Applied Structural and Mechanical Vibrations 2009 Part 14 ppsx

preferred instrument where high-quality displacement and vibrationmeasurements are needed, such as for laboratory calibration purposes or forhighly demanding applications.14.5 Relative-v

resistive elements that are cemented on a structure of which they measurethe local deformation through the variation in resistance caused by elongation

or contraction

Strain gauges function on a principle based on the expression

which gives the resistance of a uniform conductor of resistivity ρ, length L

and cross-section area A.

The fractional change in resistance is then given by

(14.1)

where v is Poisson’s ratio of the conductor material, and is the strain.According to eq (14.1), strain gauges do not actually measure displacementbut strain, i.e the average gauge elongation or contraction divided by the

gauge length The parameter K is called the gauge factor, which accounts for

the resistance variations due to dimensional changes, represented by the term

(1+2v), and for those caused by the strain-induced resistivity variations

This latter effect is called the piezoresistive effect

Depending on the material of which the strain gauge is made, the gaugefactor assumes different values, ranging from close to 2 for nickel-copper(constantan) and 2.1 nickel-chromium (karma) alloys, to about 3.5 forisoelastic, to above 100 for semiconductors

Metal alloy strain gauges are the most widely used and, as shown in Fig.14.2, they typically have the form of grid foils of various dimensions andgeometry supported by an insulating backing carrier which allows them to

be bonded to the body under test The backing carrier performs thefundamental function of transferring the strain from the specimen to thegauge with maximum fidelity

The nominal values of resistance are normally 120, 350, 700 or 1000 Ω,with strain-induced variations that are usually quite small, as low as fewparts per million (ppm), and therefore require special care in theirmeasurement Moreover, the temperature appreciably influences both thegauge resistance and the gauge factor, producing the so-called thermal output,which is due to the temperature coefficient of resistance (TCR) and of gaugefactor (TCGF) combined with the thermal expansion of the specimen

A typical solution is given by the use of the Wheatstone bridge configurationwith voltage or current excitation of either DC or AC type (Chapter 15 ).Special arrangements including multiple active and/or dummy gauges areused to maximize linearity and compensate for the thermal effects, and properwiring techniques allow for lead wire resistance cancellation in distantconnections between the bridge and the excitation and amplification circuitry.With properly employed good-quality signal conditioning circuits, strainlevels in the microstrain range can be ordinarily detected and values lower

than 1 µ ε are possible Such figures enable the use of strain gauges for stress

metal foils bonded to the transducer structure, but may be of differentmaterials and construction For instance, they may be conductors deposited

in thin- or thick-film technology, or integrated semiconductors, such as insilicon micromachined sensors

14.4.3 Linear variable differential transformers

The linear variable differential transformer (LVDT) is a transducer based onthe magnetic induction principle It is made by a transformer with one primarycoil and two secondary coils with a movable core of ferromagnetic materialwhich is placed coaxially in the coils without touching them, as shown inFig 14.3 The core terminates in a shaft or plunger which is attached to thetarget object either by threading or spring loading The transducer bodycontaining the primary and secondary coils is mounted in the referenceposition As the core moves, it produces a variation of the magnetic couplingdepending on its position along the coil axis When the primary coil is excited

with a sinusoidal voltage of amplitude VE an induced voltage Vo is collected

across the secondary coils, which is linearly related to the core position

through the mutual inductance coefficient M Since the secondary coils are connected in series opposition, M equals zero when the core is centered and

it changes sign according to the sign of the core off-centre position For therotary variable differential transformer (RVDT) the operating principle isthe same, with the difference that the rotary core movement allows themeasurement of angular rather than linear displacement

A measure of the core displacement can then be obtained by rectifying

the voltage Vo while taking into account of its phase relative to VE This

readout operation is typically carried out by dedicated electronic circuitry,and is usually obtained by employing an oscillator and a phase-sensitivedemodulation stage followed by low-pass filtering (Chapter 15)

Fig 14.3 Schematic diagram of linear variable differential transformer (LVDT).

The oscillator signal for the primary coil excitation is preferably sinusoidal

to avoid the generation of harmonics, and often a particular frequency isrecommended by the transducer manufacturer for obtaining maximumcompensation of residual phase shift at null core position In the case ofdynamic measurements, the oscillator frequency should be set at a value up

to ten times higher than the highest motion frequency, to avoid frequencyoverlapping Since the maximum recommended excitation frequencies oftypical LVDTs are in the order of 20 kHz, the useful measurement bandwidth

is generally limited to several kilohertz

The mechanical input impedance of LVDTs is mainly given by the massassociated with the core inertia in threaded-core types, with an added springeffect when spring-loaded plunger-type cores are used Friction is generallyalmost absent, since there is no contact during the core motion within thetransducer This fact offers an ideally unlimited resolution and virtually nohysteresis, which represent fundamental advantages of LVDTs compared,for instance, with resistive potentiometers Practical devices can reachresolutions better than 0.01% of the range, therefore submicron displacementsmay be appreciated with transducers with a stroke of few millimetres.These features, joined to a typically rugged construction and goodimmunity to environmental factors and electromagnetic interference, makesLVDTs the first choice for transducers in many precision measurementapplications They are often considered the electrical equivalent of themechanical dial gauge, or micrometer

As a drawback, LVDTs are not generally cheap and require dedicatedsignal conditioning electronics, which is comparatively costly In this respect,some devices which include the excitation and amplification electronics withinthe transducer case are particularly advantageous, providing a DC voltageoutput signal ready to be acquired

As a precaution, nonmagnetic materials such as aluminium or plasticshould be used for the mounting fixture in order not to alter the sensitivity.Similar in shape to the LVDTs is a type of variable-inductance transducerwhich includes only two series-connected coils in an autotransformerconfiguration, i.e analogous to the secondary windings of the LVDTs withthe primary absent When the core is halfway between the coils their respectiveinductances are equal, while they become different according to the amountand sign of the core off-centre displacement The transducer then essentiallyworks as an inductive potentiometer whose cursor is represented by the core,with the advantage of no internal electrical contact The inductance imbalancecan be measured by connecting the transducer in an AC-excited bridge andreading the correspondent bridge output

14.4.4 Inductive transducers

The functioning principle is based on the variable inductance of a coil wound

on a core caused by the changes in the magnetic flux reluctance when the

distance from a ferromagnetic target varies If the measured object is ferrous itcan act as the target, otherwise a ferromagnetic target must be attached to theobject The inductance changes are usually measured in an AC bridge circuit,

or by making it part of a resonant circuit and detecting the resonant frequencyshift Two geometries may be used, namely the closed-loop and open-loopmagnetic system, as shown in Fig 14.4 In both cases, provided that the magneticpermeability of both the core and the target are much greater than that of air

requirement for a ferromagnetic target) the coil inductance L may be

approximated by

(14.2)

where A is the area of the core facing the target, N is the number of coil turns and d is the distance from the target.

It can be observed that the relationship between L and d is nonlinear,

therefore these transducers are best suited for use as proximity sensors ratherthan distance measuring devices To obtain linear operation, electroniclinearization circuitry is generally added, often within the sensor housing,and typical residual linearity errors are in the order of ±1% FS

This type of transducer is inherently sensitive to stray magnetic fields,and to ferromagnetic materials in proximity of the sensing coil (especially inthe open-loop magnetic geometry), therefore attention should be paid tothis aspect in its positioning and mounting

14.4.5 Eddy-current transducers

When a coil is driven by an alternating voltage it generates an electromagneticfield If an electroconductive object is placed in proximity of the coil, the

Fig 14.4 Schematic diagrams of inductive displacement transducers based on

variable reluctances: (a) closed magnetic loop design, (b) open magentic loop design.

electromagnetic field induces eddy currents (the name comes from the circular

nature of their flow within the object) Such eddy currents in turn produce

an electromagnetic field which opposes to the original field, and has theultimate effect of changing both the inductance and the quality factor, i.e.the losses, of the coil Therefore, from the measurement of the coil impedance,the distance from the object can be derived

The principle is generally applied by making use of a two-coil arrangement,which includes a driving coil and a sensing coil both oriented with their axisperpendicular to the target, as shown in Fig 14.5 Both shielded andunshielded constructions exist, which differ in that the former provides amore directional field that ensures a higher immunity from stray effects caused

by metallic objects near the sides of the transducer When two or moretransducers need to be mounted in close proximity, the shielded construction

is preferred to minimize mutual interference

Eddy-current inductive sensors are responsive to both the magnetic

permeability µ and the electrical conductivity s of the target material but, as

opposed to the variable-reluctance type, they do not require that the magneticpermeability is high Therefore, they operate properly even with non-ferromagnetic yet conductive target materials, or at temperatures higher thanthe Curie temperature of ferromagnetic materials

As a drawback, different target materials give different sensitivities and,therefore, require different calibrations Moreover, the thickness of the target

is also influential on the sensitivity, since the eddy currents have a finitepenetration depth δ in the material which depends on its µ and s and on the

field frequency f through the relationship

(14.3)

Fig 14.5 Inductive displacement transducer based on eddy currents: (a) schematic

diagram of the two-coil configuration; (b) shielded design; (c) unshielded design ([4, p 279], reproduced with permission).

On the basis of this formula it is possible to use eddy-current probes tomeasure the thickness of metal foils or coatings and to detect material cracks.Eddy-current sensors are almost always provided with built-in or externalsignal-conditioning electronic circuits which drive the coil, amplify the signaland linearize it to a typical value of ±1% FS.

A resolution as high as ±0.1% FS can be obtained Typically, with size short-range devices with 5 mm of probe diameter and 0–1 mm measuringdistance, submicron resolutions can be achieved The time and temperaturestability can be very high making eddy-current sensors very suitable for long-term operation even in harsh and dirty environments

small-A frequency response typically ranging from zero to several kilohertz or

a few tens of kilohertz and the absence of mechanical loading because oftheir noncontact operation make eddy-current sensors ideal for themeasurement of vibration For instance, they are well suited and widely usedfor measuring the vibrations and the eccentricity of rotating shafts, or thelooseness of bearings

As far as the mounting is concerned, attention should be paid to ensuringthat the lateral dimensions of the target are at least two to three times theprobe diameter and the target surface is as flat as possible Especially for theunshielded version, the side-mounting of more transducers or the use of metallicfixtures may perturb the sensitivity and, therefore, the recommendations of themanufacturer should be followed to keep distances at safe values

14.4.6 Capacitive transducers

Capacitive transducers are based on the principle that the capacitance oftwo electrical conductive bodies (armatures) separated by a dielectric mediumvaries if either the dielectric constant of the medium or the system geometryvary The change in the dielectric properties of the separating medium isexploited, for instance, in liquid level or air humidity sensors The change ofgeometry is well suited to use in dimensional measurements, such as linearand rotational displacement sensing For instance, the principle can be applied

in devices where an armature terminating in a shaft is guided to move betweenfixed armatures in a cylindrical geometry, giving rise to a capacitive linearpotentiometer

The capacitive effect is well suited to noncontact displacement

measurements according to the expression of the capacitance C of two parallel plates of area A separated by an air gap d given by

(14.4)

where ε is the dielectric constant of air practically equal to that of vacuum

With reference to the above formula, two alternative methods may be

used, as illustrated in Fig 14.6, namely the variation of the distance d between the armatures and the variation of the area of overlap A.

The noncontact capacitive probes are based on the former principle, i.e

they use capacitance variations to measure the air-gap d between two parallel

conductive plates, of which one is the fixed reference and the second isattached to the moving object If the object material is conductive it may act

as the electrode, otherwise it may be equipped with a metal target or madeconductive with the aid of conductive paint or rubbed graphite As opposed

to the eddy-current probes, the conductivity value of the target is notinfluential on the sensitivity Capacitive transducers for nonconductive targetsare also on the market, but their sensitivity typically depends on the targetmaterial and are mostly suited to proximity detection

Equation (14.4) shows that the capacitance between two conductor plates

varies nonlinearly with the plate spacing d The problem may be partially

overcome by operating the transducer over a reduced portion of its usablerange to approach linearity A better and elegant solution is given by employing

an electronic readout scheme which provides an output signal proportional

to the modulus of the transducer impedance |Z| Since

the output signal is proportional to d at a fixed frequency ω.

A further limitation to linearity comes from the fringing effect caused bythe electric field lines diverging from parallel at the border of the plates due

to their finite extension As shown in Fig 14.7, this problem may be solved

with the help of the so-called guard electrode which encircles the moving

armature and, by an electronic active driving circuitry, is kept at its samepotential without, however, establishing any physical short-circuit betweenthe two In this way, the fringing effect is moved to the external border ofthe guard electrode, while the inner field lines in the region facing the sensitiveelectrode are steered to be perfectly parallel

Fig 14.6 Variable-distance and variable-area methods for measuring displacement

along the direction x making use of a parallel-plate capacitive transducer.

electromagnetic interference On the other hand, they are typically sensitive

to the optical properties of the target object, such as colour, reflectivity, surfaceroughness, presence of dirt or dust, and of the optical path between thesensor and the object Therefore, their use is generally confined to cleanenvironments

A very simple principle makes use of a light source, such as a light-emittingdiode (LED), coupled to a light detector in a side-by-side arrangementcontained in a single unit which is positioned in front of the target object.The amount of reflected light collected by the photosensor depends on thetarget distance The method is simple and cheap but gives a limited range oflinearity, and suffers from a significant dependence on the optical properties

of the target

Transducers based on the triangulation method employ the configurationdepicted in Fig 14.8 The light beam emitted by a visible or infrared lightsource, such as an LED or a semiconductor laser diode, is reflected by thetarget object and reaches a linear position-sensitive detector (PSD) at a

particular point x of its length Such a point is related to the target distance

by trigonometric relationships, therefore the properly processed signal fromthe PSD gives a measure of the distance Triangulation transducers typicallyhave a working range of few millimetres around a stand-off distance thatcan be as high as several centimetres The resolution is in the micron rangewith a frequency response no wider than few hundred hertz which is generallyinversely dependent on the resolution

The method with the highest performance and cost is that based on thelaser interferometer A Michelson configuration is generally adopted in whichthe laser light beam is split into two beams which travel along differentpaths One path has a fixed length and works as the reference, while theother one comprises the distance from the light source to the measurandobject usually equipped with a mirrored reflecting target The two beamsrecombine in a photodetector and, due to the high coherence of the laserlight, produce a neat interference pattern whose number of fringes can becounted and related to the target distance The achievable resolution can be

as high as 1 nm, and the frequency response extends from DC to tens ofkilohertz or more Such performance makes the laser interferometer the

Fig 14.8 The optical triangular method for noncontact displacement measurement.

preferred instrument where high-quality displacement and vibrationmeasurements are needed, such as for laboratory calibration purposes or forhighly demanding applications.

14.5 Relative-velocity measurement

14.5.1 Differentiation of displacement

In principle, the output signal from any displacement sensor can be differentiatedwith respect to time to obtain a velocity signal This may be done eitherelectronically, by making use of differentiation circuits cascaded to the transduceroutput, or as a postprocessing step on the recorded data This indirect approach,however, has possible problems with the fact that the process of differentiationinherently enhances the high-frequency components in a signal, since theamplitude of each sinusoidal component at a frequency ω results multiplied by

a factor ω Therefore, any spurious high-frequency component in the displacementsignal is amplified to a level which may impair the detectability of the truevelocity signal The situation may be critical, for instance, with wire woundpotentiometers, due to their staircase characteristic which causes stepping outputsignals under rapid cursor movement, as well as with AC excited transducers,such as the LVDTs, due to possible residual ripple in the output signal.Although the differentiating method may prove satisfying in severalnoncritical situations, the choice which provides a more general applicabilityand is therefore often preferred is to make use of velocity measuringtransducers

14.5.2 Electrodynamic transducers

The operating principle is based on the Faraday-Lenz law of magnetic

induction for which the electromotive force (EMF), i.e the voltage E,

generated in a closed circuit is equal to the time derivative of the magneticflux Φ linked with such a circuit That is, with the minus signrepresenting the fact that the magnetic flux generated by the induced current

caused by E opposes to the original flux Φ On this principle, it is possible

to develop self-generating sensors where velocity is converted into variations

of the magnetic flux concatenated with a coil, and therefore produces aproportional output voltage signal

A simple and effective method is that of making use of a permanent barmagnet positioned inside a coil and free to move relative to it There are twoalternatives, called the moving-coil and the moving-magnet designs, differing

in that the former has the element fixed to the reference point, while thelatter has it attached to the moving object

The moving-magnet geometry is widely used for linear velocity transducerswhich generally base on the two-coil configuration shown in Fig 14.9 The

14.5.3 Laser velocimeters

The laser velocimeters are based on the Doppler effect, for which a light wave

of frequency f reflected by a target moving at a velocity υ relative to the light

source becomes shifted to a frequency where ∆f depends on the ratio v/

sign of v can then be determined by measuring the difference between the

frequencies of the emitted and reflected laser beams Laser velocimeters arespecialized and costly instruments which have the advantage of providing anoncontact measuring method They are used for instance to measure vibrations

in rotating blades or acoustic emitting surfaces, such as loudspeakers

14.6 Relative-acceleration measurement

Ideally, acceleration may be derived by time-differentiating velocity or bydoubly differentiating displacement, either electronically on the signals ornumerically on the recorded data However, it should be noted that thedouble-differentiation process is still more sensitive to high-frequency spuriouscomponents than the single differentiation is Therefore, obtaining reliableacceleration data from displacement readings is hardly feasible except forvery smooth signals Starting from velocity signals may be less problematicdepending on the particular case

In general, given the availability of reliable and high-performancetransducers which directly measure acceleration, as illustrated in the followingsection, it is common practice to make use of such devices, and the application

of the differentiation techniques can be considered as an exception

14.7 Absolute-motion measurement

14.7.1 The seismic instrument

Consider a single-degree-of-freedom mass-spring-damper system mountedwithin a case, as shown in Fig 14.10, which is subject to motion whenrigidly attached to a vibrating structure Motion is assumed to be directedalong the instrument sensitive axis, which is oriented vertically in the figure

The coordinate x0 gives the position of the transducer base with respect to

a motionless absolute reference, indicated as ground, while z0 and y0 givethe position of the mass with respect to the same reference and to the base,

respectively The variations of x0, z0 and y0 from their initial unperturbed

values are indicated by x, z and y, which therefore represent the displacements referenced to ground, for x and z, and to the transducer base, for y.

The mass m is called the proof or seismic mass The term seismic comes

from the fact that vibrating the transducer base puts in motion the mass, in thesame way as buildings are acted on through their foundations by earthquakes

approximates it well when, as generally happens, the transducer seismic mass

m is negligible compared to that of the measuring structure.

Equation(14.6) shows that the transducer absolute displacement X causes

a mass-base relative displacement Yx which depends on frequency and on

the transducer mechanical parameters

Additionally, Y includes the term Yg caused by the force due to gravity If, as

assumed in Fig 14.10, the sensitive axis is oriented vertically and the transducer

spans a limited altitude, Fg is nothing but the transform of the constant proof mass weight mg, where g=0.981m/s2 is the gravity acceleration, i.e

Therefore, the resulting term Yg seen in the time domain produces only a static

displacement (For the mathematical details on the fact that the

transform of a constant k is given by kδ(ω) the reader can refer to Chapter

2.) If, however, the transducer’s sensitive axis has a different orientation, fg and

in turn yg change accordingly both in magnitude and sign, being zero for a

perfectly horizontal orientation This implies that if the transducer happens toexperience a motion with a nonzero rotational component in a vertical plane,

then yg is no longer a constant but depends on time and, as such, it becomes a signal source indistinguishable from that due to the absolute displacement x(t).

Failing to recognize such an orientation-dependent effect may cause significanterrors in measurement using seismic transducers

The transducers working on the seismic principle employ an internal

method to measure the relative displacement y(t), or some related quantity, and infer the absolute motion represented by x(t) or its time derivatives on

the basis of eq (14.6) The internal relative displacement is measured by asecondary sensor, the type of which can vary depending on the constructiontechnology, and which provides the electrical output of the whole transducer

14.7.2 Seismic displacement transducers

Assume that the contribution due to gravity is constant and directed in the

negative x0 direction (Fig 14.10), so that the antitransform of the term Yg in

eq (14.6) reduces to the static displacement

With reference to eq (14.6), the expression of Yx is

where Td ( ω) is given by

(14.7)

T d ( ω) can be called the displacement frequency response function The magnitude and phase curves of Td ( ω) are shown in Fig 14.11 The damping

ratio ζ controls the peaking of the magnitude and the steepness of the phasearound ω0 For frequencies higher than approaches unity and

therefore the relative displacement yx becomes equal to the displacement x Since z=x+y, this means that above ω0 the mass remains still at z=–yg

Considering x(t) as a pure sinusoidal displacement represented by the

complex exponential then

The sinusoidal transfer function or sensitivity function of thedisplacement secondary sensor can now be introduced, where the minus sign

is taken for notation convenience According to eq (14.6) the electrical output

e(t) of the transducer is then given by

(14.8)

Provided that ke ( ω) is constant in the region where Td ( ω) is unitary,

the first term results in an output time signal which, in such a frequency

range, is proportional to x(t) Since Td(0)=0, the system is not sensitive to

static displacements, therefore it is only capable of motion measurements.The second term represents the gravity-induced electrical signal Since fornonvertical orientations such a signal would vary, in the absence of motion

the transducer behaves as an inclinometer referenced to the gravity axis It

is worth noting that if ke(0)=0, i.e if the secondary transducer is not sensitive

to static displacement (as for instance in piezoelectric elements), then thegravity term is cancelled This occurrence has no detrimental consequences

on the sensitivity to x(t), which is already zero at DC due to Td(0)=0 The displacement sensitivity function Sd ( ω) of the overall transducer is

therefore given by

(14.9)

In practical transducer designs, the secondary sensing element which convertsdisplacement into an electrical signal may be based on resistive potentiometers,LVDTs or noncontact displacement sensors Alternatively, strain gauges bonded

to a flexible member which behaves as the spring element, may be used

To extend the useful bandwidth towards the low frequencies, the naturalfrequency ω0 should be as low as possible This requires low stiffness andhigh mass, which however imply reduced robustness and possibly highercross-sensitivity, and unavoidably increase size and loading effect

14.7.3 Seismic velocimeters

One possible way to measure velocity is by electronically differentiating thesignal from a seismic displacement transducer The possible problems related

to the time differentiation operation are essentially the same as those alreadydiscussed for the case of the relative-displacement transducers in Section 14.5.1.

A more efficient and widely used solution is that of equipping the seismicsystem of Fig 14.10 with a secondary sensor inherently sensitive to relativevelocity, rather than displacement

To analyse this solution, assume a purely sinusoidal displacement represented

fixed orientation in space, so that the gravity term yg is constant Therefore,

the relative velocity is given by

Introducing the sensitivity function of the velocity sensor, the

electrical output e(t) of the transducer is given by

(14.10)

Therefore, the velocity sensitivity function Sv ( ω) of the overall transducer

results

(14.11)

It can be observed that as long as the secondary sensor sensitivity ke is

independent of frequency, the frequency response of the system as a velocity

transducer is again given by the displacement response function Td ( ω), and

therefore it may be analysed by referring back to Fig 14.11 In particular,

even in the case of ke extending to DC, that is the secondary sensor being responsive to constant velocity, the overall sensitivity Sv ( ω) has a low-frequency

cutoff given by the natural frequency ω0, which poses an inferior limit to theusable measurement bandwidth

The most widely used method to convert relative velocity into an electricalsignal is the electrodynamic principle, either in the moving coil or movingmagnet variant Seismic velocimeters of this kind are often called vibrometers

or, in the case of transducers with very low resonant frequency (typically

below 1 Hz), seismometers Electrodynamic transducers have their main

advantages in their self-generating nature with a high-level output, significantsensitivity (typical values range from 1 to 10V/(m/s) and reliability ofoperation due to the absence of electrical contact and friction in the secondarysensor

The drawbacks are that they are susceptible to magnetic fields, even ifmagnetically shielded versions are on the market, and tend to be sensitive totheir orientation, since the damping and stiffness of the seismic system aresomewhat influenced by the gravity force

Regarding the frequency response, extending the bandwidth towards thelow frequencies requires ω0 to be as low as possible, but, as discussed for theseismic displacement transducers, this is usually obtained at the expense of

an increased mass loading and reduced robustness By setting the dampingfactor ζ typically close to 0.7, with oil filling or some other means, the usable

bandwidth can be extended slightly below the resonant frequency ?0 withoutappreciable phase distortion The high-frequency limit is often posed by thefirst contact resonance between the transducer case and the structure and it

is therefore dependent on the mounting method Overall, the typical extension

of the usable bandwidth is from few hertz to few kilohertz

Seismic velocimeters are becoming less widely used due to the availability

of reliable and high-performance accelerometers, but still find their way intothose applications where low-frequency signals of interest are mixed withextraneous high-frequency components which would possibly cause anoverrange condition in an accelerometer

14.7.4 Seismic accelerometers

Suppose we are now interested in measuring the absolute acceleration (t),

whose transform is given by With reference to eq (14.6),

the term Yx can be written as where Ta ( ω) is given by

(14.12)

T a ( ω) can be called the acceleration frequency response function, which can

be recognized as the characteristic response function of a second-order system

The magnitude and phase curves of Ta ( ω) for various values of the damping

ratio ζ are shown in Fig 14.12 For frequencies from zero to around isequal to (1/ω0)2 and, therefore, in this frequency region the relative displacement

y x , except for the sign reversal, is proportional to the absolute acceleration

Assume a purely sinusoidal acceleration (t) represented by the complex

contribution due to gravity is constant, so that the antitransform of the term

Y g in eq (14.6) reduces to the static displacement

The secondary sensor is chosen to be responsive to displacement, with

being its sensitivity function In this way, the electrical output e(t) of

the transducer results to be given by

(14.13)

Provided that ke ( ω) is constant in the region where Ta ( ω) is

constant, the first term results in an output signal which in such a frequency

range is proportional to the measurand acceleration (t) The second term

is due to the effect of gravity on the relative displacement y If ke(0)=0 the

transducer is not responsive to gravity and static accelerations If on thecontrary in the absence of motion it works as an inclinometer

The acceleration sensitivity function Sa ( ω) of the overall transducer is given by

(14.14)

Due to the fact that the low-frequency value of Ta ( ω) decreases with increasing

ω0, a trade-off between sensitivity and high-frequency response is required.

Depending on the damping value, the flat-band region, i.e the range offrequencies where the sensitivity is constant and therefore the shape of theinput signal spectrum is not distorted, extends more or less towards ω0 Since

the damping ratio depends on the particular accelerometer construction, themanufacturer’s specification must be consulted for the case at hand Typically,operation below 0.2ω0 would ensure negligible sensitivity variations even forlightly damped transducers

Usually, the unit used for acceleration is not the m/s2 SI unit, but the

gravitational acceleration g=9.81 m/s2 Therefore, in the case of an accelerometer

with voltage output, the sensitivity may be expressed in volts per g.

Seismic accelerometers are probably the most widely used transducer forthe measurement of vibrations, and they are well suited to the measurement

of both continuous vibrations and transients, or shocks Their functioningprinciple requires a high resonant frequency to obtain a wide measuringbandwidth, therefore they tend to have high stiffness and small mass Thismakes them typically rugged and small size devices, especially in the high-frequency versions, thereby ensuring a reduced loading effect

The demand for a high-sensitivity secondary sensor to detect thecorrespondingly small internal relative displacements is satisfied by several sensingmethods, which result in diverse accelerometer types and construction technologies,

as discussed in the following section Depending on such sensing methods, thereadout of the electrical output may require some precautions and special circuitry,but generally high-level signals with good linearity over a wide dynamic rangemay be obtained at moderate price This, in turn, makes affordable the ever-increasing use of several devices in a multipoint test on the same structure

In many cases, accelerometers prove useful in providing displacement andvelocity signals by means of time integration, which is a technique intrinsicallyinsensitive to noise and high-frequency disturbances due to its averaging nature

14.8 Accelerometer types and technologies

14.8.1 Piezoelectric

Piezoelectric elements are insulators which become electrically polarized whensubject to mechanical stress (direct piezoelectric effect), and conversely

elongate or contract when electrically excited by the application of a voltage(reverse piezoelectric effect) [5].

Materials exhibiting the piezoelectric effect fall in two categories Thenatural piezoelectrics, such as quartz (SiO2), are intrinsically piezoelectricdue to their structure The artificial piezoelectrics are ferroelectric materials

in which the piezoelectric effect is permanently induced by a poling process

at the manufacturing stage Poling consists of applying a high-intensity electricfield to align the electric dipoles

The reverse piezoelectric effect can be used for actuating purposes, such as

in the case of ultrasound generation transducers The direct piezoelectric effectcan be exploited for sensing all those mechanical quantities which ultimatelyproduce a stress on the piezoelectric element, such as force, torque or pressure.Piezoelectric sensors are self-generating In piezoelectric accelerometers, thepiezoelectric material has the role of the relative displacement secondary sensor

It behaves as a continuous elastic element generating an output charge Q

proportional to the strain induced in the element itself by the inertial force of

an overlying seismic mass In some designs the seismic mass may be missing,leaving only the distributed mass of the piezoelectric element itself, whichtherefore embodies both the seismic system and the secondary sensor.The proportionality factor between relative displacement and output charge

is given by the charge sensitivity function kQ ( ω), which depends on the elastic

properties and dimensions of the piezoelectric element, and by its intrinsic

charge piezoelectric coefficient d, usually expressed in picocoulombs per

newton As will be discussed shortly, different piezoelectric materials have

different values of d Generally, in the region of interest, the charge sensitivity function kQ ( ω) can be considered as independent of frequency Therefore, according to eq (14.14), the overall charge sensitivity SQa ( ω) is given by

(14.15)

The charge sensitivity is usually expressed in picocoulombs per g (pC/g) and its frequency behaviour is determined by Ta ( ω).

As shown in the equivalent circuit of Fig 14.13, the charge

generated by an acceleration ( ω) is developed across the capacitor C, which

is formed by the portion of the piezoelectric material delimited by the twoelectrode faces Piezoelectrically generated charges do not last indefinitely but

Fig 14.13 Equivalent circuit of a piezoelectric accelerometer ( ω) is the

acceleration, S Qa ( ω) and S Va ( ω) indicate the charge and voltage

sensitivity respectively.

tend to neutralize due to the fact that the material is an imperfect insulator,

with losses represented by the resistance R in the equivalent circuit These losses are responsible for an intrinsic discharging effect of the capacitor C.

As a consequence, if the voltage V is taken as the electrical output quantity and the voltage sensitivity function kV ( ω) is introduced, it follows that

Similarly to SQa ( ω), the overall voltage sensitivity

S Va ( ω) can then be defined, expressed in volt per g, which obeys the expression

(14.16)

From the comparison of the expressions for SQa ( ω) and S Va ( ω) follows a

very important fact regarding piezoelectric accelerometers A low-frequencylimit exists given by under which the voltage sensitivity SVa ( ω) drops, and at zero frequency SVa(0)=0 That is, piezoelectric accelerometers

are not able to respond with their output voltage to DC acceleration Inparticular, they are not sensitive to gravity acceleration and orientation Seen

in the time domain, this implies that the response to a step-changingacceleration is a decreasing exponential with a discharge time constant (DTC)given by

The problem is virtually absent if the output charge is considered However,

in practice, the charge has to be extracted from the piezoelectric material insome way to be measured, and this in turn necessarily involves the presence

of a time constant which is nothing but the product of the equivalent Req and Ceq of the electronic circuit used for the charge readout (Section 15.4.2).

By properly choosing Req and Ceq the low-frequency cutoff limit can be varied

with respect to

Charge- and voltage-output readout schemes also differ when the presence

of the connecting cable is considered, which contributes with its equivalent

capacitance CS in parallel to the sensor capacitance C As will be discussed

in more detail in Chapter 15, the presence of CS has essentially no consequences on the sensitivity SQa ( ω) when charge amplification is employed.

On the other hand, when voltage readout is adopted, the shunting action of

C S determines a diminution in the voltage sensitivity from its open-circuit expression SVa ( ω) of a factor proportional to C S /C, and a corresponding

variation in the low-frequency limit In order to keep the value of CS constant

and as low as possible, the voltage readout is generally accomplished byinserting the amplifier within the transducer case, giving rise to the so-calledlow-impedance voltage-output transducers Both charge- and low-impedancevoltage-output piezoelectric accelerometers are on the market, with the formerindicated more for laboratory use or where high operating temperatureswould damage the built-in electronics, and the latter commonly used morefor general purpose field applications

At the high-frequency end, both SQa ( ω) and S Va ( ω) are limited in the same way by the behaviour of Ta ( ω), i.e by the natural frequency The

damping is typically low, resulting in a narrow region of resonance and in asteep phase change Usually, the flat-band region is individuated by the upperand lower limiting frequencies where the voltage sensitivity is within 5% ofits midband value given by When proper mounting is adopted,

operation up to f0/3 and f0/5 typically ensures a deviation from the midbandsensitivity of 12% (1 dB) and 6% (0.5 dB) respectively

Piezoelectric materials typically used in acceleration sensors are eitherquartz, or poled ceramics mainly of the lead zirconate-titanate or bariumtitanate family For high temperatures tourmaline or lithium niobate are used.Quartz has a crystalline structure, is highly stable both thermally and overtime, and it offers good repeatability It represents, therefore, the best solutionfor transducers used for continuous monitoring over prolonged durations attemperatures between –190 and 240°C and for calibration reference

standards It has a piezoelectric coefficient d of 2.3 pC/N, which is a rather low value and, as such, the charge sensitivity SQa is not very high On the

other hand, since its dielectric constant is comparatively low, such a charge

produces a rather high value of the open-circuit voltage sensitivity SVa

Poled ceramics have a piezoelectric coefficient which is much higher thanquartz, reaching a typical value of 350 pC/N for lead zirconate-titanate (PZT).This causes a very high charge sensitivity, even if, by properly tailoring thematerial composition to keep its dielectric constant low, significant values ofthe voltage sensitivity can also be obtained However, poled ceramics arepolycrystalline materials and, as such, they are typically less stable then quartz.They tend to depolarize over time in a process called time depoling, andtypically suffer from a significant sensitivity to temperature Moreover, theygenerally suffer from a significant pyroelectric effect, consisting of thegeneration of charge under temperature variations which adds unwantedly

to the piezoelectric signal of interest In general, they lack stability whenexposed to extreme mechanical or thermal shocks, even thoughaccelerometers based on ceramics can operate between –190 and 400°C.Irrespective of the material employed, the piezoelectric accelerometerconstruction basically conforms to one of the three configurations shown inFig 14.14 The compression mounting is the traditional and simplestconstruction where the piezoelement operates with the electrodes placedperpendicular to the transducer sensitivity axis Generally, it includes apreloading spring and offers a moderately high sensitivity-to-mass ratio.However, it is rather sensitive to base bending and thermally induced inputs.Its typical use is in shock tests, where the signals are high, or in controlledlaboratory environments for calibration purposes

In the shear mounting the piezoelement operates with the electrodes placedparallel to the transducer sensitivity axis This configuration generally offers

a high sensitivity-to-mass ratio and a good thermal stability Moreover, itgives a reduced sensitivity to base strain and a minimal cross-sensitivity,which makes the shear configuration the best choice as a general purposeaccelerometer

and dynamic ranges as wide as 1:1000 are attainable, which are especiallyhelpful around structural antiresonances where the motion is virtually absentand the signal is low compared to background noise As a limitation, theyare not sensitive to static acceleration and have a low frequency cutoff, butthis in turn may be an advantage in those applications where it is required

to have independence from sensor spatial orientation

General purpose accelerometers may have a frequency range from 1 to 5

kHz or more, with a voltage sensitivity of up to 100 mV/g High-frequency

and shock types are available that work well beyond 10 kHz; however, these

yield few millivolts per g Low-frequency and low-acceleration level applications

such as in vibration of bridges and large structures are addressed by the range accelerometers, which have large size and weight but can provide typically

seismic-1 V/g from a tenth of a hertz up to several hundred hertz.

Some manufacturers sell multifunction transducers which incorporate apiezoelectric accelerometer plus an additional sensing function, such astemperature or velocity/displacement obtained through electronic integration,

in a single unit

When talking about piezoelectric materials it is worth mentioning thepiezofilms They are polymers mostly derived from polyvinylidene fluoride(PVDF) shaped in thin flexible layers coated with metal electrodes The

piezoelectric coefficient d of PVDF is about 20 pC/N which is ten times greater

than quartz and an order of magnitude lower than PZT Piezofilms are verylight and can be bent and deformed a great amount, which makes it possible

to directly bond them even to small and irregularly shaped structures to detectvibrations However, since the deformation pattern can be very complex, it isnot very easy to determine exactly the motion direction Though piezofilmshave great potential in specialized applications as embedded sensors and haveeven been used as sensing elements in lightweight low-cost accelerometers,they are hardly suitable to be employed in precision vibration measurements

14.8.2 Combined linear-angular

The flexural bending of piezoelectric beams can be exploited for thesimultaneous measurement of both translational and rotational accelerationswith a single device The Translational-Angular PiezoBEAM® (TAP) sensor1

accomplishes this task by making use of two clamped-free beams [7] Eachbeam is actually a flexure bimorph made by two bonded piezoelectric layerselectrically connected The signals from the two beams are both summed andsubtracted, providing two separate outputs The beams are mounted in such

a geometry that a translational acceleration produces a signal at the summingoutput, while a rotational acceleration determines a signal at the differenceoutput, thereby producing separate information on both components withoutany appreciable cross-sensitivity

1 PiezoBEAM ® is a trademark of Kistler Instrumentation.