B&K Mobility

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A Guide to Mechanical Impedanceand Structural Response Techniques


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A Guide to Mechanical Impedance andStructural Response Techniques

by H. P. Olesen and R. B. Randall

Introduction

In recent years there has been a that the comfo rt of passengers is en- 1. Determinati on of natural fre quen -rapidly developing interest in the sur ed. cies and mode shapes.fiel d of mechani cal dynamics for a 2. Measur ement of specific materia lvariety of reasons. An exampl e of a diff eren t kind is properties such as damping ca-

given by the mach ine tool indu stry , parit y or dynamic stiff ness.Firstl y, the developmen t of whe re excessive vib rat ion can se- 3. As a basis of an analytic al

stronger materi als and greater econ- verely limi t the quali ty of mach ini ng model. From measurem ents ofomy in design has led to increa s- and grin ding opera tion s. the impedances of indivi dual com-ingly lighter str uctu res, more prone ponents or subst ructu res it is pos-to vibra tion probl ems. At the same The overall resul t is that the dy- sibie to predict the behaviour oftim e, increasi ng rotatio nal speeds namic behavio ur of a mach ine or combined systems, in a manneralso give increasi ng likeli hood of stru ctur e is now an impo rtan t factor completely analogous to thehaving to deal wi th stru ctur al reso- in design and devel opment along study of complex electri cal cir-nances. wi th the analysi s of static stresses cuits .

and deflections, and is normally Another impor tant factor is the re- studie d in its ow n rig ht, rather than The concepts of mechanical im-

cent upsurge of interest in envi ron- jus t being all owed for in an exces- pedance and mobility wer e devel-ment ai questio ns since the improve - sive "safet y factor", or treated as oped from electro-mechanical andment of both noisy and vibrat ing en- an afte rtho ught wh en probl ems electro-acoust ic analogies in thevironment s often can be simplif ied have been encoun tered . 19 20 s. Since then the usefulnessto a quest ion of reducing the me- of these concepts in forced vibr atio nchanicaf vibr ati on, either at its One very usefu l expe rimenta l techni ques and in the theore tica lsource or somewh ere along the tech nique for the study of dynami c evaluat ion of stru ctur es has im-transmission path. behaviour of mach ines and stru c- proved considerab ly. This is due

ture s concer ns the mea sur eme nt of partly to more sophisti cated vibra-Typical examples are provided by wha t is loosely ter med "me cha nic al tion trans ducer s, vibra tion exciters

the transportation industries, wher e impe dance" . Broadly speaking, this and analysis equipment and partlyin the development of for example defines the rela tion ship s b etwe en to the acceptance in mechani cal andairc raft , automobi les and ship s, forces an d mot ions at various civil engineer ing of mechani cal im-care has to be taken not only that point s, both wi th respect to amp ii - pedance and related concepts sothe various components can wi th- tude and phase. Ref. 10 lists the that they could be handled on theirstand the dynamic loadings to thre e mai n appli cati ons of imped - own wit hout resorti ng to a previouswh ic h they are subje cted, but also ance testi ng as: conversi on to electri cal circ uits.

Mechanical impedance and mobility

The mechanic al impedance and not be given here. The units after of motion relative to the direct ion ofmobil ity (for simple harmonic mo- each ratio are Sl -u ni ts *. force whe n this is not obvious fromtion) are defined as the complex ra- the measur ement conditi ons ortios of force vector to velocity vec- As both force and moti on are vec- from the calcu latio ns.

tor, and veloci ty vector to for ce vector s in space as we ll as in ti me care * mtemat.onai Organ.zat.on for standardstor respectively. This is sh own in shoul d be take n to defi ne dire ctio ns tion (ISO).Table 1 where, in addition, the simi- , _ _ ^ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ^ ^ ^ _ _ ^ _ ^iar ratios involving acceleration anddisplacement are given. p v * ^ s F N ( A c c e ( e r a t j o n h r h F o r c e ) | , JIL ]

(Apparent Weight} a l m ! M F Ns^

The terms given in the tab le are Mechanical Impedance | ~ j Mobility JL < H itaken from the American Standard v m (Mechanical Admittance) F NsUSAS S2.6- 1963: Specifying the F N d mMechanical Impedance of Struc- S t i f f n e s s j [ - | Compliance T ' N~ ]

tures (1). Other terms have been ' "~~ ~USed by di ff er en t au th or s bu t wi ll Table 1. Terminology for complex dynamic ratios of force and motion

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Dynamic Mass(Apparent Weight}

Fa

Ns 21 m s (Acceleration through Force)

aF

m ,1 Ns2 J

Mechanical Impedance _FV

NsMobility(Mechanical Admittance)

V\ i[ Ns J

Stiffnessi

Fd 1 m ' , Compliance

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F 1 11 N J


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Whe n f orce and motion values into motion . By measuring , for ex- ledge about the response ability ofare measured at the same point and ample, the mechanical impedance of the struc utres involved, and of thein the same directio n the ratios are points on a struc ture, knowledge is actual responses or forces. Aftertermed driving point values, or in gained about its response to vibra- combinati on of this infor matio n theshort, point values, e.g., point im- tion al forces at different frequencie s. need or the possibilit y of correctivepedance. Simil arly, a measurement of the mo- measures may be evaluat ed.

tio n of the struc ture, after it hasWhen force and motion are meas- been placed on a vibrating support. In the fol low ing sections, after a

ured at differe nt points or at the may be compared to its mechanical brief discussion of narro w band fre-same point wit h an angle between impedance to obtain inform ation quency analysis, rules are given forthem they are termed transfer val- about the forces whic h act on the evaluating mechanical impedanceues e.g. , transfer impedance. stru ctur e. data by graphical means, the inst ru

mentation used for practical meas-The ratios given in the firs t and To solve vibrational probl ems, ureme nt is discussed and a fe w

second column of Table 1 really re- there fore, both a mechanical imped- practical examples are given, as wel lpresent, as func tion s of frequen cy, ance may have to be measured, and as references to furth er literaturethe diffic ulty or ease, respectively, a narro w band frequency analysis about Mechanic al impedance appli-wi th whic h a stru cture can be set carried out to obtain detailed know- cations.

Narrow Band Frequency Analysis

One of the major reasons for st u- from the source exc itat ion, or in Wher e the Q-factors (see laterdying dynamic phenomena as func- modifying the struct ure to "d et un e" for definition) are greater than abouttions of frequency is the simplici ty or damp that particular frequency re- 50 , however, it may be necessary tothat this introduces for linear sys- gion. go to constant bandwi dth filt ers,

terns, since many actual struc tures _. , , , purely in order to obtain a bandwi dth. . . . The type of f requency ana lysi s , i U - ft / - * xhave approx imately linear parame- , 7 ... ;_ \ - n l e s s t n a n 1% m certain frequency, /: . . , _ performed wi ll perhaps be influ-ters (stiffness, damping, mass). One _ , . . u u ^ ranges.x , , ,. enced by the approach chos en, and a

important property of such l inear , _- ,_ , . . . ,. . / , thu s a brief discus sion is given of _ . . ._. , . ,systems is that of super posit ion. In . . , ., , , , , Constant bandwi dth analysis (par-

. , K K . , the methods available for frequency . . , ,. , *particu lar, an input at a given fre - ticularl y on a linear frequency scale)quency gives an out put at the same is also benef icia l for the analysi s offrequenc y, though modified in ampl i- One of the first decisions to be excitations w it h a high harmonictude and phase according to the fre - made is between constant band- content , since the ha rmonics arequency response fun ct ion , and the wid th and constant proportional then unifor mly separated.behaviour at this frequency is thus bandw idth analysis, it is oftenindependent of wha t is going on at claime d that "na rro w band analy sis" The overall consideration inother frequenc ies. A related advan- is synonymous wit h narrow constant choice of analysis method is that ittage is that combinat ion of cascaded bandw idth analysis, but this is not should everywher e give suffici entsystems involves only mult ipli cat ion necessarily the case. For example resolu tion, wit hou t giving too muchof their characterist ics at each fre - the Analyzer Type 21 20 has con- inform ation in other areas, becausequency, and this in tur n is simpl ified stant percentage bandwidths down of the detrim ental effect of the lat-to addition whe n logarithm ic (dB) to 1% , and this wil l often give ade- ter on analysis speed and effi-amplitude sca\es are used. Even quate resolut ion. In fact, the re- ciency. Frequency Analysis isthou gh excitatio n is rarely sinuso idal sponse of mechanical struct ures covered in depth in Ref .17 .at a single frequenc y, th e use of tends to be similar in principle to aFourier analysis (narrow band fre - constant percentage bandwidth filter Perhaps the best compromise isquency analysis) makes it possible to (a certa in amplifica tion factor Q the Heterodyne Analyzer Type 20 10break down a more complex signal corresponding to a certain percen- whi ch has both linear and logarit h-into its components at various fre - tage bandwi dth). Thus, whe re the mic frequency sweeps covering thequencies, thus considerably sim pli - excit ation is fairly broadband it may range from 2 Hz to 2 00 kHz. Al -fying its inter preta tion. be most efficient to analyze the re- though it is primari ly a constant

sponse wi th constant percentage bandwidt h inst rument it can be pro- A typ ica l dynamic pro ble m wo ul d ba ndwi dt h fi lt er s. grammed to step up automatical ly in

involve obtaining the frequency bandwidth wit h increasing fre-spectrum of the input to a mech ani- Anoth er advantage of constant quency, thus approxima ting a con-cal system (be it force or motion) and perce ntage analys is is that it gives stant percentage bandwidth analysisby comparing this wit h the meas- unif orm resolution on a logarithmic (where the percentage can be con-ured response character istics to de- frequenc y scale, and thus can be siderably lower tha n 1%). The mainterm ine whethe r a problem wi ll arise used over a wide frequency range. disadvantage of such an analyzer,due to coincidence of peaks in the As explained later, logarithmic viz. long analysis ti me, can be obvi-excit ation and mobi lit y. The sol ut ion scales are moreover advantageous ated by use of the Digi tal Event Re-of such a problem wou ld consist for the interpret ation of mechanical corder Type 75 02 as described ineither in elimi natin g that compo nent impedance data. Ref. 1 1 . The large freque ncy trans-

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The impedances and the mobi li- thi s repr esent atio n the impedanc es tance on both abcissae and ordi-ties of the elements are best illu- and mobi lit ies are given as stra ight nates and wher eby the slopes forstrated in log-log diagrams wi th fre- lines . (See Figs.2 and 3 whe re a mass and spring impeda nce linesquency f = co/ln as the abcissae. In factor of 10 equals the same dis- are + 1 and 1 respectively).

Combination of elements

A mass sup por ted on spr ings is acommon case in practice, e.g., in vibration isolation. In simple systemsthe mass can be considered to beplaced on one spring which hasa stiffness value equal to the sumof stiffnesses of the supports.(Damped systems will be consideredlater).

In the literature this basic systemis very often symbolized as inFig.4a for force applied to the massas for instance wi th a motor placedon springs. However, this representation may lead to the misconcep-

ion that the system is a so-cal led from equation 7 but it is less timeseries system whi le it is, in fact , a cons umin g to combine th e curvesparallel system whe re the force is graphic ally. (Remember that theshared betw een the mass and the mass impedance has a positivespring as indic ated clearly in Fig.4 b. phase angle of 90 (j) and theHere the force is appl ied to a mov- spri ng impedance has a negati veng plane to wh ic h both the mass phase angle of 90 (j or 1/j) rela-md the sprin g are attac hed. (See tive to the force). The curves can be^ef.2) . obtained by subtr acti ng the lowest

value from the highest value at As the mo ti on is co mmon to the each frequency but a mor e st raight

wo elements their impedances forw ard method is to constr uct a so-from Table 2) can be added to ob- called impedance skel eton, as givenain the point impeda nce. by Salter (2). This is sho wn in

- 7 - 7 . - 7 , I / Fig.5b where the spring and massZ = Z m + Z K = jtom + k/jto a . , . -rn

(7) l ines are combined up to thei r inter- ;

= j (com k/co) sect ion at fp wher e they coun teract At low frequenc ies to is very each other to produc e a ver tical line

>mall and Z equals k/jaj as jojm can for Z - 0.)e neglected. At high frequencies tos large and Z equals jw m. The impedance curve can then be

drawn to the desired accuracy by de- At a fr equency fR wh er e jo>m = te rm in in g tw o or mor e poi nts fr om

k /j w a resonance occ urs wh er e Z the spr ing and mass cur ves and= 0 and o) = co 0 = v k /m . drawing a curve throug h the points

from the skeleton values at 0,1 fR,The impeda nce can be plotte d fR and 10 fp (Fig.5c).

Fig.4. A mass supported on a spring s hownin an often used representation (a) andin the correct way (b)

Base excited system

If the system of Fig.4 is excited atthe base as shown in Fig.6 it isseen that the velocities at the baseand at the mass are differen t, i.e.,both point and transfer values do exist. As the force on the mass isequal to the force at the base thesystem can best be evaluated fromthe mobilities of the mass and the

Fig.5. Graphica l const ruct ion of the mechanical impedance of a mass supporte d on a spring

spring. These are directly taken byinversion of the impedance curves ofFig.5 (see Fig.7a).

From these curves the point mobility skeleton and the point mobilitycurve can be constructed in a similarmanner to the impedance curve ofFig.5c (seeFig.7b). Fig.6. Base exci ted syste m

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However, it is a mobility plot andthe minimum at fA represents anantiresonance, i.e. an infinite forcewould be required to produce anymotion at all.

The transfer mobility, on theother hand, experiences no discontinuit y. As the force transmi ttedthrough the spring remains constant and equal to F the velocity Vtof the mass remains the same as itwould be for a mass suspended inspace and hence the transfer mobility is a straight line with the sameslope and position as a point mobility curve for the mass alone. Themotion of the mass being reducedrapidly suggests that at high fre-

Fig. 7. Point mobi lity and transfer mob ility f or a base excited system

quenci es to all practic al purposes placed on a rigid support as that ofthe sprin g can be consider ed as Fig. 1 b.

The mass-spring-mass system An exampl e of fu rt her extension

of the model is given by the evaluation of the mass-spring-mass system which is often encountered.This system is shown in Fig.8, andit is seen that the force is dividedbetween the mass and the springsupported mass (the sprung mass).Hence, the point impedance mustbe found from the combination ofthe mass impedance line (shown forthree different masses in Fig.9),and the point impedance skeleton ofthe sprung mass which has antiresonance at fA The point i mpedance

skeleton is obtained by inversion ofthe point mobility skeleton of Fig.7band is shown in Fig.10. The resulting point impedance skeletons and

Fig.8. A mass-s pring- mass system

curves for the three values of rri2are given in Fig. 1 1.

It is seen that the impedance isobtained by the combination of thecurves in Figs.9 and 10 by keepingthe highest value and by letting the

Fig. 10. Point impedanc e skeleton of thesprungmass shown in Fig.6

Fig.9. Mass imped ance li nes for three val ues of m2 dr awn to the same scales as used in Figs.10 and 11

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Fig. 1 1 . Point impedance for the mass-spring-mass system for three differen t ratios of m2 /m -|

values go to infi nity at the antir eso- to mi + m2 and after the peak- Disreg arding for the moment thenance and to zero at the reson ance. notch they cont inue as impedanc es point impedance curve it can beThereby the so-catled peak-n otch re- wi th respect to nri2 as m-j is now seen that to main tai n a constantsponse curve is obta ined . That thi s decoupl ed. Between the peak and trans fer velocity Vt belo w and overis the case can be derived fro m the the notch th ere is an inter val in the anti res onanc e, the force F mustfact that at the antir esonan ce the whi ch the impedance is spri ngli ke. be jw( mi + 1H2) V t and, hence, thepoint impedance switches instan- transfer impedance value continuestaneously from an infinitely high T n e frequency of antiresonance is a s a straight line with slope + 1 ac-mass value to an infi nite ly high stiff- equal to r o s s t n e anti reso nance (see Fig. 12).ness value which is negative ^ ^(180phase shift) with respect to the f A = ( 1 /2 ff ) / ] ^


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Fig.14. Damped impedance curves

* For systems with closely spaced antfresonances and resonances the Q value should be applied to the antiresonant subsystems before combinationwith other elements.

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Fig. 13 Damp ed mobili ty curves

nance. As the total force required pie conver sion as above. In this respect reference (2)to keep V p cons tant, and thereby _ by Salter is valuable as it extendskeep the force input to the sprung the discuss ion to larger systems

mass consta nt, must experience a By adding subsys tems to the sys- and to systems wi th more than onesimilar change of slope the trans fer tern in Fig.8 or by letti ng itself be axis, and describes the incl usion ofimpedance slope wi ll increase from part of a larger model the total re- simpl e rotary systems . Similar ly it+ 1 to + 3. From the impedance skel- sponse can be evalua ted by com bin - discusses the influ ence of dampi ngetons in Figs. 11 and 12 , the mobii- ing either impedanc e or mobil ity skel- on the impedance and mobil ityity skelet ons can be obtained by sim- etons fol lo win g the rules given curves whi ch is treated belo w.

The influence of dampingOn most s truc tures wh ic h have cation factor s Q. In simpl e viscoelas- name impl ies, represents the factor

not been treat ed specifical ly to be tic systems only wi th wh ic h to mult iply or divid e*thehigh ly damped one must expect Q = c/ /"km (14) inter secti on values betwe en massrather low dampi ng values , and con- and stiffn ess lines in the mobili ty di-seque ntly high mechan ical ampl ifi - The ampl ifi cati on factor, as its agram to obtain the mobili ty values


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of the resonan ces and the anti reso n- for med fro m one piece of raw mate-ances respecti vely, and vice versa rial the Q values foun d may rangefor the impedance value s. This is il - to more tha n a hund red.lustr ated in Figs. 13 and 14 wh erethe mobil ity curve fro m Fig.7b and On the other hand , sand wich con-the impedance curve from Fig.11b structions wit h viscoelastic layers inhave been red raw n for Q values of shear, special ly designed dampers4, 10 and 25. or even materials wit h high integral

damping, may provide Q values con-Q values of 4 to 10 are often siderably lower tha n 4. Wh en the Q

experi enced for e.g. masses placed value is 0,5 or smal ler, the systemon rubber isola tors. Other isolat ing is said to be crit icall y damped (seematerials used in compressi on may Fig.15) i.e. after a forcing functi onprovide Q factors around 10 whi le has been disc ontin ued the vibr atio nmany other mechanical engineer ing amplitude wil l die out wit hout anyor civil engi neering construct ions oscillations . All systems wit h higherare found wit h Q values in the Q wilt oscill ate at their resonancerange from 10 to 25. However , for frequ ency for a shorter or longer pe-integr al metal cons truc tion s as for riod after exci tatio n depending onexample cas tings or parts cut or the O valu e.

Phase relationships in mechanical impedance and mobilityIn the above sections the phases bilit y or th e impedance as it is

of the different impedances and mo- either + 90 or 9 0 corresponden ce s have only been briefly men - ing to a positiv e slope (+ 1) or a ne-tion ed. Howev er, it may be usefu l gative slope (1) respectively of theto consider the phase rela tion shi ps skeleto n line s. The sudden shift ofas they may prove imp orta nt in slope by a positive or negative valuesome applic ations . of 2 at antir esonan ces and reson

ances corresponds to phase shifts ofIf the phase of the excit ing forc e 18 0 .

is taken as reference it is seen fromthe unity vector diagr am Fig.1 6 that From thi s it can be conclude d thatthe velocities of the mass, the the point impedances of Fig.1 1

spring and the damper of Fig.1 re- have + 90 phase below the antir es-spectively have 9 0 , + 90 and onance. 9 0 between the antires-0 phase shift (j, + j , + 1). onance and the resonance and

again + 90 above the reso nance.The tra nsfer impedances of Fig. 12

As the impedances and mo bi li ti es have + 90 phase below the reso-are given by nance and 9 0 above the reso

nance as the slope of the curve2 = F/vand M = v/F = 1/Z (15 ) changes by + 2. For point and tra ns

fer mobilities similar rules are valid

Fig. 16. The phase r elatio nships for singleelements

i.e. as the mobility curve is changedby inversion, the phase changesfrom positive to negative or viceversa.

their phases are found from

LZ = LF -LM (16}

and

LM=Lv-L? = -LZ (17)

As an exampl e the ang le of th eimpedance of the mass is (seeFig.17)

Z _ Z m = 0 - ( - 9 0 V + 90 (18)

In all undamped cases it is verysimple to find the phase of the mo-

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Fig. 17. The phases of the impedances and mobiliti es of single element s

Fig.1 5. Critic ally damped impedance curve


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In the damped case the phase series damper, whi ch provides a Qdoes not change imme diate ly but va- - 4 is there fore chosen to allow dines gradu ally betwe en + 90 and rect addition of the three mobil itie s.

90 as the frequ ency is swept The response curve for the systemover an antir esonan ce or reso- is given by Fig. 13a.nance, the direction of variation being dependent on wh ic h represen ta- It is seen that at 0, 4f A the mobi l-ti on is cho sen . For the point imped- ity of the mass is the l argest atance and the point mobility the 9 0 . The +90 mobility is sub-phase is zero at the resonance and tract ed from the mass mobilit y andthe antiresonanc e, whereby these the remaining 90 mobility isfreque ncies can be deter mine d accu- added vectoriail y to the mobili ty ofra f ely by phase meas urements even the damper (whi ch is 0,2 5) to ob-for highly damped str uctu res . tai n a resultan t of app. 2 wi th a

phase of 82,4.The phase relationships are given

by vector diagr ams in Fig. 19 for an At the antires onance the 90 antir esonan t system (Fig.18) wit h Q and + 9 0 mobil ities of the mass- 4 at three freq uenci es. The sys- and the spri ng compe nsate eachtern is equival ent to that of Fig.6 other exactly and the resul tingwi th a damper added. As the damp- mobility is that of the damper at 0ing is only evaluated aroun d the an- phase. At 2 ,5 fA the spring mobilitytire sonanc e the most suitabl e dam- is the largest resulti ng in a positiveper confi gurat ion can be chos en. A phase angle of 82 ,4 .

Fig.18. Dampe d i intiresonant system

For transfer impedances thephase angle may turn several timesthrough 360 depending on thecomplexity of the system. The direction would be positive for positivechanges in slope and negative fornegative changes in slop*; (a changein slope of 2 being equal to a phasechange of 180' - and a change inslope of 1 being equal to 90 phasechange).

F*9.19. The phase relati onshi ps arou nd the antireson ance of Fig.1 3a. See also Fig.18 for the mathe mat ica l mo del

Practical considerations in the measurement andevaluation of mechanical impedance, mobility,and other ratios of force and motion

To meas ure mechani cal imped - In Fig. 20 is shown an example of The arra ngement was first used inance it is necessary to have a force a measu remen t arrangement wh ic h the meas urem ent of sttftness of as-sourc e, force and motion tra nsd u- provides the various func tion s phait bars to provide h e complexcers as wel l as analysing and re- whi ch may be needed for most im- modul us of asphalt at freque nciescordi ng equip ment pedance or mobili ty meas urem ents . below the fir st bending resonan ce

(Ref.4).

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Fig. 20. Measur ement arran gement for mechani cal impedance measurement s

This particul ar applicat ion de- any other of the ratios ment ion ed in tion confi gurat ions from economi calmanded that the test specimen was Table 1. or technic al reasons and it may beexcited wit h a constant displace- Alth ough the set-up sho wn may useful to examine each functi on ofment over the frequenc y r ange in be used for a great numbe r of appli - the system to fin d the demands and,ques tio n. However, the measu re- cations it may sometim es be an ad- thereby make the correct choice ofment arrangement is suitable for vantage to use other inst rumen ta- instr umentat ion.

The Vibration Exciter and the Power Amplifier

The Vibration Exciter and thePower Amplifier should be considered as an inseparable pair. In certain circumstances, naturally, a

larger Power Amplifier may bechosen to drive two or more Vibration Exciters in series or parallelfrom the same amplifier, or a Vibration Exciter may be driven by an inferior amplifier for non-demandingpurposes. However, in most casesthe Power Amplifier should bechosen according to the vibration exciter for example as given in Table 2which shows the present range ofBriiei & Kjaer Vibration Exciters.

j Ta bl e 2. Sp ec if ic at io ns fo r Vi br at io n Ex ci ter s an d Po we r Am pl if ie rs 073040

The lim iti ng para meter for the desired is the more impor tant pail choice of Vib rati on Exciter is nor- ramete r, and the inter change abil ity3 mally the max. force required . This of Exciter Heads of the 4 8 0 1 , 48 02B is also the parame ter of highes t eco- and 48 03 fami ly may provide the

nomic impor tance as it puts re- optimal solut ion of large stroke or2 quir emen ts on both the Vib rat ion Ex- max. force applied to the payloadf citer and the Power Amp lif ier . How- for any given size of Vib rati on Exci-

ever , in some cases the max. strok e ter (See Ref.5) .

VibrationExctier No.

ForceNpeak

I

Stroke Velocitym/s peak

Max. FrequencykHz

Power Ampl ifi er No.

PowerVA

48014802480348094810

3 8 0 - 4451450- 17805340 - 66 70

44.57

12,7 -25,419 - 3 827,9 - 55,9

86

1,01 -1,271,271,271,65

5,4 -1 04, 5- 5,52,9- 3,5

2018

27072708270927062706

12012006000

7575

Table 2. Sp lecifications for Vibration Excit


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I Type No. I 6200 j 8201 [ most any type of Bruei & Kjaer accel- Alt hou gh at! the acc eleromet ersMax Tensile icoo N 4coo N erometer may be used alt houg h the ment ione d may be used down to

"M^~C7^. |~ SCOVN j ZO.CSON " r a n 9 e of Uni -G ai n types are pre- 1 Hz extra care shoul d be executed~ c h l ^ 7 s l ^ r ~ AJCV I 4 P C N f e r r e d (charge sensitivity = below app. 5 Hz especi ally at tow~te^^H^~T^ z ' ^T' 1 P c / m s ~ 2 o r 1 0 p C / m s - 2 Types sign al level s. This is due to the fact

( 5 g l o a d > j J , 4 3 7 1 , 43 70 respectively, and vol- that most piezoelectric transducersM s t e n a l stainless steel t a g e s e n s j t j v j t v 1 m V/ms ~~ 2 Type are sensit ive to temp erat ure tran-He | 9 ht I 1 3 m m I 5 6 8 m m I 830 1). sients whi ch may be induced in

Tab le 3. Specif ications for Force Transducers t h e a c c e l e r o m e t er s b y S l i g h t a i

For applic ation on very light struc - move ment s in the room wher eFor st ruct ures larger than app. tur es, for very high levels of vibra - meas urem ents are taking place. To

1 kg mass Force Transduc ers Types tion and for hi gh freque ncies the reduce this effect the accelerometers82 00 or 82 01 should be used ac- range of Min iat ure Accele romet ers may be covered by insu lati ng mate-cording to the force demands (see Types 43 44 , 830 7, or 83 09 may rial or alternatively the above-men-Table 3 and Ref.6). be used to avoid loading the speci- tion ed Type 83 06 or Quartz Accefer-

men and to ensure correct measu re- ometer Type 83 05 may be used.The motion is normal ly best ment s.

measured by an Acce lerom eter. The Quartz Acce lerometer may beThis is due to the large dynamic For applicati ons wi th low signal ie- used dow n to virtua lly DC becausrang e, the large frequenc y ra nge, vels Type 83 06 whi ch has a sens i- of the stabili ty of the quartz crystal .and the relia bilit y provided by these tivi ty of 1 V / ms ~~ 2 may be used. It may be scr ewed to the Forcetran sduc ers. Howeve r, some consi d- The latter acceler ometer also pro- Transducer Type 82 0 0 to form aneration should be given to the vides stable oper ation do wn to Impedance Head if so des ire d.choice of acceler ometer type. For 0,3 Hz.most non-demanding purposes at-

" i I Type No. I B200 j 8201I

Max Ten si le 1C00 M j . .

4C0O N

Max. Compr. 5C00 N j 20.C30 N

Charge Sens. A pC N ] 4 pC N _ ^J- - -. *

Resonant Freq. .(5 g load)

L

35 kHz 20 kHz

Material Stainless Stee

Height1 * ' * " " "" H J "

13 mm 36 8 mm

*

Table 3. Spec ifi catio ns for Force Transducers

Preamplifiers

As Ac ce le romete rs and Force si tivi ty due to diffe rent cable For accele rometers wi th other sensi-Transd ucers have very high elec tri - lengt hs. tivi ties and Force Transduc ers,cal output impedances, a preampii - preamplifiers Types 26 26 , 26 28fier must be insert ed after each Type 26 34 is a smal l unit only and 26 50 provide both accuratetransd ucer in order to provide a high 21 mm * 34,5 mm * 100 mm sensitivity adjus tment and adjus-input impedance to the trans ducer whi ch can be placed near the mea- table gain. Type 2650 has 4 digitsignal and a low output impedance suri ng point, and whi ch is operated adju stme nt compared to 3 digits forto the fofi owing electronic instr u- from an external 28 V source. The the other two , and Type 26 28 pro-ment s. Thereby low frequenc y and Type 26 51 is especially intended vides very low frequ ency operationlow noise operation is made possi- for use wit h Uni- Gai nc har ge cali- to 10 Hz.bie. brated acceler ometers such as

Types 43 71 and 4 37 0 , and gives In addi tion , both Type 26 26 andCharge preamplif iers automat i- then a calibr ated output propor- Type 26 28 provide adjustable low

cally compens ate for change in sen- tiona ! to accel eration or velocity . pass and high pass fil teri ng of thesignal.

Exciter Control

The Exciter Control Type 10 47 or displac ement over the frequen cy the speci men in ques tio n. The mostcontains a fully electronically con- range. It has no bui lt- in filter s but obvious ones wou ld be the Hetero-trol led oscil lator secti on to drive the control s external fitters and record- dyne Analyzer Type 20 10 or per-Power Ampl ifi er in the frequenc y ing devices. haps the Sine Random Generatorrange fro m 5 Hz to 10 kHz. It con- Type 10 27 . The forme r in additiontain s one meas urement and contr ol Simi larly a numbe r of other ins tru - gives excel lent frequ ency analysischannel, all owin g two preset c on- ments may be used to obtain a con- capability fro m 2 Hz to 20 0 kHz.trol values of acce lera tion, velocity trol led force or vibr ation level on

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Recording Device

The Level Recorders Types 2306or 2307 may be used with precali-brated recording paper as shown inFigs.21 and 22. Recording paperscan have a 50 mm , or a 1 00 mmordinate (2307 only} which can beused for logarithmic scales of 10,25, 50 or 75 dB accordin g to theRecorder Potentiometer used (50dB

being the most usual) or for a linearscale (Potentiometer ZR 0002 for2307) which must be used forphase recordings. The frequencyscale of Fig.2 1 has a decade lengt hof 50 mm whi ch prov ides the possibility of drawing the skeletons as described in the theoretical section.(The slopes being 20dB for 1 decade or multiples of that).

To obtain better accuracy withclosely spaced resonances or antire-sonances the recordings can also

be made with an enlarged frequency scale (either logarithmic orlinear). For the latter a redrawingmust be made to have log-log representation for final evaluation.

Similarly X-Y recorders may beused for recording of the graphs provided either the recorder outputused (Heterodyne Analyzer Type2010 or Measuring Amplifier Type2607) has a logarithmic output, orthe X-Y recorder has a logarithmicpreamplifier.

If acceleration or displacement ismeasured or used for control instead of velocity the graph can berecalibrated to mobility or impedance by drawing a reference line onthe log-log graph and reading thevalues above this line for each frequency. See Fig.23.

h F ig .21 . Recordi ng paper wit h precalibrated frequenc y scate and an ordinate calibrated in dB

Fig.22. Recordi ng paper wi th calibrated ordinate

Fig.2 3. Convers ion of recorded curves to mobility and impedance curves


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Fig.2 4 Nomo grap h of the relations of sinusoidal accelerati on, velocity and displacement as functi ons of freque ncy

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As acce lera ti on for sinuso idal vi - imped ance lin es are given wi th posi- the above relation ship or from thebration a = jwv the velocity is re- tive slope 20 dB /d ec ad e in Fig .23b . graph in Fig .24 .duced for increasing frequency andlines wi th negative slope The curves are draw n thr ough Whe n displace ment has been20 dB /d ec ad e wou ld represent con- points at one frequency for whi ch measured the calibrat ion is carriedstant mobility lines in the represen- the mobil ity or impedance respec- out in a similar way but wit h oppo-tat ion of Fig. 23a. Simil arly constant tively have been deter mined fro m site signs for the slopes.

Phase measurementsThe phase is best measu red by tion bet ween the ir resonance s or accel erat ion or veloci ty at a given

means of a phase mete r. For many anti reso nanc es, it proves very use- point may be the resul t of cont rib u-purposes however, phase monitor- ful to know the phases of the sys- tions f rom tran slator y mo tion anding may be suffic iently well carried tern responses. Whe n the reson- one or tw o torsio nal motion s, andout by means of an oscilloscope on ances or antiresonances of highly phase measu rement wit h respect towh ic h either the beam is defle cted damped systems must be meas ured forc e or to other poin ts may be nec-in the X and Y directio n by the two accurately, phase indication is vital. essary to deter mine the mut ual in-signals in quest ion, or wher e one Whe n the damping of a system fluen ce of these vibration modes.signal is displayed below the other. must be determ ined outside reson- Simila rly in mode studies on sys-

In many cases, however the need ance accurate phase measu remen t terns wi th varying stiffness thefor phase mea sure men ts does not by a phase meter is absol utely ne- wav ele ngt h of vib rat ion at a givenarise. From good recordi ngs of point cessary. freq uenc y may vary considerabl yimpedan ces and tra nsfe r impe d- fro m one part of the system toances it is possible to evaluat e the Under many other circ umst ance s anoth er and phase meas urem entsresponse fro m the amplitu de the phase measure ments provide provide an extra control that all an-curves . However, wh en tw o sys- an extra check on the measu re- tin odes and nodes have been de-tems are to be joined together and ments. For example in measure- tected .there is no large frequency separa- ment on complicated systems, the

Examples of Application

In Fig.25 is shown an arrangement which was used to determinethe point impedances of the surfaceof a hermetically sealed containerfor a refrigerator compressor(Fig.26). The purpose of the investigation was to find the optimalpoints to secure the springs inwhich the compressor was to be suspended, in order to reduce thetransmission of vibration to a minimum and thereby to minimize thesound emitted from the container.The measurements revealed that atmost points the impedance curveshowed numerous resonances andantiresonan ces (see Fig27a) whe reas four points had relatively highimpedance and very small variations over the entire frequencyrange (Fig.27b).

"Y Fig.2 5. Measu ring arran gemen t suitable to dete rmine the point imped ance at variou s points ofthe surface of a steel contai ner for a refrige rator compr essor

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the modulus of elasticity of bars inlongitudinal vibration. The bar isfixed onto the combination of aForce Transducer Type 8200 and aStandard Quartz acceierometerType 8305 and the point impedance curve is obtained (Fig. 2 9),From the exact frequencies of resonance or antiresonance and the formof the curve the modulus of elasticityand the damping coefficient can be

obtained (see Ref.16). In Fig.30 theantiresonant peaks have been enlarged by using a reduced paperspeed and a 10d B potentiometer.

In Fig.31 is given an improvedversion of the instrumentation usedfor the measurements described inRef .7 . The measurements weretaken to obtain knowledge aboutthe response of a prefabricatedbuilding structure by means of mobility measurements. The acceleration signal is integrated by the Con

ditioning Amplifier Type 2635 andled through the Tracking Filter Type5716 to exclude ambient vibrationfrom nearby punch presses. The mobility recording obtained is shown inFig.32.

Further examples of applicationsare given in the Briiel & Kjaer Appli-cation Note No. 1 31 20 "Measurement of the Dynamic Properties ofMaterials and Structures" (Ref.8).

Fig.28. Measurement configuration for determination of the modulus of elasticity of a PVC bar by impedancemeasurements

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f ig .27 . Point impedance recordin gs obtained fro m measureme nts on the steel container , a) asrecorded at most poin ts, b) as recorded in four points

Fig.26. Cross-sec tion of a refrigerat or com

pressor suspended in a hermeticallysealed container te

These nodal points would giveminimum transmission of vibrationenergy when combined with thelow impedance of the springs andwere, therefore, selected as fixingpoints. (There would be a high transfer impedance from the compressorto the point; compare with the system of Fig.6 and with the inverse ofits mobility curve in Fig.7).

Fig.28 shows a very similar system used for the determination of


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Concluding Remarks

Measur ement of the mechanic alimpedance, mobility or any other ofthe complex dynamic ratios givenin Table 1 may provide useful knowledge about a structure. By comparison of the obtained recordings withlines for masslike or springlike response one will gain insight into thedynamic characteristics of the struc

ture to aid in further development orin corrective measures.

The existing forces may be derived by comparison with e.g., themobility plot, and the need for correction may often be directly decided upon, by frequency analysisof signals generated during operation.

in more complicated applicationsthe measurements may be used tofind the elements of a mode!. Thencomparison between calculated values for the model and measurementresults may be used for correctingthe parameters of the model untilsufficient accuracy is obtained.

Similarly comparisons with frequency analyses may be used to calculate all relevant forces or couplesin the system.

In other cases the response ofsubsystems may be measured andthe properties of the total system becalculated before final assembly (SeeRef. 12).

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Fig .32 . Mobi lit y recording obtained for a concre te beam

Fig .31 . Measur ement arrang ement for measurement of the mobility of a concrete beam

Fig.3 0. Extended recording s of the antires onant peaks of Fig.29

Fig.2 9. The mechanica l impedance record ing obtained for a PVC bar


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References

1. USAS S 2.6 -1 96 3, "Speci fying 8. H. P. Olesen, "Mea sure ment of 13. N. F. Hunter et a!., "The meas-the Mechani cal impedance of the Dynamic Properties of Mate- urement of Mechanic al Imped-Str uct ure s" rials and Str uctu res" Bruel & ance and its use in Vibrati on

Kjaer App lic atio n Note No. Te st in g" . U.S. Nav. Res. Lab. r2. J. -P . Salter, "Steady State Vi- 17 180 Shock Vib. Bull . 42 , Pt. 1 , Jan.

orat ion". Kenneth Mason 196 9 19 72, P.55699. C. M. Harr is and C. E. Crede,

3. Jens Trampe Broch , "M ec han i- "Shock and Vibration Hand- 14. E. F. Ludwig et al. , "Me asur e-cal Vibrat ion and Shock Meas- book". Mc Gr aw- Hi iM 961 ment and Applic ation of Me-

ure men ts" Bruel & Kjaer, 19 72 chanicai Imped ance" . U.S. Nav.10 . D. J . Ewin s, "So me why s and Res. Lab., Shock Vib . Bull. 424. K. Zaveri and H. P. Ole sen , wher efo res of impedance test- Pt. 1, Ja n. 1 972 P. 43 4 5

"Me asu rem ent of Elastic Mod u- in g" . See Dynamic Testing Sym-lus and Loss Factor of As pha lt ". posium , Jan . 56 , 7 1 , London 15. M. E. Szendrei et al, , "Road Re-Bruet & Kjaer Techn ical Review sponses to Vib rati on Tes ts ".No. 4- 19 72 1 1. R. B. Randall, "High Speed Nar- Journal of the Soil Mechani cs

row Band Analys is using the and Found ation Div isi on, No-5 Gait Boot h, "In ter chan geab le Digital Event Recorder Type vembe r, 19 70

Head Vibrati on Exci ters" Bruel & 7 5 02 " , Bruel & Kjaer TechnicalKjaar Techni cal Review No. 2- Review No. 2, 1 973 1 6. Mea sur eme nt of t he Complex197 1 Modulu s of Elasticity: A Brief

12 . A. L. Kiost erman, "A combined Survey. Bruel & Kjaer Applic a-6. Will y Braender, "Hi gh Frequency Experimental and Analyti cal tion Note No. 17 0 51

Response of Force Tra nsd u- Procedure for Improvi ng Aut o-cer s" . Briiei & Kjaer Technic al motive System Dynam ics" . SAE, 17. R. B. Randall, "Frequenc y Ana -Review No. 3- 19 72 Automo tive Engineering Con- lysi s" Briie l & Kjaer, 19 77

gress, Detroit Mich., January7.To rben Licht, "Meas ureme nt of 10 14, 197 2

Low Level Vibrations in Buildings" Bruel & Kjasr TechnicalReview No. 3-1972

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