15.4 Performance Specification

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Proper selection and integration of sensors and transducers are crucial in “instrumenting” a vibrating

system. The response variable that is being measured (for example, acceleration) is termed the

measurand. A measuring device passes through two stages in making a measurement. First, the

measurand is sensed. Then, the measured signal is transduced (converted) into a form that is particularly

suitable for signal conditioning, processing, or recording. Often, the output from the transducer stage is

an electrical signal. It is common practice to identify the combined sensor– transducer unit as either a

sensor or a transducer.

The measuring device itself might contain some of the signal-conditioning circuitry and recording (or

display) devices or meters. These are components of an overall measuring system. For our purposes, we

shall consider these components separately.

In most applications, the following four variables are particularly useful in determining the response

and structural integrity of a vibrating system (in each case the usual measuring devices are indicated in

parentheses):

1. Displacement (potentiometer or LVDT)

2. Velocity (tachometer)

3. Acceleration (accelerometer)

4. Stress and strain (strain gage)

It is somewhat common practice to measure acceleration first and then determine velocity and

displacement by direct integration. Any noise and DC components in the measurement, however, could

give rise to erroneous results in such cases. Consequently, it is good practice to measure displacement,

velocity, and acceleration by using separate sensors, particularly when the measurements are employed

in feedback control of the vibratory system. It is not recommended to differentiate a displacement

(or velocity) signal to obtain velocity (or acceleration), because this process would amplify any noise

present in the measured signal. Consider, for example, a sinusoidal signal give by A sin vt: Since

d=dtðA sin vtÞ ¼ Av cos vt; it follows that any high-frequency noise would be amplified by a factor

proportional to its frequency. Also, any discontinuities in noise components would produce large

deviations in the results. Using the same argument, it may be concluded that the acceleration

measurements are desirable for high-frequency signals and the displacement measurements are desirable

for low-frequency signals. It follows that the selection of a particular measurement transducer should

depend on the frequency content of the useful portion of the measured signal.

Transducers are divided into two broad categories: active transducers and passive transducers. Passive

transducers do not require an external electric source for activation. Some examples are electromagnetic,

piezoelectric, and photovoltaic transducers. Active transducers, however, do not possess selfcontained

energy sources and thus need external activation. A good example is a resistive transducer, such as a

potentiometer.

In selecting a particular transducer (measuring device) for a specific vibration application, special

attention should be give to its ratings, which usually are provided by the manufacturer, and the required

performance specifications as provided by the customer (or developed by the system designer).

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15.4.1 Parameters for Performance Specification

A perfect measuring device can be defined as one that possesses the following characteristics:

1. Output instantly reaches the measured value (fast response).

2. Transducer output is sufficiently large (high gain, low output impedance, high sensitivity).

3. Output remains at the measured value (without drifting or being affected by environmental effects

and other undesirable disturbances and noise) unless the measurand itself changes (stability and

robustness).

4. The output signal level of the transducer varies in proportion to the signal level of the measurand

(static linearity).

5. Connection of a measuring device does not distort the measurand itself (loading effects are absent

and impedances are matched).

6. Power consumption is small (high input impedance).

All these properties are based on dynamic characteristics and therefore can be explained in terms

of dynamic behavior of the measuring device. In particular, items 1 to 4 can be specified in terms of the

device (response), either in the time domain or in the frequency domain. Items 2, 5, and 6 can be specified

using the impedance characteristics of a device. First, we shall discuss response characteristics that are

important in performance specification of a sensor/transducer unit.

15.4.1.1 Time-Domain Specifications

Several parameters that are useful for the time-domain performance specification of a device are as

follows:

1. Rise time ðTrÞ: This is the time taken to pass the steady-state value of the response for the first time.

In overdamped systems, the response is nonoscillatory; consequently, there is no overshoot. So

that the definition is valid for all systems, rise time is often defined as the time taken to pass 90% of

the steady-state value for the first time. Rise time is often measured from 10% of the steady-state

value in order to leave out irregularities occurring at start-up and time lags that might be present

in a system. Rise time represents the speed of response of a device: a small rise time indicates a fast

response.

2. Delay time (Td): This is usually defined as the time taken to reach 50% of the steady-state value for

the first time. This parameter is also a measure of the speed of response.

3. Peak time (Tp): This is the time at the first peak. This parameter also represents the speed of

response of the device.

4. Settling time (Ts): This is the time taken for the device response to settle down within a certain

percentage (e.g., ^2%) of the steady-state value. This parameter is related to the degree of

damping present in the device as well as the degree of stability.

5. Percentage overshoot (PO): This is defined as

PO ¼ 100ðMp 2 1Þ% ð15:30Þ

using the normalized-to-unity step response curve, where Mp is the peak value. Percentage

overshoot is a measure of damping or relative stability in the device.

6. Steady-state error: This is the deviation of the actual steady-state value from the desired value.

Steady-state error may be expressed as a percentage with respect to the (desired) steady-state

value. In a measuring device, steady-state error manifests itself as an offset. This is a systematic

(deterministic) error that normally can be corrected by recalibration. In servo-controlled

devices, steady-state error can be reduced by increasing the loop gain or by introducing a lag

compensation. Steady-state error can be completely eliminated using the integral control (reset)

action.

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For the best performance of a measuring device, we wish to have the values of all the foregoing

parameters as small as possible. In actual practice, however, it might be difficult to meet all specifications,

particularly under conflicting requirements. For instance, Tr can be decreased by increasing the dominant

natural frequency vn of the device. This, however, increases the PO and sometimes the Ts. On the other

hand, the PO and Ts can be decreased by increasing device damping, but this has the undesirable effect of

increasing Tr.

15.4.1.2 Frequency-Domain Specifications

Since any time signal can be decomposed into sinusoidal components through Fourier transformation, it

is clear that the response of a system to an arbitrary input excitation also can be determined using

transfer-function (frequency response-function) information for that system. For this reason, one could

argue that it is redundant to use both time-domain specifications and frequency-domain specifications,

as they carry the same information. Often, however, both specifications are used simultaneously, because

this can provide a better understanding of the system performance. Frequency-domain parameters are

more suitable in representing some characteristics of a system under some types of excitation.

Consider a device with the frequency-response function (transfer function) Gð jvÞ: Some useful

parameters for performance specification of the device in the frequency domain are:

1. Useful frequency range (operating interval): This is given by the flat region of the frequency

response magnitude, lGð jvÞl; of the device.

2. Bandwidth (speed of response): This may be represented by the primary natural frequency (or

resonant frequency) of the device.

3. Static gain (steady-state performance): Since static conditions correspond to zero frequencies; this

is given by Gð0Þ:

4. Resonant frequency (speed and critical frequency region) vr: This corresponds to the lowest

frequency at which lGð jvÞl peaks.

5. Magnitude at resonance (stability): This is given by lGðjvr Þl:

6. Input impedance (loading, efficiency, interconnectability): This represents the dynamic resistance

as felt at the input terminals of the device. This parameter will be discussed in more detail under

component interconnection and matching.

7. Output impedance (loading, efficiency, interconnectability): This represents the dynamic

resistance as felt at the output terminals of the device.

8. Gain margin (stability): This is the amount by which the device gain could be increased before the

system becomes unstable.

9. Phase margin (stability): This is the amount by which the device phase lead could be decreased

(i.e., phase lag increased) before the system becomes unstable.

15.4.2 Linearity

A device is considered linear if it can be modeled by linear differential equations, with time t as the

independent variable. Nonlinear devices are often analyzed using linear techniques by considering small

excursions about an operating point. This linearization is accomplished by introducing incremental

variables for the excitations (inputs) and responses (outputs). If one increment can cover the entire

operating range of a device with sufficient accuracy, it is an indication that the device is linear. If the

input/output relations are nonlinear algebraic equations, that represents a static nonlinearity. Such a

situation can be handled simply by using nonlinear calibration curves, which linearize the device without

introducing nonlinearity errors. If, on the other hand, the input/output relations are nonlinear

differential equations, analysis usually becomes quite complex. This situation represents a dynamic

nonlinearity.

Transfer-function representation is a “linear” model of an instrument. Hence, it implicitly assumes

linearity. According to industrial terminology, a linear measuring instrument provides a measured value

that varies linearly with the value of the measurand. This is consistent with the definition of static linearity.

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All physical devices are nonlinear to some degree. This stems from any deviation from the ideal behavior,

due to causes such as saturation, deviation from Hooke’s Law in elastic elements, Coulomb friction, creep

at joints, aerodynamic damping, backlash in gears and other loose components, and component wearout.

Nonlinearities in devices are often manifested as some peculiar characteristics. In particular, the following

properties are important in detecting nonlinear behavior in dynamic systems:

1. Saturation: The response does not increase when the excitation is increased beyond some level.

This may result from such causes as magnetic saturation, which is common in transformer devices

such as differential transformers, plasticity in mechanical components, or nonlinear deformation

in springs.

2. Hysteresis: In this case, the input/output curve changes depending on the direction of motion,

resulting in a hysteresis loop. This is common in: loose components such as gears, which have

backlash; in components with nonlinear damping, such as Coulomb friction; and in magnetic

devices with ferromagnetic media and various dissipative mechanisms (e.g., eddy current

dissipation).

3. The jump phenomenon: Some nonlinear devices exhibit an instability known as the jump

phenomenon (or fold catastrophe). Here, the frequency response (transfer) function curve suddenly

jumps in magnitude at a particular frequency, while the excitation frequency is increased

or decreased. A device with this nonlinearity will exhibit a characteristic “tilt” of its resonant

peak either to the left (softening nonlinearity) or to the right (hardening nonlinearity).

Furthermore, the transfer function itself may change with the level of input excitation in the case

of nonlinear devices.

4. Limit cycles: A limit cycle is a closed trajectory in the state space that corresponds to sustained

oscillations without decay or growth. The amplitude of these oscillations is independent of the

location at which the response began. In the case of a stable limit cycle, the response will return to the

limit cycle irrespective of the location near the limit cycle from which the response was initiated. In

the case of an unstable limit cycle, the response will steadily move away from the location with the

slightest disturbance.

5. Frequency creation: At steady state, nonlinear devices can create frequencies that are not present in

the excitation signals. These frequencies might be harmonics (integer multiples of the excitation

frequency), subharmonics (integer fractions of the excitation frequency), or nonharmonics (usually

rational fractions of the excitation frequency).

Several methods are available to reduce or eliminate nonlinear behavior in vibrating systems. They

include calibration (in the static case), use of linearizing elements, such as resistors and amplifiers to

neutralize the nonlinear effects, and the use of nonlinear feedback. It is also good practice to take the

following precautions:

1. Avoid operating the device over a wide range of signal levels.

2. Avoid operation over a wide frequency band.

3. Use devices that do not generate large mechanical motions.

4. Minimize Coulomb friction.

5. Avoid loose joints and gear coupling (i.e., use direct-drive mechanisms).

15.4.3 Instrument Ratings

Instrument manufacturers do not usually provide complete dynamic information for their products. In

most cases, it is unrealistic to expect complete dynamic models (in the time or the frequency domain)

and associated parameter values for complex instruments. Performance characteristics provided by

manufacturers and vendors are primarily static parameters. Known as instrument ratings, these are

available as parameter values, tables, charts, calibration curves, and empirical equations. Dynamic

characteristics such as transfer functions (e.g., transmissibility curves expressed with respect to excitation

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frequency) might also be provided for more sophisticated instruments, but the available dynamic

information is never complete. Furthermore, the definitions of rating parameters used by manufacturers

and vendors of instruments are in some cases not the same as analytical definitions used in textbooks.

This is particularly true in relation to the term linearity. Nevertheless, instrument ratings provided by

manufacturers and vendors are very useful in the selection, installation, operation, and maintenance of

instruments. Some of these performance parameters are indicated below.

15.4.3.1 Rating Parameters

Typical rating parameters supplied by instrument manufacturers are:

1. Sensitivity

2. Dynamic range

3. Resolution

4. Linearity

5. Zero drift and full-scale drift

6. Useful frequency range

7. Bandwidth

8. Input and output impedances

The conventional definitions given by instrument manufacturers and vendors are summarized below.

Sensitivity of a transducer is measured by the magnitude (peak, root-mean-square [RMS] value, etc.)

of the output signals corresponding to a unit input of the measurand. This may be expressed as the ratio

of (incremental output)/(incremental input) or, analytically, as the corresponding partial derivative. In

the case of vectorial or tensorial signals (e.g., displacement, velocity, acceleration, strain, force), the

direction of sensitivity should be specified.

Cross-sensitivity is the sensitivity along directions that are orthogonal to the direction of primary

sensitivity; it is expressed as a percentage of the direct sensitivity. High sensitivity and low crosssensitivity

are desirable for measuring instruments. Sensitivity to parameter changes, disturbances, and

noise has to be small in any device, however; this is an indication of its robustness. Often, sensitivity

and robustness are conflicting requirements.

Dynamic range of an instrument is determined by the allowed lower and upper limits of its input or

output (response) so as to maintain a required level of measurement accuracy. This range is usually

expressed as a ratio, in decibels. In many situations, the lower limit of the dynamic range is equal to

the resolution of the device. Hence, the dynamic range is usually expressed as the ratio (range of

operation)/(resolution), in decibels.

Resolution is the smallest change in a signal that can be detected and accurately indicated by a

transducer, a display unit, or other instrument. It is usually expressed as a percentage of the maximum

range of the instrument or as the inverse of the dynamic range ratio, as defined above. It follows that

dynamic range and resolution are very closely related.

Linearity is determined by the calibration curve of an instrument. The curve of output amplitude

(a peak or rms value) vs. input amplitude under static conditions within the dynamic range of an

instrument is known as the static calibration curve. Its closeness to a straight line measures the degree of

linearity. Manufacturers provide this information either as the maximum deviation of the calibration

curve from the least squares straight-line fit of the calibration curve or from some other reference straight

line. If the least squares fit is used as the reference straight line, the maximum deviation is called

independent linearity (or more correctly, the independent nonlinearity, because the larger the deviation,

the greater the nonlinearity). Nonlinearity may be expressed as a percentage of either the actual reading at

an operating point or the full-scale reading.

Zero drift is defined as the drift from the null reading of the instrument when the measurand is

maintained steady for a long period. Note that in this case, the measurand is kept at zero or any other

level that corresponds to null reading of the instrument. Similarly, full-scale drift is defined with respect to

the full-scale reading (the measurand is maintained at the full-scale value). Usual causes of drift include

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instrument instability (e.g., instability in amplifiers), ambient changes (e.g., changes in temperature,

pressure, humidity, and vibration level), changes in power supply (e.g., changes in reference DC voltage

or alternating current [AC] line voltage), and parameter changes in an instrument (due to aging,

wearout, nonlinearities, etc.). Drift due to parameter changes that are caused by instrument

nonlinearities is known as parametric drift, sensitivity drift, or scale-factor drift. For example, a change

in spring stiffness or electrical resistance due to changes in ambient temperature results in a parametric

drift. Note that the parametric drift depends on the measurand level. Zero drift, however, is assumed to

be the same at any measurand level if the other conditions are kept constant. For example, a change in

reading caused by thermal expansion of the readout mechanism due to changes in the ambient

temperature is considered a zero drift. In electronic devices, drift can be reduced by using AC circuitry

rather than direct current (DC) circuitry. For example, AC-coupled amplifiers have fewer drift problems

than DC amplifiers. Intermittent checking for the instrument response level for zero input is a popular

way to calibrate for zero drift. In digital devices, this can be done automatically and intermittently,

between sample points, when the input signal can be bypassed without affecting the system operation.

Useful frequency range corresponds to the interval of both flat gain and zero phase in the frequency

response characteristics of an instrument. The maximum frequency in this band is typically less than half

(say, one fifth of) the dominant resonant frequency of the instrument. This is a measure of instrument

bandwidth.

Bandwidth of an instrument determines the maximum speed or frequency at which the instrument is

capable of operating. High bandwidth implies faster speed of response. Bandwidth is determined by the

dominant natural frequency, vn; or the dominant resonant frequency, vr; of the transducer. (Note: For

low damping, vr is approximately equal to vn.) It is inversely proportional to the rise time and the

dominant time constant. Half-power bandwidth is also a useful parameter. Instrument bandwidth must

be several times greater than the maximum frequency of interest in the measured signal. The bandwidth

of a measuring device is important, particularly when measuring transient signals. Note that the

bandwidth is directly related to the useful frequency range.

15.4.4 Accuracy and Precision

The instrument ratings mentioned above affect the overall accuracy of an instrument. Accuracy can be

assigned either to a particular reading or to an instrument. Note that instrument accuracy depends not

only on the physical hardware of the instrument but also on the operating conditions (e.g., design

conditions that are the normal, steady operating conditions or extreme transient conditions, such

as emergency start-up and shutdown). Measurement accuracy determines the closeness of the

measured value to the true value. Instrument accuracy is related to the worst accuracy obtainable

within the dynamic range of the instrument in a specific operating environment. Measurement error is

defined as

Error ¼ ðmeasured valueÞ 2 ðtrue valueÞ ð15:31Þ

Correction, which is the negative of error, is defined as

Correction ¼ ðtrue valueÞ 2 ðmeasured valueÞ ð15:32Þ

Each of these can also be expressed as a percentage of the true value. The accuracy of an instrument may

be determined by measuring a parameter whose true value is known, and is near the extremes of the

dynamic range of the instrument, under certain operating conditions. For this purpose, standard

parameters or signals that can be generated at very high levels of accuracy would be needed. The National

Institute for Standards and Testing (NIST) is usually responsible for the generation of these standards.

Nevertheless, accuracy and error values cannot be determined to 100% exactness in typical applications,

because the true value is not known. In a given situation, we can only make estimates for accuracy,

by using ratings provided by the instrument manufacturer or by analyzing data from previous

measurements and models.

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Causes of error include instrument instability, external noise (disturbances), poor calibration,

inaccurate information (e.g., poor analytical models, inaccurate control), parameter changes (e.g., due

to environmental changes, aging, and wearout), unknown nonlinearities, and improper use of the

instrument.

Errors can be classified as deterministic (or systematic) and random (or stochastic). Deterministic errors

are those caused by well-defined factors, including nonlinearities and offsets in readings. These usually

can be removed by applying proper calibration and analytical practices. Error ratings and calibration

charts are used to remove systematic errors from instrument readings. Random errors are caused by

uncertain factors entering into the instrument response. These include device noise, line noise, and the

effects of unknown random variations in the operating environment. A statistical analysis using

sufficiently large amounts of data is necessary to estimate random errors. The results are usually

expressed as a mean error, which is the systematic part of random error, and a standard deviation or

confidence interval for instrument response.

Precision is not synonymous with accuracy. Reproducibility (or repeatability) of an instrument reading

determines the precision of an instrument. Two or more identical instruments that have the same high

offset error might be able to generate responses at high precision, even though these readings are clearly

inaccurate. For example, consider a timing device (clock) that very accurately indicates time increments

(say, up to the nearest microsecond). If the reference time (starting time) is set incorrectly, the time

readings will be in error, even though the clock has a very high precision.

Instrument error may be represented by a random variable that has a mean value me and a

standard deviation se. If the standard deviation is zero, the variable is considered deterministic. In

that case, the error is said to be deterministic or repeatable. Otherwise, the error is said to be random.

The precision of an instrument is determined by the standard deviation of error in the instrument

response. Readings of an instrument may have a large mean value of error (e.g., large offset), but if

the standard deviation is small, the instrument has a high precision. Hence, a quantitative definition

for precision is

Precision ¼ ðmeasurement rangeÞ=se ð15:33Þ

Lack of precision originates from random causes and poor construction practices. It cannot be

compensated for by recalibration, just as the precision of a clock cannot be improved by resetting the

time. On the other hand, accuracy can be improved by recalibration. Repeatable (deterministic)

accuracy is inversely proportional to the magnitude of the mean error me.

In selecting instruments for a particular application, in addition to matching instrument ratings with

specifications, several additional features should be considered. These include geometric limitations

(size, shape, etc.); environmental conditions (e.g., chemical reactions including corrosion, extreme

temperatures, light, dirt accumulation, electromagnetic fields, radioactive environments, shock and

vibration); power requirements; operational simplicity; availability; the past record and reputation

of the manufacturer and of the particular instrument; and cost-related economic aspects (initial cost,

maintenance cost, cost of supplementary components such as signal-conditioning and processing

devices, design life and associated frequency of replacement, and cost of disposal and replacement).

Often, these considerations become the ultimate deciding factors in the selection process.

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