1061

7.22 Vibration, Shock, and Acceleration

A. BRODGESELL

(1969, 1982)

B. G. LIPTÁK

(1995)

H. EREN

(2003)

Types:

A. Seismic (Inertial) sensorsB. Piezoelectric sensorsC. Piezoresistive and strain gagesD. Electromechanical sensorsE. Capacitive and electrostatic sensorsF. Micro- and nanosensorsG. Velocity sensorsH. Noncontact proximity sensorsI. Mechanical switchesJ. Optical sensors

Operating Temperature:

A.

−

20

°

C to 70

°

CB. Generally

−

60 to 500

°

F (

−

50 to 260

°

C); special designs available for

−

450 to1500

°

F (

−

268 to 816

°

C)C.

−

40 to 250

°

F (

−

40 to 121

°

C)F.

−

40 to 120

°

C

Typical Range of Vibration

(Most rotary equipment vibrates at frequencies between 1 and 20,000 Hz.)

Frequency (Hz):

A. DC to 50 Hz.B. From 1 to 15,000 Hz; special designs from under 1 Hz up to 30,000 HzC. From 0 to about 1000 HzD. Between 10 and 1000 HzF. 0 to 3500 HzH. 0 to 5000 Hz

Typical Range of Vibration

(Most rotary equipment vibrates at amplitudes of 0.01 to 100 g.)

Amplitude (g):

A.

±

0.5 or

±

2.0 gB. Generally from under 1 to 1000 g; special sensors are available to detect shock

in ranges exceeding 10,000

gD. 0.02 to 10 in./s (0.5 to 254 mm/s)F.

±

5,000 gH. 2 to 600 g for capacitance, 5 to 160 mil for eddy current

Nonlinearity:

B. 1% of straight lineC. 1%

Sensitivity:

A. 0.02 g/gB. For amplitude ranges up to 5 g about 1000 mV/g; for ranges around 1000 g about

3 mV/g or 5 pico C/gC. From 0.25 to 25 mV/g corresponding to ranges of 5 to 1000 gD. 250 mV/in./sF. 0.05 mV/g

XSEquipment

Vibration

Flow Sheet Symbol

ToReceiver

© 2003 by Béla Lipták

1062

Safety and Miscellaneous Sensors

Partial List of Suppliers:

ACT/TechkorANCO Engineers, Inc.ATA SensorsBalmac Inc.Bruel & Kjaer (www.bkhome.com)Campbell Scientific, Inc.CEC Vibration ProductsDytran Instruments Inc. (www.dytran.com)Endevco Corporation (www.endevco.com)Entran Sensors and Electronics (www.entran.com)Hardy Instruments Inc.Hewlett-Packard Co. (www.hp.com)Indikon Company, Inc.Metrix Instrument Co.Kistler Instruments Corp. (www.kistler.com)L.A.B. Equipment, Inc. (www.labequipment.com)MB Dynamics, Inc. (www.mbdynamics.com)MEMS Equipment (www. xactix.com)MetroLaser, Inc. (www.metrolaserinc.com)Micro-EpsilonMontronix, Inc.MTI Instruments Inc.Murata ElectronicsNanoPositioning & Piezo TechnoOceana Sensor Technologies, Inc.Ometron, Inc. (www.ometron.com)Ono Sokki Co. Ltd.Optodyne, Inc.PCB Piezotronics, Inc.PI Polytec KK (www.physikinstrumente.com)R.C. Electronics, Inc. (www.rcelectronics.com)Sencera Company Ltd.Shinkawa Electric Co., Ltd.Silicon Designs, Inc.Spectral Dynamics, Inc. (www.sd-corp.com)Sprengnether Instruments, Inc. (www.Sprengnether.com)Syscom InstrumentsTDK Corporation of AmericaTrig-tek Inc.Trilion Quality SystemsVibroSystM Inc.Walesch Electronic GmbHWilcoxon Research, Inc. (www.wilcoxon.com)Zygo Corp. (www.zygo.com)

INTRODUCTION

Displacement, velocity, and acceleration are products ofmotion and they are related to each other by time. Becauseof ease in measurements, acceleration is particularly animportant section of kinematic quantities. Other quantitiesare the position, velocity, acceleration, and jerk, whichbear linear relationships with the neighboring ones. Thisindicates that all the kinematic quantities can be derivedfrom a single known quantity. For example, accelerationcan be obtained by differentiating the corresponding veloc-ity or by integrating the jerk. Likewise, velocity can beobtained by differentiating the position or by integrating

the acceleration. In practice, only the integration is widelyused since it provides better noise characteristics andattenuation.

Acceleration is an important parameter for general-purposeabsolute motion measurements, vibration, and shock sensing.Many different accelerometers are commercially available ina wide a range and type to meet diverse application require-ments, mainly in three areas:

1. Commercial applications: automobiles, ships, appli-ances, sports, and other hobbies

2. Industrial applications: robotics, machine control,vibration testing, and instrumentation

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration

1063

3. High reliability applications: military, space and aero-space, seismic monitoring, tilt measurements, vibra-tion, and shock measurements.

The unit of acceleration is the Earth’s gravitational accel-eration,

g

, which equals 32.3 ft/sec

2

in English units, or 9.8 m/s

2

in metric units. It is this unit that is used to express theamplitude of acceleration.

There are two classes of acceleration measurements tech-niques:

direct measurements

by specific accelerometers and

indirect measurements

where velocity is differentiated. Theapplicability of these techniques depends on the type ofmotion (rectilinear, angular, curvilinear motions) or equilib-rium centered vibration. For rectilinear and curvilinearmotions, the direct measurement accelerometers are pre-ferred. However, the angular acceleration is usually measuredby indirect methods. Accelerometers that measure the accel-eration of an object can be classified in a number of ways,including mechanical or electrical types, active or passivetypes, deflection or null-balance types, and others. Acceler-ometers used in vibration and shock measurements are usu-ally the deflection types, while those used for the measure-ment of motions of vehicles, aircraft, etc. for navigationpurposes may be either deflection or null-balanced.

The main theme of this section is vibration, shock, andacceleration sensing. We will largely concentrate on the directtechniques that are generally achieved by the accelerometersof the following types:

1.

Seismic (Inertial)

2.

Piezoelectric

3.

Piezoresistive and strain gages

4.

Electromechanical

5.

Capacitive and electrostatic

6.

Micro- and nanosensors

7.

Velocity sensors

8.

Noncontact proximity sensors

9.

Mechanical switches

10.

Optical sensors

In recent years, the Integrated Micro-Electro-MechanicalSystems technology is used to produce a sizable portion of

sensors and accelerometers along with well-establishedpiezoelectric and bulk-micromachined devices. This articledeals with the new technology as well as the traditional estab-lished ones.

In the preceding sections, acceleration and vibration asphenomenon are explained to make the users of thesedevices fully aware of some important concepts for a fruit-ful and meaningful use and interpretation of their testresults.

ACCELERATION AS A PHENOMENONAND DYNAMIC CHARACTERISTICS

Acceleration is related to motion, a vector quantity, exhibitinga direction as well as magnitude. The direction of motion isdescribed in terms of some arbitrary Cartesian or orthogonalcoordinate systems. Typical rectilinear, angular, and curvilin-ear motions are illustrated in Figures 7.22a, parts (a) to (c),respectively. The governing equations of these motions areas follows:

Rectilinear acceleration:

7.22(1)

Angular acceleration:

7.22(2)

Curvilinear acceleration:

7.22(3)

where

a

and

α

are the accelerations;

v

and

ω

are the speeds;

s

the distance;

θ

is the angle;

i

,

j

, and

k

are the unit vectorsin

x

,

y

, and

z

directions, respectively.

a Limvt

dvdt

d ds dtdt

d sdtt

= = = =→∆

∆∆0

2

2

( / )

a Lim tddt

d d dtdt

ddtt

= = = =→∆

∆∆0

2

2

ω ω θ θ( / )

advdt

d xdt

id ydt

jd zdt

k= = + +2

2

2

2

2

2

FIG. 7.22a

Types of motions to which accelerometers are commonly applied.

y

Pat

x

z

y

Pat

x

z

+S−S

Μ

Μ

(a)(b)

θ

y

Patx

z

Μ

(c)

θ

© 2003 by Béla Lipták

1064

Safety and Miscellaneous Sensors

The majority of accelerometers can be viewed and ana-lyzed as a single-degree-of-freedom seismic instrument thatcan be characterized by a mass, a spring, and a damperarrangement as shown in Figure 7.22b. In the case of multi-degrees-of-freedom systems, the principles of curvilinearmotion can be applied as in Equation 7.22(3) and multipletransducers must be used to create uniaxial, biaxial, or triaxialsensing points of the measurements.

If a single-degree-of-freedom system behaves linearly ina time invariant manner, the basic second-order differentialequation describing the motion of the forced mass-springsystem can be written as

7.22(4)

where

f

(

t

) is the force,

m

is the mass,

c

is the velocityconstant, and

k

is the spring constant.The base of the accelerometer is also in motion. When

the base is in motion, the force is transmitted through thespring to the suspended mass depending on the transmissi-bility of the force to the mass. Equation 7.22(4) may begeneralized by taking the effect motion of the base intoaccount as

7.22(5)

where

z

=

x

2

−

x

1

is the relative motion between the massand the base,

x

1

is the displacement of the base,

x

2

is thedisplacement of mass, and

θ

is the angle between the senseaxis and gravity.

The complete solution to Equation 7.22(5) can beobtained by applying the superposition principle. The super-position principle states that if there are simultaneouslysuperimposed actions on a body, the total effect can beobtained by summing the effects of each individual action.

Using the superposition and applying Laplace transformgives

7.22(6)

or

7.22(7)

where

s

is the Laplace operator,

K

=

1

/

k

is the static sensitivity,

ω

n

=

is the undamped critical frequency rad/s, and

ζ

=

(

c

/

2) is the damping ratio.As can be seen in the performance of accelerometers, the

important parameters are the static sensitivity, the naturalfrequency, and the damping ratio, which are all functions ofmass, velocity, and spring constants. Accelerometers aredesigned and manufactured to have different characteristicsby suitable selection of these parameters.

VIBRATION AND SHOCK

Vibration can be defined in terms of displacement, velocity,or acceleration. It is essentially an oscillatory motion result-ing from application of varying forces to a structure. Vibra-tions can be periodic, stationary random, nonstationary ran-dom, or transient. Traditional mechanical tripout devices,such as the spring-loaded magnetic holding types, are allacceleration sensors. These are actuated by changes in thevibration force at the measurement. At low frequencies thesedevices are severely limited because acceleration decreaseswith frequency, and large displacement is required to producesufficient acceleration to trip the device.

In rotary machinery, mechanical degradation causesvibration, and the mechanical health of the equipment ismonitored by measuring vibration. Vibration is measured byits frequency and amplitude. The frequency of vibration is afunction of the mass of the vibrating body, and the range ofvibration that occurs in compressors, turbines, and otherrotary equipment is generally between 10 and 2000 Hz, withextreme values falling between 1 and 20,000 Hz. The ampli-tude of vibration is a function of the design of the rotaryequipment, where at one extreme one might measure a vibra-tion amplitude of only 0.01

g

in a smoothly operating electricmotor, while a high-speed gearbox might have a vibrationamplitude of 100

g

at 5000 Hz or above. In shock detectionthe amplitude of acceleration can reach 100,000

g

.If the sensor (velocity or acceleration) is mounted onto

the surface of the vibrating object, it will provide a readingof absolute vibration relative to a fixed point in space. If themass of the rotating object (and therefore its resonant fre-quency) would be affected by physically attaching the sensorto it, a hole can be drilled through the housing of the bearings.

FIG. 7.22b

A typical seismic accelerometer.

Damper

DisplacementTransducer

Housing

Seismic Massm

x2(t)

Spring

Workpiece

x1= x0 Sinω1t

x1= x0 Cosω1t

f t md xdt

cdxdt kx( ) = + +

2

2

md zdt

cdzdt kz mg m

d xdt

2

2

2

21+ + = −cosθ

X s

F s mscs k

( )( )

= + +12

X s

F s

K

s sn n

( )( ) / /

=+ +( )2 2 2 1ω ζ ω

k m/km

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration

1065

A noncontacting proximity probe can detect the relativemotion between shaft and bearing. The limitation of thisapproach is that if the shaft and bearing housing are both invibration because some tie-down bolts are loose, the prox-imity probe will not detect this.

The electronics associated with the sensor can be integralor remote. The integral designs are less costly and less sensitiveto noise, but are more limited in their operating temperaturesand their vibration amplitude ranges. The main advantage ofself-generating sensors is that they do not require excitationpower supplies, while the main limitation is that they are notsuited for the measurement of constant acceleration, such asthat generated in a centrifuge.

Periodic Vibrations

In periodic vibrations, the motion of an object repeats itselfin an oscillatory manner. This can be represented by a sinu-soidal waveform

7.22(8)

where

x

(

t

)

=

time dependent displacement,

ω

=

2πft is theangular frequency, and Xp is the maximum displacement froma reference point.

The velocity of the object is the time rate of change ofdisplacement

7.22(9)

where v(t) is the time-dependent velocity and Vp = ωXp is themaximum velocity.

The acceleration of the object is the time rate change ofvelocity

7.22(10)

where a(t) is the time-dependent acceleration, and Ap = ω2Xp

= ωVp is the maximum acceleration.From the preceding equations it can be seen that the basic

form and the period of vibration remain the same in accel-eration, velocity, and displacement. But velocity leads dis-placement by a phase angle of 90° and acceleration leadsvelocity by another 90°.

In nature, vibrations can be periodic but not necessarilysinusoidal. If they are periodic but nonsinusoidal, they canbe expressed as a combination of a number of pure sinusoidalcurves, determined by the Fourier analysis as

7.22(11)

where ω 1, ω 2,…,ω n are the frequencies (rad/s), X0, X1,…,Xn

are the maximum amplitudes of respective frequencies, andφ1, φ 2,…,φn are the phase angles.

The number of terms may be infinite, and the higher thenumber of elements the better the approximation. These ele-ments constitute the frequency spectrum. The vibrations canbe represented in the time domain or frequency domain, bothof which are extremely useful in the analysis.

Stationary Random Vibrations These random vibrations aremet often in nature, where they constitute irregular cycles ofmotion that never repeat themselves exactly. Theoretically,an infinitely long time record is necessary to obtain a com-plete description of these vibrations. However, statisticalmethods and probability theory can be used for the analysisby taking representative samples. Mathematical tools such asprobability distributions, probability densities, frequencyspectra, cross-correlations, auto-correlations, digital Fouriertransforms (DFTs), fast Fourier transforms (FFTs), auto-spectral-analysis, root-mean-square values, and digital filteranalysis are some of the techniques that can be employed.

Nonstationary Random Vibrations In this case, the statisti-cal properties of vibrations vary in time. Methods such astime averaging and other statistical techniques can beemployed.

Transients and Shocks Short duration and sudden occur-rence vibrations need to be measured often. Shock and tran-sient vibrations may be described in terms of force, acceler-ation, velocity, or displacement. As in the case of randomtransients and shocks, statistical methods and FFTs are usedin the analysis.

SEISMIC (INERTIAL) SENSORS

Inertial acceleration and vibration sensors make use of aseismic mass that is suspended by a spring or a lever insidea rigid frame as shown in Figure 7.22b. The frame carryingthe seismic mass is connected firmly to the vibrating sourcewhose characteristics are to be measured. As the systemvibrates, the mass tends to remain fixed in its position, sothat the motion can be registered as a relative displacementbetween the mass and the frame. An appropriate transducersenses this displacement and the output signal is processedfurther. In practice, the seismic mass does not remain abso-lutely steady, but it can satisfactorily act as a reference posi-tion for selected frequencies.

By proper selection of mass, spring, and damper combi-nations, the seismic instrument may be used for either accel-eration or displacement measurements. In general, a largemass and soft spring are suitable for vibration and displace-ment measurement, while relatively small mass and a stiff

x t X tp( ) = sinω

v tdxdt X t V tp p( ) ( ) ( / )= = = +ω ω ω πcos sin 2

a tdvdt

d xdt

X t A tp p( ) sin( ) ( )= = = − = +2

22ω ω ω πsin

x t X X t X t

X tn n n

( ) ( ) ( )

( )

= + + + +

+ + +

0 1 1 1 2 2 2sin sin

sin

ω ω

ω

Φ Φ

ΦL

© 2003 by Béla Lipták

1066 Safety and Miscellaneous Sensors

spring are used in accelerometers. However, the term seismicis commonly applied to instruments, which sense very lowlevels of vibration in the ground or structures.

In order to describe the response of seismic accelerom-eter, from Newton’s second law the following equation ofmotion may be written as:

7.22(12)

where x1 is the displacement of the vibration frame, x2 is thedisplacement of the seismic mass, c is the velocity constant,θ is the angle between the sense axis and gravity, and k isthe spring constant.

Taking md2x1/dt2 from both sides of the equation andrearranging gives

7.22(13)

where z = x2 − x1 is the relative motion between the massand the base.

In Equation 7.22(13), it is assumed that the dampingforce on the seismic mass is proportional to velocity only. Ifa harmonic vibratory motion is impressed on the instrument:

7.22(14)

where ω1 is the frequency of vibration (rad/s). Writing

modifies Equation 7.22(13) as

7.22(15)

where a1 = mX0ω12.

Equation 7.22(15) will have transient and steady statesolutions. The steady-state solution of this differential equa-tion may be determined as

7.22(16)

Rearranging Equation 7.22(16) results in

7.22(17)

where ωn (= ) is the natural frequency of the seismicmass; ς (= c/2 ) is the damping ratio, which also can bewritten in terms of critical damping ratio as ς = c/cc wherecc = 2 ; φ (=tan1[cω 1/(kmω1

2)]) is the phase angle; and r(=ω 1/ωn) is the frequency ratio.

For satisfactory system performance, the instrument con-stant c/cc and ωn should carefully be calculated or obtainedfrom calibrations. In this way the anticipated accuracy ofmeasurement may be predicted for frequencies of interest. Acomprehensive treatment of the analysis is by McConnell;1

interested readers should refer to this text for further details.Seismic instruments are constructed in a variety of ways.

In a potentiometric instrument, a voltage divider potentiom-eter is used for sensing the relative displacement between theframe and the seismic mass. In the majority of potentiometricaccelerometers, the device is filled with a viscous liquid thatinteracts continuously with the frame and the seismic massto provide damping. These accelerometers have a low fre-quency of operation (less than 100 Hz) and are mainlyintended for slowly varying acceleration and low-frequencyvibrations. A typical family of such instruments offers manydifferent models, covering the range of ±1 to ±50 g full scale.The natural frequency ranges from 12 to 89 Hz, and thedamping ratio ζ can be kept between 0.5 to 0.8 by using atemperature compensated liquid-damping arrangement.Potentiometer resistance may be selected in the range of1,000 to 10,000 Ω, with a corresponding resolution of 0.45to 0.25% of full scale. The cross-axis sensitivity is less than±1%. The overall accuracy is ±1% of full scale or less atroom temperatures. The size is about a 50 mm3 with a massof about 1/2 kg.

Linear variable differential transformers (LVDTs) offeranother convenient means of measuring the relative displace-ment between the seismic mass and the accelerometer housing.These devices have higher natural frequencies than potentiom-eter devices (up to 300 Hz). Since the LVDT has lowerresistance to motion, it offers much better resolution. A typ-ical family of liquid-damped, differential-transformer accel-erometers exhibits the following characteristics: the full scaleranges from ±2 to ±700 g, the natural frequency from 35 to620 Hz, the nonlinearity 1% of full-scale. The full-scaleoutput is about 1 V with an LVDT excitation of 10 V at 2,000Hz, the damping ratio ranges from 0.6 to 0.7, the residualvoltage at the null position is less than 1%, and the hysteresisis less than 1% full scale. The size is about 50 mm3, with amass of about 120 g.

Electrical resistance strain gages are also used for dis-placement sensing of the seismic mass. In this case, theseismic mass is mounted on a cantilever beam rather than onsprings. Resistance strain gauges are bonded on each side ofthe beam to sense the strain in the beam resulting from thevibrational displacement of the mass. A viscous liquid thatentirely fills the housing provides damping of the system.The output of the strain gauges is connected to an appropriatebridge circuit. The natural frequency of such a system isabout 300 Hz. The low natural frequency is due to the need

md xdt

cdxdt kx c

dxdt kx mg

22

22

21

1+ + = + + cos( )θ

md zdt

cdzdt kz mg m

d xdt

2

2

21

2+ + = −cos( )θ

x t X t1 0 1( ) sin( )= ω

− =md xdt

mX t2

12 0 1sinω

md zdt

cdzdt kz mg ma t

2

2 1 1+ + = +cos sin( )θ ω

zmg

k

ma t

k m jc= +

− +( )cos sinθ ω

ω ω1 1

12

1

zmg a t

r rn n

= +−

− +( )cos sinθω

ω φ

ω ζ1 1

2 2 2 21 2

( )

( ) ( )

k m/km

km

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration 1067

for a sufficiently large cantilever beam to accommodate themounting of the strain gauges.

One serious drawback of the seismic instruments is tem-perature effects requiring additional compensation circuits.The damping of the instrument may also be affected bychanges in the viscosity of the fluid due to temperature. Forinstance, the viscosity of silicone oil, often used in theseinstruments, is strongly dependent on temperature.

Piezoelectric Sensors

The piezoelectric sensor is a self-generating device that isideal for the measurement of such dynamic events as shockand vibration. They are basically motion transducers withlarge output signals and comparatively small sizes. They areavailable with very high natural frequencies and are suitablefor high-frequency applications and shock measurements.Their sensing element is sandwiched between the transducerbody and a seismic mass (usually tungsten). Because theseismic mass is constant, the force acting on the sensingelement (F = ma) is proportional to acceleration. If the sens-ing element is a quartz crystal or lead-zirconate-titanate(PZT), an electric charge is generated (150 times more forPZT than for quartz) that is in proportion to the experiencedforce and therefore to acceleration.

These devices utilize a mass in direct contact with thepiezoelectric component or crystal. A separation of charge isproduced on the opposite faces of the crystal when it issubjected to acceleration forces. The magnitude of the volt-age produced is in proportion to mechanical deformation and,hence, acceleration. Generally, these units are provided astwo components: a sensor (Figure 7.22c) that mounts on theequipment and includes a solid-state amplifier and alarmswitch, and a control unit that contains the readout meter,power supply, alarm reset, and facilities for vibration record-ing or waveform analysis by oscilloscope. In the event ofapplication of varying motion to the sensor, the crystal expe-riences a varying force excitation (F = ma), causing a pro-portional electric charge q to be developed across it.

7.22(18)

where q is the charge developed and dij is the piezoelectriccoefficient of the material.

As this equation shows, the output from the piezoelectricmaterial is dependent on its mechanical properties, dij. Twocommonly used piezoelectric crystals are PZT and crystallinequartz. They are both self-generating materials and produce alarge electric charge for their size. As a result, PZTs are moresensitive and smaller in size than quartz counterparts. Theseaccelerometers are useful for high-frequency applications.

Mathematically, their transfer function approximates toa third-order system that can be expressed as:

7.22(19)

where Kq is the piezoelectric constant related to charge (Ccm), τ is the time constant of the crystal, and s is the Laplacevariable. It is worth noting that the crystal itself does nothave a time constant τ, but the time constant is observedwhen the accelerometer is connected into an electric circuit,for example, an RC circuit.

The low-frequency response is limited by the piezoelec-tric characteristic τs/(τs + 1), while the high-frequencyresponse is related to mechanical response. The dampingfactor ζ is very small, usually less than 0.01 or near zero.Accurate low-frequency response requires large τ, which isusually achieved by use of high-impedance voltage amplifi-ers. At very low frequencies, thermal effects can have severeinfluences on the operation characteristics.

In piezoelectric accelerometers, two basic design config-urations are used: compression types and shear-stress types.In compression-type accelerometers, the crystal is held incompression by a preload element; therefore, the vibrationvaries the stress in compressed mode. In a shear-stress accel-erometer, vibration simply deforms the crystal in shear mode.The compression accelerometer has a relatively good mass-to-sensitivity ratio and exhibits better performance. Since thehousing acts as an integral part of the spring-mass system, itmay produce spurious interfaces in the accelerometer outputif excited around its natural frequency.

Piezoelectric accelerometers are available in a very widerange of specifications and are offered by a large number ofmanufacturers. For example, the specifications of a shockaccelerometer may have 0.004 pC/g in sensitivity and a nat-ural frequency of up to 250,000 Hz, while a unit designedfor low-level seismic measurements might have 1,000 pC/gin sensitivity and only 7,000 Hz natural frequency. They aremanufactured as small as 3 × 3 mm2 in dimension with about0.5 g in mass, including cables. They have excellent temper-ature ranges and some of them are designed to survive theintensive radiation environment of nuclear reactors. However,piezoelectric accelerometers tend to have larger cross-axissensitivity than other types, about 2 to 4%. In some cases, largecross-axis sensitivity may be minimized during installationsby the correct orientation of the device. These accelerometers

FIG.7.22c Piezoelectric sensor.

q d F d maij ij= =

x1 = x0 Sinω1t

Alarm TripLevel

AdjustmentKnob

Circuitry Mass

PiezoelectricCrystal

e s

a s

K s

C s s w sq

n n n

02 2 21 2 1

( )

( ) ( ) / /=

+ + +( )τ

ω τ ζ ω

© 2003 by Béla Lipták

1068 Safety and Miscellaneous Sensors

may be mounted with threaded studs, cement or wax adhe-sives, or magnetic holders.

Piezoelectric crystals are affected in their output by tem-perature variations; however, quartz and some of the newerpiezoelectric ceramics are superior to such ceramics as bar-ium titanate or PZT in their reduced sensitivity to tempera-ture. In noisy electrical environments, electrical shielding ofthe crystal is recommended to reduce noise pickup.

Acceleration can expose the crystal element to compres-sion, shear, or bending. When the vibration lifts the sensorin the compression accelerometer, the inertial mass movesdownward, further compressing the already preloaded ele-ment. Conversely, downward acceleration reduces theamount of compression. In the shear accelerometer, the vibra-tion deforms the crystal in shear. A wide variety of piezo-electric sensors are available. Some are very small in sizeand weight (from 1 g up), while others are designed for high-temperature service or with abilities for measuring accelera-tion and vibration at frequencies up to 30,000 Hz and ampli-tudes of up to 100,000 g. PZT elements have high mechanicalspring constants and small inertial masses; therefore, theytend to be small and suitable for high frequencies.

The accelerometer can be mechanically connected to thevibrating surface (Figure 7.22d) or can be secured by adhe-sives or by magnetic means. The lighter-weight sensors areusually attached by glue, while when larger elements areused or if the vibrating surface is irregular, a mounting padis used to provide a flat mounting surface and ground iso-lation to prevent ground loops. On ferromagnetic surfaces,magnetic mounts can be used when only a quick test isneeded, but only up to about 1000 Hz frequency. Triaxialmounting pads are also available when three accelerometersare to be installed for three-dimensional (orthogonal) vibra-tion analysis.

Other factors affecting the performance of piezoelectricsensors include cable, magnetic, and dynamic effects. Inhigh-impedance accelerometers, the movement or strain onthe connecting cable can cause triboelectric (charge trappingdue to relative motion) noise. The straining or flexing of theaccelerometer base or housing due to thermal expansion orother forces can also transmit error-causing forces to the

crystal element. Because the output signal carried by thecable is low level, it needs to be protected against noise. Straycapacitive coupling can cause electrostatic noise; therefore,it should be fully shielded. One should also protect againstground loops by grounding the system at only one point.Common mode noise caused by magnetic coupling can beminimized by using twisted pairs and by not routing the cablenear conductors that carry power.

PIEZORESISTIVE AND STRAIN GAUGE SENSORS

Piezoresistive Sensors

Piezoresistive sensors are essentially semiconductor straingauges with large gauge factors. High gauge factors areobtained since the material resistivity is dependent primarilyon the stress not only on the dimensions. This effect can begreatly enhanced by appropriate doping of semiconductorssuch as silicon. Most piezoresistive accelerometers use twoor four active gauges arranged in a Wheatstone bridge. Extraprecision resistors are used, as part of the circuit, in serieswith the input to control the sensitivity, for balancing, andfor offsetting temperature effects.

Piezoresistive sensors are not self-generating and they dorequire a source of excitation power. They are superior topiezo-electronic crystals when low-frequency (under 10 Hz)vibrations, long duration shocks (over 10 ms), or constantaccelerations are to be measured. In the wire strain design, thespring wires are usually Nichrome and the frequency range isunder 1000 Hz. The bonded-foil-type accelerometer gaugeshave stable zero outputs and their temperature sensitivity islow. In the piezoresistive design, the wire elements are replacedby higher-sensitivity semiconductor (metallic oxide) chips.

The piezoresistive units are more rugged (but still fragile)and can operate at frequencies of 2000 Hz. Because of theirhigh sensitivity, they are frequently damped by oil filling tosuppress their resonant response. While damping makes thedesign more rugged, changes in the temperature of the fillingcan effect its frequency response. When excited at its resonantfrequency, the piezoresistive element can be damaged.

FIG. 7.22d The main components of a piezoelectric accelerometer are the body, the sensing element, and the seismic mass. (Courtesy of KistlerInstrument Co.)

Preload Sleeve

TransducerBody

Seismic Mass

Quartz CrystalsSensing Elements

BasePlate

Tape or Clamp to RelieveStress on Connector

Wrong

Correct

x1 = x0 Sinω1t

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration 1069

Strain-Gauge Sensors

Strain-gauge sensors are based on resistance properties ofelectrical conductors. If a conductor is stretched or com-pressed, its resistance alters due to both dimensional changesand resistance changes. This indicates that the resistivity ρof the conductor depends on the mechanical strain appliedonto it. The dependence is expressed as the gauge factor

7.22(20)

where 1 indicates the resistance change due to length, 2vindicates resistance change due to area, and (dρ/ρ)/(dL/L)indicates the resistance change due to piezoresistivity.

Strain gauge accelerometers measure a change in elec-trical resistance, which is in proportion to the force due toacceleration. The semiconductor strain gauge type consistsof semiconductors bonded to a mass whose deformationunder acceleration forces is reflected as a change of resis-tance. The resistance measurement is made by means of aWheatstone bridge or half Wheatstone bridge with the ele-ments so arranged on the mass that half of them sense tensionand the other half senses compression. A bending type ofaccelerometer with circuit diagram is shown in Figure 7.22e.

At full-scale deflection of the beam, RG1 and RG2 are amaximum and minimum, respectively, since they are subjectto equal but opposite deformations. For an initial resistanceof RG1 = RG2 = 1000 Ω and

7.22(21)

the full-scale output is given by the relationship

7.22(22)

The readout meter must have a high resistance so thatfor practical purposes the current flow through the meter iszero. In the above example, the accelerometer contained onlytwo active elements. Sensitivity can be improved by theaddition of two more elements to complete the Wheatstonebridge.

There are many types of strain-gauges: unbonded metal-wire gauges, bonded metal-wire gauges, bonded metal-foilgauges, vacuum-deposited thin-metal-film gauges, bondedsemiconductor gauges, and diffused semiconductor gauges.However, usually bonded and unbonded metal-wire gaugesfind wider applications. A section of the strain-gauge acceler-ometers (particularly bonded semiconductor types) known asthe piezoresistive transducers are used, but they suffer fromhigh temperature sensitivities, nonlinearities, and some mount-ing difficulties. Nevertheless, with the recent developments ofmicromachine technology, these sensors have been improvedconsiderably and are finding many new applications.

Unbonded-strain-gauge accelerometers use the strainwires as the spring element and as the motion transducer. Anexample of an unbonded-wire-type accelerometer is shownin Figure 7.22f. Mass m is attached to the base by cantileversprings, and the resistance elements are connected in the formof the Wheatstone bridge. Usually the entire assembly iscontained in an oil-filled case for viscous damping. They areuseful for general-purpose motion and vibration measure-ments from low to medium frequencies. They are availablein wide ranges and characteristics: typically ±5 to ±200 gfull scale, a natural frequency of 17 to 800 Hz, a 10 Vexcitation voltage AC or DC, full scale output ±20 to ±50 mV,a resolution less than 0.1%, an inaccuracy less than 1% fullscale, and a cross-axis sensitivity less than 2%. The damping

FIG. 7.22e Strain gauge vibration sensor (acceleration sensitive).

dR R

dL L

d

dL L

//

//

= + +1 2ν ρ ρ

∆ R

RG

G

= 0 1.

V ER R

R R R R

R

R ROG G

G G G G

=+

+ + − +

1

1 2

4

3 4

∆∆ ∆( ) ( )

–

VO =

1011002000

10002000

–

Mass Strain Gauges

SensitiveAxis

CantileverSprings Base

InsulatedPosts

x1 = x0 Sinω1t

FIG. 7.22fBending-type vibration sensor (acceleration sensitive).

x1 = x0 Sinω1t

Sens

itive

Axi

s

Cover

Strain ElementRG1

Cantilevered Mass

ElectricalConnectionRG2

RG1

RG2

Base Physical Construction

Sensor

C

OutputTerminals

RegulatedSensor

Power Supply

+

−

Capacitor to Eliminate DCfrom the Output Signal

Circuit Diagram

Vo

R3

R4

© 2003 by Béla Lipták

1070 Safety and Miscellaneous Sensors

ratio (using silicone oil damping) is 0.6 to 0.8 at room tem-perature. These instruments are small and light, usually witha mass less than 25 g.

Bonded-strain-gauge accelerometers generally use amass supported by a thin flexure beam. The strain gauges arecemented onto the beam to achieve maximum sensitivity,temperature compensation, and sensitivity to both cross-axisand angular accelerations. Their characteristics are similar tothe unbonded-strain-gauge accelerometers but have greatersizes and weights. Often silicone oil is used for damping.Semiconductor strain gauges are widely used as strain sen-sors in cantilever-beam and mass types of accelerometers.They allow high outputs (0.2 to 0.5 V full scale). Typically,a ±25 g acceleration unit has a flat response from 0 to 750 Hz,a damping ratio of 0.7, a mass of about 28 g, and an opera-tional temperature of −18 to 93°C. A triaxial ±20,000 g modelhas a flat response from 0 to 15 kHz, a damping ratio of 0.01,and a compensation temperature range of 0 to 45°C, and is13 × 10 × 13 mm3 in size and 10 g in mass.

ELECTROMECHANICAL SENSORS

Electromechanical acceleration and vibration sensors, essen-tially servo or null-balance types, rely on the principle offeedback. In these instruments, an acceleration-sensitivemass is kept very close to a neutral position or zero displace-ment point by sensing the displacement and feeding back theeffect of this displacement. A proportional magnetic force isgenerated to oppose the motion of the mass displaced fromthe neutral position, restoring this position just as a mechan-ical spring in a conventional accelerometer would do. Theadvantages of this approach are better linearity and elimina-tion of hysteresis effects as compared to the mechanicalsprings. Also, in some cases, electrical damping can be pro-vided, which is much less sensitive to temperature variations.

One very important feature of electromechanical accel-erometers is the capability of testing the static and dynamicperformances of the devices by introducing electricallyexcited test forces into the system. This remote self-checkingfeature can be quite convenient in complex and expensivetests where accuracy is essential. These instruments are alsouseful in acceleration control systems, since the referencevalue of acceleration can be introduced by means of a pro-portional current from an external source. They are used forgeneral-purpose motion measurements and monitoring low-frequency vibrations.

There are a number of different electromechanical accel-erometers, including coil-and-magnetic types, inductiontypes, and others.

Coil-and-Magnetic Accelerometers

These accelerometers are based on Ampere’s law, that is, “acurrent-carrying conductor disposed within a magnetic fieldexperiences a force proportional to the current, the length of

the conductor within the field, the magnetic field density, andthe sine of the angle between the conductor and the field.”

The coils of these accelerometers are located within thecylindrical gap defined by a permanent magnet and a cylin-drical soft iron flux return path. They are mounted by meansof an arm situated on a minimum friction bearing or flexureso as to constitute an acceleration-sensitive seismic mass. Apickoff mechanism senses the displacement of the coil underacceleration and causes the coil to be supplied with a directcurrent via a suitable servo controller to restore or maintaina null condition. The electrical currents in the restoring circuitis linearly proportional to acceleration, provided (1) armaturereaction affects are negligible and fully neutralized by a com-pensating coil in opposition to the moving coil, and (2) thegain of the servo system is large enough to prevent displace-ment of the coil from the region in which the magnetic fieldis constant.

In these accelerometers, the magnetic structure must beshielded adequately to make the system insensitive to exter-nal disturbances or the earth’s magnetic field. Also, in thepresence of acceleration there will be a temperature rise dueto i2R losses. The effects of these i2R losses on the perfor-mance are determined by the thermal design and heat-transferproperties of the accelerometers.

Induction Accelerometers

The cross-product relationship of current, magnetic field, andforce is the basis for induction-type electromagnetic acceler-ometers. These accelerometers are essentially generatorsrather than motors. One type of instrument, the cup-and-magnet design, includes a pendulous element with a pickoffmechanism and a servo controller driving a tachometer cou-pling. A permanent magnet and a flux return ring, closelyspaced with respect to an electrically conductive cylinder, areattached to the pendulous element. A rate-proportional dragforce is obtained by the electromagnetic induction effectbetween the magnet and the conductor. The pickoff mecha-nism senses pendulum deflection under acceleration andcauses the servo controller to turn the rotor in a sense to dragthe pendulous element toward the null position. Understeady-state conditions motor speed is a measure of the accel-eration acting on the instrument. Stable servo operation isachieved by employing a time-lead network to compensatethe inertial time lag of the motor and magnet combination.The accuracy of the servo-type accelerometers is ultimatelylimited by consistency and stability of scale factors of cou-pling and cup-and-magnet devices as a function of time andtemperature.

Another accelerometer based on induction design uses theeddy-current induction torque generation. The force-generatingmechanism of an induction accelerometer consists of a stablemagnetic field, usually supplied by a permanent magnet,which penetrates orthogonally through a uniform conductionsheet. The movement of the conducting sheet relative to themagnetic field in response to acceleration results in a generated

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration 1071

electromotive potential in each circuit in the conductor. Thisaction is in accordance with the Faraday’s principle. In induc-tion-type accelerometers, the induced eddy currents are con-fined to the conductor sheet, making the system essentially adrag coupling. Since angular rate is proportional to accelera-tion, angular position represents change in velocity. This is auseful feature particularly in navigation applications.

A typical commercial instrument based on the servo-accelerometer principle might have a micromachined quartzflexure suspension, differential capacitance angle pickoff,air squeeze film plus servo lead compensation for systemdamping. Of the various available models, as an example,a typical 30-g unit has a threshold and resolution of 1 µg,a frequency response that is flat to within 0.05% at 10 Hzand 2% at 100 Hz, a natural frequency of 1,500 Hz, adamping ratio from 0.3 to 0.8, and transverse or cross-axissensitivity of 0.1%. If, for example, the output current wereabout 1.3 mA/g, a 250 Ω readout resistor would give about±10 V full scale at 30 g. These accelerometers are good forprecision work and used in many applications such as air-craft and missile control systems, measurement of tilt anglesfor borehole navigation, and axle angular bending in aircraftweight and balance systems.

CAPACITIVE AND ELECTROSTATIC SENSORS

Capacitive and electrostatic sensors are based on Coulomb’slaw between two charged electrodes; therefore, they arecapacitive types. Depending on the operation principles andexternal circuits, they can be broadly classified as electrostatic-force-feedback accelerometers or capacitive accelerometers.

Electrostatic-Force-Feedback Accelerometers

An electrostatic-force-feedback accelerometer consists of anelectrode, with mass m and area S, mounted on a light pivotedarm that moves relative to some fixed electrodes. The nom-inal gap h between the pivoted and fixed electrodes is main-tained by means of a force-balancing servo system, which iscapable of varying the electrode potential in response tosignals from a pickoff mechanism that senses relativechanges in the gaps. Mathematically, the field between theelectrodes may be expressed by

7.22(23)

where E is the intensity or potential gradient (dV/dx), Q isthe charge, S is the area of the conductor, and k is the dielec-tric constant of the space outside the conductor.

From this expression, it can be shown that the force perunit area of the charged conductor (in N/m2) is given by

7.22(24)

Considering one movable and one stationary electrodesand assuming that the movable electrode is maintained at abias potential V1 and the stationary one at a potential V2. Theelectrical intensity E in the gap can be expressed as

7.22(25)

so that the force of attraction may be found as

7.22(26)

In the presence of acceleration, if V2 is adjusted torestrain the movable electrode to the null position, theexpression relating acceleration and electrical potential maybe given by

7.22(27)

The device so far described can measure acceleration inone direction only, and the output is quadratic character, that is,

7.22(28)

where D is the constant of proportionality.The output may be linearized in a number of ways, one

of which is the quarter-square method. If the servo controllerapplies a potential −V2 to the other fixed electrode, the forceof attraction between this electrode and the movable electrodebecomes

7.22(29)

and the force-balance equation of the movable electrodewhen the instrument experiences a downward acceleration anow is

or

7.22(30)

Hence, if the bias potential V1 is held constant and the gainsof the control loop are high so that variations in the gap arenegligible, the acceleration becomes a linear function of thecontroller output voltage V2.

EQ

kS=

ε

F

S

Q

kS

kE= =2

2

2

2 2εε

EV V

h11 2=

−

FkE S

h

k V V S

h1

2

21 2

2

22 2= =

−ε ε ( )

aF

m

k V V S

h m= =

−1 1 22

22

ε ( )

( )V V D a1 2− =

aF

m

k V V S

h m= =

+2 1 22

22

ε ( )

ma F FkS V V V V

h= − =

+ − −[ ]2 1

1 22

1 22

22

ε ( ) ( )

ma F FkSV V

h= − =2 1

1 22

2ε

© 2003 by Béla Lipták

1072 Safety and Miscellaneous Sensors

The principal difficulty in mechanizing the electrostaticforce accelerometer is the relatively high electric field inten-sity required to obtain an adequate force. Damping can beprovided electrically or by viscosity of the gaseous atmo-sphere in the interelectrode space if the gap h is sufficientlysmall. The scheme works best with micromachined instru-ments. Nonlinearity in the voltage breakdown phenomenonpermits larger gradients in very small gaps.

The main advantages of electrostatic accelerometers aretheir extreme mechanical simplicity, low power require-ments, absence of inherent sources of hysteresis errors, zerotemperature coefficients, and ease of shielding from strayfields.

Capacitive Accelerometers

There are various types of capacitive accelerometers. Thedifferential-capacitance accelerometers are based on the prin-ciple of the change of capacitance in proportion to appliedacceleration. In one type, the seismic mass of the accelerom-eter is made as the movable element of an electrical oscillatoras shown in Figure 7.22g. The seismic mass is supported bya resilient parallel-motion beam arrangement from the base.The system is set to have a certain defined nominal frequencywhen undisturbed. If the instrument is accelerated, the fre-quency varies above and below the nominal value dependingon the direction of acceleration.

The seismic mass carries an electrode located in opposi-tion to a number of base-fixed electrodes that define variablecapacitors. The base-fixed electrodes are resistances coupledin the feedback path of a wide-band, phase-inverting ampli-fier. The gain of the amplifier is predetermined to ensuremaintenance of oscillations over the range of variation of thecapacitance determined by the applied acceleration. The

value of the capacitance C for each of the variable capacitorsis given by

7.22(31)

where k is the dielectric constant, ε is the permittivity of freespace, S is the area of electrode, and h is the variable gap.

Denoting the magnitude of the gap for zero accelerationas h0, the value of h in the presence of acceleration a maybe written

7.22(32)

where m is the value of the proof mass and K is the springconstant. Thus,

7.22(33)

For example, the frequency of oscillation of the resistance-capacitance type circuit is given by the expression

7.22(34)

Substituting this value of C in Eq. 2.23(34) gives

7.22(35)

Denote the constant quantity /2πRεkS as B and rewritethe Equation 7.22(35) as

7.22(36)

The first term on the right-hand side expresses the fixedbias frequency f0 and the second term denotes the change infrequency resulting from acceleration, so that the expressionmay be written as

7.22(37)

If the output frequency is compared with an independentsource of a constant frequency of f0, the fa can be determinedeasily.

A commonly used example of a capacitive-type acceler-ometer is based on a thin diaphragm with spiral flexures thatprovide the spring, proof mass, and moving plate necessaryfor the differential capacitor. Plate motion between the elec-trodes pumps air parallel to the plate surface and throughholes in the plate to provide squeeze film damping. Since air

FIG. 7.22g A differential-capacitance-type accelerometer.

Profit Mass

C C C

R R R

Amplifier

Output

CkS

h= ε

h hma

K= +0

CkS

h ma K=

+ε

0 ( / )

fRC

= 62π

fh ma K

R kS=

+6

20[ ( / )]

π ε

6

f BhBma

K= +0

f f fa= +0

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration 1073

viscosity is less temperature sensitive than oil, the desireddamping ratio of 0.7 hardly changes more than 15%. Familiesof such instruments are easily available with full-scale rangesfrom ±0.2 g (4 Hz flat response) to ±1000 g (3000 Hz), across-axis sensitivity less than 1%, and a full-scale output of±1.5 V. The size of a typical device is about 25 mm3 with amass of 50 g.

MICRO- AND NANOSENSORS

By the end of the 1970s, it became apparent that the essen-tially planar processing integrated-circuit (IC) technologycould be modified to fabricate three-dimensional electrome-chanical structures by the micromachining process. Acceler-ometers and pressure sensors were among the first IC sensors.The first accelerometer was developed in 1979. Since then,the technology has been progressing steadily and now anextremely diverse range of accelerometers are readily avail-able. Most sensors use bulk micromachining, rather thansurface micromachining techniques. In bulk micromachining,the flexures, resonant beams, and all other critical compo-nents of the accelerometer are made from bulk silicon inorder to exploit the full mechanical properties of siliconcrystals. With proper design and film process, bulk microma-chining yields an extremely stable and robust accelerometer.

The selective etching of multiple layers of depositedthin films, or surface micromachining, allows movablemicrostructures to be fabricated on silicon wafers. Withsurface micromachining, layers of structure material aredisposed and patterned as shown in Figure 7.22h. Thesestructures are formed by polysilicons and sacrificial mate-rials such as silicon dioxides. The sacrificial material actsas an intermediate spacer layer and is etched away to pro-duce a freestanding structure. Surface machining technol-ogy also allows smaller and more complex structures to bebuilt in multiple layers on a single substrate. A typicalexample of a modern micromachined accelerometer is givenin Figure 7.22i. Multiple accelerometers can be mountedon a single chip, sensing accelerations in x, y, and z direc-tions. The primary signal conditioning is also provided inthe same chip. The output from the chip is usually read inthe digital form.

The operational principles of microaccelerometers arevery similar to capacitive force-balance or vibrating-beamaccelerometers, discussed earlier. Manufacturing techniquesmay change from one manufacturer to another. However, ingeneral, vibrating-beam accelerometers are preferred becauseof better air-gap properties and improved bias performancecharacteristics.

Vibrating-beam accelerometers, also termed resonant-beam force transducers, are made in such a way that anacceleration along a positive input axis places the vibratingbeam in tension. Thus the resonant frequency of the vibratingbeam increases or decreases with the applied acceleration.

In double-end tuning fork (DETF) accelerometers, anelectronic oscillator capacitively couples energy into twovibrating beams to keep them oscillating at their resonantfrequency. The beams vibrate 180° out of phase to cancelreaction forces at the ends. The dynamic cancellation effectof the DETF design prevents energy from being lost throughthe ends of the beam. Hence, the dynamically balanced DETFresonator has a high Q factor, which leads to a stable oscil-lator circuit. The acceleration signal is output from the oscil-lator as a frequency-modulated square wave that can be usedfor a digital interface.

The frequency of resonance of the system must be muchhigher than any input acceleration, and this limits the mea-surable range. In a micromachined accelerometer, used inmilitary applications, the following characteristics are given:a range of ±1,200 g, a sensitivity of 1.11 Hz/g, a bandwidthof 2,500 Hz, and an unloaded DETF frequency of 9,952 Hz.The frequency at +1,200 g is 11,221 Hz, the frequency at−1,200 g is 8,544 Hz, and the temperature sensitivity is 5mg/°C. The accelerometer size is 6 mm diameter by 4.3 mmlength, with a mass of about 9 g. It has a turn on time less

FIG. 7.22h Steps of micromachining to manufacture micro and nano acceler-ometers.

IsolationLayer

SpacerLayer

Substrate

BodyLayer

Substrate

Substrate

FreestandingMicromechanicalStructure

© 2003 by Béla Lipták

1074 Safety and Miscellaneous Sensors

then 60 s, the accelerometer is powered with +9 to +16 V DC,and the nominal output is 9,000 Hz square wave.

Surface micromachining has also been used to manufac-ture specific application accelerometers, such as air-bagapplications in the automotive industry. In one type, a three-layer differential capacitor is created by alternate layers ofpolysilicon and phosphosilicate glass on a 0.38 mm thick,100 mm long wafer. A silicon wafer serves as the substratefor the mechanical structure. The trampoline-shaped middlelayer is suspended by four supporting arms. This movablestructure is the seismic mass for the accelerometer. The upperand lower polysilicon layers are fixed plates for the differen-tial capacitors. The glass is sacrificially etched by hydroflu-oric acid.

VELOCITY SENSORS

Acceleration is the time differential of velocity and can bemeasured by using velocity sensors. A typical velocity sensorconsists of a housing that is attached to the vibrating surfaceand a magnetic core suspended inside it. Relative motionbetween the two induces a voltage that is proportional tovelocity. The advantages of this sensor include its self-generating nature, strong output signal (250 mV/in./sec),insensitivity to mounting or torque, and low noise pickupdue to its low impedance. Its disadvantages are its relativelylarge size and mass, inability to be suitable for frequenciesbelow 10 Hz, and low output signals at frequencies above1000 Hz.

Velocity sensors consist of an electrical coil suspendedby a spring or springs in the field of a permanent magnet asin Figure 7.22j. The entire assembly may be contained insilicone oil, or, for low-mass coils, the reactive forces caused

by the induced currents can be used for damping. The springcoil mass system has a natural frequency below, and the coilmoves with the enclosure. Above the natural frequency, thecoil remains stationary, and the relative motion between coiland magnet induces a voltage in the coil. This voltage hasan amplitude in proportion to the vibration velocity and afrequency equal to the frequency of vibration. With velocitypickup, the output can be in the form of mils (thousandthsof an inch) displacement by integration of velocity. A controlunit mounted separately is required for these pickups; it con-tains the readout, amplifier, and power supply. Electronicfiltering can be provided where it is desired to look at onlyspecific frequency bands. These devices are fairly rugged,but the oil-damped types are temperature-sensitive becauseof viscosity changes of the filling oil.

FIG. 7.22i Multiple accelerometers in a single chip.

SiliconWafer

IndividualAccelerometer

Sawing Lines

FIG. 7.22j Schematic representation of a velocity sensor.

CoilMass

PermanentMagnet

Oil Fillfor

Damping

Springs

OutputConnections

Sens

itive

Axi

s

N S

© 2003 by Béla Lipták

7.22 Vibration, Shock, and Acceleration 1075

NONCONTACT OR PROXIMITY SENSORS

As the proximity detectors measure the distance betweenobjects, they can also be used to measure the frequency andamplitude of vibrations. Several proximity sensor principlescan be used. The advantage of capacitance types is theirsimplicity and low temperature sensitivity. Potentiometricunits have the advantage of accommodating large amplitudesof movement and strong outputs, while their disadvantage istheir low natural frequency. LVDTs and variable inductanceunits are used in telemetry.

To detect the proximity of conducting materials, an eddycurrent probe can be used. The schematic arrangement ofsuch a probe is shown in Figure 7.22k. Two identical coilsare wound on the probe, and these, together with the resis-tances, complete a bridge circuit. With no conducting surfacenear the probe, the bridge is in balance. When a conductingobject is brought near the probe, the bridge becomes unbal-anced and the output signal is in proportion to the proximityof the object. The excitation is a high-frequency signal, whichinduces eddy currents in the test object. These currents pro-duce losses in the bridge circuit in such a way that bridgeimbalance is related to the proximity of the object. The outputsignal amplitude is related to the vibration or displacementamplitude, and the frequency of amplitude variations is thefrequency of vibration. The bridge is powered by a high-frequency (100 kHz to 2 MHz) oscillator that supplies therapidly changing magnetic field required for sensor operation.

MECHANICAL-MAGNETIC SWITCHES

The detector mechanism of this switch consists of an arma-ture suspended on a flexure pivot and restrained from movingby a permanent magnet as shown in Figure 7.22l. A com-pression spring provides an adjustable force opposing themagnet. The armature acts as a lever where the adjustingspring force is balanced by the magnetic force. The armatureis constrained to move in only one plane by the flexure pivot.When the entire assembly is subjected to vibration perpen-dicular to the base, the product of acceleration and armaturemass produces an inertial force trying to pull the armatureaway from the magnet. Since the spring force aids the inertial

force, the spring loading is selected to trip the mechanism atthe desired acceleration. When the peak acceleration exceedsthe preset level, the armature leaves the stop pin and movesto the latch magnet, which holds the armature in the excessivevibration position. Motion of the armature actuates a snapacting switch that opens or closes an electrical circuit. Oncethe switch has been tripped by excessive vibration, the arma-ture must be returned to the stop pin. Resetting can be accom-plished remotely by energizing the reset coil momentarily orby depressing the reset button mounted on the switch. Thereset coil requires direct current for operation. If DC poweris not available, a rectifier can be supplied with the switch.The rectifier output is sufficient to power several reset coils.

OPTICAL SENSORS

Optical sensors are often used in industrial and laboratoryenvironment for displacement, vibration, and accelerationsensing. They are mainly noncontact devices and are appliedwhere precision is necessary, but they tend to be more expen-sive than the other devices. There are many different tech-niques available to suit a particular application. They canbroadly be classified as fiber optic sensors, laser optic, inter-ferometric sensors, time-of-flight sensors, Doppler-effectsensors, encoders, and digital displacement sensors. Further,there are many different types in each class. For example, theinterferometric sensors include the optical fiber, laser, holo-graphic, and monochromatic light types. In these sensors,Michelson, Sagnac, or Fizeau effects of interference of twoor more light beams is used by suitable optical arrangements.The most commonly used optical arrangements are beamsplitters, polarizing prisms, and diffraction gratings. Some ofthese include fringe counting inteferometers, frequency mod-ulation inteferometers, heterodyne interferometers, phase-locked interferometers, laser Doppler and laser feedbackinterferometers, and fiber interferometers.

FIG. 7.22k Eddy current probe schematic.

Resistor

Probe

ACProbe

Excitation Resistor

OutputSignal

FIG. 7.22l Mechanical-magnetic switch.

SensitivityAdjusting

Screw SensitiveAxis

ResetButton

Switch

Armature

LeafSpring

FlexurePivot

ResetCoil

StopPin

Magnet

LatchMagnet

© 2003 by Béla Lipták

1076 Safety and Miscellaneous Sensors

CONCLUSIONS

Acceleration, vibration, and shock measurements are veryimportant in all types of industry. Because of the vast market,there are many vendors producing sensors and devices forthese measurements. Also, there are many different types. Themechanical sensors are limited to detection of lower frequen-cies of vibrations. Output signals for waveform analysis arenot available with these units. They are used as trip devicesto shut down equipment or an alarm on excessive vibration.If vibration sensing is for the purpose of detecting impendingmalfunction, devices with continuous outputs are required.PZT and peizoresistive acceleration sensors are commonlyused for their ruggedness and compactness and their suitabilityin high frequency applications. Today, mass produced micro-and nano-accelerometers are gaining popularity because ofthe price, small size, and ease in signal processing require-ments. Velocity sensors, however, offer a direct measurementof the energy dissipated in vibration, and thus indicate a quan-tity directly related to efficiency and the destructive forces.

Reference

1. McConnell, K.G., Vibration Testing: Theory and Practice, New York:John Wiley & Sons, 1995.

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© 2003 by Béla Lipták