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15.6 Torque, Force, and Other Sensors

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The forced vibrations in a mechanical system depend on the forces and torques (excitations) applied

to the system. Also, the performance of the system may be specified in terms of forces and torques

that are generated, as for machine-tool operations such as grinding, cutting, forging, extrusion, and

rolling. Performance monitoring and evaluation, failure detection and diagnosis, and vibration

testing may depend considerably on the accurate measurement of associated forces and torques. In

mechanical applications such as parts assembly, slight errors in motion can generate large forces and

torques. These observations highlight the importance of measuring forces and torques. The strain

gage is a sensor that is commonly used in this context. There are numerous other types of sensors

and transducers that are useful in the context of mechanical vibration. In this section, we will

outline several of these sensors.

15.6.1 Strain Gage Sensors

Many types of force and torque sensors, as well as motion sensors such as accelerometers, are based on

strain gage measurements. Hence, strain gages are very useful in vibration instrumentation. Although

strain gages measure strain, the measurements can be directly related to stress and force. Note, however,

that strain gages may be used in a somewhat indirect manner, using auxiliary front-end elements, to

measure other types of variables, including displacement and acceleration.

15.6.1.1 Equations for Strain Gage Measurements

The change of electrical resistance in material when it is mechanically deformed is the property used in

resistance-type strain gages. The resistance R of a conductor that has length ‘ and area of cross section A,

is given by

R ¼ r

‘

A ð15:68Þ

15-50 Vibration and Shock Handbook

where r denotes the resistivity of the material. Taking the logarithm of Equation 15.68, we attain

log R ¼ log r þ logð‘=AÞ: Now, taking the differential, we obtain

R ¼

r þ

dð‘=AÞ

‘=A ð15:69Þ

The first term on the right-hand side of Equation 15.69 depends on the change in resistivity, and the

second term represents deformation. It follows that the change in resistance comes from the change in

shape as well as from the change in the resistivity of the material. For linear deformations, the two terms

on the right-hand side of Equation 15.69 are linear functions of strain, 1; the proportionality constant of

the second term, in particular, depends on Poisson’s ratio of the material. Hence, the following

relationship can be written for a strain gage element:

R ¼ Ss1 ð15:70Þ

The constant Ss is known as the sensitivity or gage factor of the strain gage element. The numerical value of

this constant ranges from 2 to 6 for most metallic strain gage elements and from 40 to 200 for SC strain

gages. These two types of strain gage will be discussed later. The change in resistance of a strain gage

element, which determines the associated strain (Equation 15.70), is measured using a suitable electrical

circuit.

Resistance strain gages are based on resistance change due to strain, or the piezoresistive property

of materials. Early strain gages were fine metal filaments. Modern strain gages are manufactured

primarily as metallic foil (for example, using the copper– nickel alloy known as constantan) or SC

elements (e.g., silicon with trace impurity boron). They are manufactured by first forming a thin

film (foil) of metal or a single crystal of SC material and then cutting it into a suitable grid pattern,

either mechanically or by using photoetching (chemical) techniques. This process is much more

economical and is more precise than making strain gages with metal filaments. The strain gage

element is formed on a backing film of electrically insulated material (e.g., plastic). This element is

cemented onto the member whose strain is to be measured. Alternatively, a thin film of insulating

ceramic substrate is melted onto the measurement surface, on which the strain gage is mounted

directly. The direction of sensitivity is the major direction of elongation of the strain gage element

(Figure 15.33(a)). To measure strains in more than one direction, multiple strain gages (e.g., various

rosette configurations) are available as single units. These units have more than one direction of

sensitivity. Principal strains in a given plane (the surface of the object on which the strain gage is

mounted) can be determined by using these multiple strain gage units. Typical foil-type strain gages

produce a relatively large output signal. A large accelerometer mass results in several disadvantages,

however. In particular:

1. The accelerometer mass distorts the measured motion variable (mechanical loading effect).

2. A heavier accelerometer has a lower resonant frequency and, hence, a lower useful frequency range

(Figure 15.31).

A direct way to obtain strain gage measurement is to apply a constant DC voltage across a seriesconnected

strain gage element and a suitable resistor, and to measure the output voltage vo across the

strain gage under open-circuit conditions using a voltmeter with high input impedance. It is known as a

potentiometer circuit or ballast circuit (see Figure 15.34(a)). This arrangement has several weaknesses. Any

ambient temperature variation will directly introduce some error because of associated change in the

strain gage resistance and the resistance of the connecting circuitry. Also, measurement accuracy will be

affected by possible variations in the supply voltage vref. Furthermore, the electrical loading error will be

significant unless the load impedance is very high. Perhaps, the most serious disadvantage of this circuit

is that the change in signal due to strain is usually a very small percentage of the total signal level in the

circuit output.

Vibration Instrumentation 15-51

A more favorable circuit for use in strain gage measurements is the Wheatstone bridge, shown in

Figure 15.34(b). One or more of the four resistors R1, R2, R3, and R4 in the circuit may represent strain

gages. To obtain the output relationship for the Wheatstone bridge circuit, assume that the load

impedance RL is very high. Hence, the load current, i; is negligibly small. Then, the potentials at nodes

A and B are

vA ¼

ðR1 þ R2Þ

vref and vB ¼

ðR3 þ R4Þ

vref

and the output voltage vo ¼ vA 2 vB is given by

vo ¼

ðR1 þ R2Þ

ðR3 þ R4Þ

􀀒 􀀓

vref ð15:71Þ

FIGURE 15.33 (a) Strain gage nomenclature; (b) typical foil-type strain gages; (c) a SC strain gage.

15-52 Vibration and Shock Handbook

By using straightforward algebra, we obtain

vo ¼ ðR1R4 2 R2R3Þ

ðR1 þ R2ÞðR3 þ R4Þ

vref ð15:72Þ

When this output voltage is zero, the bridge is said

to be “balanced.” It follows from Equation 15.72

that, for a balanced bridge

R2 ¼

R4 ð15:73Þ

Note that Equation 15.73 is valid for any value of

RL, not just for large RL, because when the bridge is

balanced, current i will be zero, even for small RL.

15.6.1.2 Bridge Sensitivity

Strain gage measurements are calibrated with

respect to a balanced bridge. When the strain

gages in the bridge deform, the balance is upset. If

one of the arms of the bridge has a variable

resistor, it can be changed to restore the balance.

The amount of this change corresponds to the

amount by which the resistance of the strain gages

changed, thereby measuring the applied strain.

This is known as the null-balance method of

strain measurement. This method is inherently

slow because of the time required to balance the

bridge each time a reading is taken. Hence, the

null-balance method is generally not suitable for

dynamic (time-varying) measurements. This

approach to strain measurement can be sped up

by using servo balancing, whereby the output error

signal is fed back into an actuator that automatically adjusts the variable resistance so as to restore

the balance.

A more common method, which is particularly suitable for making dynamic readings from a strain

gage bridge, is to measure the output voltage resulting from the imbalance caused by the deformation of

active strain gages in the bridge. To determine the calibration constant of a strain gage bridge, the

sensitivity of the bridge output to changes in the four resistors in the bridge should be known. For small

changes in resistance, this may be determined using the differential relation (or, equivalently, the firstorder

approximation for the Taylor series expansion):

dvo ¼

i¼1

›vo

›Ri

dRi ð15:74Þ

The partial derivatives are obtained directly from Equation 15.71. Specifically,

›vo

›R1 ¼

ðR1 þ R2Þ2 vref ð15:75Þ

›vo

›R2 ¼

ðR1 þ R2Þ2 vref ð15:76Þ

›vo

›R3 ¼

ðR1 þ R4Þ2 vref ð15:77Þ

R1 R2

(b)

Supply vref

Strain

Gage

Output

Signal

Resistor

−

Load

Output

−

vref

− +

(a)

FIGURE 15.34 (a) A potentiometer circuit (ballast

circuit) for strain gage measurements; (b) a Wheatstone

bridge circuit for strain gage measurements.

Vibration Instrumentation 15-53

›vo

›R4 ¼

ðR3 þ R4Þ2 vref ð15:78Þ

The required relationship is obtained by substituting the equations from Equation 15.75 to

Equation 15.78 into Equation 15.74; thus

dvo

vref ¼ ðR2dR1 2 R1dR2Þ

ðR1 þ R2Þ2 2 ðR4dR3 2 R3dR4Þ

ðR3 þ R4Þ2 ð15:79Þ

This result is subject to Equation 15.73, because changes are measured from the balanced condition.

Note that, from Equation 15.79, if all four resistors are identical (in value and material), resistance

changes due to ambient effects cancel out among the first-order terms ðdR1; dR2; dR3; dR4Þ; producing no

net effect on the output voltage from the bridge. Closer examination of Equation 15.79 will reveal that

only the adjacent pairs of resistors (e.g., R1 with R2 and R3 with R4) have to be identical in order to

achieve this environmental compensation. Even this requirement can be relaxed. Compensation is

achieved if R1 and R2 have the same temperature coefficient and if R3 and R4 have the same temperature

coefficient.

15.6.1.3 The Bridge Constant

Numerous activating combinations of strain gages are possible in a bridge circuit. For example, there

might be tension in R1 and compression in R2, as in the case of two strain gages mounted symmetrically

at 458 about the axis of a shaft in torsion. In this manner, the overall sensitivity of a strain gage bridge can

be increased. It is clear from Equation 15.79 that, if all four resistors in the bridge are active, the best

sensitivity is obtained if all four differential terms have the same sign, for example, when R1 and R4 are in

tension and R2 and R3 are in compression. If more than one strain gage is active, the bridge output may

be expressed as

dvo

vref ¼ k

4R ð15:80Þ

where

k ¼

bridge output in the general case

bridge output if only one strain gage is active

This constant is known as the bridge constant. The

larger the bridge constant is, the better the

sensitivity of the bridge.

Example 15.4

A strain gage load cell (force sensor) consists of

four identical strain gages, which form a Wheatstone

bridge and are mounted on a rod that has a

square cross section. One opposite pair of strain

gages is mounted axially and the other pair is

mounted in the transverse direction, as shown in

Figure 15.35(a). To maximize the bridge sensitivity,

the strain gages are connected to the

bridge as shown in Figure 15.35(b). Determine

the bridge constant k in terms of Poisson’s ratio y of

the rod material.

vref

− +

−

1 2

3 4

(b)

Axial

Gage

2 Transverse

Gage

Cross Section

of Sensing

Member

(a)

FIGURE 15.35 A strain-gage force sensor: (a) mounting

configuration; (b) bridge circuit.

15-54 Vibration and Shock Handbook

Solution

Suppose that dR1 ¼ dR: Then, for the given configuration, we have

dR2 ¼ 2y dR

dR3 ¼ 2y dR

dR4 ¼ dR

Note that, from the definition of Poisson’s ratio, transverse strain ¼ (2y ) £ longitudinal strain. Now, it

follows from Equation 15.79 that

dvo

vref ¼ 2ð1 þ y Þ

4R ð15:81Þ

according to which the bridge constant is given by

k ¼ 2ð1 þ y Þ

15.6.1.4 The Calibration Constant

The calibration constant, C; of a strain gage bridge relates the strain that is measured to the output of the

bridge. Specifically,

dvo

vref ¼ C1 ð15:82Þ

Now, in view of Equation 15.70 and Equation 15.80, the calibration constant may be expressed as

C ¼

Ss ð15:83Þ

where k is the bridge constant and Ss is the

sensitivity or gage factor of the strain gage. Ideally,

the calibration constant should remain constant

over the measurement range of the bridge (i.e.,

independent of strain 1 and time t) and should be

stable with respect to ambient conditions. In

particular, there should not be any creep, nonlinearities

such as hysteresis, or thermal effects.

Example 15.5

A schematic diagram of a strain gage accelerometer

is shown in Figure 15.36(a). A point mass

of weight W is used as the acceleration sensing

element, and a light cantilever with a rectangular

cross section, which is mounted inside the

accelerometer casing, converts the inertia force of

the mass into a strain. The maximum bending

strain at the root of the cantilever is measured

using four identical active SC strain gages. Two of

the strain gages (A and B) are mounted axially on

the top surface of the cantilever, and the remaining

two (C and D) are mounted on the bottom surface,

as shown in Figure 15.36(b). In order to maximize

the sensitivity of the accelerometer, indicate the

vref

− +

−

dvo

(a)

(b)

(c)

FIGURE 15.36 A strain gage accelerometer: (a) schematic

diagram; (b) strain gage mounting configuration;

Vibration Instrumentation 15-55

manner in which the four strain gages (A, B, C, and D) should be connected to a Wheatstone bridge

circuit. What is the bridge constant of the resulting circuit?

Obtain an expression relating the applied acceleration a (in units of g, which denotes acceleration due

to gravity) to the bridge output dvo (measured using a bridge balanced at zero acceleration) in terms of

the following parameters:

W ¼ weight of the seismic mass at the free end of the cantilever element

E ¼ Young’s modulus of the cantilever

‘ ¼ length of the cantilever

b ¼ cross-sectional width of the cantilever

h ¼ cross-sectional height of the cantilever

Ss ¼ sensitivity (gage factor) of each strain gage

vref ¼ supply voltage to the bridge

If W ¼ 0.02 lb, E ¼ 10 £ 106 lbf/in.2, ‘ ¼ 1 in., b ¼ 0.1 in., h ¼ 0.05 in., Ss ¼ 200, and vref ¼ 20 V,

determine the sensitivity of the accelerometer in mV/g.

If the yield strength of the cantilever element is 10 £ l03 lbf/in.2, what is the maximum acceleration

that could be measured using the accelerometer?

Is the cross-sensitivity (i.e., the sensitivity in the two directions orthogonal to the direction of

sensitivity shown in Figure 15.36(a)) small given your arrangement of the strain gage bridge? Explain.

Note: For a cantilever subjected to force F at the free end, the maximum stress at the root is given by

s ¼

6F‘

bh2 ð15:84Þ

with the present notation.

Solution

The bridge sensitivity is maximized by connecting the strain gages A, B, C, and D to the bridge as shown

in Figure 15.36(c). This follows from Equation 15.79, noting that the contributions from all four strain

gages are positive when dR1 and dR4 are positive and dR2 and dR3 are negative. The bridge constant for

the resulting arrangement is k ¼ 4. Hence, from Equation 15.80, we have

dvo

vref ¼

or, from Equation 15.82 and Equation 15.83

dvo

vref ¼ Ss1

Also,

1 ¼

E ¼

6F‘

Ebh2

where F denotes the inertia force

F ¼

x€ ¼ Wa

Note that x€ is the acceleration in the direction of sensitivity and x€=g ¼ a is the acceleration in units of g.

Thus,

1 ¼

6W ‘

Ebh2 a ð15:85Þ

15-56 Vibration and Shock Handbook

dvo ¼

6W ‘

Ebh2 Ssvref a ð15:86Þ

Now, with the given values

dvo

a ¼

6 £ 0:02 £ 1 £ 200 £ 20

10 £ 106 £ 0:1 £ ð0:05Þ2 V =g ¼ 0:192 V =g ¼ 192 mV=g

a ¼

Ssvref

dvo

a ¼

0:192

200 £ 20

strain=g

yield strain ¼

yield strength

E ¼

10 £ 103

10 £ 106 ¼ 1 £ 1023 strain

Hence,

number of gs to yielding ¼

1 £ 1023

48 £ 1026 g ¼ 20:8g

Cross-sensitivity comes from accelerations in the two directions (y and z) orthogonal to the direction of

sensitivity (x). In the lateral (y) direction, the inertia force causes lateral bending. This will produce equal

tensile (or compressive) strains in B and D, and equal compressive (or tensile) strains in A and C.

According to the bridge circuit, we see that these contributions cancel each other. In the axial (z)

direction, the inertia force causes equal tensile (or compressive) stresses in all four strain gages. These also

will cancel out, as is clear from the following relationship for the bridge:

dvo

vref ¼ ðRCdRA 2 RAdRCÞ

ðRA þ RCÞ2 2 ðRBdRD 2 RDdRBÞ

ðRD þ RBÞ2 ð15:87Þ

with

RA ¼ RB ¼ RC ¼ RD ¼ R

which gives

dvo

vref ¼ ðdRA 2 dRC 2 dRD þ dRBÞ

4R ð15:88Þ

It follows that this arrangement is good with respect to cross-sensitivity problems.

15.6.1.5 Data Acquisition

As noted earlier, the two common methods of measuring strains using a Wheatstone bridge circuit are

(1) the null-balance method and (2) the imbalance output method. One possible scheme for using the

first method is shown in Figure 15.37(a). In this particular arrangement, two bridge circuits are used.

The active bridge contains the active strain gages, dummy gages, and bridge-completion resistors. The

reference bridge has four resistors, one of which is micro-adjustable, either manually or automatically.

The output from the each of the two bridges is fed into a difference amplifier, which provides an

amplified difference of the two signals. This error signal is indicated on a null detector, such as a

galvanometer. Initially, both bridges are balanced. When the measurement system is in use, the active

gages are subjected to the strain that is being measured. This upsets the balance, giving a net output that

is indicated on the null detector. In manual operation of the null-balance mechanism, the resistance knob

in the reference bridge is adjusted carefully until the galvanometer indicates a null reading. The knob can

be calibrated to indicate the measured strain directly. In servo operation, which is much faster than the

manual method, the error signal is fed into an actuator that automatically adjusts the variable resistor in

the reference bridge until the null balance is achieved. Actuator movement measures the strain.

For measuring dynamic strains in vibrating systems, either the servo null-balance method or the

imbalance output method should be employed. A schematic diagram for the imbalance output method is

Vibration Instrumentation 15-57

shown in Figure 15.37(b). In this method, the output from the active bridge is directly measured as a

voltage signal and calibrated to provide the measured strain. An AC bridge may be used, where the bridge

is powered by an AC voltage. The supply frequency should be about ten times the maximum frequency of

interest in the dynamic strain signal (bandwidth). A supply frequency on the order of 1 kHz is typical.

This signal is generated by an oscillator and is fed into the bridge. The transient component of the output

from the bridge is very small (typically less than 1 mV and sometimes a few microvolts). This signal must

be amplified, demodulated (especially if the signals are transient), and filtered to provide the strain

reading. The calibration constant of the bridge should be known in order to convert the output voltage to

strain.

Strain gage bridges powered by DC voltages are very common. They have the advantages of portability

and simplicity with regard to necessary circuitry. The advantages of AC bridges include improved

stability (reduced drift), improved accuracy, and reduced power consumption.

15.6.1.6 Accuracy Considerations

Foil gages are available with resistances as low as 50 V and as high as several kilohms. The power

consumption of the bridge decreases with increased resistance. This has the added advantage of decreased

heat generation. Bridges with a high range of measurement (e.g., a maximum strain of 0.01 m/m) are

available. The accuracy depends on the linearity of the bridge, environmental (particularly temperature)

effects, and mounting techniques. For example, a calibration error occurs in the case of zero shift, due to

the strains produced when the cement that is used to mount the strain gage dries. Creep will introduce

errors during static and low-frequency measurements. Flexibility and hysteresis of the bonding cement

will bring about errors during high-frequency strain measurements. Resolutions on the order of 1 mm/m

(i.e., one microstrain) are common. The cross-sensitivity should be small (say, less than 1% of the direct

sensitivity). Manufacturers usually provide the values of the cross-sensitivity factors for their strain gages.

This factor, when multiplied by the cross strain present in a given application, gives the error in the strain

reading due to cross-sensitivity.

Active

Bridge

Power

Supply

Reference

Bridge

Null

Adjustment

(Manual or

Automatic)

Difference

Amplifier

Null Detector

(Galvanometer)

Strain

Error Feedback (Manual or Servo)

Bridge

Circuit

Calibration

Power Supply

Amplifier/

Filter

Dynamic

Strain

Measurement

(a)

(b)

FIGURE 15.37 Strain gage bridge measurement: (a) null-balance method; (b) imbalance output method.

15-58 Vibration and Shock Handbook

Often, measurements of strains in moving members are needed, for example, in real-time monitoring

and failure detection in machine tools. If the motion is small or the device has a limited stroke, strain

gages mounted on the moving member can be connected to the signal-conditioning circuitry and the

power source using coiled flexible cables. For large motions, particularly in rotating shafts, some form of

commutating arrangement must be used. Slip rings and brushes are commonly used for this purpose.

When AC bridges are used, a mutual-induction device (rotary transformer) can be used, with one coil

located on the moving member and the other coil stationary. To accommodate and compensate for errors

caused by commutation (e.g., losses and glitches in the output signal), it is desirable to place all four arms

of the bridge, rather than just the active arms, on the moving member.

15.6.1.7 Semiconductor Strain Gages

In some low-strain applications (e.g., dynamic

torque measurement), the sensitivity of foil gages

is not adequate to produce an acceptable strain

gage signal. SC strain gages are particularly

useful in such situations. The strain element of a

SC strain gage is made of a single crystal of

piezoresistive material such as silicon, doped with a

trace impurity such as boron. A typical construction

is shown in Figure 15.38. The sensitivity

(gage factor) of a SC strain gage is about two

orders of magnitude higher than that of a metallic

foil gage (typically, 40 to 200). The resistivity is

also higher, providing reduced power consumption

and heat generation. Another advantage of SC

strain gages is that they deform elastically until

fracture. In particular, mechanical hysteresis is negligible. Furthermore, they are smaller and lighter,

providing less cross-sensitivity, reduced distribution error (i.e., improved spatial resolution), and

negligible error due to mechanical loading. The maximum strain that is measurable using a SC strain

gage is typically 0.003 m/m (i.e., 3000 m 1). Strain gage resistance can be several hundred ohms (typically,

120 V or 350 V).

There are several disadvantages associated with SC strain gages, however, which can be interpreted as

advantages of foil gages. Undesirable characteristics of SC gages include the following:

1. The strain – resistance relationship is more nonlinear.

2. They are brittle and difficult to mount on curved surfaces.

3. The maximum strain that can be measured is an order of magnitude smaller (typically, less than

0.01 m/m).

4. They are more costly.

5. They have a much higher temperature sensitivity.

The first disadvantage is illustrated in Figure 15.39. There are two types of SC strain gages: the P-type

and the N-type. In P-type strain gages, the direction of sensitivity is along the ð1; 1; 1Þ crystal axis, and the

element produces a “positive” (P) change in resistance in response to a positive strain. In N-type strain

gages, the direction of sensitivity is along the ð1; 0; 0Þ crystal axis, and the element responds with a

“negative” (N) change in resistance to a positive strain. In both types, the response is nonlinear and can

be approximated by the quadratic relationship

R ¼ S11 þ S212 ð15:89Þ

The parameter S1 represents the linear sensitivity, which is positive for P-type gages and negative for

N-type gages. Its magnitude is usually somewhat larger for P-type gages, thereby providing

Single Crystal of

Semiconductor

Gold Leads

Conductor

Ribbons

Phenolic Glass

Backing Plate

FIGURE 15.38 Details of a semiconductor strain gage.

Vibration Instrumentation 15-59

better sensitivity. The parameter S2 represents the

degree of nonlinearity, which is usually positive for

both types of gage. Its magnitude, however, is

typically a little smaller for P-type gages. It follows

that P-type gages are less nonlinear and have

higher strain sensitivities. The nonlinear relationship

given by Equation 15.89 or the nonlinear

characteristic curve (Figure 15.39) should be used

when measuring moderate to large strains with SC

strain gages. Otherwise, the nonlinearity error will

be excessive.

15.6.1.8 Force and Torque Sensors

Torque and force sensing is useful in vibration

applications, including the following:

1. In vibration control of machinery where a

small motion error can cause large damaging

forces or performance degradation.

2. In high-speed vibration control when

motion feedback alone is not fast enough

(here, force feedback and feedforward force

control can be used to improve the accuracy

and bandwidth).

3. In vibration testing, monitoring, an diagnostic

applications, where torque and force

sensing can detect, predict, and identify

abnormal operation, malfunction, component

failure, or excessive wear (e.g., in

monitoring machine tools such as milling

machines and drills).

4. In experimental modal analysis where both excitation forces and response motioning may be

needed to experimentally determine the system model.

In most applications, torque (or force) is sensed by detecting either an effect or the cause of torque

(or force). There are also methods for measuring torque (or force) directly. Common methods of torque

sensing include the following:

1. Measuring the strain in a sensing member between the drive element and the driven load, using a

strain gage bridge.

2. Measuring the displacement in a sensing member (as in the first method), either directly, using a

displacement sensor, or indirectly, by measuring a variable, such as magnetic inductance or

capacitance, that varies with displacement.

3. Measuring the reaction in the support structure or housing (by measuring a force) and the

associated lever arm length.

4. In electric motors, measuring the field or armature current that produces motor torque; in

hydraulic or pneumatic actuators, measuring the actuator pressure.

5. Measuring the torque directly, for example, using piezoelectric sensors.

6. Employing the servo method to balance the unknown torque with a feedback torque generated by

an active device (say, a servomotor) whose torque characteristics are known precisely.

7. Measuring the angular acceleration in a known inertia element when the unknown torque is

applied.

−0.2

−0.1

0.1

0.2

0.3

0.4

−0.3

Strain

×103 me

Resistance

Change

= Strain of 1×10−6

(a)

(b)

−3 −2 −1 1 2 3

−0.2

−0.1

0.1

0.2

0.3

0.4

−0.3

Strain

Resistance

Change

δR

me = 1 Microstrain

FIGURE 15.39 Nonlinear behavior of a semiconductor

(silicon/boron) strain gage: (a) a P-type gage; (b) an

N-type gage.

15-60 Vibration and Shock Handbook

Note that force sensing may be accomplished by essentially the same techniques. Some types of force

sensor (e.g., the strain gage force sensor) have been introduced before. Now, we will limit our discussion

primarily to torque sensing. The extension of torque-sensing techniques to the task of force sensing is

somewhat straightforward.

15.6.1.9 Strain Gage Torque Sensors

The most straightforward method of torque sensing is to connect a torsion member between the drive

unit and the load-in series, and to measure the torque in the torsion member. If a circular shaft (solid or

hollow) is used as the torsion member, the torque – strain relationship is relatively simple. A complete

development of the relationship is found in standard textbooks on elasticity, solid mechanics, or strength

of materials. With reference to Figure 15.40, it can be shown that the torque, T, may be expressed in terms

of the direct strain, 1; on the shaft surface along a principal stress direction (i.e., at 458 to the shaft axis) as

T ¼

2GJ

1 ð15:90Þ

where G ¼ shear modulus of the shaft material, J ¼ polar moment of area of the shaft, and r ¼ shaft

radius (outer). This is the basis of torque sensing using strain measurements.

Using the general bridge Equation 15.82 along with Equation 15.83 in Equation 15.90, we can obtain

torque, T, from bridge output, dvo:

T ¼

8GJ

kSsr

dvo

vref ð15:91Þ

where Ss is the gage factor (or sensitivity) of the strain gages. The bridge constant, k, depends on the

number of active strain gages used. Strain gages are assumed to be mounted along a principal direction.

45°

Torque

rmax tmax

Circular Shaft

(Solid)

(a)

(b)

(c)

Shear

Stress

Tensile

Stress

−s s

−t

t B

C A

A = Stress Along Principal Direction x

B = Circumferential Stress

C = Stress Along Principal Direction y

D = Axial (Longitudinal) Stress

FIGURE 15.40 (a) Linear distribution of shear stress in a circular shaft under pure torsion; (b) pure shear state of

stress and principal directions x and y; (c) Mohr’s circle.

Vibration Instrumentation 15-61

Three possible configurations are shown in Figure 15.41. In configurations (a) and (b), only two strain

gages are used, and the bridge constant, k, is equal to 2. Note that both axial loads and bending are

compensated with the given configurations because resistance in both gages will be changed by the same

amount (the same sign and same magnitude) that cancels out, up to first order, for the bridge circuit

connection shown in Figure 15.41. Configuration (c) has two pairs of gages, mounted on the two

opposite surfaces of the shaft. The bridge constant is doubled in this configuration, and here again, the

sensor clearly selfcompensates for axial and bending loads up to first order ½OðdRÞ􀀉:

For a circular-shaft torque sensor that uses SC strain gages, design criteria for obtaining a suitable value

for the polar moment of area (J) are listed in Table 15.3. Note that f is a safety factor.

Although the manner in which strain gages are configured on a torque sensor can be exploited to

compensate for cross-sensitivity effects arising from factors such as tensile and bending loads, it is

advisable to use a torque-sensing element that inherently possesses low sensitivity to these factors that

cause error in a torque measurement. A tubular torsion element is convenient for analytical purposes

because of the simplicity of the associated expressions for design parameters. Unfortunately, such an

R1 R2

dvo

−

T T T vref

2 1

2 2

Bridge Constant (k): 2 2 4

Axial Loads Compensated: Yes Yes Yes

Bending Loads Compensated: Yes Yes Yes

(a) (b) (c)

Strain Gage Bridge

FIGURE 15.41 Strain gage configurations for a circular shaft torque sensor.

TABLE 15.3 Design Criteria for a Strain Gage Torque-Sensing Element

Criterion Specification Governing Formula for the Polar

Moment of Area (J)

Strain capacity of strain gage

element

1max

Tmax

1max

Strain gage nonlinearity Np ¼

Max strain error

Strain range £ 100% $

25frS2

GS1

Tmax

Sensor sensitivity (output voltage) vo ¼ Kadvo where Ka ¼ transducer gain #

Ka kSs rvref

Tmax

Sensor stiffness (system bandwidth

and gain)

K ¼

Torque

Twist angle

15-62 Vibration and Shock Handbook

element is not very rigid to bending and tensile

loading. Alternative shapes and structural arrangements

have to be considered if inherent rigidity

(insensitivity) to cross-loads is needed. Furthermore,

a tubular element has the same strain at all

locations on the element surface. This does not

provide a choice with respect to mounting

locations of strain gages in order to maximize

the torque sensor sensitivity. Another disadvantage

of the basic tubular element is that the surface is

curved; therefore, much care is needed in mounting

fragile SC gages, which could be easily

damaged by even slight bending. Hence, a sensor

element that has flat surfaces to mount the strain

gages would be desirable. A torque-sensing

element that has the aforementioned desirable

characteristics (i.e., inherent insensitivity to crossloading,

nonuniform strain distribution on the

surface, and the availability of flat surfaces to

mount strain gages) is shown in Figure 15.42. Note

that two sensing elements are connected radially

between the drive unit and the driven member.

The sensing elements undergo bending while

transmitting a torque between the driver and the

driven member. Bending strains are measured at locations of high sensitivity and are taken to be

proportional to the transmitted torque. Analytical determination of the calibration constant is not easy

for such complex sensing elements, but experimental determination is straightforward. Note that the

strain gage torque sensor measures the direction as well as the magnitude of the torque transmitted

through it.

15.6.1.10 Deflection Torque Sensors

Instead of measuring strain in the sensor element, the actual deflection (twisting or bending) can be

measured and used to determine torque, through a suitable calibration constant. For a circular-shaft

(solid or hollow) torsion element, the governing relationship is given by

T ¼

u ð15:92Þ

The calibration constant GJ/L must be small in order to achieve high sensitivity. This means that the

element stiffness should be low. This will limit the bandwidth (which measures speed of response) and

gain (which determines steady-state error) of the overall system. The twist angle, u, is very small (e.g., a

fraction of a degree) in systems with high bandwidth. This requires very accurate measurement of u in

order to determine the torque T. A type of displacement sensor that could be used is described as follows.

Two ferromagnetic gear wheels are splined at two axial locations of the torsion element. Two stationary

proximity probes of the magnetic induction type (selfinduction or mutual induction) are placed radially,

facing the gear teeth, at the two locations. As the shaft rotates, the gear teeth change the flux linkage of the

proximity sensor coils. The resulting output signals of the two probes are pulse sequences, shaped

somewhat like sine waves. The phase shift of one signal with respect to the other determines the relative

angular deflection of one gear wheel with respect to the other, assuming that the two probes are

synchronized under no-torque conditions. Both the magnitude and the direction of the transmitted

torque are determined using this method. A 3608 phase shift corresponds to a relative deflection by an

integer multiple of the gear pitch. It follows that deflections less than half the pitch can be measured

(a)

(b)

Strain

Gage

Connected to

Drive Member

Connected to

Driven Member

A = Torque Sensing

Elements

FIGURE 15.42 Use of a bending element in torque

sensing: (a) sensing element; (b) element configuration.

Vibration Instrumentation 15-63

without ambiguity. Assuming that the output signals of the two probes are sine waves (narrow-band

filtering can be used to achieve this), the phase shift will be proportional to the angle of twist, u.

15.6.1.11 Variable-Reluctance Torque Sensor

A torque sensor that is based on the sensor element deformation and that does not require a

contacting commutator is a variable-reluctance device that operates like a differential transformer

(RVDT or LVDT). The torque-sensing element is a ferromagnetic tube that has two sets of slits,

typically oriented along the two principal stress directions of the tube (458) under torsion. When a

torque is applied to the torsion element, one set of gaps closes and the other set opens as a result of

the principal stresses normal to the slit axes. Primary and secondary coils are placed around the slitted

tube, and they remain stationary. One segment of the secondary coil is placed around one set of slits,

and the second segment is placed around the other, perpendicular, set. The primary coil is excited by

an AC supply, and the induced voltage, vo, in the secondary coil is measured. As the tube deforms, it

changes the magnetic reluctance in the flux linkage path, thus changing the induced voltage. The two

segments of the secondary coil should be connected so that the induced voltages are absolutely

additive (algebraically subtractive), because one voltage increases and the other decreases, to obtain the

best sensitivity. The output signal should be demodulated, by removing the carrier frequency

component, to measure transient torques effectively. Note that the direction of torque is given by the

sign of the demodulated signal.

15.6.1.12 Reaction Torque Sensors

The foregoing methods of torque sensing use a sensing element that is connected between the drive

member and the driven member. A major drawback of such an arrangement is that the sensing

element modifies the original system in an undesirable manner, particularly by decreasing the system

stiffness and adding inertia. Not only will the overall bandwidth of the system decrease, but the

original torque will also be changed (mechanical loading) because of the inclusion of an auxiliary

sensing element. Furthermore, under dynamic conditions, the sensing element will be in motion,

thereby making the torque measurement more difficult. The reaction method of torque sensing

eliminates these problems to a large degree. This method can be used to measure torque in a rotating

machine. The supporting structure (or housing) of the rotating machine (e.g., a motor, pump,

compressor, turbine, or generator) is cradled by releasing its fixtures, and the effort necessary to

keep the structure from moving is measured. A schematic representation of the method is shown in

Figure 15.43(a). Ideally, a lever arm is mounted on the cradled housing, and the force required to fix

the housing is measured using a force sensor (load cell). The reaction torque on the housing is

given by

TR ¼ FR·L ð15:93Þ

where

FR ¼ reaction force measured using load cell

L ¼ lever arm length

Alternatively, strain gages or other types of force sensors could be mounted directly at the fixture

locations (e.g., at the mounting bolts) of the housing to measure the reaction forces without cradling the

housing. Then, the reaction torque is determined with a knowledge of the distance of the fixture locations

from the shaft axis.

The reaction-torque method of torque sensing is widely used in dynamometers (reaction

dynamometers) that determine the transmitted power in rotating machinery through torque and

shaft speed measurements. A drawback of reaction-type torque sensors can be explained using Figure

15.43(b). A motor with rotor inertia, J, which rotates at angular acceleration, u€; is shown. By Newton’s

Third Law (action equals reaction), the electromagnetic torque generated at the rotor of the motor, Tm,

and the frictional torques, Tf1 and Tf2, will be reacted back onto the stator and housing. By applying

15-64 Vibration and Shock Handbook

Newton’s Second Law to the motor rotor and the housing combination, we obtain

TL ¼ TR 2 Ju€ ð15:94Þ

Note that TL is the variable which must be measured. Under accelerating or decelerating conditions, the

reaction torque, Tr, is not equal to the actual torque, TL, that is transmitted. One method of

compensating for this error is to measure the shaft acceleration, compute the inertia torque, and adjust

the measured reaction torque using this inertia torque. Note that the frictional torque in the bearings

does not enter the final Equation 15.94. This is an advantage of this method.

15.6.2 Miscellaneous Sensors

Motion and force/torque sensors of the types described thus far are widely used in vibration

instrumentation. Several other types of sensors are also useful. A few of them are indicated now.

15.6.2.1 Stroboscope

Consider an object that executes periodic motions such as vibrations or rotations in a fairly dark

environment. Suppose that a light is flashed at the object at the same frequency as the moving object.

Since the object completes a full cycle of motion during the time period between two adjacent flashes,

Motor Housing

(Stator)

Lever

Arm

F Frictionless

Bearing

Force Sensor

(Load Cell)

(a)

(b)

Reaction

Torque

Motor

Torque

Frictional

Torque

Tf1

Tf1 Tf 2

Frictional

Torque

Tf 2

To Load

Stator

Housing

Rotor

Load

Torque

Bearings

FIGURE 15.43 (a) Schematic representation of a reaction torque sensor setup (reaction dynamometer); (b) various

torque components.

Vibration Instrumentation 15-65

the object will appear to be stationary. This is the principle of operation of a stroboscope. The main

components of a stroboscope are a high-intensity strobe lamp and circuitry to vary the frequency of

the electrical pulse signal that energizes the lamp. The flashing frequency may be varied either

manually using a knob or according to the frequency of an external periodic signal (trigger signal)

that is applied to the stroboscope.

It is clear that by synchronizing the stroboscope with a moving (vibrating, rotating) object so that the

object appears stationary, and then noting the flashing (strobe) frequency, the frequency of vibration or

speed of rotation of the object can be measured. In this sense, the stroboscope is a noncontacting

vibration frequency sensor or a tachometer (rotating speed sensor). Note that the object appears

stationary for any integer multiple of the synchronous flashing frequency. Hence, once the strobe is

synchronized with the moving object, it is good practice to check whether the strobe also synchronizes

at an integer fraction of that flashing frequency. (Typically, trying 1/2, 1/3, 1/5, and 1/7 the original

synchronous frequency is adequate.) The lowest synchronous frequency thus obtained is the correct

speed (frequency) of the object. Since the frequency of visual persistence of a human is about 15 Hz, the

stationary appearance will not be possible using a stroboscope below this frequency. Hence, the lowfrequency

limit for a stroboscope is about 15 Hz.

In addition to serving as a sensor for vibration frequency and rotating speed, the stroboscope has

many other applications. For example, by maintaining the strobe (flashing) frequency close (but not

equal) to the object frequency, the object will appear to move very slowly. In this manner, visual

inspection of objects that execute periodic motions at high speed is possible. Also, stroboscopes are

widely used in dynamic balancing of rotating machinery. In this case, it is important to measure the

phase angle of the resultant imbalance force with respect to a coordinate axis (direction) that is

fixed to the rotor. Suppose that a radial line is marked on the rotor. If we synchronize a stroboscope

with the rotor such that the marked line appears not only stationary but also oriented in a fixed

direction (e.g., horizontal or vertical), we in effect make the strobe signal in phase with the rotation

of the rotor. Then by comparing the imbalance force signal of the rotor (obtained, for example, by

an accelerometer or a force sensor at the bearings of the rotor) with the synchronized strobe signal

(with a fixed reference), by means of an oscilloscope or a phase meter, it is possible to determine

the orientation of the imbalance force with respect to a fixed body reference of the rotating

machine.

15.6.2.2 Fiber Optic Sensors and Lasers

The characteristic component in a fiber optic sensor is a bundle of glass fibers (typically a few

hundred) that can carry light. Each optical fiber may have a diameter on the order of 0.01 mm. There

are two basic types of fiber optic sensors. In one type, the “indirect” or the extrinsic type, the optical

fiber acts only as the medium in which the sensed light is transmitted. In this type, the sensing

element itself does not consist of optical fibers. In the second type, the “direct” or the intrinsic type,

the optical fiber bundle itself acts as the sensing element. When the conditions of the sensed medium

change, the light-propagation properties of the optical fibers change, providing a measurement of the

change in the conditions. Examples of the first (extrinsic) type of sensor include fiber optic position

sensors and tactile (distributed touch) sensors. The second (intrinsic) type of sensor is found, for

example, in fiber optic gyroscopes, fiber optic hydrophones, and some types of micro-displacement or

force sensors.

A schematic representation of a fiber optic position sensor (or proximity sensor or displacement

sensor) is shown in Figure 15.44(a). The optical fiber bundle is divided into two groups:

transmitting fibers and receiving fibers. Light from the light source is transmitted along the first

bundle of fibers to the target object whose position is being measured. Light reflected onto the

receiving fibers by the surface of the target object is carried to a photodetector. The intensity of the

light received by the photodetector will depend on the position, x, of the target object. In particular,

if x ¼ 0, the transmitting bundle will be completely blocked off and the light intensity at the receiver

will be zero. As x is increased, the received light intensity will increase, because more light will be

15-66 Vibration and Shock Handbook

reflected onto the receiving bundle tip. This will reach a peak at some value of x. When x is

increased beyond that value, more light will be reflected outside the receiving bundle; hence, the

intensity of the received light will decrease. Hence, in general, the proximity – intensity curve for an

optical proximity sensor will be nonlinear and will have the shape shown in Figure 15.44(b). Using

this (calibration) curve, we can determine the position (x) once the intensity of the light received at

the photosensor is known. The light source could be a laser (or light amplification by stimulated

emission of radiation; structured light), infrared light-source, or some other type, such as a lightemitting

diode (LED). The light sensor (photodetector) could be some light-sensitive discrete SC

element such as a photodiode or a photo field effect transistor (photo FET). Very fine resolutions,

better than 1 £ 1026 cm, can be obtained using a fiber optic position sensor. An optical encoder is a

digital (or pulse-generating) motion transducer. Here, a light beam is intercepted by a moving disk

that has a pattern of transparent windows. The light that passes through, as detected by a

photosensor, provides the transducer output. These sensors may also be considered in the extrinsic

category.

The advantages of fiber optics include insensitivity to electrical and magnetic noise (due to optical

coupling), safe operation in explosive, high-temperature, hazardous environments and high sensitivity.

Furthermore, mechanical loading and wear problems do not exist because fiber optic position sensors are

noncontacting devices with stationary sensor heads. The disadvantages include direct sensitivity to

variations in the intensity of the light source and dependence on ambient conditions (ambient light, dirt,

moisture, smoke, etc.).

As an example of an intrinsic application of fiber optics in sensing, consider a straight optical fiber

element that is supported at each end. In this configuration almost 100% of the light at the source end

will transmit through the optical fiber and will reach the detector (receiver) end. Then, suppose that a

slight load is applied to the optical fiber segment at its mid span. The fiber will deflect slightly due to the

FIGURE 15.44 (a) A fiber-optic proximity sensor; (b) nonlinear characteristic curve.

Vibration Instrumentation 15-67

load, and as a result the amount of light received at the detector can significantly drop. For example, a

deflection of just 50 mm can result in a drop in intensity at the detector by a factor of 25. Such an

arrangement may be used in deflection, force, and tactile sensing. Another intrinsic application is the

fiber optic gyroscope, as described below.

15.6.2.3 Fiber-Optic Gyroscope

This is an angular speed sensor that uses fiber optics. Contrary to the implication of its name, however,

it is not a gyroscope in the conventional sense. Two loops of optical fibers wrapped around a cylinder

are used in this sensor. One loop carries a monochromatic light (or laser) beam in the clockwise

direction, and the other loop carries a beam from the same light (or laser) source in the

counterclockwise direction. Since the laser beam traveling in the direction of rotation of the cylinder

has a higher frequency than that of the other beam, the difference in frequencies of the two laser beams

received at a common location will measure the angular speed of the cylinder. This may be

accomplished through interferometry, as the light and dark patterns of the detected light will measure

the frequency difference. Note that the length of the optical fiber in each loop can exceed 100 m.

Angular displacements can be measured with the same sensor simply by counting the number of cycles

and clocking the fractions of cycles. Acceleration can be determined by digitally determining the rate of

change of speed.

15.6.2.4 Laser Doppler Interferometer

The laser produces electromagnetic radiation in the ultraviolet, visible, or infrared bands of the spectrum.

A laser can provide a single-frequency (monochromatic) light source. Furthermore, the electromagnetic

radiation in a laser is coherent in the sense that all waves generated have constant phase angles. The laser

uses oscillations of atoms or molecules of various elements. The helium – neon (HeNe) laser and the SC

laser are commonly used in industrial applications.

As noted earlier, the laser is useful in fiber optics, but it can also be used directly in sensing and

gaging applications. The laser Doppler interferometer is one such sensor. It is useful in the accurate

measurement of small displacements, for example, in strain measurements. To explain the operation

of this device, we should explain two phenomena: the Doppler effect and light wave interference.

Consider a wave source (e.g., a light source or sound source) that is moving with respect to a

receiver (observer). If the source moves toward the receiver, the frequency of the received wave

appears to have increased; if the source moves away from the receiver, the frequency of the received

wave appears to have decreased. The change in frequency is proportional to the velocity of the source

relative to the receiver. This phenomenon is known as the Doppler effect. Now consider a

monochromatic (single-frequency) light wave of frequency, f (say, 5 £ 1014 Hz), emitted by a laser

source. If this ray is reflected by a target object and received by a light detector, the frequency of the

received wave is

f2 ¼ f þ Df ð15:95Þ

The frequency increase Df will be proportional to the velocity, v, of the target object, which is

assumed to be positive when moving toward the light source. Hence,

Df ¼ cv ð15:96Þ

Now by comparing the frequency, f2, of the reflected wave, with the frequency

f1 ¼ f ð15:97Þ

of the original wave, we can determine Df and, hence, the velocity, v, of the target object.

The change in frequency Df due to the Doppler effect can be determined by observing the fringe

pattern due to light wave interference. To understand this, consider the two waves

v1 ¼ a sin 2pf1t ð15:98Þ

15-68 Vibration and Shock Handbook

and

v2 ¼ a sin 2pf2t ð15:99Þ

If we add these two waves, the resulting wave is

v ¼ v1 þ v2 ¼ aðsin 2pf1t þ sin 2pf2tÞ

which can be expressed as

v ¼ 2a sin pðf2 þ f1Þt cos pðf2 2 f1Þt ð15:100Þ

It follows that the combined signal will beat at the

beat frequency Df =2: When f2 is very close to f1

(i.e., when Df is small compared with f), these

beats will appear as dark and light lines (fringes)

in the resulting light wave. This is known as

wave interference. Note that Df can be determined

by two methods:

1. By measuring the spacing of the fringes

2. By counting the beats in a given time

interval or by timing successive beats using

a high-frequency clock signal

The velocity of the target object is determined in

this manner. Displacement can be obtained simply

by digital integration or by accumulating the count. A schematic diagram for the laser Doppler

interferometer is shown in Figure 15.45. Industrial interferometers usually employ a HeNe laser that has

waves of two frequencies close together. In that case, the arrangement shown in Figure 15.45 has to be

modified to take into account the two frequency components.

Note that there are laser interferometers that directly measure displacement rather than speed. They are

based on measuring phase difference between the direct and the returning laser, not the Doppler

effect (frequency difference). In this case, integration is not needed to obtain displacement from a

measured velocity.

15.6.2.5 Ultrasonic Sensors

Audible sound waves have frequencies in the range of 20 Hz to 20 kHz. Ultrasound waves are

pressure waves, just like sound waves, but their frequencies are higher than the audible frequencies.

Ultrasonic sensors are used in many applications, including displacement and vibration sensing,

medical imaging, ranging for cameras with autofocusing capability, level sensing, machine

monitoring, and speed sensing. For example, in medical applications, ultrasound probes of

frequencies 40 kHz, 75 kHz, 7.5 MHz and 10 MHz are commonly used. Ultrasound can be generated

according to several principles. For example, high-frequency (gigahertz) oscillations in piezoelectric

crystals subjected to electrical potentials are used to generate very high-frequency ultrasound.

Another method is to use the magnetostrictive property of ferromagnetic material. Ferromagnetic

materials deform when subjected to magnetic fields. Respondent oscillations generated by this

principle can produce ultrasonic waves. Another method of generating ultrasound is to apply a

high-frequency voltage to a metal-film capacitor. A microphone can serve as an ultrasound detector

(receiver).

Analogous to the case of fiber-optic sensing, there are two common ways of employing ultrasound in a

sensor. In one approach, the intrinsic method, the ultrasound signal undergoes change as it passes

through an object, due to acoustic impedance and the absorption characteristics of the object. The

resulting signal (image) may be interpreted to determine properties of the object, such as texture,

firmness, and deformation. This approach is utilized, for example, in machine monitoring and object

Laser

Beam

Splitter

Speed v

Reflector

Photosensor

Signal

Processor

Target

Object

Speed, Position

Readings

FIGURE 15.45 A laser Doppler interferometer for

measuring velocity and displacement.

Vibration Instrumentation 15-69

firmness sensing. In the other approach, the

extrinsic method, the time for an ultrasound

burst to travel from its source to some object

and then back to a receiver is measured. This

approach is used in distance, position, and

vibration measurement and in dimensional

gauging.

In distance (vibration, proximity, displacement)

measurement using ultrasound, a burst of

ultrasound is projected at the target object, and

the time taken for the echo to be received is

clocked. A signal processor computes the

position of the target object, possibly compensating for environmental conditions. This

configuration is shown in Figure 15.46. Alternatively, the velocity of the target object can be

measured, using the Doppler effect, by measuring (clocking) the change in frequency between the

transmitted wave and the received wave. The “beat” phenomenon may be employed here. Position

measurements with fine resolution (e.g., a fraction of a millimeter) can be achieved using the

ultrasonic method. Since the speed of ultrasonic wave propagation depends on the temperature of

the medium (typically air), errors will enter into the ultrasonic readings unless the sensor is adjusted

to compensate for temperature variations.

15.6.2.6 Gyroscopic Sensors

Consider a rigid body spinning about an axis at angular speed, v: If the moment of inertia of the body

about that axis is J, the angular momentum H about the same axis is given by

H ¼ Jv ð15:101Þ

Newton’s Second Law (torque ¼ rate of change of angular momentum) tells us that to rotate (precess)

the spinning axis slightly, a torque has to be applied, because precession causes a change in the spinning

angular momentum vector (the magnitude remains constant but the direction changes), as shown in

Figure 15.47(a). This is the principle of operation of a gyroscope. Gyroscopic sensors are commonly used

in control systems for stabilizing vehicle systems.

Ultrasound

Generator

Transmitter/

Receiver

Signal

Processor

Distance

Reading

Target

Object

FIGURE 15.46 An ultrasonic position sensor.

Spinning

Disk

Frictionless

Bearings

Gimbal

Spin

Axis

Gimbal

Axis

Torque

Motor

H1 = Jw

H2 = Jw

Δq = Angle of Procession

H2 ΔH

Δq

(a) (b)

FIGURE 15.47 (a) Illustration of the gyroscopic torque needed to change the direction of an angular momentum

vector; (b) a simple single-axis gyroscope for sensing angular displacements.

15-70 Vibration and Shock Handbook

TABLE 15.4 Rating Parameters of Several Sensors and Transducers

Transducer Measurand Measurand Frequency Output Impedance Typical Resolution Accuracy Sensitivity

Max/Min

Potentiometer Displacement 10 Hz/DC Low 0.1 mm 0.1% 200 mV/mm

LVDT Displacement 2500 Hz/DC Moderate 0.001 mm or less 0.3% 50 mV/mm

Resolver Angular displacement 500 Hz/DC

(limited by excitation

frequency)

Low 2min 0.2% 10 mV/deg

Tachometer Velocity 700 Hz/DC Moderate (50 V) 0.2 mm/sec 0.5% 5 mV/mm/sec;

75 mV/rad/sec

Eddy current proximity sensor Displacement 100 kHz/DC Moderate 0.001 mm 0.05% full scale 0.5% 5 V/mm

Piezoelectric accelerometer Acceleration (and velocity, etc.) 25 kHz/1 Hz High 1 mm/sec2 1% 0.5 mV/m/sec2

Semiconductor strain gage Strain (displacement,

acceleration, etc.)

1 kHz/DC

(limited by fatigue)

200 1 to 10 msec(1 msec ¼ 1026

unity strain)

1% 1 V/1, 2000

msec max

Loadcell Force (10 – 1000 N) 500 Hz/DC Moderate 0.01N 0.05% 1 mV/N

Laser Displacement/shape 1 kHz/DC 100 V 1.0 mm 0.5% 1 V/mm

Optical encoder Motion 100 kHz/DC 500 V 10 bit ^ 1/2 bit 104/rev

Vibration Instrumentation 15-71

Consider the gyroscope shown in Figure 15.47(b). The disk is spun about frictionless bearings using a

torque motor. Since the gimbal (the framework on which the disk is supported) is free to turn about the

frictionless bearings on the vertical axis, it will remain fixed with respect to an inertial frame, even if the

bearing housing (the main structure in which the gyroscope is located) rotates. Hence, the relative angle

between the gimbal and the bearing housing (angle u in the figure) can be measured, and this gives the

angle of rotation of the main structure. In this manner, angular displacements in systems such as aircraft,

space vehicles, ships, and land vehicles can be measured and stabilized with respect to an inertial frame.

Note that bearing friction introduces an error that must be compensated for, perhaps by recalibration

before a reading is taken.

The rate gyro, which has the same arrangement as shown in Figure 15.47(b), except with a slight

modification, can be used to measure angular speeds. In this case, the gimbal is not free but is restrained

by a torsional spring. A viscous damper is provided to suppress any oscillations. By analyzing this gyro as

a mechanical tachometer, we will note that the relative angle of rotation, u; gives the angular speed of the

structure about the gimbal axis.

Several areas can be identified where new developments and innovations are being made in sensor

technology:

1. Microminiature sensors: IC-based, with built-in signal processing.

2. Intelligent sensors: built-in reasoning or information preprocessing to provide high-level

knowledge.

3. Integrated and distributed sensors: sensors are integral with the components and agents of the

overall multiagent system that communicate with each other.

4. Hierarchical sensory architectures: low level sensory information is preprocessed to match higher

level requirements.

These four areas of activity are also representative of future trends in sensor technology

development. To summarize, rating parameters of a selected set of sensors/transducers are listed in

Table 15.4.

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