a guide to mechanical impedance and structural response techniques

8/20/2019 A Guide to Mechanical Impedance and Structural Response Techniques

1/18

17—179

A Guide to Mechanical Impedanceand Structural Response Techniques


2/18

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. Determ ination of natural fre quen -rapidly developing interest in the sured. cies and mode shapes.

fiel d of mechanical dynamics for a 2. Measur ement of specific materia l

variety of reasons. An example 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 analytical

stronger materials and greater econ- verely limi t the quali ty of mach ining model. From measurem ents of

omy in design has led to increa s- and grin ding opera tion s. the impedances of indivi dual com-

ingly lighter structu res, more prone ponents or subst ructu res it is pos-

to vibra tion problems. At the same The overall resul t is that the dy- sibie to predict the behaviour of

tim e, increasing rotatio nal speeds namic behavio ur of a mach ine or combined systems, in a manner

also give increasi ng likeli hood of stru ctur e is now an impo rtan t factor completely analogous to the

having to deal wi th stru ctural 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-

mentai questio ns since the improve - sive "safet y factor", or treated as oped from electro-mechanical and

ment of both noisy and vibrat ing en- an afte rtho ught wh en problems electro-acoust ic analogies in the

vironment s often can be simplif ied have been encoun tered . 19 20 s. Since then the usefulness

to a quest ion of reducing the me- of these concepts in forced vibratio n

chanicaf vibr ation, either at its One very usefu l expe rimenta l techniques and in the theore tica l

source or somewh ere along the tech nique for the study of dynamic evaluat ion of stru ctures has im-

transmission path. behaviour of mach ines and stru c- proved considerab ly. This is due

ture s concerns the mea sureme nt of partly to more sophisti cated vibra-

Typical examples are provided by wha t is loosely ter med "me cha nic al tion transducers, vibra tion exciters

the transportation industries, wher e impe dance" . Broadly speaking, this and analysis equipment and partly

in the development of for example defines the rela tion ship s b etwe en to the acceptance in mechani cal and

airc 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 so

the various components can wi th- tude and phase. Ref. 10 lists the that they could be handled on their

stand the dynamic loadings to thre e main appli cations of imped - own wit hout resorting to a previous

wh ic h they are subje cted, but also ance testing as: conversion to electri cal circuits.

Mechanical impedance and mobility

The mechanic al impedance and not be given here. The units after of motion relative to the direct ion of

mobil ity (for simple harmonic mo- each ratio are Sl -u ni ts *. force whe n this is not obvious from

tion) are defined as the complex ra- the measurement conditi ons or

tios of force vector to velocity vec- As both force and moti on are vec- from the calcu latio ns.

tor, and velocity vector to force vector s in space as we ll as in time 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 Ns

USAS 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

3

Dynamic Mass(Apparent Weight}

Fa

Ns2

1 m s(Acceleration through Force)

aF

m ,1 Ns2 J

Mechanical Impedance _F

V

NsMobility(Mechanical Admittance)

V

\ — i[ Ns J

Stiffnessi

Fd 1 m '

, Compliance6

F 1 — 11 N J


3/18

When force and motion values into motion . By measuring , for ex- ledge about the response ability of

are measured at the same point and ample, the mechanical impedance of the strucutres involved, and of the

in the same direction the ratios are points on a struc ture, knowledge is actual responses or forces. After

termed driving point values, or in gained about its response to vibra- combinati on of this information the

short, point values, e.g., point im- tional forces at different frequencies. need or the possibilit y of corrective

pedance. Simil arly, a measurement of the mo- measures may be evaluated.

tio n of the struc ture, after it has

When force and motion are meas- been placed on a vibrating support. In the fol low ing sections, after a

ured at different points or at the may be compared to its mechanical brief discussion of narrow band fre-

same point wit h an angle between impedance to obtain information quency analysis, rules are given for

them they are termed transfer val- about the forces which act on the evaluating mechanical impedance

ues e.g. , transfer impedance. structur e. data by graphical means, the inst ru

mentation used for practical meas-The ratios given in the firs t and To solve vibrational problems, urement is discussed and a few

second column of Table 1 really re- there fore, both a mechanical imped- practical examples are given, as wel l

present, as func tions of frequency, ance may have to be measured, and as references to further literature

the difficulty or ease, respectively, a narro w band frequency analysis about Mechanical impedance appli-

wi th whic h a structure can be set carried out to obtain detailed know- cations.

Narrow Band Frequency Analysis

One of the major reasons for stu- from the source exc itat ion, or in Where the Q-factors (see later

dying dynamic phenomena as func- modifying the structure to "d etune" for definition) are greater than about

tions of frequency is the simplici ty or damp that particular frequency re- 50 , however, it may be necessary to

that 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 bandwidth. . . . The type of f requency ana lysis , 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 aimportant property of such l inear , _- ,_ , . . . ,

. . / , thu s a brief discussion is given of „ _ . . ._. , . ,systems is that of superposit 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 output at the same is also benef icia l for the analysis offrequency, 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 are

quency response fun ct ion , and the wid th and constant proportional then uniformly separated.

behaviour at this frequency is thus bandw idth analysis, it is often

independent of wha t is going on at claimed that "na rrow band analy sis" The overall consideration in

other frequenc ies. A related advan- is synonymous wit h narrow constant choice of analysis method is that it

tage is that combinat ion of cascaded bandw idth analysis, but this is not should everywhere give suffici ent

systems 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- information in other areas, because

quency, and this in tur n is simpl ified stant percentage bandwidths down of the detrimental 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 is

though excitation is rarely sinuso idal sponse of mechanical structures covered in depth in Ref .17 .

at a single frequenc y, the use of tends to be similar in principle to a

Fourier analysis (narrow band fre - constant percentage bandwidth filter Perhaps the best compromise is

quency analysis) makes it possible to (a certa in amplifica tion factor Q the Heterodyne Analyzer Type 20 10

break down a more complex signal corresponding to a certain percen- whi ch has both linear and logarith-

into its components at various fre - tage bandwidth). Thus, whe re the mic frequency sweeps covering the

quencies, thus considerably sim pli - excitation is fairly broadband it may range from 2 Hz to 2 00 kHz. Al -

fying its interpreta tion. be most efficient to analyze the re- though it is primari ly a constant

sponse wi th constant percentage bandwidth inst rument it can be pro-

A typ ica l dynamic problem wo ul d bandwi dth fi lt ers. grammed to step up automatical ly ininvolve obtaining the frequency bandwidth wit h increasing fre-

spectrum of the input to a mechani- Another advantage of constant quency, thus approxima ting a con-

cal system (be it force or motion) and percentage analys is is that it gives stant percentage bandwidth analysis

by comparing this with the meas- unif orm resolution on a logarithmic (where the percentage can be con-

ured response character istics to de- frequency scale, and thus can be siderably lower than 1%). The main

term 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 time, can be obvi-

excitation and mobi lity. The solut ion scales are moreover advantageous ated by use of the Digi tal Event Re-

of such a problem wou ld consist for the interpretation of mechanical corder Type 75 02 as described in

either in eliminating that component impedance data. Ref. 1 1 . The large frequency trans-

4


4/18

Fig.3. Mobi lity for mass, spring and damper

5

' "format ions available wi th this inst ru

ment (which incidentally can equally

well be used with the Analyzer Type

2120) allow reduction of analysis

time to the order of a minute or so,

and also allow the effective band

width to be made smaller than the

minimum available on the 2010

(3,16Hz).

When even faster analysis is re

quired, or where a very large num-

Impedance and mobility of struct

The vibrational response of struc

tures may in many cases be repre

sented by a theoretical model which

consists of masses, springs and

dampers. If the structure is compli

cated and if the response must be

duplicated exactly over a large fre

quency range the number of ele

ments needed may be very large.

However, for simple systems, and

even for complicated structures in a

limited frequency range, the re

sponse may be represented sufficiently well by a few elements.

The force F needed to set a pure

mass m into vibration is propor

tional to the acceleration a.

F - m a [Nl (1}

The force required to deflect a

spring with stiffness k is propor

tional to the relative displacement d

of the two ends of the spring

F = kd [N] (2)

Finally the force is proportional to

the relative velocity v of the two

ends of a damper with damping co

efficient c for pure viscous damp

ing.

F ^ cv [N] (3)

For sinusoidal motions, accelera

t ion, velocity and displacement mea

sured at a given point are related by

the relationships

a = JOJV = -u)2d (4)

v = \OJ6 ~ (1/jo>)a (5)

d = (1/jw) v - (1/-w 2)a (6)

here OJ = 2ni

and f is the frequency of vibration

The graphical signatures for the

ber of spectra must be averaged to AM the analyzers ment ion ed can

give a stable result , the Real-Time wr ite out a "har d copy" of the analy-

Nar row Band Analyzer Type 33 48 sis resul ts on a Level Recorder Type

wil l often be preferred. The real-tim e 23 06 , or 23 07 wh ic h, as described

capab ilit y also gives the possibi lity of later, can also be used for recor ding

visually foll owing non-stationar y the impedance amplitud e and phase

phenom ena e.g., seeing how the re- characteristi cs. Output in digital

sponse of a machi ne varies as it for m (ASCII-coded BCD) on punch

runs up or down in speed, and thu s tape is also possible.

quickly establishing "dangerous"

operating areas.

:tural elements

thr ee elem ents are given in Fig.1 to- end to correspond to the relati ve mo-

gether wi th their mechanical imped- tion betw een their ends. They are

ance and mobilit y. considered massless. in cases wher e

both ends of a spring or damper

Note that the mass is free in move, the n it is the diff erenc e be-

space and that the spring and dam- twe en the absolute motions of their

per requi re one end fixed in order for ends wh ic h must be subs tituted into

the absolute moti on of the excited equa tion s (2) and (3).


5/18

The impedances and the mobi li- thi s representatio n the impedances tance on both abcissae and ordi-

ties of the elements are best il lu- and mobi lit ies are given as stra ight nates and wher eby the slopes for

strated in log-log diagrams wi th fre- lines . (See Figs.2 and 3 whe re a mass and spring impeda nce lines

quency 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 a

common case in practice, e.g., in vi

bration isolation. In simple systems

the mass can be considered to be

placed on one spring which has

a stiffness value equal to the sum

of stiffnesses of the supports.

(Damped systems will be considered

later).

In the literature this basic system

is very often symbolized as in

Fig.4a for force applied to the mass

as for instance wi th a motor placed

on springs. However, this represen

tation may lead to the misconcep-

ion that the system is a so-cal led from equation 7 but it is less time

series system whi le it is, in fact , a consumin g to combine th e curves

parallel system whe re the force is graphically. (Remember that the

shared betw een the mass and the mass impedance has a positive

spring as indicated clearly in Fig.4 b. phase angle of 90 ° (j) and the

Here the force is appl ied to a mov- spri ng impedance has a negati ve

ng plane to wh ic h both the mass phase angle of 90 ° (—j or 1/j) rela-

md the sprin g are attached. (See tive to the force). The curves can be

^ef.2) . obtained by subtracting the lowest

value from the highest value at

As the motion is co mmon to the each frequency but a more stra ight

wo elements their impedances forw ard method is to construct a so-

from Table 2) can be added to ob- called impedance skeleton, as given

ain 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 where they coun teract

At low frequenc ies to is very each other to produce a ver tical line

>mall and Z equals k/jaj as jojm can for Z - 0.

)e neglected. At high frequencies to

s large and Z equals jw m. The impedance curve can then be

drawn to the desired accuracy by de-

At a frequency fR wh er e jo>m = te rm in in g tw o or more points fr om

—k/jw a resonance occurs wh er e Z the spr ing and mass curves and

= 0 and o) = co0 = 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 hown

in an often used representation (a) and

in the correct way (b)

Base excited system

If the system of Fig.4 is excited at

the base as shown in Fig.6 it is

seen that the velocities at the base

and at the mass are differen t, i.e.,

both point and transfer values do ex

ist. As the force on the mass is

equal to the force at the base the

system can best be evaluated from

the 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 by

inversion of the impedance curves of

Fig.5 (see Fig.7a).

From these curves the point mobil

ity skeleton and the point mobility

curve can be constructed in a similar

manner to the impedance curve of

Fig.5c (seeFig.7b). Fig.6. Base exci ted syste m

6


6/18

However, it is a mobility plot and

the minimum at fA represents an

antiresonance, i.e. an infinite force

would be required to produce any

motion at all.

The transfer mobility, on the

other hand, experiences no discon

tinuit y. As the force transmi tted

through the spring remains con

stant and equal to F the velocity Vt

of the mass remains the same as it

would be for a mass suspended in

space and hence the transfer mobil

ity is a straight line with the same

slope and position as a point mobil

ity curve for the mass alone. The

motion of the mass being reduced

rapidly suggests that at high fre-

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

quenci es to all practical purposes placed on a rigid support as that of

the sprin g can be considered as Fig. 1 b.

The mass-spring-mass system

An example of fu rt her extension

of the model is given by the evaluation of the mass-spring-mass sys

tem which is often encountered.

This system is shown in Fig.8, and

it is seen that the force is divided

between the mass and the spring

supported mass (the sprung mass).

Hence, the point impedance must

be found from the combination of

the mass impedance line (shown for

three different masses in Fig.9),

and the point impedance skeleton of

the sprung mass which has antire

sonance at fA The point impedance

skeleton is obtained by inversion ofthe point mobility skeleton of Fig.7b

and is shown in Fig.10. The result

ing point impedance skeletons and

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

curves for the three values of rri2

are given in Fig. 1 1.

It is seen that the impedance isobtained by the combination of the

curves in Figs.9 and 10 by keeping

the highest value and by letting theFig. 10. Point impedanc e skeleton of the

sprungmass 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

7


7/18

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 m i + m2 and after the peak- Disreg arding for the moment the

nance and to zero at the reson ance. notch they cont inue as impedanc es point impedance curve it can be

Thereby the so-catled peak-n otch re- wi th respect to nri2 as m-j is now seen that to main tain a constant

sponse curve is obta ined . That thi s decoupled. Between the peak and trans fer velocity Vt belo w and over

is the case can be derived fro m the the notch th ere is an inter val in the anti resonanc e, the force F must

fact that at the antiresonan ce the whi ch the impedance is spri ngli ke. be jw( mi + 1H2) V t and, hence, the

point 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 ^ „ ^

(180°phase shift) with respect to the f A = ( 1 /2 ff ) / ] ^


8/18

Fig.14. Damped impedance curves

* For systems with closely spaced antfresonances and resonances the Q value should be applied to the antiresonant subsystems before combination

with other elements.

9

Fig. 13 Damp ed mobil i 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 extends

keep 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 onesim ilar change of slope the trans fer tern in Fig.8 or by letti ng itself be axis, and describes the incl usion of

impedance slope wi ll increase from part of a larger model the total re- simple 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 ng

etons in Figs. 11 and 12 , the mobii- ing either impedance or mobil ity skel- on the impedance and mobil ity

ity skeletons can be obtained by sim- etons fol lo win g the rules given curves whi ch is treated belo w.

The influence of damping

On most s truc tures wh ic h have cation factors Q. In simpl e viscoelas- name impl ies, represents the factor

not been treated specifical ly to be tic systems only wi th wh ic h to mult iply or divid e*the

high ly damped one must expect Q = c/ /"km (14) intersecti 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


9/18

of the resonan ces and the anti reso n- formed fro m one piece of raw mate-

ances respecti vely, and vice versa rial the Q values foun d may range

for the impedance value s. This is il - to more tha n a hund red.

lustr ated in Figs. 13 and 14 wh ere

the 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 in

have been red raw n for Q values of shear, special ly designed dampers

4, 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

experienced for e.g. masses placed value is 0,5 or smal ler, the system

on rubber isola tors. Other isolat ing is said to be crit icall y damped (see

materials used in compressi on may Fig.15) i.e. after a forcing functi on

provide Q factors around 10 whi le has been discontin ued the vibratio n

many other mechanical engineer ing amplitude wil l die out wit hout any

or civil engi neering construct ions oscillations . All systems wit h higher

are found wit h Q values in the Q wilt oscill ate at their resonance

range from 10 to 25. However , for frequ ency for a shorter or longer pe-

integral metal cons truc tion s as for riod after exci tatio n depending on

example 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° correspond

en ce s have only been briefly men - ing to a positive slope (+ 1) or a ne-

tion ed. Howev er, it may be usefu l gative slope (—1) respectively of the

to consider the phase rela tion ships skeleto n line s. The sudden shift of

as they may prove imp orta nt in slope by a positive or negative value

some applications . of 2 at antir esonan ces and reson

ances corresponds to phase shifts of

If the phase of the excit ing force 18 0° .

is taken as reference it is seen from

the unity vector diagr am Fig.1 6 that From this it can be conclude d that

the 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 antires-spectively have —90° , + 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 curve

2 = 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 single

elements

i.e. as the mobility curve is changed

by inversion, the phase changes

from positive to negative or vice

versa.

their phases are found from

LZ = LF -LM (16}

and

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

As an example the ang le of th e

impedance of the mass is (seeFig.17)

Z _ Z m = 0 ° - ( - 9 0 V + 90° (18)

In all undamped cases it is very

simple to find the phase of the mo-

10

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

Fig.1 5. Critic ally damped impedance curve


10/18

In the damped case the phase series damper, which provides a Q

does not change imme diate ly but va- - 4 is there fore chosen to allow di

nes 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 system

over an antir esonan ce or reso- is given by Fig. 13a.

nance, the direction of variation be

ing 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 largest at

ance and the point mobility the —90° . The +90° mobility is sub-

phase is zero at the resonance and tracted from the mass mobilit y and

the antiresonanc e, whereby these the remaining —90° mobility is

freque ncies can be deter mine d accu- added vectoriail y to the mobili ty of

ra f ely by phase meas urements even the damper (which is 0,2 5) to ob-for highly damped structu res . tain a resultan t of app. 2 wi th a

phase of —82,4°.The phase relationships are given

by vector diagrams in Fig. 19 for an At the antiresonance 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 spring compe nsate each

tern is equivalent to that of Fig.6 other exactly and the resul ting

wi th a damper added. As the damp- mobility is that of the damper at 0°

ing is only evaluated aroun d the an- phase. At 2 ,5 fA the spring mobility

tire sonance the most suitable dam- is the largest resulti ng in a positive

per confi gurat ion can be chos en. A phase angle of 82 ,4 °.

Fig.18. Dampe d i int iresonant system

For transfer impedances the

phase angle may turn several times

through 360° depending on the

complexity of the system. The direc

tion would be positive for positive

changes in slope and negative for

negative changes in slop*; (a change

in slope of 2 being equal to a phase

change of 180' - and a change in

slope of 1 being equal to 90 ° phase

change).

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 measure mechanical imped - In Fig. 20 is shown an example of The arra ngement was first used in

ance 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 complex

cers as wel l as analysing and re- whi ch may be needed for most im- modulus of asphalt at freque ncies

cording equip ment pedance or mobili ty measurem ents . below the fir st bending resonan ce

(Ref.4).

11


11/18

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

This particular applicat ion de- any other of the ratios ment ion ed in tion configurat ions from economi cal

manded that the test specimen was Table 1. or technical reasons and it may be

excited wit h a constant displace- Alth ough the set-up sho wn may useful to examine each functi on of

ment over the frequency range 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 of

ment 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 the

Power Amplifier should be consid

ered as an inseparable pair. In cer

tain circumstances, naturally, a

larger Power Amplifier may bechosen to drive two or more Vibra

tion Exciters in series or parallel

from the same amplifier, or a Vibra

tion Exciter may be driven by an in

ferior amplifier for non-demanding

purposes. However, in most cases

the Power Amplifier should be

chosen according to the vibration ex

citer for example as given in Table 2

which shows the present range of

Briiei & Kjaer Vibration Exciters.

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

The lim iti ng para meter for the desired is the more impor tant pa

il choice of Vib rati on Exciter is nor- ramete r, and the interchange abil ity

3 mally the max. force required . This of Exciter Heads of the 4 8 0 1 , 48 02

B 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 or

2 quir emen ts on both the Vib rat ion Ex- max. force applied to the payload

f citer and the Power Amp lif ier . How- for any given size of Vib rati on Exci-

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

Vibration

Exctier No.

Force

Npeak

I

Stroke Velocity

m/s peak

Max. Frequency

kHz

Power

Ampl ifi er No.

Power

VA

4801

4802

4803

48094810

3 8 0 - 445

1450- 1780

5340 - 66 70

44.57

12,7 -25,4

19 - 3 8

27,9 - 55,9

86

1,01 -1,27

1,27

1,27

1,65

5,4 -1 0

4, 5- 5,5

2,9- 3,5

2018

2707

2708

2709

27062706

120

1200

6000

7575

Table 2. Splecif ications for Vibration Excit


12/18

I Type No. I 6200 j 8201 [ most any type of Bruei & Kjaer accel- Alt hou gh at! the accelerometers

Max 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 -Gain® 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 especially 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 erature 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 icat ions for Force Transducers t h e

a c c e l e r o m e t er s b y S l i gh t a

For application on very light struc - move ments in the room where

For st ructures larger than app. tur es, for very high levels of vibra - measurem ents are taking place. To

1 kg mass Force Transducers Types tion and for high freque ncies the reduce this effect the accelerometers

82 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 ments.

measured by an Acce lerom eter. The Quartz Acce lerometer may be

This is due to the large dynamic For applications wi th low signal ie- used dow n to virtua lly DC becaus

rang e, the large frequency 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 screwed to the Force

tran sduc ers. Howeve r, some consi d- The latter accelerometer also pro- Transducer Type 82 0 0 to form an

eration should be given to the vides stable oper ation do wn to Impedance Head if so des ire d.

choice of accelerometer type. For 0,3 Hz.

most non-demanding purposes at-

" iI Type No. I B200 j 8201I

Max Tensi 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 Acce le rometers and Force si tivi ty due to diffe rent cable For accele rometers wi th other sensi-

Transd ucers have very high elec tri - lengths. tivi ties and Force Transduc ers,

cal output impedances, a preampii - preamplifiers Types 26 26 , 26 28

fier must be insert ed after each Type 26 34 is a smal l unit only and 26 50 provide both accurate

transd ucer in order to provide a high 21 mm * 34,5 mm * 100 mm sensitivity adjus tment and adjus-

input impedance to the transducer whi ch can be placed near the mea- table gain. Type 2650 has 4 digit

signal and a low output impedance suring point, and whi ch is operated adju stme nt compared to 3 digits for

to 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 frequency and Type 26 51 is especially intended vides very low frequ ency operation

low noise operation is made possi- for use wit h Uni- Gai n®c har ge cali- to 10 Hz.

bie. brated accelerometers such as

Types 43 71 and 4 37 0 , and gives In addi tion , both Type 26 26 and

Charge preamplif iers automat i- then a calibrated output propor- Type 26 28 provide adjustable low

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

signal.

Exciter Control

The Exciter Control Type 10 47 or displacement over the frequen cy the speci men in ques tio n. The most

contains 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 section to drive the controls external fitters and record- dyne Analyzer Type 20 10 or per-

Power Ampl ifi er in the frequency ing devices. haps the Sine Random Generator

range fro m 5 Hz to 10 kHz. It con- Type 10 27 . The forme r in addition

tain s one measurement and control Simi larly a numbe r of other ins tru - gives excel lent frequ ency analysis

channel, 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 vibration level on

13


13/18

14

Recording Device

The Level Recorders Types 2306

or 2307 may be used with precali-

brated recording paper as shown in

Figs.21 and 22. Recording papers

can have a 50 mm , or a 1 00 mm

ordinate (2307 only} which can be

used for logarithmic scales of 10,

25, 50 or 75 dB accordin g to the

Recorder Potentiometer used (50dB

being the most usual) or for a linearscale (Potentiometer ZR 0002 for

2307) which must be used for

phase recordings. The frequency

scale of Fig.2 1 has a decade lengt h

of 50 mm whi ch prov ides the possi

bility of drawing the skeletons as de

scribed in the theoretical section.

(The slopes being ±20dB for 1 de

cade or multiples of that).

To obtain better accuracy with

closely spaced resonances or antire-

sonances the recordings can also

be made with an enlarged frequency scale (either logarithmic or

linear). For the latter a redrawing

must be made to have log-log re

presentation for final evaluation.

Similarly X-Y recorders may be

used for recording of the graphs pro

vided either the recorder output

used (Heterodyne Analyzer Type

2010 or Measuring Amplifier Type

2607) has a logarithmic output, or

the X-Y recorder has a logarithmic

preamplifier.

If acceleration or displacement is

measured or used for control in

stead of velocity the graph can be

recalibrated to mobility or imped

ance by drawing a reference line on

the log-log graph and reading the

values above this line for each fre

quency. 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


14/18

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

15


15/18

As acce lera tion for sinuso idal vi - impedance lines are given wi th posi- the above relation ship or from the

bration 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 and

lines wi th negative slope The curves are draw n thr ough Whe n displace ment has been

20 dB /d ec ad e wou ld represent con- points at one frequency for whi ch measured the calibrat ion is carried

stant 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 measurements

The phase is best measu red by tion between the ir resonances or accelerat ion or veloci ty at a given

means of a phase mete r. For many anti resonanc es, it proves very use- point may be the resul t of cont ribu-

purposes however, phase monitor- ful to know the phases of the sys- tions f rom tran slator y mo tion and

ing may be suffic iently well carried tern responses. Whe n the reson- one or tw o torsio nal motion s, and

out by means of an oscilloscope on ances or antiresonances of highly phase measu rement wit h respect to

wh ic h either the beam is deflected 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 measure men ts does not by a phase meter is absolutely ne- wav ele ngt h of vib rat ion at a given

arise. From good recordi ngs of point cessary. freq uenc y may vary considerabl y

impedances and transfe r imped- fro m one part of the system to

ances it is possible to evaluate the Under many other circ umst ances anoth er and phase meas urem ents

response 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- tinodes 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 arrange

ment which was used to determine

the point impedances of the surface

of a hermetically sealed container

for a refrigerator compressor

(Fig.26). The purpose of the investi

gation was to find the optimal

points to secure the springs in

which the compressor was to be sus

pended, in order to reduce the

transmission of vibration to a minimum and thereby to minimize the

sound emitted from the container.

The measurements revealed that at

most points the impedance curve

showed numerous resonances and

antiresonan ces (see Fig27a) whe re

as four points had relatively high

impedance and very small varia

tions over the entire frequency

range (Fig.27b)."Y Fig.2 5. Measu ring arran gemen t suitable to dete rmine the point imped ance at variou s points of

the surface of a steel contai ner for a refrige rator compr essor

16


16/18

the modulus of elasticity of bars in

longitudinal vibration. The bar is

fixed onto the combination of a

Force Transducer Type 8200 and a

Standard Quartz acceierometer

Type 8305 and the point imped

ance curve is obtained (Fig. 2 9),

From the exact frequencies of reso

nance or antiresonance and the form

of the curve the modulus of elasticity

and the damping coefficient can be

obtained (see Ref.16). In Fig.30 theantiresonant peaks have been en

larged by using a reduced paper

speed and a 10d B potentiometer.

In Fig.31 is given an improved

version of the instrumentation used

for the measurements described in

Ref .7 . The measurements were

taken to obtain knowledge about

the response of a prefabricated

building structure by means of mo

bility measurements. The accelera

tion signal is integrated by the Con

ditioning Amplifier Type 2635 andled through the Tracking Filter Type

5716 to exclude ambient vibration

from nearby punch presses. The mo

bility recording obtained is shown in

Fig.32.

Further examples of applications

are given in the Briiel & Kjaer Appli-

cation Note No. 1 3—1 20 "Measure

ment of the Dynamic Properties of

Materials and Structures" (Ref.8).

Fig.28. Measurement configuration for de

termination of the modulus of elas

ticity of a PVC bar by impedance

measurements

17

f ig .27. Point impedance recordin gs obtained fro m measureme nts on the steel container , a) as

recorded 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 give

minimum transmission of vibration

energy when combined with the

low impedance of the springs and

were, therefore, selected as fixing

points. (There would be a high trans

fer impedance from the compressor

to the point; compare with the sys

tem of Fig.6 and with the inverse of

its mobility curve in Fig.7).

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


17/18

Concluding Remarks

Measur ement of the mechanic al

impedance, mobility or any other of

the complex dynamic ratios given

in Table 1 may provide useful know

ledge about a structure. By compar

ison of the obtained recordings with

lines for masslike or springlike re

sponse one will gain insight into the

dynamic characteristics of the struc

ture to aid in further development orin corrective measures.

The existing forces may be de

rived by comparison with e.g., the

mobility plot, and the need for cor

rection may often be directly de

cided upon, by frequency analysis

of signals generated during opera

tion.

in more complicated applications

the measurements may be used to

find the elements of a mode!. Then

comparison between calculated val

ues for the model and measurement

results may be used for correcting

the parameters of the model until

sufficient accuracy is obtained.

Similarly comparisons with fre

quency analyses may be used to cal

culate all relevant forces or couples

in the system.

In other cases the response of

subsystems may be measured and

the properties of the total system be

calculated before final assembly (See

Ref. 12).

18

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


18/18

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.55—69

9. 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. 42

4. 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 . 5—6 , 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 Booth, "In terchan 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 Improving 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., January

7.To rben Licht, "Meas ureme nt of 10— 14, 197 2

Low Level Vibrations in Build

ings" Bruel & Kjasr Technical

Review No. 3-1972

19

a guide to mechanical impedance and structural response techniques

Documents