shock test carried out on a shaker to cancel the velocity and

displacement of the shaker table at the end of its movement (return to the rest position with a zero

velocity).

Potential energy (PE)

Energy due to deformation of a spring or other flexible member (elastic PE) or due to elevated state

(gravitational PE).

Potentiometer (Pot)

A displacement sensor consisting of a conductor with a constant voltage applied to it and a slider moving

along it. The voltage of the slider is proportional to the displacement.

Power conversion factor

The ratio is sound power generated from a machine divided by the total power of the machine. This

factor ranges from 1028 to 1025 for both mechanical and electrical machinery.

Power flow

Acoustic power translation from a subsystem to the next connected subsystem.

Power spectral density

The Fourier transform of the autocorrelation function of a signal. Also called power spectrum. It is a

measure of how energy is distributed as a function of frequency for a random signal. A cross spectrum is

similarly defined corresponding to the cross-correlation function of two signals.

Glossary G-49

Power supply

A device that supplies DC power to instruments and circuits. In a regulated power supply, the output

voltage does not drop when connected to an external instrument.

Predictive maintenance

An equipment maintenance strategy based on measuring the condition of equipment in order to assess

whether it will fail during some future period, and then taking appropriate action to avoid the

consequences of that failure. The performance of the equipment is assessed using such approaches as

condition monitoring, statistical process control, or the human experience. See also Preventive

maintenance.

Preferred noise criteria (PNC)

See Noise criteria (NC). PNC curves are used to modify NC curves and are utilized in exactly the same

manner as NC curves.

Preprocessing

The process of building up the geometric and finite element model for a physical system.

Preshock

Shock pulses added before the specified shock test carried out on a shaker to cancel the velocity and

displacement of the shaker table at the end of its movement (return to the rest position with a zero velocity).

Pressure of sound

Incremental variation of pressure due to a sound wave, about the ambient atmospheric pressure, with an

accompanying variation of the air density.

Pressure pulsations

The oscillatory pressure variations that occur at a point in a fluid system.

Preventive maintenance

An equipment maintenance strategy based on replacing, overhauling or refurbishing machine parts and

components at a fixed interval, regardless of its state or condition at that time. See also Predictive

maintenance.

Primary adaptation

In feature classification: the process of adapting each weight of the single-category-based classifier

(SCBC) closer to the current value of the feature if the corresponding feature is classified as normal.

Primary (or initial) negative shock response spectrum

Shock response spectrum (SRS) plotted using the highest negative response observed during the shock.

Primary (or initial) positive shock response spectrum

Shock response spectrum (SRS) plotted using the highest positive response observed during the shock.

Principal axes of inertia

Orthogonal axes relative to which no cross inertia exist. The inertia matrix will be diagonal in this

coordinate frame.

Principal axes of stiffness

Axes along which any deflection will not create resultant loading in any other direction. The stiffness

matrix will be diagonal in this coordinate frame.

G-50 Vibration and Shock Handbook

Principal directions

See Principal axes.

Principal inertia

Moment of inertia about a principal axis where the cross inertia are zero and the inertia matrix is

diagonal.

Principal stresses

Stresses along the principal axes where the shear stresses are zero.

Probability density function

A function that defines the likeliness (or probability) that a random parameter will take a particular value

of the function variable. Derivative of the “probability distribution function.” This function completely

defines the random parameter. In vibration, this can be used to determine the probability of realizing a

particular instantaneous amplitude value from a vibration signal within a certain amplitude range.

Probability density moments

See Expected value. Single number indices expressed as moments weighted by the probability density

function.

Probability distribution function

A function that defines the probability that a random parameter will take a value less than or equal to a

particular value of the function variable. Integral of the probability density function. This function

completely defines the random parameter.

Process damping

In machining, damping due to the rubbing between the back side of tool and the surface of the machined

part. This decreases chatter.

Propeller

Rotor with blades or a screw like device that imparts kinetic energy to a fluid (which can be converted

into pressure and thrust). Used in compressors, pumps, jet engines. Provides thrust to push a boat

through the water, or an aircraft through air.

Proportional damping

The damping matrix of a mechanical (vibrating) system is such that the modal vectors are real. The

damping matrix is proportional to the mass matrix, stiffness matrix or both.

Pseudoacceleration

Product of the squared natural frequency ð2pfnÞ2 and the highest relative displacement response of a

linear one degree of freedom system to a shock input. This product has the dimension of acceleration but

does not represent the acceleration of the mass, except when damping is zero. In such cases, this term is

then strictly equal to the absolute acceleration of the mass.

Pseudovelocity

Product of the natural frequency ð2pfnÞ2 and the highest relative displacement response of the mass of a

linear one degree of freedom system to a shock input. This product has the dimension of velocity but

does not represent the velocity of the mass.

Pulse rise time

Interval of time required for a shock pulse to rise from some specified small fraction to a large fraction

(typically 10% to 90%) of its maximum value.

Glossary G-51

Pyrotechnic shock (or pyroshock)

Transient response of a structure to loading induced by the ignition of pyrotechnic (explosive or

propellant activated) devices. Pyroshocks are characterized by high peak acceleration, short duration and

high-frequency content.

Quadrature error

The secondary signal that is at 908 phase to the primary signal, produced by an LVDT when the slider is

symmetrically placed, while in theory it should be zero.

Quadrature signals

The two signals from a shaft encoder that are 908 out of phase, which are used to determine the direction

of motion.

Quality factor (Q-factor)

A measure of the sharpness of a resonance or frequency selectivity of a vibratory linear one-degree-offreedom

system. Assuming a lightly damped single-DoF system, the Q-factor is related to damping ratio

by Q ¼ 1=ð2z Þ: Other approximate relations for Q are:

1. Transmissibility at resonance

2. p/logarithmic decrement

3. 2pW =DW where W is the stored energy and DW the energy dissipation per cycle

4. fr=Df where fr is the resonant frequency and Df is the bandwidth between the half-power points

Nonlinear damping causes it to vary with time. The concept is applicable to both mechanical and

electrical systems.

Quantization error

The error created by the finite representation of numbers (i.e., finite number of digits or bits) within a

computer (i.e., for a digital quantity).

Quefrency

See Cepstrum. Similar to frequency, but in the Quefrency domain. The independent variable of a

cepstrum (similar to spectrum).

Quefrency domain

See Cepstrum. The domain of a cepstrum — inverse Fourier transform of a log of the Fourier transform

of a signal.

Radiational sound

Sound in an acoustic field generated from a radiating source.

Random vibration

A vibration whose magnitude cannot be precisely predicted for any given instant of time.

Range of normal hearing

Human hearing is most sensitive in the range 2000 to 5000 Hz, where the threshold response is very close to

0 dB. At the other end of the scale, we have the threshold of pain, which is usually about 135 to 140 dB.

Reactive muffler

A muffler where there are one or more chambers, resonators, or finite sections of pipe, which collectively

provide impedance mismatch for the acoustic energy traveling along the duct, causing noise suppression.

See Muffler and Dissipative muffler.

G-52 Vibration and Shock Handbook

Real-time

Refers to the relative response time for a particular process under computer control to occur after an

action is taken. Should be consistent with the response time of the physical system.

Receptance

The transfer function of displacement or force in the frequency domain.

Rectangular shock

Simple shock for which the acceleration – time curve increases instantaneously up to a given value,

remains constant throughout the signal and decreases instantaneously to zero. Practically, trapezoidal

shocks are rather carried out.

Rectilinear

Along a straight line. Often (incorrectly) called “linear.”

Recurrence relation

Discrete equation (difference equation) that allows the computation of numbers in a series one by

one.

Regenerative chatter

Unstable, self-excited interaction between machining forces and structural vibrations.

Relative displacement shock spectrum

See Shock response spectrum. SRS plotted, taking its ordinate (y-axis) the pseudoacceleration of the mass

of a linear one DoF system.

Relaxation

Process characterized by an exponential variation, for example, sample strain response to a constant load.

Reluctance

Magnetic resistance. Resistance to magnetic field in a magnetic circuit (magnetic path).

Repeatability

The ability of a transducer to reproduce output readings accurately when the same measured

value is applied to it repeatedly, under the same conditions (in the same direction, etc.).

Repeatability is expressed as the maximum difference between output readings as a percent of full

scale. This concept is also applicable to the operation of other instruments and machinery such as

robots.

Required response spectrum

See RRS.

Residual base displacement

A permanent base displacement of an isolated structure after an earthquake.

Resilient friction base isolator (RFBI)

One type of sliding isolator, which is made of a central rubber core and Teflon-coated steel plates.

Resilient mechanism

An automatic recentering mechanism of isolators by which the isolated structure can return to its

original position after an earthquake.

Glossary G-53

Resolution

The smallest change (in a parameter, signal, instrument output, etc.) that can be meaningfully detected

and used.

Resonance

A resonance occurs in a mechanical system excited by a harmonic force if the excitation frequency is

equal to the resonant frequency of the system. A small change in the frequency of excitation in either

direction will cause the system response to decrease rapidly if the system damping is low. At a resonance,

very little energy input into a structure results in a very large displacement (for low damping levels).

Resonant frequency is almost equal to (but less than) the undamped natural frequency when the

damping is low. Although the resonant frequency is nearly equal to natural frequency, the two concepts

are not the same. Resonant frequency is excited by a harmonic forcing function at that frequency. The

natural frequency is excited by an initial condition (corresponding the particular mode shape) followed

by a free (unforced) response.

Resonance search

A vibration testing procedure whose objective is to identify and observe various resonances in a test

object. A harmonic excitation with variable frequency may be used for this. See Sine sweep.

Resonator absorber

A method in which a vibration absorber (see Vibration absorber) is tuned to have a natural frequency

equal to the external excitation frequency to mimic optimally the vibratory energy from the primary

system (system of concern). The absorber then “absorbs” the energy of vibration of the primary system,

thereby suppressing those vibrations. Actually, the absorber generates a force in the opposite direction to

the force of vibration of the primary system.

Response

The response of a device or system is the motion (or other output) resulting from an excitation (force,

stimulus, or input) under specified conditions.

Response spectrum (RS)

A plot of maximum amplitudes (acceleration, velocity, or displacement) of a single-DoF oscillator

(Single-DOF) as the natural period (or natural frequency) of the Single-DOF is varied across a spectrum

of engineering interest (typically, for natural periods from 0.03 to 3 or more seconds, or frequencies of

0.3 to 30 Hz; in seismic applications). One RS curve is obtained for a specified damping ratio of the

Single-DOF system.

Response spectrum analysis

Computation of the response spectrum (RS) of a given signal (see Response spectrum). This is quite

different from spectral analysis (or spectrum analysis), which is to compute the Fourier spectrum of a

signal. These are different from spectral density analysis, which determines the power spectral density

function of a “random” signal.

Reverse whirling

Whirling is the rotation of a spinning rotor center of mass, which has an offset (due to shaft deflection)

with respect to the axis through the bearings of its shaft. Reverse whirling occurs if the direction of this

rotation about the bearing axis is in the opposite direction to the angular speed of spinning of the rotor

with respect to a fixed frame of reference.

Reynolds number

A dimensionless parameter (that depends on viscosity and velocity), whose value determines if a fluid

flow is laminar or turbulent.

G-54 Vibration and Shock Handbook

Ridge

Curve of local maxima in a mean-square wavelet map.

Rigid bodies

Bodies exhibiting no deformation under acceleration or forces.

Rigid-body mode

A mode of motion that corresponds to zero natural frequency. Such a mode will exist in an unrestrained

structure (no end fixtures) that is free to move as a rigid body without any deformation.

Rigid-body motion

A structural motion that results in no deformations or strains. By definition, these are zero frequency

motions.

Rigid – perfectly plastic force – deformation characteristics

Characteristics of hysteretic loops produced by an ideal friction-damping device.

Rigid structure

A structure that cannot be deformed and whose shape cannot be changed.

RMS value

Root-mean-square value. Square root of the arithmetical average of a set of squared instantaneous values

of a response quantity.

Rolling element bearings

A rolling element bearing generally consists of two rings (races or raceways) with a set of rolling elements

running in their tracks. The rolling elements take the form of balls or various types of rollers. The outer

race is located in a housing and the inner race on the shaft.

Root cause failure

The basic cause for a given failure.

Root Mean Square

See RMS value.

Rotating machinery

The group of electrical or mechanical machinery composed of a rotating component that rides on some

form of bearings encased in a stationary casing (housing).

Rotor

The rotating part (as opposed to the stator) of an electrical or mechanical machine.

Rotor instability

An excitation mode, which is self-induced and does not require a sustained forcing phenomenon to

initiate or maintain the motion. It occurs in machines operating at speeds well above the critical speeds of

the rotor, and the rotor whirling speed is identical to the critical speed irrespective of the rotor (spinning)

speed.

Rotor internal damping

A damping phenomenon, which occurs within the internal fibers of a material due to friction when there

is relative movement between the fibers.

Glossary G-55

Rotor track

The vertical position of the blades of a rotor, usually measured by optical sensors.

Rotor tuning

The process of adjusting the rotor blades of a jet engine to reduce the system vibration (e.g., aircraft) and

the spread of rotors.

Rotordynamics

The dynamics of rotating machinery.

Round-off error

The error created by the finite representation of numbers (i.e., finite number of digits) within a

computer. Similar to Quantization error.

RRS

In a dynamic test, the response spectrum specification that has to be satisfied (enveloped) by the

excitation signal.

Rub

A condition when a moving part comes into contact with, and slides against, a stationary part.

Run-to-failure maintenance

A strategy where maintenance is performed only when machinery has failed. See Scheduled maintenance,

Predictive maintenance, Preventive maintenance, and Condition-based maintenance.

Runge – Kutta

Numerical methods (integration schemes) based on approximating the solution by a Taylor series, and

weighted averages of corresponding differences.

RVDT

Rotary variable differential transformer. See LVDT. Similar to an LVDT, but for measuring angular

displacements.

Safe shutdown earthquake (SSE)

An earthquake representation (e.g., by a response spectrum) for which a nuclear reactor can be safely

shutdown without causing damage or fault in equipment due to the earthquake.

Sampling

The process of converting continuous signals into discrete values for digital processing.

Sampling frequency

The rate at which samples of a continuous time signal is recorded in discrete fixed time periods.

Sampling theorem

When a signal is sampled, its spectrum has no meaning beyond half the sampling frequency, called

Nyquist frequency.

Saturation

A nonlinearity state in a device where a further increase in input no longer changes the output.

Scaling

To stretch or compress a function.

G-56 Vibration and Shock Handbook

Scheduled maintenance

A strategy where maintenance is performed at set time intervals (or duty cycles) in order to maintain a

significant margin between machine capacity and actual duty.

SEA

Statistical energy analysis.

Second moment of area

Similar to moment of inertia, except uses area elements rather than mass elements. (The two are

proportional for uniform material density and constant thickness along the axis.)

Secondary (or Residual) negative shock response spectrum

SRS plotted, considering the highest negative response observed after the end of the shock.

Secondary (or Residual) positive shock response spectrum

SRS plotted, considering the highest positive response observed after the end of the shock.

Secondary system

A general term for equipment attached to a primary structural system.

Seismic

Measurements and activity concerning earth motion, such as earthquakes.

Seismic gap

A gap provided between a seismically isolated building and the ground so that the ground-level story can

move in relation to the ground during earthquake excitations.

Seismic hazards

The phenomena or expectation of an earthquake-related agent of damage, such as fault rupture,

vibratory ground motion (i.e., shaking), inundation (e.g., tsunami, seiche, dam failure), various kinds of

permanent ground failure (e.g., liquefaction), fire, or hazardous material release.

Seismic isolation

Application of base isolation technology to structures subjected to earthquake motions.

Seismic risk

The product of the hazard and the vulnerability (i.e., the expected damage or loss, or associated full

probability distribution).

Seismicity

The geographic and historical distribution of earthquakes.

Self excited instability

Same as Rotor instability.

Self-induced

Created within itself and does not require an external influence.

Self-organizing map (SOM)

A type of unsupervised neural network that yields a graphical representation of input data where similar

input vectors are placed in close proximity.

Glossary G-57

Semiactive vibration control

A vibration control system in which both active and passive (see Active vibration control and Passive

vibration control) treatments are combined to reduce the amount of external power necessary to achieve

the desired vibration control performance.

Shaker

See Exciter.

Shape function

Is a spatial function (i.e., function of position vectors) describing the change of a variable within an

element domain.

Shear balance method

An analysis method for sliding isolated structures, which uses the force balance condition of total shear

under the isolation system to compute the friction force.

Shear force

An internal force acting normal to the longitudinal axis given by the algebraic sum of all forces to one side

of the section chosen.

Shear wall

A structural wall between stories of a building to increase shear strength but at the expense of increasing

stiffness.

Shifting

The process of translating (moving) a function with respect to some reference (coordinate system).

Shock

Transient condition where the equilibrium of a system is disrupted by a sudden applied force or by a

sudden change in the direction or magnitude of a velocity vector. The time duration of this change is

shorter than about twice the natural period of the excited mechanical system.

Shock duration

Interval of time during which the shock pulse has a higher acceleration value than some specified small

fraction of its maximum value (usually 10%).

Shock machine (or shock test machine)

Device for subjecting a system to controlled and reproducible mechanical shock. Two principal categories

are usually distinguished. Impulse machines: increase the velocity of the test item during the shock. The

initial velocity is generally zero. Impact machines: decrease the velocity of the test item throughout the

shock or change its direction.

Shock motion

An excitation involving motion of a foundation.

Shock programmer

Same as Shock simulator. Device on a shock machine that is used to format the applied shock waveform

according to a desired shape during impact (usually half-sine, terminal peak sawtooth, and rectangular

pulses).

G-58 Vibration and Shock Handbook

Shock pulse

A substantial disturbance characterized by a rise of acceleration from a constant value to a substantial

peak value, and decay of acceleration to the constant value in a very short period of time. Shock pulses are

normally displayed graphically as curves of acceleration as functions of time.

Shock pulse index

An accelerometer tuned to have its resonant frequency excited by a specific shock can provide early

warning of fault (e.g., deteriorating bearings). The sensor output level is the shock pulse index (also

known as spike energy).

Shock response spectrum (SRS)

The response of an array of single-DoF systems to a shock input represented as the maximum

response versus the natural frequency of single-DoF systems for a specified damping ratio. This

response may be expressed in units of acceleration, velocity, or displacement.

Shock simulator

Same as shock programmer. Device on a shock machine that is used to format the applied shock

waveform during impact according to a desired shape (usually half-sine, terminal peak sawtooth, and

rectangular pulses).

Shock spectrum

See shock response spectrum. A plot (amplitude vs. natural frequency) of the response of a naturalfrequency-

variable single-DoF system to an applied shock for a specified damping ratio. This response

may be expressed in units of acceleration, velocity, or displacement.

Shock wave

A strong pressure wave produced by explosions or other phenomena that create violent changes in

pressure.

Shoring

Temporary support at the base floors of a structure during replacement of its originally fixed support to

seismically isolated support.

Short-term statistics

Statistics of a random process that are only good for a few hours or a few days at the most. It is assumed

that the significant wave height and the wave spectrum are invariant with time.

Short time FFT

Time sequence of concatenated FFT of short-period records of a signal.

Shroud of impeller

The circular section that forms a sidewall of a pump or compressor impeller.

Signal-to-noise ratio (SNR)

Determined by both electronic and mechanical noise (but not thermal noise due to the fluctuation –

dissipation theorem).

Signature spectrum

A base-line frequency spectrum taken from new or recently overhauled machinery.

Glossary G-59

Significant wave height

A statistical description of random waves. Average of the highest one third of all waves or four times the

variance. Consistently very close to the observed wave height.

Simple beam

See Simply supported beam. A beam restrained at its end only against vertical movement.

Simple harmonic motion

See Simple harmonic vibration.

Simple harmonic vibration

A motion such that the displacement, velocity and acceleration are pure sinusoidal functions of time.

Sometimes called “harmonic motion.”

Simple oscillator

A single-DoF mass – spring – damper system (or an LRC electric circuit), which can produce damped

oscillations at one frequency (damped natural frequency). An undamped simple oscillator is a harmonic

oscillator, which can produce a periodic oscillation described mathematically by either of the harmonic

trigonometric functions (sine or cosine), and has only one frequency component (undamped natural

frequency).

Simple (or perfect) shock

Shock whose signal can be represented exactly in simple mathematical terms. Standards specify generally

one of the three following: half-sine (approached by a versed sine waveform); terminal peak sawtooth;

and rectangular shock (approached by a trapezoid waveform).

Simply supported beam

A beam pinned (or simply supported) at its ends so that the displacement and bending moment at these

ends will be zero. Slopes and shear forces will not be zero.

Sine beat

A signal formed by adding two sine signals whose frequencies nearly equal, so as to generate beats. Such a

signal is used for excitation in some types of vibration testing.

Sine-dwell

A vibration test where a shake applies a sinusoidal excitation of a specific frequency to the test object over

a specific duration, changes the frequency, and repeats the dwell in specific frequency steps.

Sine sweep

A vibration test procedure where a harmonic excitation is applied to a test object (using an exciter or

shaker) and the frequency of excitation is slowly and continuously varied at a specified rate.

Single-category-based classifier (SCBC)

The pattern classifier that performs abnormality scaling by comparing the value of its feature with its

no-fault value.

Single-degree-of-freedom (single-DoF) system

A system with only one coordinate (strictly, one incremental coordinate to include nonholonomic

systems) to define completely the configuration of the system at any instant.

Single plane balancing

See Static balancing, which is synonymous.

G-60 Vibration and Shock Handbook

SISO

Single-input – single-output system. A system with just one input and one output.

Skeleton

Wavelet transform values along a ridge.

Sliding bearing

A type of base isolator that protects a structure from ground excitations by using a sliding frictional

interface at the base.

SNR

Signal-to-noise ratio, determined by both electronic and mechanical noise (but not thermal noise due to

the fluctuation – dissipation theorem).

Soft foot

A condition that occurs when one of a machine’s hold-down bolts are not tight enough to resist the

dynamic forces exerted by the machine.

Soft story

A story of a building that is more flexible than the other stories so that the stories above this story are

isolated from the ground shaking.

Softened compression stress – strain relationship

Viewing the shear action on a membrane element as a two-dimensional problem, the compressive

strength of concrete in one direction is reduced by cracking due to tension in the perpendicular direction.

Softening parameter

When a membrane element is subjected to shear, the compressive strength of concrete is reduced to

approximately 40 to 60%.

Soil – structure interaction

A dynamic interaction between a structure and its supporting soil, in which dynamic properties of one

will affect the response of the other.

Solenoid

A linear (rectilinear) two-position (on – off) actuator. A ferromagnetic rod surrounded by a coil of many

turns. An electric current through the coil will generate a flux that will move the rod, thereby turning on

the solenoid. A version (linear motor) is able to provide continuous movements.

Sone

Unit of the loudness that is defined as the loudness of a 1000 jHz pure tone with a sound pressure level of

40 dB. This relationship is expressed as S ¼ 2ðLL 240Þ=10; S (sone) and L (phons).

Sound absorbing material

Porous materials such as glass fiber blankets used to attenuate sound propagating in ducts to improve

sound transmission loss and to reduce standing waves in air spaces.

Sound bridge

Structures such as studs or stud bolts connecting the plates added to a noise source in insulation lagging

or the plates of a double wall. They can considerably reduce insertion loss of the lagging or transmission

loss of the double wall.

Glossary G-61

Sound insulation lagging

Thin plates separated by airspace or sound absorbing materials added to noise sources such as pipes,

ducts and machines to isolate their noise.

Sound intensity

The time-average rate of energy flow per unit area, denoted by the vector I (W/m2).

Sound level

Sound pressure level that has been weighted according to particular weighting curves: A- B-, C-weighting

networks and associated levels. The frequency response and decibel conversions from a flat response for

each of these weightings are available.

Sound power level

The acoustic power radiated by a source with respect to the international reference of 10212 (watts),

expressed in decibels (dB).

Sound pressure

The difference between the instantaneous pressure at a fixed point in a sound field and the pressure at the

same point without the sound (Pa). Alternatively, this may be defined as the ratio (dB) of the

instantaneous pressure at a fixed point in a sound field with respect to a reference of 2 £ 1026 (Pa), for

which a pure tone of sound is assumed.

Span

The distance between two consecutive supports of a beam, truss, bridge, guideway, etc.

Spatial coupling

Effect of multiple support motions occurring when the attached equipment is mounted at multiple

locations of the structure.

Specific fan noise level

Ratio (dB) of sound power generated from a fan due to its pressurized flow with respect to some standard

power level.

Specifications

Conditions that must be satisfied by operating or design parameters during operation or design of a

product in order to satisfy the objectives of the product.

Spectral density

See Power spectral density. Fourier transform of the autocorrelation function of a random signal

(random process). It shows how the variance of the system is distributed over frequency. It may

also be expressed as a “spectral energy density” since the unit of the variance often has the unit of

energy.

Spectrogram

A graph of time vs. frequency, e.g., the Gabor distribution.

Spectrum

A description of a quantity as a function of frequency or wavelength. Fourier transform of a signal is

called the Fourier spectrum (or simply a spectrum).

G-62 Vibration and Shock Handbook

Spherical wave

When propagation of sound wave is described by radial (polar) coordinates, the sound field is

represented by a “spherical wave,” which is generated by a point monopole.

Spike energy meter

An accelerometer tuned to have its resonant frequency excited by the impacts in a deteriorating bearing,

thus giving an early warning of an impending failure.

Spindle speed selection

In machining: a technique where the spindle speed is chosen to avoid or suppress chatter.

Spindle speed variation

In machining: a technique where the spindle speed is continuously adjusted to suppress chatter.

Spring

The basic flexibility element characterized by its stiffness, force, or deflection. Stores elastic potential

energy. In three-dimensional problems, a spring may assume three orthogonal stiffness parameters.

SRSS

Square-root of sum of squares of a set of values.

SSE

See Safe shutdown earthquake.

Stability

Stability of a dynamic system: If the natural (unforced) response of a system decays to zero, then the

system is asymptotically stable. If for a finite excitation, the system response is bounded, we have

bounded-input – bounded-output (BIBO) input. Stability of a numerical method: Numerical errors do

not grow without bound.

Stability lobe diagram

Chatter stability chart in the process parameter space.

Standardized shock response spectrum

See Shock response spectrum (SRS). Standardized or reduced SRS plotted in dimensionless coordinates.

The abscissa is the product of natural frequency and shock duration. The ordinate is the ratio spectrum

value or shock pulse amplitude.

Standoff distance

Distance away from a specific structure that determines the amount of damage that would occur if an

explosion of a specific size was detonated. For standoff threats, the distance away from a potential vantage

point with direct line of sight to the asset.

State space formulation

A generalized analytical model for a dynamic system expressed as a minimum set of first-order ordinary

differential equations (state equations). The associate variables are the state variables that may include

structural displacements and velocities, which will represent the dynamic state of the system.

Static balancing

See Balancing. Balancing achieved on a single plane of vibration perpendicular to the axis of

rotation.

Glossary G-63

Static loading

Constant loading of a member.

Stationary process

A random process where the statistics can be considered independent of time. Stationary process is used

synonymously with weakly stationary process where the first-order statistics such as the mean and the

variance are independent of time, and the second-order statistics such as the autocorrelation function

depend on the difference in time.

Statistically determinate

A structure is considered statistically determinate if all the support reactions and internal forces in the

members can be determined solely by the equations of static equilibrium.

Stator

The stationary structure surrounding the rotating member (rotor) of a rotating electrical or mechanical

machine.

Steady-state error

Offset. The error in the response of a system under steady-state conditions.

Steady-state vibration

Steady-state vibration exists in a system if the velocity of each particle is a continuous periodic quantity.

Stiffness

Characteristic parameter of a spring element. Ratio of change of force (or torque) to the corresponding

change in translational (or rotational) deflection of an elastic element, i.e., force required to deflect an

elastic element through a unit distance. Typical units are Newtons/meter (N/m).

Stiffness asymmetry

The stiffness (or flexibility) is not symmetrical in two or more orthogonal directions.

Stiffness law

Transmission loss characteristic of a single plate that the transmission loss depends on frequency and

equivalent stiffness of the plate below the first-order natural frequency of the plate.

Stiffness matrix

The matrix multiplied by the displacement vector in the Newtonian form of differential equations of

motion of a mechanical (vibrating) system, expressed in the vector-matrix form.

Stochastic process

An indexed collection of correlated (in general) random variables.

Stochastic resonance

Counterintuitive increase of SNR when the noise level is raised. Only possible in nonlinear systems.

Stokes law

Viscous friction force on a sphere of radius R, given by 6mRv, where m is the fluid’ viscosity.

Strain

Change in length divided by length. A nondimensional measure of the physical deformation or deflection

resulting from stress (force per unit area).

G-64 Vibration and Shock Handbook

Strain gauge

A miniature resistive element (metal foil or semiconductor) whose resistance changes when stretched

(deformed), which can be measured using a Wheatstone bridge circuit. When properly mounted on

(i.e., bonded to) a structural member, it can measure the local direct strain in the member.

Stress – strain curve of a steel bar in concrete

A steel bar in concrete is stiffened by tensile stress of concrete. It relates the average stress to the average

strain of a large length of bar crossing several cracks.

Stretched exponentials

Discovered with Leyden jars (capacitors), a multiexponential model used by some to explain features of

anelasticity.

Stribeck damping

A sophisticated friction model where the coefficient of friction drops rapidly and then increases gradually

(somewhat linearly) as the relative velocity increases from zero.

Stroke

The maximum rectilinear movement that is possible in a device.

Strouhal frequency

Frequency at which the vortices are shed from a fixed cylinder subjected to a constant velocity flow. It is

proportional to the diameter of the structure and inversely proportional to the stream velocity.

Structural influence

The coefficient that represents the proximity effect of the accelerometer generating a feature of a machine

(gearbox) component.

Structural wall

An in-plane wall to reduce the relative interstory distortions of a building caused by seismic-induced

motions.

Structure-based connectionist network (SBCN)

In machine monitoring for fault detection, this network takes advantage of the integration capability of

neural networks, but avoids the need for supervised training. In the SBCN, the network weights

are defined according to the “structural” knowledge of the monitored machine (e.g., gearbox) and

the type of fault represented by various vibration features. The structural influences in the SBCN,

which represent the proximity effect of faults on accelerometers, are determined according to the

RMS value of the frequency response of a simplified lumped-mass model of the monitored machine

(e.g., gearbox).

Structure-borne noise

Type of noise that is based on the generation and propagation of timewise varying motions and forces in

solid bodies, as well as on the associated sound radiation.

Subharmonic

Sinusoidal quantity of a frequency that is an integral submultiple of a fundamental frequency.

Subharmonic resonancy

A phenomenon observed in sliding isolation where the isolated system possesses extra harmonic resonant

frequencies in the low-frequency range in addition to the original ones as a result.

Glossary G-65

Substructured online hybrid test

A test in which only inelastic parts of a structure are experimentally tested, while the remaining elastic

parts are modeled in a computer. Both parts are interconnected using online communications so that the

complete structural responses are obtained.

Substructuring

Modeling, analysis and design development method where subsystems are developed first and assembled

together (interconnected) through appropriate connecting elements. The analysis is simplified by using

interconnection laws and making use of the results for the subsystems.

Subsystem structure

A structure that is regarded as a subsystem in the analyzing process of statistical energy analysis (SEA).

Sweep rate

The rate at which the frequency is changed in a sine-sweep test.

Symmetric matrix

The corresponding elements of a symmetric matrix about its main diagonal are identical. Specifically,

aij ¼ aji where a is an element, i is the row number, and j is the column number. The transpose does not

change the matrix in this case.

Synchronous averaging

Averaging of the vibration signal synchronous with the running speed of the machinery being monitored.

Synchronous whirling

See Whirling. The angular velocity of whirling rotation of the rotor geometric center is equal to the

rotational speed (spin) of the rotor shaft.

Tachometer

A speed sensor. See DC tachometer. AC tachometers have a primary coil with an AC signal (carrier

signal), a secondary coil whose induced signal is demodulated to determine the speed, and a rotor that is

either a permanent magnet of a closed conduction (as in induction motor).

Thermal bearings

Flexible bearings used in long span bridges for accommodating the changes of deck length due to varying

temperature to protect bridge piers from receiving large lateral forces.

Terminal peak sawtooth shock (TPS)

Same as Final peak sawtooth shock (FPS). A simple shock for which the acceleration – time curve has the

shape of a triangle where acceleration increases linearly up to a maximum value and then instantly

decreases to zero.

Test factor

Where the shock test specification is derived from measurements of the real environment, this is the

coefficient applied to mean SRS of the measured shock pulses to take into account the variability of the

resistance of the test item and the very small number of specimens tested (often only one specimen).

Test response spectrum

See TRS.

Thermoelastic damping

Energy dissipation (damping) due to the anharmonic, interatomic potential independent of defects.

G-66 Vibration and Shock Handbook

Threshold

The minimum value a stimulus may have to create a desired effect.

Threshold of hearing

This is defined for binaural listening as the lowest sound pressure in free field that one can just hear as the

signal reduced.

Threshold speed

The speed at which a stimulus is perceptible or can produce an effect.

Throttling

Reducing the fluid flow in a system by a valve or regulator.

Time constant

The inverse of the real part (disregarding the negative sign) of an eigenvalue. Inversely proportional to the

speed of response and the level of stability.

Time-dependent effects

Normally referred to material properties that are influenced by motion history.

Time domain

Domain of analysis of a dynamic (vibrating) system where the independent variable is time (t). In this

domain, vibration data (and systems) are represented, displayed or analyzed as a function of time.

Time-domain analysis

Analysis of dynamic (vibration) systems and response (motion) results, which are described with respect

to time.

Time-domain simulation

Computer simulation of the response of a dynamic system as a function of time. Technique to analyze

chatter where delay-differential equations defining the machining forces and structural vibrations are

numerically simulated in the time domain.

Time-history analysis

The analysis of a response of a dynamic system (e.g., structure) with respect to time.

Time waveform index

A single number representative of a waveform, calculated using a raw vibration signal and used for

trending and comparisons.

TNT equivalent

The weight of trinitrotoluene (TNT) that has an equivalent energy output to that of a specified weight of

another explosive compound.

Tool chatter

Self-excited vibration between the cutting tool and the workpiece due to a cutting tooth machining over a

portion of the workpiece that was machined by a previous tooth of the cutter. This causes uneven chip

thickness and consequently, excessive self-excited vibrations.

Tool condition monitoring (TCM)

A method by which the condition or state of cutting tools can be monitored.

Glossary G-67

Tool condition monitoring system (TCMS)

A combination of sensors and algorithms to achieve TCM.

Tool wear

The loss of material on the edges of a cutting tool due to machining.

Tooth passing frequency

Frequency at which the cutting tool teeth pass a given point during a spindle revolution.

Tooth passing period

Time interval for two successive cutting tool teeth to pass a given point during a spindle revolution.

Tornado

A violently rotating column of air in contact with, and extending between, a convective cloud and the

surface of the Earth. It is the most destructive of all storm-scale atmospheric phenomena. They can occur

anywhere in the world given the right conditions, especially after landfall of hurricanes.

Torque whirl

A whirling instability that can occur in overhung, long, slender, rotors with high load torque values. See

Whirling.

Torsional response

Response causing a twisting motion.

Torsional stiffness

Torque applied to a member divided by the resulting angular displacement (twist) of the member.

Torsional vibrations

Oscillatory angular motion causing twisting of a shaft (carrying a rotor) in a rotating machine.

Tracking analyzer

Devise used to record and analyze data from machines with variable speed.

Transducer

A device that senses a physical quantity is called a sensor. A transducer is a device that converts the sensed

signal (measurand) into a form (typically, an electrical output signal of sufficient value) proportional to

the measured variable. Often, the terms sensor and transducer are used interchangeably.

Transfer function

The ratio of the output of a dynamic system to its input in the Laplace domain or frequency domain

(see Frequency domain and FRF) assuming zero initial conditions.

Transient-response analysis

Sometimes called time-history analysis, this determines the dynamic response of a dynamic system

(e.g., structure, machine) under the action of time-varying excitation. Excitation forces are explicitly

defined in the time domain. Computed response usually includes the time-varying displacements,

velocities, accelerations, strains, and stresses.

Transient vibration

The vibratory motion of a mechanical system as it changes from one state to another as a function of

time. This may be a decaying or growing signal but not a steady vibration. It may consist of forced or free

vibration or both.

G-68 Vibration and Shock Handbook

Transmissibiliy

Transmitted vibration (force or motion) divided by an applied vibration (force or motion). Inverse of

Isolation.

Transmission loss

The difference between incident and transmitted acoustic intensity levels. It is used to rate the sound

insulating properties of a wall or some other structure or device. For a muffler, this is the ratio between

the energy in incident sound wave at the muffler inlet and energy transmitted at the muffler outlet into

perfectly absorbing termination.

Transmitted power

An acoustic power that is transmitted from a subsystem to another subsystem.

Trapezoidal shock

Simple shock for which the acceleration – time curve grows linearly up to a given value, remains constant

during a certain time, and then decreases linearly to zero.

Trim tab adjustments

Trim tabs are adjustable surfaces on the trailing edge of the rotor blades of a rotating machine (e.g., jet

engine). They affect the aerodynamic pitch moment of their foils and consequently, their vibration

characteristics. Tab adjustments are measured in thousandths of an inch (thou), with positive and

negative changes representing upward and downward tabbing, respectively.

TRS

Test-response spectrum. The response spectrum of the excitation signal that is applied to a test object

during testing.

Truncation error

The error created by deleting a signal after a finite time period to obtain a finite segment before digitally

analyzing it. This is equivalent to multiplying the signal by a rectangular window. The error can be

reduced (or ripples in the frequency spectrum of the truncated signal can be suppressed) by using other

types of window functions with smoothing at the two ends.

Tuning

The process of adjusting the controller parameters in order to achieve the required performance (e.g.,

vibration control).

Unbalance (or imbalance)

A condition when the center of mass of a rotating component is not coincident with the center of

rotation.

Unbalance response

The displacement vibratory response of a rotor to its mass imbalance as a function of rotor speed.

Uncertainty factor

Where the shock test specification is derived from measurements of a real environment, this is the

coefficient applied to the mean SRS of the measured shock pulses to take into account the variability of

the shock environment and the variability of the resistance of the material.

Glossary G-69

Uncertainty principle

The product of uncertainties of conjugate variables is greater than or equal to a constant, which in

quantum mechanics is Planck’s constant. Also known by the name Heisenberg.

Undamped natural frequency

The frequency of free vibration resulting from only elastic and inertial forces of the system while

neglecting damping.

Undamped system

A system where there is no energy dissipation. It has energy storage capability (e.g., inertia elements to

store kinetic energy and springs to store elastic potential energy). If there is no external force applied on

the system, then the system will experience free steady vibrations. Motion of the system are established by

an initial disturbance (i.e., initial conditions).

Undetermined coefficient method

Use a trial solution consisting of the sum of possible set of functions multiplied by unknown

coefficients. Substitute in the equation to be solved and equate like terms to determine the unknown

coefficients.

Uplift force

Force perpendicular to the flow.

Van der Pol oscilaltor

Classic nonlinear electronic oscillator.

Variable frequency drives

A controlling device to vary the frequency of AC supplied to an electric motor in order to vary

proportionately the speed of rotation of the motor.

Variable structure control (VSC)

A class of nonlinear switching controllers whose structure consists of different phases. Typically

associated with a discontinuity function (e.g., signum function). When the system response enters a

different phase, a different controller (corresponding to the particular phase) is switched on.

VCC

Voltage to current converter.

Velocity

A vector quantity that specifies the magnitude and direction of the rate of change of displacement with

time along a specified axis with respect to a reference frame (m/sec). First derivative of displacement with

respect to time. It may refer to angular motion as well as to uniaxial (rectilinear) motion. Where the

reference frame is not inertial, the velocity is often designated “relative velocity.”

Velocity change

Variation of the velocity during a shock, i.e., integral of the acceleration time-history signal over the

shock duration.

Velocity of sound

Velocity of sound wave propagation. In air, it is proportional to the square root of ambient pressure and

of the inverse of ambient density, or it is proportional to the square root of the absolute temperature. For

example, in stationary air, the velocity is 340.4 m/sec at 158C.

G-70 Vibration and Shock Handbook

Velocity shock

Shock motion characterized by a sudden velocity change of the support (base) of a system.

Versed-sine (or Haversine) shock

Simple shock for which the acceleration – time curve has the form of one period of the curve

representative of the function [1 2 cos( )], with this period starting from the zero value of the function.

It is thus a signal ranging between two minima of a sine wave.

VFC

Voltage to frequency converter.

Vibration

The oscillating, reciprocating, or other periodic motion of a rigid or elastic body or medium (such as a

floor) when its position or state of equilibrium has been changed.

Vibration absorber

A secondary system (usually a mass – spring – damper trio with zero or low damping) to protect the

primary device from vibrating. Typically, the secondary system vibrates at its natural frequency

(resonance) generating an opposing force to suppress vibration in the primary system.

Vibration absorption

Addition of a vibration absorber to protect the primary device from vibrating.

Vibration analyzer

The device that could monitor and analyze the machine conditions using the machine’s vibration signals.

Vibration control

Means of controlling vibration using external or internal forces and loads.

Vibration features

Aspects of vibration that can illuminate the effect of individual faulty components of a machine (e.g.,

gearbox) on vibration. Usually, vibration features are defined as functions of specific frequencies affected

by the components.

Vibration isolation

See Isolation. Isolating the source of vibration from the concerned system protecting a device from

vibration of its point of attachment. Motion isolation and force isolation are of interest.

Vibration measurements

For a bladed rotating machine (e.g., jet engine, turbine, compressor), obtained at the frequency of once

per blade revolution (1 per rev) by two accelerometers, A and B, attached to the sides of the cockpit. The

vibration data is vectorially combined into two components: A þ B, representing the vertical (heave)

vibration of the system (e.g., aircraft), and A 2 B, representing its roll vibration.

Vibration meter

Small, handheld, inexpensive, simple-to-use, self-contained device that gives an overall vibration level

reading.

Vibration mode

See Mode shape. For a flexible plate, the vibration represents the particular spatial configuration of

vibration amplitude (i.e., mode shape) with phase information of the moving elements, and the

associated natural frequency. The actual vibration of the plate is obtained by the superposition of the

Glossary G-71

distinct vibration modes. For continuous (i.e., distributed-parameter) systems (e.g., plate), there are

infinite distinct modes, many of which at the high-frequency end can be neglected depending on the

frequency range of operation.

Vibration modeling

The process of converting an engineering vibration problem into a mathematical model, whereby the

major vibration characteristics of the original problem can be accurately predicted.

Vibration mounts

Devices that are placed at the base (or legs) of a machine or equipment to reduce its vibration problems

and specifically, to reduce the vibration forces transmitted from the machine or equipment to the

support structure and its environment. A key purpose is vibration isolation. Inertia blocks may be

attached to the machine base and the mounts.

Vibration response

The output of a vibrating system for an arbitrary input.

Vibration signature

A unique frequency response or pattern that corresponds to a particular input or excitation signal.

Vibration standards

National and international vibration standards and guidelines published by recognized professional

organizations and regulatory agencies. The objective of these publications is to establish and control

quality, safety, durability and reliable performance of rotating machinery.

Viscoelasticity

Model of anelasticity where an elastic solid also has fluid like (viscous) properties.

Viscous damping

The dissipation of energy that occurs when a particle in a vibrating system is resisted by a force that has a

magnitude proportional to the magnitude of the relative velocity of the particle and a direction opposite

to the direction of the particle.

Volume coefficient of elasticity of air

Generally, isothermal conditions prevail for frequencies below 100 Hz, whereas adiabatic conditions

prevail above 1 kHz. In between these limits, there is a volume coefficient of elasticity (similar to bulk

modulus) that varies between 1.0 £ 105 and 1.4 £ 105 N/m2.

Volute

One of the major components of a centrifugal pump forming the casing around the impeller, which

channels the flow exiting the impeller into the outlet of the pump while converting the velocity energy

generated by the impeller into pressure energy.

Vortex

Any circular or rotary flow in the atmosphere that possesses vorticity.

Vulnerability

The expected damage given a specified value of a hazard parameter.

G-72 Vibration and Shock Handbook

Wake

Region surrounded by the shear layers originating from separation points.

Wake excitation

Excitation caused by the vortices in the wake.

Waterfall plots

Successive spectra plotted with respect to time and displayed in a three-dimensional manner. Also known

as Cascade plots.

Wavelength

In acoustics, the distance between successive two peaks in the repetitive spatial variation of the pressure

or the particle local velocity.

Wavelet

Functions having representations in the time and frequency domains of small support set.

Wavelet analysis

Similar to Fourier analysis, wavelet analysis is the breaking up of a time waveform into shifted and scaled

versions of a wavelet. A wavelet is a waveform of limited duration with an average value of zero.

Wavelet transform

Representation of a function as a series of wavelets.

Wear-in failures

Failures that occur at the beginning of a machine’s useful life, usually characterized by a relatively high

rate of failure.

Wearout failures

Failures that occur as a machine reaches the end of its design life.

WECPNL

Acronym of weighted equivalent continuous perceived noise level. It is very close to LAeq in concept and it

would not be difficult to change the assessment index from WECPNL to LAeq. WECPNL is used for the

assessment of aircraft noise.

Whipping

A term used to describe relatively large motion associated with whirling of a rotor. Typically, occurs when

the whirling speed matches the critical speed of the rotor.

Whirling

Condition when the rotor shaft bends so that the center of mass of the rotor rotates about the fixed axis

through the bearings while spinning about the shaft geometric axis. See Synchronous whirl.

White noise process

A random process where the energy is equally distributed over the entire frequency domain. It is a

mathematical simplification rather than a real process.

Wideband

Corresponding to a wide (large) frequency band.

Glossary G-73

Wideband process

A random process where most of the energy is in a wideband of frequency.

Wiener– Khinchine relationship

Relates the autocorrelation function to the spectral density of a random process.

Young’s modulus

See Modulus of elasticity.

Zener model

Classic description of damping that involves relaxation processes — named after the individual who also

contributed to a well-known voltage controlling diode.

Zero drift

See Drift. The drift in a device under conditions of zero output.

Zero period acceleration

See ZPA.

Zero shift

Slow variation of the average of a signal due to the fact that high accelerations make the crystal of the

accelerometer work in a temporarily nonlinear field. This shift can affect the calculation of the shock

response spectrum.

Zeros

The roots of the numerator polynomial equation of a transfer function. Together with poles, they

represent the entire system (transfer function) and govern the system behavior (stability, phase, speed of

response, percentage overshot, rise time, etc.).

ZPA

Zero period acceleration. The acceleration shown on a RS or shock SRS at infinite frequency. This can be

used as a representative parameter of a signal (e.g., earthquake ground motion) for testing and designing

a system.

G-74 Vibration and Shock Handbook

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