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Reprinted From

THESHOCK AND VIBRATION

BULLETIN

Part 1Keynote Address, Invited Papers, Panel Sessions,

Modal Test and Analysis

September 1978

A Publication ofTHE SHOCK AND VIBRATION

INFORMATION CENTERNaval Research Laboratory, Washington, D.C.

Office ofThe Director of Defense

Research and Engineering

Approved Tor public release: distribution unlimited.

ON THE DISTRIBUTION OF SHAKER FORCES

IN MULTIPLE-SHAKER MODAL TESTING

W. L. Hallauer, Jr. and J. F. StaffordVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

The method proposed by Asher for structural dynamic modal testing by multiple-shaker sinusoidal excitation is reviewed, and its theory and application arediscussed in detail. Numerical results from simulated modal testing on mathe-matical structural models are presented to illustrate the strengths and weak-nesses of the method. The characteristics of these models include dampingwhich couples the normal modes and closely spaced modes. Numerical techniquesrequired for implementation of the method are described. A procedure is sug-gested for replacing actual mechanical tuning with calculations employingtransfer function 'data.

I. INTRODUCTION

Structural dynamic modal testing by meansof multiple-shaker sinusoidal excitation re-quires some method for specifying shaker forcesso as to isolate or tune individual modes ofvibration. Asher [1], expressing doubt thatother methods available or proposed at thetime (1958) were capable of separating closelyspaced modes, proposed a quantitative methodwhich both detects natural frequencies andprovides the force distributions for multiple-shaker tuning. The method uses only experi-mental transfer function data as input and, inprinciple, is capable of tuning any mode, re-gardless of the degree of modal density.

Asher was' unable to apply his method suc-cessfully in modal testing due to equipmentlimitations. Furthermore, he stated that imple-mentation of the method would require thedevelopment of data acquisition and processingequipment to handle experimental transfer func-tion data. These practical obstacles evidentlydiscouraged attempts to apply Asher's method inthe laboratory throughout the 1960's. Nonethe-less, the method was considered promising andwas discussed extensively by several authors,notably Bishop and Gladwell [2] and Craig andSu [3]. Finally, in 1974 the practical useful-ness of Asher's method was demonstrated byapplication in the MODALAB test facility, asdescribed by Smith et a! [4], Stroud et al [5],and Hamma et a! [6],

The objectives of this paper are (i) toreview and discuss in detail the theory under-lying Asher's method, (ii) to present numericalresults developed from computer simulation ofmodal testing on relatively simple structural

models, some of which have closely spaced modes,and (iii) to discuss some aspects of the prac-tical application of the method. Our computersimulation models are similar to those ofCraig and Su [3]; however, whereas they repre-sented structural damping as the proportionalor non-coupling type, we have accounted forthe more realistic type of damping which couplesthe normal modes of vibration.

The computer simulation results demonstrateinstances of both success and failure of Asher'smethod. In many cases, it detects all modes andprovides force distributions which effectivelyseparate the modes so that mode shapes may bemeasured accurately. However, it is also pos-sible that Asher's method may provide poortuning, miss modes, or even introduce falsemodes.

The distribution of excitation on a testspecimen encompasses both spatial distributionor position (location and direction) and force-amplitude distribution of the mechanicalshakers. However, Asher's method addressesdirectly only the problem of force-amplitudedistribution; the shaker positions must alreadyhave been selected before Asher's method isapplied. The positioning of exciters is usuallybased on the judgment and experience of testand analysis engineers. To date, very littlequantitative study has been devoted to thepositioning problem and, although important, itis not the primary concern of this paper.

II. THEORETICAL BACKGROUND

Consider a linear structural system havingn degrees of freedom, the time-dependent

49

responses of which make .up .the n x 1 columnmatrix x. (Notation_is_11s tedat the end of__

the paper.) The system is excited by a set ofsinusoidally varying forces having coherentphases, i.e., all having 0° or 180° phase,

f = F cos cot = Re {F e1a)t} (la)

It is assumed that all starting transientshave decayed away, so the response is steady-state sinusoidal,

x = Re .{X e1ut}

Response amplitude vector X is generally com-

plex, reflecting the facts that response is usual-ly not in phase with excitation and that eachresponse may have a different phase. Havingequations (la, b), we may treat the problem asone in which F is the input, X the output, andtheir linear relationship is completely definedby the n x n'transfer function matrix [H(to)],

X = [H(co)] F (2)

It will be necessary to distinguish betweenthe in-phase (0° or 180°) portion and the out-of-phase (±90°) portion of response in theforms, respectively, of the coincident-responsematrix [C(to}] and the quadrature-response matrix[Q(o>)] defined by

[HU)] = [C(co)] (3)

All structures considered in this paperare assumed to be discretized to n degrees offreedom, so that the governing matrix equationof motion for steady-state sinusoidal excita-tion and response is

[m]x + [c(u)]x + [k]x = F cos ut (4)

System inertia, damping and stiffness matrices,[m], [c(u)] and [k] respectively, are assumedto be real,'• symmetric, and positive-definite.The representation of damping by an arbitrarilyfrequency-dependent term is rather general. Inparticular, it encompasses any combination ofthe standard viscous and hysteretic types; if[b] and [d] are, respectively, the constantviscous and hysteretic damping matrices, then[c(co)] = [b] + l/o) [d]. Damping matrix [c(o>)]is not necessarily diagonalized by the standardnormal mode transformation; i.e., if [j] is themodal matrix of normal modes, then [^l^tclCiti]is not necessarily diagonal. If the product isnot diagonal, then [c] is called a couplingdamping matrix since it couples the otherwiseuncoupled normal equations of motion.

The frequency response solution for arbi-trary linear damping [c(oi)] has been estab-lished by application of the characteristicphase-lag theory of Fraeijs de Veubeke [7,8].The summary here follows primarily the develop-ments in English by Bishop and Gladwell [2] andBishop et al [9].

To solve equation (4) for' frequency re-sponse, references_F2'LancLr-9-1-deve1op-a-frequency-dependent eigenvalue problem,

2<o[c(w)]i|i - tan e ([k] - u [m])ij( = 0 (5a)

At each driving frequency to there are n'eigen-solutions, each consisting of real mode shapei|>r(i») and real phase-lag angle 9 U), which is

calculated from the actual eigenvalue tan e .

With these quantities known, the associatedforce-amplitude vectors r (to), r = 1 ,2 , ...,n, are calculated from ~

r = cos e ([k] -co [m]) 4> + co sin e [c(u>)] 41 (5b)

Each vector r (co) is that amplitude distribu-

tion of coherently phased forces,

f = r cos tot

which produces responses in all degrees offreedom having the same phase lag e ,

x = 41 cos (cot - e )

Solutions 4) (u), e (co) and r (to) may be termed

collectivefy the rth frequency-dependent "char-acteristic phase-lag mode".

Equation (5a) is valid provided e f 5-.

However, if e = % for the sth mode, then

equations (5a,b) reduce, respectively, to

( [k] - c o [ m ] )r = to [c(to)]

= 0 (6a)

(6b)

The only solution to equation (6a) is thenormal mode solution. Hence if co = u , the

sth undamped natural frequency, then the sthcharacteristic phase-lag mode shape becomesthe sth normal mode shape,

IS(7a)

and the responses of all degrees of freedomin that mode are tin quadrature with the excita-tion,

(7b)

and, finally, the excitation has the particularform,

[c((os)] (7c)

The characteristic phase-lag modes repre-sent a specific type of forced response, sothey do not give directly an equation of thetype (2) expressing response due to arbitraryexcitation, from which [H(u)] may be written.However, at each excitation frequency the setof vectors r , r = 1, 2, ..., n, is linearlyindependent, so any arbitrary force amplitudedistribution F of coherently phased forces may

50

be expressed as a linear sum of the r ,

By using this expansion, orthogonality condi- ',tions which are easily derived from equations(5a,b), and the principle of superposition forlinear systems, references [2] and [9] derivethe frequency response transfer function matrixin the form

S ( R r + « r > (8a)

where

(8b)

[m])

It is worth repeating for emphasis that themodal vectors t|> are frequency-dependent, quiteunlike modal vectors for normal or complexmodes.- Equation (8a) applies for almost anytype of linear damping that one might wish toconsider, so it is quite useful for discussionof the nature of solutions. However, it issomewhat impractical for numerical studies be-cause it requires the solution of "eigenvalueproblem (5a) at each excitation frequency ofinterest. None of the numerical results inthis paper have been calculated from equation(8a).

The coincident-response matrix defined by.equation (3) is of particular interest. Itmay be expressed either as a sum, from equation(8a),

(9a)

(9b)

'

or as a matrix product,

[CM] = [*] [R] [*]'

where [ij>] is the square modal matrix of ti vec-tors and [R] is the diagonal matrix whose kthdiagonal element is Rk- From equation (9b) thedeterminant of [C(u)] is

det [CM] = (det [*])2 n Rp (10)

An important conclusion may be inferredfrom equation (10), namely,

det [C(us)] = 0, s = 1, 2 n- (11)

That is, the coincident-response matrix is sin-gular at each undamped natural frequency. Thisis so because, from equations (6a), (7a), and(8b), BS(OJS) = 0 so RS(US) = 0. Another inter-pretation of this singularity is seen by expres-ing the kth column of the coincident-responsematrix as

(12)

where i|/. is the kth element of y . When u fa) , r•= 1, 2, ..., n, each column C, is a sumof the n linearly independent vectors i|» , sothat [C] is not singular. However, when ta =u , the sth undamped natural frequency, thenRS(O>S) = 0 so equation (12) becomes

n

' S8' (13)

In this case, each column C^ is a sum of onlyn - 1 1 inearly independent n-dimensional vec-tors, so the n columns of [C(u )] are linearlydependent and the matrix is singular.

Another important conclusion is

0. s= l , 2, ....CC(«S)] rs(us) (14)

This can be proved by substituting equations(9a) and (7c) into the left-hand side of (14),applying the following orthogonality conditionderived in references [2] and [9],

[c(iii)] ,(/s 0 for r

and, finally, using R (u ) = 0.

In view of results (11) and (14), theequation

[CM]F = 0 (15)

may be regarded as a non-standard eigenvalueproblem having n eigensolutions. The rth eigen-value is the rth undamped natural frequency,and the associated eigenvector is the amplitudedistribution of coherently phased forces re-quired to tune exactly the rth undamped normalmode in quadrature phase with the excitation.

Asher [1] first stated results (11) and(14) and proposed the interpretation of equa-tion (15) as an eigenvalue problem. Bishopand Gladwell [2] rigorously established themathematical validity of those results. Craigand Su [3] established the same conclusionswith a short and direct derivation. We havepresented this review rather than simply re-stating the conclusions for the following rea-sons: to provide a summary for readers un-familiar with the characteristic phase-lag

51

theory; to extend the scope of Bishop and Glad-j«tllLjj5_4ieS-U.l-ts_to_i.nc.lude_a^inore-geneFa-l—form—

circle on a co-quad plot in the complex

of damping (They regarded their results .asapplicable only for viscous or hysteretic damp-ing, but the extension is almost trivial..);finally, to represent the coincident-responsematrix in the new forms (9b) and (12), whichexpedite interpretation of results.

III. . ASHER'S METHOD

. Asher's method is. based on the theoreticalresults contained in equations. (11). and (14).The method can be described with reference tothose equations in the context of an idealizedmodal test on a discrete n-degree-of- freedom .linear structure with arbitrary linear struc- ,tural damping. The test begins with measurementof the .n x n frequency response transfer func-tion [H(o))] over the entire modal spectrum ofthe. structure. This' data may be assembled byapplying excitation and measuring response ateach degree of freedom,, with the use of eitherincremental sine-sweep excitation for directmeasurement or broadband excitation and calcula-tion by digital time series analysis. (SeeHamma et al [6] and Brown e ta ! [10] for con-trasting views on the merits of these two test-ing methods.) Next, the coincident-responsematrix [C(u)] is extracted as the real part of[H(u)], and det [C(u)] is plotted over theentire modal spectrum. Exactly n zero cross-ings of det [C(u)] are found, and the frequen-cies at which they occur are, from equation (11),the undamped natural frequencies of the struc-ture, even though damping is present. Next,the relative force-amplitude distributions fortuning all modes are determined as the solutionsr (u 5, s = 1, 2, ..., n, of equation (14).Each vector r (<D ) is equal to any one of theidentical (to within constant multiples) columnsof the adjoint matrix adj [C(u )].. This rela-tively simple solution procedure is predicatedon the availability of [C] at the exact zerosof det [C]; otherwise, interpolation is required.Finally, each of the n modes is tuned exactlyin a multiple-shaker sine dwell. For the sthmode, the shakers are set .to produce the forces

f = r (o) ) cos a) t.

Then the response is measured to be

x = cos

The response-amplitude distribution $ is the •exact mode shape of the sth undamped normalmode, again, even though the structure isdamped..

A short digression on the tuning step isin order. Normally a narrow-band sine sweepabout a) with fixed force-amplitude distribu-tion is more useful than a sine dwell. Ifdamping is hysteretic and non-coupling, thenthe locus of each complex response is a perfect

-fronrwhictrmodal^damping as well as the modeshape component may be determined (Kennedy andPancu [11]). Moreover, if damping is viscousand non-coupling, the co-quad plot is stillnearly circular. Even if damping couples thenormal modes, a co-quad plot is useful forindicating the presence of a true mode; how-ever, as shown by the simulation study resultsof Section V, damping coupling of closelyspaced modes may give co-quad plots which aresomewhat difficult to interpret in light of'ourexpectations based on non-coupling damping.

Returning to Asher's method, we may statethat in principle it is capable of measuringexactly every undamped natural frequency andthe associated real mode shape of an ideal dis-crete structure with n degrees of freedom andlinear damping. The method uses only experi-mental data as input; no estimated data such asa mass matrix is required.

Actual modal testing of real hardware is,of course, much more complicated than theidealized, testing of a perfect discrete struc-ture described above. The practical realitiesare that any continuous structure has an in-definitely large number of degrees of freedom,and that sensing and exciting equipment can beapplied at only a relatively small number oflocations on the structure. It is feasibletherefore to measure only a p x p incompletecoincident-response matrix [C*(u)], wherep < n. It is clearly not possible to satisfythe exact equations (11) and (14), so the mostreasonable alternative course for practicalapplication of Asher 's method is to seek usefulsolutions to the analogous equations,

det [C*(u)] '= 0[C*(u)] r* = 0

(16a)(16b)

where r* is a p x 1 vector of force amplitudescorresponding to the p available shaker loca-tions. As we shall .demonstrate with the simula-tion study results, use of equations (16a,b)can lead to the following possible consequences,listed in order of decreasing desirability:(i) a zero of det [C*] may occur close, to butnot exactly at a natural frequency, and theforce vector r* will effectively tune thecorrespond ing~mode; (ii) a zero of det [C*]may occur close to a natural frequency, butthe vector r* will not adequately isolate themode from interfering modes; (iii) a zero ofdet [C*] may be totally unrelated to any of thenatural frequencies; (iv) det [C*] may have nozero in the vicinity of a natural frequency. •

Mathematical justification is quite limit-ed for the proposition that the solutions ofequations (16a,b) will represent accurate nat-ural frequencies and effective tuning force-amplitude distributions. With the definitionof the characteristic phase-lag p x 1 trun-cated modal vectors r = 1 ,?, n, and

5Z

the corresponding p x n modal matrix [i|<*],equation (9b) gives

The degrees of freedom deleted from each <i>r toform i>r* are, of course, the degrees of freedommissing from [C*]. Since [i|/ *] is not square,it is not possible to write a simple equationfor.det [C*] analogous to equation (10) fordet;[C]. Hence, it is difficult if not im-possible to predict, in a precise mathematicalsense, how closely the zeros of det [C*] appro-ximate the true natural frequencies. It isperhaps even more difficult to assess mathe-matically the tuning effectiveness of a force-amplitude solution r* to equation (16b). Bishopand -Gladwell [2] stated without proof severalinteresting results regarding the numbers ofzeros of det [C*] and their lower and upperbounds (u, and u ). These results are possiblythe most precise mathematical statements thatcan be made regarding the solutions of equations(16a,b).

Despite the absence of firm mathematicaljustification, application of equations (16a,b)often leads to surprisingly good results. Thisfrequent success can be partially explained,at least physically if not mathematically, onthe basis of the following result from equations(7): forced vibration in a pure normal mode atits undamped natural frequency is characterizedby all response being in quadrature phase withthe excitation. The physical consequence ofequations (16a,b) being satisfied at some fre-quency oo' is that the p degrees of freedomrepresented in [C*] are constrained to respondexactly in quadrature phase with the p forces.The n - p unforced degrees of freedom are, ofcourse, not so constrained. But if u1 is closeto a natural frequency and if the p shakersare located at positions appropriate to tunethe mode, then it is reasonable to expect, andit often happens, that the unforced degrees offreedom having significant modal energy will"cooperate" by responding nearly in quadraturephase with the forces. In such a case, theentire structure vibrates in a close approxima-tion to the true normal mode.

In discussing equations (16a,b), we haveassumed that the p response locations areidentical to the p excitation locations. How-ever, in the spirit of the quadrature-responseexplanation given above, it is permissiblethat the response and excitation locationsdiffer. Stroud et al [5] and Ibanez [12] madethis observation and, in addition, describedapproaches for which the number of responsesmeasured need not equal the number of exciters.

IV. COUPLING DAMPING MATRICES

Although most analytical studies usingstandard modal analysis assume that damping

does not couple the normal modes, there seemsto be no reason to expect that the dissipativemechanisms of the actual structures should havethis very special and desirable characteristic.Hence, damping matrices which couple the normalmodes have been used in calculations of theresults to be presented in Section V.

But how does one specify numerically theelements of coupling damping matrix [c], whichis required for calculation of the frequencyresponse solution? Since structural damping isalways dissipative and since dashpot elementsare analogous to spring elements in mechanicalmodels, it seems reasonable to expect that [c]should be symmetric and positive-definite.Beyond that, however, there is apparently noinformation available to guide one in selectingphysically reasonable element values for damp-ing matrices of continuous structures. Wetherefore have used the concept of a coupledmodal damping matrix with unit diagonal elements(Hasselman [13]) to derive an equation for cal-culating a coupling damping matrix. The equa-tion is based on an assumption about the natureof damping rather than on actual measurementsof damping, so the resulting matrices may beregarded as plausible but not necessarilyphysically representative.

For a damping matrix [c ] which does notcouple the undamped normal modes, we have bydefinition

[cn] = [G]

where [G] is the diagonal generalized dampingmatrix. For example, if damping is a combina-tion of non-coupling viscous and hysteretictypes, then

so that

[cn] - [b] + [d]

[G] = [BJ + 1 CD] (17)

where the rth diagonal elements are calculatedfrom

Br = 2 Mr <

°r = Mr 4 (18)

Mr = [m]

and r, and g are, respectively, the standardviscous damping ratio and hysteretic dampingconstant, for the rth mode. Although [G] isdiagonal for non-coupling damping, in generalthe modal vectors are not normalized so as tomake it the unit or identity matrix. However,if the modal vectors are re-normalized in theform

1 = *r. r = 1, 2 n

53

.or

then there follows

i-7-it [cni [*] = (20)

We may regard the'unit matrix in this equationas the uncoupled unit modal damping matrix.

Consider now the inverse problem, namely,'calculating the damping matrix given the un-damped normal modes and the modal damping con-stants, which would normally be e and/or g ,

r = 1, 2, ..., n. In this case, [G] is calcu-lated from equations (17) and (18), and equa:,tion (20) is post- and pre-multiplied by [$]"and its transpose, respectively, to give

[cnJ=[V]<

where

[V] =.[*]-1

(21)

(22)

Equation (21) by itself has no particular valuesince a damping matrix is certainly not re-quired for a frequency response solution ifdamping does not couple the normal modes andif modal data, including structural dampingconstants, are given. However, the form ofequation (21) does provide the model for calcu-lation of a plausible coupling damping matrix. -

If damping couples the normal modes, thenthe modal damping matrix will certainly not bethe unit matrix of equation (20). It is reason-able to expect, though, that the modal dampingmatrix should have the appearance of a properlynormalized experimental mass orthogonalitymatrix (See, for example, Stroud et al [5].);that is, the coupled modal damping matrix, de-noted [lc], should be symmetric and should haveunit diagonal elements, with at least some non-zero Off -diagonal elements to represent coupl-ing between modes. Then, by analogy with equa-tion (21), we define the physical couplingdamping matrix associated with the degree ofmodal coupling in [Ic] and with the uncoupledmodal damping in a given [G] to be

[c] = [V] (23)

where [V] is evaluated from equations (17), (18)and (22) . We have no rational basis at presentfor choosing the off-diagonal elements of [lc].Our usual procedure is to specify positivenumbers less than 1 and then judge our choiceto be satisfactory or unsatisfactory on thebasis of the physical plausibility of the con-sequent frequency response calculations.

Methods are available for calculating thefrequency response of a system with couplingdamping which is either purely viscous or purelyhysteretic. For viscous damping, we have used

the solution of Foss [14] based on a complexeigenva_lye_pj^oblem devised by Frazenjyyjl^n Si-Mi ramand et a! [16] recently presented an effi-cient computational form of this solution. Forhysteretic damping, we have used the solutionestablished by Bishop and Johnson [17].

V. NUMERICAL SIMULATION OF MODAL TESTING

• Simulation studies with mathematical modelshaving non-coupling damping were conductedpreviously by Bishop and Gladwell [2], Craigand Su [3], and Stafford [18] (an unpublishedthesis). Bishop and Gladwell examined onlythe zeros of det [C*] and demonstrated theoccurrence of a good mode indication (a zerovery close to a natural frequency), a falsemode indication (a zero distant from all natu-ral frequencies), and a missed mode (no zeroin the vicinity of a natural frequency). In amore detailed study, Craig and Su examined thezeros of det [C*] and the quality of sine dwelltuning achieved by r* force-amplitude distribu-

tions. Stafford included plots of det [C*(<»)]and examined the quality of narrow-band sine-sweep tuning achieved by r* distributions;

tuning quality was demonstrated by both co-quad response plots and modal energy calcula-tions.

Although not included in previously pub-lished simulation studies, plots of det [C*(u)]are quire informative, certainly much more sothan listings of zero-crossing frequencies. Afew determinant plots calculated from experi-mental data appear in the papers by Smith et al[4] and Stroud et al [5]. But those plotsdemonstrate only good mode indications. Sincefalse mode indications and missed modes alsoexhibit informative characteristic forms oncoincident-response determinant plots, severalexamples of such plots are included in thepresent results.

All results presented here have been cal- ,culated for coupling damping. Both hystereticand viscous damping were considered, but thereappear to be no distinguishing differences inthe determinant or response plots. The effectsof substantial coupling appear very clearly onco-quad response plots, which also indicatethe quality of modal tuning.

A particular advantage of computer simula-tion of modal testing is that the parametersof the mathematical structural model are knownprecisely and can even be designed to producedesired characteristics such as high modaldensity and nodes at certain degrees of free-dom. Hence, an absolute standard is availableagainst which to judge the simulated test re-sults and, furthermore, failures of the testmethod can often be attributed directly tosome specific cause. The models described andthe results presented here have been selectedto demonstrate the consequences of applyingAsher's method in situations which are typical

54

of actual hardware testing; our expectation isthat these results will provide guidance forthe interpretation of real data.

V.I MODELS 5H AND 5V

These two models are structurally iden-tical but have different damping matrices. Thestructure, a cantilevered plane grid with fivedegrees of freedom, is illustrated in Figure 1.It is similar to Craig and Su's [3] model struc-ture. Five discrete masses are located in thea - B plane at rigid right-angle joints con-necting the bars, and the degrees of freedomare vertical (y) translations of the masses.The mass values, which are the elements of thediagonal mass matrix, are listed as follows inorder from 1 to 5: 1.259 kg (0.007191 lb-sec2/in), 1.385 (0.007911), 2.096 (0.01197), 0.3447(0.001968), 1.613 (0.009213). (All calcula-tions for models 5H and 5V were made in theprimary units of pounds, inches and seconds.)The eight identical bars are assumed masslessand flexible only in vertical bending and tor-sion. Each bar has 0.254 m (10.0 inch) lengthand 6.35 mm (0.25 inch) diameter. Bar materialhas Young's modulus E = 68.9 GPa (10.0 x 106

psi) and shear modulus G = 27.6 GPa (4.0 x 105

psi).

Fig. 1 Structure of Models 5H and 5V

The undamped normal mode solution is list-ed in Table 1. The structure was designed tohave its third and fourth natural frequencieswithin 4% of each other by means of an optimi-zation procedure (Hallauer et a! [19]). Damp-ing values gp and t for the third and fourth

modes were selected (see below) so as to makethese modes closely spaced in the sense thatthey be separated by less than one half-powerbandwidth.

Coupling damping matrices for these modelswere calculated from equation (23). Model 5Hhas hysteretic damping, and its [d] matrix wascalculated from a [lc] matrix with all off-diagonal elements equal to 0.15. The hyster-etic modal damping values g , listed in order

from 1 to 5, are 0.025, 0.035, 0.05, 0.04, 0.02.Model 5V has viscous damping with ? = 0.025

for each mode, and its [b] matrix was calculatedfrom the following coupled modal damping matrix:

1.0 0.31.0

SYM

0.10.31.0

0.00.10.3

0.00.00.10.31.0 J

V.2 MODEL 11H

This model consists of a beam subject tovertical bending and torsion and an attachedspring-mass oscillator. It is illustrated inFigure 2. A massless flexible beam of lengthi is attached to supports at points A and B.These supports provide boundary restraint suchthat the beam is simply-supported relative tobending in the y direction and clamped relativeto torsion. Discrete masses 1 - 5 are uniform-ly spaced along the beam at intervals of t/6;masses 6 - 1 0 are suspended from the beam inthe o - 6 plane by rigid massless links, eachhaving length e; mass 11 is suspended vertical-ly from mass 4 by a spring with stiffness con-stant k. The eleven degrees of freedom arevertical (y) translations of the masses. Fornumerical evaluation, the equations describing

TABLE 1

Undamped normal modes of models 5H and 5V

1

(rad/sec)

10.908 32.055 53.500 55.409 94.924

(mass)

(D(2)(3)(4)(5)

0.1260.2320.3370.7731.000

0.9430.479

-0.1651.000

-0.281

-0.380-0.075-0.7031.0000.195

-0.5950.4780.3201.000

-0.342

0.121-0.712

0.2171.000

-0.130

55

this model were written in dimensionless form.The dimensionless mass_yaUies, stated_re.la.t.1.v.e_

-to-the~summed^tota1~M of the beam and link

masses, are 0.18 for masses 1 - 5, 0.02 formasses 6 - 10, and 0.07 for mass 11. The other

dimensionless parameters are y = / 0.1, FT =

V.3 RESULTS AND DISCUSSION

0.01 and k.3

IT = 5.5, in which El and GJ are

beam bending and torsional stiffnesses, respec-tively.

Fig. 2 Structure of Model 11H

The first five normal modes of modelare listed in Table 2. The dimensionlessnatural frequencies are defined as

11 H

The hysteretic damping matrix for thismodel was calculated from equation (23) with a[Ic] matrix having all off-diagonal elementsequal to 0.2 and with g for each mode equal

to 0.02.

Coincident-response determinant plots formodel 5H are shown on Figures 3a, b, c, in solidcurves for all masses excited and in dashedcurves for only masses 1 and 5 excited. In thevicinity of natural frequencies, these curvesusually have the form'typical of the coincidentcomponent of a transfer function. Each curveis normalized to the determinant having largestabsolute value in the frequency range of thegraph. With this normalization scheme, it istypical that the lowest mode dominates andhigher modes may be lost within the amplituderesolution of the graph. Thus the 94.9 rad/secmode is not evident on Figure 3a, so a re-normalized expanded-scale plot, Figure 3b, mustbe made. (A simple way to indicate on Figure3a at least the zero crossing of the high modeis to plot -the reciprocal of the determinant,which for data at discrete frequencies wouldhave finite positive and negative peaks adja-cent to the crossing.) The det [C] curves onFigures 3a, b have zero crossings exactly atthe undamped natural frequencies, in accordancewith theory.

Figure 4 represents the consequences ofapplying Asher's method with complete excita-tion (i.e., all masses forced) to isolate thehigher of the two close modes of model 5H. Theriprmalized tuning force-amplitude distributionF (= r in this case) was calculated from

[C(io,)]r = 0. This distribution was then applied

to the model in a 3 rad/sec narrow-band sinesweep to_determine the co-quad curves of com-pliance X. of each mass. (X, was omitted to

reduce plot clutter.) Conjugates were plottedwhere appropriate to reduce graph size. Thestarting frequency in the sweep is indicated

TABLE 2

First five undamped normal modes of model 11H

1

7.366 8.870 11.456 16.969 24.215

:r

(mass)

(D(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)

0.1550.2700.3160.2780.1610.4960.8621.0000.8710:5040.892

0.0090.0130.0090.001

-0.0010.5070.8741.0000.8580.493

-0.984

-0.335-0.575-0.655-0.558-0.3200.486

• 0.8511.0000.8820.5140.828

0.0270.026

-0.002-0.029-0.0281.0001.0000.001

-0.999-0.9990.011

0.0100.000

-0.0100.0000.01010

-1

000000000

0.0001.0000.000

56

•• 1.0

-i.o5.00 20.83 36.66 52.50 . 68.33 84.16 100.00

• u(rad/sec)

90.00 11.66 93.33 9^.00

u (rad/sec)

96.66 «.33 100.00

1.0

E

I

-1.0

(c)

Masses 1. 5 farced

40.00 43.33 46.66 50.00u trad/sec)

53.33 56.66 60.00

Fig. 3 Determinant plots for complete andincomplete excitation of model 5H:(a) 5-100 rad/sec, (b) 90-100 rad/sec,(c) 40-60 rad/sec

by an arrow for each response, and tic marks onresponse curves indicate increments of equalfrequency. The test mode shape vector <rf was

calculated from the quadrature components ofresponse at the determinant zero-crossing fre-quency; in this case it is the exact mode shape,in accordance with theory.

If damping had been non-coupling, eachcurve of the co-quad plot would be an arc of aperfect circle centered on the imaginary axiswith its highest point passing through theorigin (Kennedy and Pancu [11]). But coupling

\ Re fX . )

-4

1.000-0.705-0.743-0.7080.775

-0.5950.4780.3201.000

-0.342

o Central Frequency: 55.409 rad/secFrequency Increment: 0.150 rad/sec

Response Scale: 2.86 ^ (0.50 {£-) per divisionN ID

Fig. 4 Perfect tuning with complete excitation:co-quad plot for the fourth mode ofmodel 5H with all masses forced in a3 rad/sec narrow-band sine sweep

J ' '. '

damping causes deviation from circular shape.In this case, coupling introduces obvious inter-ference from the lower of-the closely spacedmodes. It should be noted that for couplingdamping, the theory of Section II guaranteesperfect mode tuning only at the natural fre- ,quency, with response in quadrature phase re-,lative to excitation, even though all degreesof freedom are excited.

It is not unusual for the tuning force-amplitude distributions given by Asher'smethod to defy physical intuition. For example,the fundamental mode of model 5H has a typicalfirst bending mode shape (Table 1) with allresponses in phase, so we might expect that alltuning forces should also be in phase. But thenormalized'force vector for perfect tuning is

F =

-0.214-0.125

0.1331.000

L 0.431

The explanation for this anomaly is that exciterforces appropriate for tuning a mode must canceldamping forces, which are not generally distri-buted spatially in accordance with mode shape.

The det [C*] dashed curves of Figures 3a,b, c exhibit good mode crossings at 10.909,32.070 and 55.327 rad/sec, very close to thenatural frequencies of the first, second andfourth modes, respectively. There is also afalse mode crossing at 44.524 rad/sec. It ischaracteristic of lightly damped structuresthat good mode crossings have steep slopes andfalse mode crossings have relatively gentle

57

slopes. Hence, with the use of a determinantplot, distinguishing between good and false

-mode-indications"!s usually a very easy matter.

Figures 3b, c show clearly that this det[C*] curve misses the third and fifth modes.But note that the curve has a relatively steepslope in the neighborhood of the natural fre-quency of each missed mode. This is anotheruseful characteristic for lightly damped struc-tures from which one can infer the presence ofa mode even though there is no zero crossing.

Figure 5 is an example of the excellenttuning with incomplete excitation often pro-vided by Asher's method. This tuning cor-responds to the fourth-mode zero crossing at55.327 rad/sec on Figures 3a, c for masses 1and 5 forced. The 5 x 1 vector F* on thisfigure is derived from the 2x1 r* vector, withzeros added for the unforced masses.

1.0

Re(X.)

«*•

0.9680.00.00.01.000

"-0.5%"0.4780.3211.000

-0.342

o Central Frequency: 55.409 rad/sec£ Zero-Crossing frequency .of det [c*]: 55.327 rad/sec

Frequency Increment: 0.150 rad/sec

Response Scale: 1.43 ^(0.25 £) per division

Fig. 5 Excellent tuning with incomplete exci-tation: co-quad plot for the fourth :

mode of model 5H with masses 1 and 5forced in a 3 rad/sec narrow-band sine'sweep

Coincident-response determinant curves formode! 5V are shown on Figure 6. For comparisonwith model 5H response on Figure 4, the force-amplitude distribution and narrow-band responsefor perfect tuning of the fourth mode of model5V are shown on Figure 7. Greater coupling isevident between the third and fourth modes ofmodel 5V because the appropriate damping cou-pling element is 0.3, twice that of model 5H.

For masses 1, 3 and 5 excited, Figure 6exhibits a false mode crossing at 44.764 rad/sec. The co-quad curves for a narrow-bandsweep about that frequency with the appropri-ate force-amplitude distribution are shown in -

-LO5.00 15.00 25.00 35.00 45.00 55.00 65.00

vlrtdfcec) .

Fig. 6 Determinant plots for complete and in-complete excitation of model 5V

Re'{-X,>

1.000-0.326-0.584-0.8380.794

-0.5950.4780.3201.000

-0.342

o Central Frequency: 55.409 rad/secFrequency Increment: 0.30 rad/sec

Response Scale: 2.28 ^ (0.401) per division

Fig. 7 Perfect'tuning with complete excita-tion: co-quad plot for the fourthmode of model 5V with all massesforced in a 3 rad/sec narrow-bandsine sweep

Figure 8. That response clearly bears no re-semblance to a mode.

Good but not excellent tuning of the thirdmode of model 5V with incomplete excitation isillustrated by the co-quad curves of Figure 9.This tuning corresponds to the 53.395 rad/seczero crossing on Figure 6 for masses 1, 3 and5 forced.

Mode shapes 2, 4 and 5 of model 11H list-ed on Table 2 have many node points on or neardegrees of freedom. Hence, this model is use-ful for simulating the consequences of position-ing shakers in nodal regions. Coincident-response determinant curves for model 11H areshown on Figure 10 for three different shaker

58

configurations.

• r= — T^T- ^ ' ' Re < xi>x /Cr9-^ i-

. . *3 TlS. '

r-'

1.0000.00.1540.00.724

o Central Frequency: 44.7M rad/sec:

Frequency Increment: 0.45 rad/sec

Response Scale: 0.0286 ̂ (0.0050^I per divisionN , ID

Fig. 8 Attempted tuning of a false mode: co-quad plot for model 5V with masses 1,3, 5 forced in'.a narrow-band sine sweep.Responses of masses 2 and 4 are far offscale.

ImfX.)

Masses 1, 3. 5 forcedMasses 1, 3. 5, 7 forced

8.00

Fig. 10

22.16 25.00

Determinant plots for,complete and in-complete excitation of model 11H. Thefrequency range of this graph excludesthe first natural frequency.

(but still indicates-their presence) and has acrossing at frequency 8.868.corresponding tomode 2 which one would normally dismiss as afalse mode crossing due to its gentle slope.The dominant responses in the narrow-band sweepassociated with the 8.868 crossing are shown onFigure 11. The presence of a mode is evidentin that co-quad plot, but tuning is clearlyvery poor.

F.ig. 9

o Central Frequency: 53.395 rad/secFrequency Increment: 0.25 rad/sec

Response Scale: 1.14 ~ (0.20 ^) per division

Good tuning with incomplete excitation:co-quad plot for the third mode ofmodel 5V with masses 1, 3, 5 forced ina 3 rad/sec narrow-band sine sweep

.Masses 1, 3 and 5 are essentially nodepoints for modes 2, 4 and 5. It is not surpris-ing, therefore, that excitation at those massesonly results in a determinant curve which lackszero crossings corresponding to modes 4 and 5

-.10

•KAf RefX.)

o Central Frequency: 8.868Frequency I ncrement: 0.02Response Scale: 0.025 per division

Fig. 11 Poor tuning with incomplete excitation:co-quad plot for the second mode ofmodel 11H with masses 1, 3, 5 forcedin a narrow-band sine sweep

When another exciter is added at mass 7,a good crossing results for mode 4, from which .one can obtain excellent tuning. Oh the otherhand, mode 2 is missed entirely, even thoughits-mode shape has a large component at mass ,7.Mode 5 is also missed, but this is to be ex-pected since all excitation is in nodal regions.

59

By comparing missed modes on the determi-nant plots of Figure 3b and 10 with the associ--ated-mode-shapes-l-isted-in-Tables^l-and-27~wemay infer the following rule of thumb: thepresence of an obvious.missed mode on a deter-minant plot indicates that the mode cannot betuned' with the given shakers (at"least not inaccordance with Asher's method) because' too"many of the shakers are positioned in nodalregions. Upon encountering such a situationin modal testing, one would be well advised totry a different shaker configuration. It ap-pears that:,having some shakers in nodal regionsmay be helpful, or at least not harmful,, fortuning, but clearly there must also be othershakers in regions of appreciable motion.

A circumstance not illustrated by ourexamples but which occurs occasionally is shownon Figure 12. • ;

Fig. 12 Typical poor zero crossing,

The determinant curve will have a zero crossingat a) , which from the curve shape is obviouslya.' * r • • .different than the natural frequency <o. . Theforce-amplitude vector resulting from the cross-ings. at>, will probably provide rather poortuning. Again, a different shaker configure-,,tion may.be required; , . ,

( . .Finally,,?we note that higher structural

damping manifests itself on coincident-responsedeterminant plots in the form of; gentler zero-crossing slopes, as illustrated by Figure 13. .As a consequence, it may be difficult to dis-tinguish between good and false mode crossingson the determinant plot for a heavily dampedstructure. - . ;

VI. NOTES ON APPLICATION OF ASHER'S METHOD

For determinant evaluation, we have found 5

the method of Crout (Hildebrand [20]), whichemploys triangular matrix decomposition, to bequite satisfactory. . The algorithm is easilyprogrammed,'and execution is fast for determi-nants of order'ten;or smaller. .

The usefulness of Asher's method is de- .pendent in part upon how well the calculatedforce-amplitude distributions are able-to tunemodes.. As discussed in Section-V, the methoditself fails in some cases due to its intrinsic

-1.05.00 15.00 25.00 35.00

u (rad/sec)

45.00 55.00 65.00

Fig. 13 Variation of determinant plots <withdamping level for model 5V with?„ = 0;025 and 0.05 in all modes

limitations. Another possible source 'of failurein applications is numerical inaccuracy in solu-tions of equations (16a, b). Accurate solutionof equation (16b) for r* is easy provided that, •p - » •[C*] is indeed.singular. Hence, it is essen-tial to be able to'determine the value offrequency u1 for which :D = det;[C*] = 0. and,furthermore, to have accurate values-for eachelement of [C*(u'>)•]. Our experience has beenthat errors in u' root solutions are amplifiedin the corresponding r* solutions.

If transfer function data is available inequation form, as•>i.n a numerical simulation,then accurate solution .is relatively easy. Wehave employed Newton's iterative method in theapproximate form . ' . .

(24)

The accuracy afforded, by this equation in purapplication is. limited only by computer pre-cision. • • • . ' .

On the other hand, if transfer functiondata is available only at discrete frequencies,then equation (24) is not immediately applica-ble. There are at least two approaches, inter-polation and curve fitting, that might, be usedin this circumstance. .

1. Interpolation. In this case, one should ' 'have available a graphics output displayof the determinant plot, such as thosereproduced in the paper by Stroud et al[5J. In the' neighborhood of the zerocrossing, that plot should resemble Figure14. An interpolation equation should bedeveloped for each transfer function using •the appropriate number of data points sur-rounding the determinant zero crossing, e.g.,two points for linear or four point forcubic interpolation. With the use of the

60

<u-a

Fig. 14 Typical determinant plot at discretefrequencies near a zero crossing.

interpolation equations, equation (24) maynow be applied to determine the crossingfrequency </, and -subsequently [C(u')] maybe calculated. This approach should givegood results if measured data is relativelyclean; however, less satisfactory resultscan be expected for noisy data in the,vicinity of the zero crossing. .

2. Curve Fitting. T,here are several. methodsfor fitting transfer function data, evennoisy data, to analytical equations. Afterall elements of [C*(u)] have been fit to ;

equations, then equation (24) may be ap-plied to calculate zero-crossing informa-tion. This approach obviously requiresaccurate curve fitting in the vicinity ofmodes. !

As is discussed in Section III, a narrow-band sine sweep in the tuning step of Asher'smethod is preferable to a sine dwell. However,in a multipl'e-shaker sine sweep'it is verydifficult, due to interaction of specimen andshaker dynamics, to maintain constant ampli-tudes and phases of shaker force outputs. Inview of this difficulty imposed by equipmentperformance, an acceptable alternative toactual mechanical tuning may be numericallysimulated tuning with the use of experimental;transfer function data and the excitation dis-tribution calculated by Asher's method. Forexample, if k fixed motion sensors and m exciterlocations are available on the specimen for amodal test, then one can measure with single-point excitation the corresponding k x mexperimental transfer function matrix [H (u)].

Square sub-matrices of the real part of[HeU)] are evaluated by Asher's method to

determine each test natural frequency u. and

the associated tuning force-amplitude m x 1vector F^ (which is the p x 1 vector r*, as

— J —previously defined, supplemented with m - l z e r oelements). Now a narrow-band sweep can be-simulated numerically to give calculated co-quad plots of the k responses in the jth testmode simply by evaluating

X(u,) = [He(u)]

in a narrow band about u - . This assumes either

that the frequency resolution of [H (u)] is

sufficiently fine to provide several points inthe narrow band, or that each element of [H (u)]

is represented continuously by a curve-fitequation. If the test specimen behaves linear-ly, then all of the information that one might "wish to determine about the k sensor responsesfrom a multiple-shaker sine sweep or dwell isalready contained in [H (a>)] . So a multiple-

shaker test is, in fact, redundant unless onewishes to determine additional information notavailable from the k sensors, such as an accu-rate map of nodal regions.

VII. CONCLUDING REMARKS .

Asher 's method for modal testing bymultiple-shaker sinusoidal excitation has beendescribed in detail. Modal testing has beensimulated numerically on mathematical struc-tural models to illustrate the types of resultsthat can be expected in practice. These re-sults demonstrate that the method may succeedor it may fail depending, evidently, on thenumber and locations of exciters. Finally,numerical techniques required for implementa-tion of Asher 's method have been discussed,and a numerical alternative to actual multiple-shaker mechanical tuning has been suggested.

In view of the existence of efficientmodern methods for characterizing structuralmodes by.curve fitting of transfer functiondata, it might be contended that multiple-shaker tuning is no longer necessary. This isa reasonable contention only if modal densityis low. As stated by Hamma et al [6] in theirconcluding remarks, curve fitting in frequencyregions of high modal density can lead to in-consistent results. All curve-fit methods arebased upon specific analytical models, all re-quire an estimate of the number of participat-ing modes, and most require at least initialestimates of some modal parameters. Hence,curve fitting is to some extent a subjectiveprocess, and for closely spaced modes it mightlead to incorrect modal parameters or it mighteven miss modes. Clearly, the most importantapplication of multiple-shaker testing byAsher 's method is in separating closely spacedmodes which cannot be adequately characterizedby curve-fit methods.

ACKNOWLEDGMENTS ' .

This work has been sponsored by NASALangley Research Center under Research GrantNSG 1276. A. G. Shostak calculated al l ' resul tsfor the model with viscous damping. The authorsmake no claim to originality for the previouslyunpublished ideas and concepts appearing inthis paper. For ideas related to practicalimplementation in particular, we are indebtedto the first author's former associates, R. C.Stroud, S-. Smith, G. A. Hamma, and R. C. Yee.Me wish to thank the reviewers for their

61

helpful comments. on Dynamics and Aeroelasticity, JLt-Jjorth,-I nst~of "Aeronautical Sci., 1958, pp.69-76.

BASIC NOTATION

Re { }Im { }

n*. <det [ ][C] = Re [H][C*]

Ff, F* .

[V]

xiX

[b]

f(t)

9r

[kl[m]n

Px( t )

T

[vl

[v]

[1C]

REFERENCES

column matrix .

real part of { }imaginary part of { }

transpose of [ ], ( )

complex conjugate of ( )determinant of [ ]coincident-response matrixincomplete coincident-responsematrixforce-amplitude vector

normalized F

constant defined by equation (8)

diagonal matrix of R 's

matrix defined by equations (17),(18), (22)complex response-amplitude vector

ith element of X

compl iance

viscous damping matrixstructural damping matirxhysteretic damping matrixtime-dependent force vector

hysteretic damping factor of therth mode

structural stiffness matrixstructural inertia matrixtotal number of degrees offreedomorder of [C*]time-dependent response vector

force-amplitude vector of rthcharacteristic phase-lag modetuning force-amplitude vector

viscous damping factor of therth modephase lag of rth characteristicphase-lag modemodal matrix of normal modesrth column of [<}>]

test mode shape

modal matrix of characteristicphase-lag modesrth column of [ijj]excitation frequencyrth undamped natural frequency

dimensionless frequency; seeSection V.2coupled modal damping matrix

Asher, G. W., "A Method of Normal ModeExcitation Utilizing Admittance Measure-ments", Proc. National Specialists' Meeting

2. Bishop, R. E. and Gladwell, G. M. L.,"An Investigation into the Theory ofResonance Testing", Phil. Trans, of theRoyal Society of London. Series A, Vol.255, Mathematical and Physical Sciences,1963, pp. 241-280.

3. Craig, R. R. and Su, Y. W. T., "OnMultiple Shaker Resonance Testing", AIAAJournal, Vol. 12, No. 7, 1974, pp. 924-93T

'4. Smith, S., Stroud, R. C., Hamma, G. A.,Hallauer, W. L. and Yee, R. C., "MODALAB-A Computerized Data Acquisition andAnalysis System for Structural DynamicTesting",, Proc. of the 21st InternationalInstrumentation Symposium, Vol. 12,Philadelphia, Instrument Society ofAmerica, 1975, pp. 183-189.

5. Stroud, R. C., Smith, S. and Hamma, G.A., "MODALAB-A New System for StructuralDynamic Testing", The Shock and VibrationBulletin, Bulletin 46, Part 5, 1976,pp. 153-175.

6. Hamma, G. A., Smith, S. and Stroud, R.C., "An Evaluation of Excitation andAnalysis Methods for Modal Testing", SAEPaper No. 760872, Aerospace Engineeringand Manufacturing Meeting, San Diego,1976.

7. Fraeijs de Veubeke, B., "Dephasagescaracteristiques et vibrations forceed'un systeme amorti", Academie Royalede Belgique, Bulletin de la Classe desSciences, 5e Serie-Tome XXXIV, 1948,pp. 626-641.

8. Fraeijs de Veubeke, B., "Comment on 'OnMultiple-Shaker Resonance Testing'",AIAA Journal. Vol 13, No. 5, 1975, pp.702-704.

9. Bishop, R. E. D., Gladwell, G. M. L. andMichaelson, S., The Matrix Analysis ofVibration, Cambridge University Press,1965, Chapter 5.

10. Brown, D., Carbon, G. and Ramsey, K.,"Survey of Excitation Tecniques Applica-ble to the Testing of Automotive Struc-tures", SAE Paper No. 770029, Interna-tional Automotive Engineering Congressand Exposition, Detroit, 1977.

11. Kennedy, C. C. and Pancu, C. D. P., "Useof Vectors in Vibration Measurement andAnalysis", J. Aeronautical Sci., Vol. 14,No. 11, 1947, pp. 603-625.

12. Ibanez, P., "Force Appropriation by Ex-tended Asher's Method", SAE Paper #760873,

Aerospace Engineering and ManufacturingMeeting, San Diego, 1976.

13. Hasselman, T. K., "Modal Coupling inLightly Damped Structures", AIM Journal,Vol. 14, No. 11, 1976, pp. 1627-1628.

14. Foss, K. A., "Co-Ordinates Which Uncouplethe Equations of Motion of Damped LinearDynamic Systems", Journal of AppliedMechanics. Vol. 25, 1958, pp. 361-364.

15. .' Frazer, R. A.vDuncan, W. J. and Collar,A. R., Elementary Matrices, CambridgeUniversity Press, 1946, p. 327.

16. Miramand, N., Billaud, J. F., Leleux,F. and Kernevez, J. P., "Identificationof Structural Modal Parameters byDynamic Tests at a Single Point", TheShock and Vibration Bulletin, Bulletin46, Part 5, 1976, pp. 197-212.

17. Bishop, R. E. D. and Johnson, D. C.,The Mechanics of Vibration, CambridgeUniversity Press, 1960.

18. Stafford, J. F., "A Numerical Simulationof Multiple-Shaker Modal Testing",Master's thesis, Virginia PolytechnicInstitute and State University, 1976.

19. Hallauer, W. L., Weisshaar, T. A. andShostak, A. G., "A Simple Method forDesigning Structural Models with CloselySpaced Modes of Vibration" submitted forpublication, 1978.

20. Hildebrand, F. B., Methods of AppliedMathematics, 2nd ed., Prentice-Hall,1965.

63