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Produced by the NASA Center for Aerospace Information (CASI)
OP. 

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Bureau of Engineering Research

The University of Texas at Austin

Ai o stin, ax-

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Report CAR 83-2

MULTISHAKER MODAL TESTING

by 

i Dr. Roy R. Craig, Jr.

f NASA Contract No. NAS8-33f0

October 1983

(NASA-Ct(- 17 J. 51) AULH!: RAKER MUUAL ?ESTING N84- 15521 

Final keport, Aug. 1980 - GCt. 1983 (Texas

;j,uiv j 35 p HC A03 /MP AJ 1 CSCL 14b

1 G3/35 18005

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N

MULTISHAKER MODAL 'TESTING

Final Report

Contract No. NASB-33980

August 1980 - October 1983

Prepared for

National Aeronautics and Space Administration

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

by

Dr. Roy R. Craig, Jr. 

Professor, ASE-EM 

Principal Investigator

Center for Aeronautical Research

Bureau of Engineering Research

The University of Texas at Austin

Austin, TX 78712

October 1983 

C
ABSTRACT

This report summarizes the research conducted on this contract and

previously reported in a number of technical reports, papers presented at

technical meetings, and published papers. The principal goal of the contract

was to explore procedures for improving the modal modeling of structures using

test data and to determine appropriate analytical models based on substructure

experimental data.

In the area of modal modeling using test data, two related research

topics were considered: modal modeling using several independently-acquired

columns of frequency response data, and modal modeling using simultaneous

multi-point excitation. In the area of component mode synthesis modeling, the

emphasis has been on determining the best way to employ complex modes and

residuals. This report presents abstracts of the major publications which have

been previously issued on these topics.
MULTISHAKER TESi" METHODS

The period of this contract has been a period of great activity on the

topic of multishaker modal testing. On the one hand, Vold and his colleagues at

Structural Dynamics Research Corporation developed a multi-input modal estimation algorithm for minicomputers (1-3) which simultaneously employs several

columns of frequency response function (FRF) data. On the other hand, Allemang

(4,5) developed a simultaneous multi-input method for computing (single-input)

columns of the FRF matrix. A combination of the multi-inpl.,t modal estimation

algorithm and the Allemang algorithm for using simultaneous uncorrelated multi

ple inputs to generate the FRF's required by the modal estimation algorithm has 

led to a great improvement in experimental modal modeling technology.

The goals of the present contract in the area of multishaker te

tt

methods are along lines similar to the work of Vold and Allemang mentioned

above. However, on the topic of modal modeling using several independently

acquired columns of FRF data, research has followed a different path than that

r; taken by Vold. In Report CAR 81-1 (6) and Report CAR 82-1 (7) the "Asher

; method" and a generalization of the Asher method called the "minimum coincident

response method" were employed to determine real modes and undamped natural

frequencies from independently-acquired columns of FRF data. An application of

r:LAltishaker random testing is also included in Report CAR 81-1. The modal

tuning work was also summarized in a paper presented at the March 1982 SDRC 

` Troubleshooting9 oftware User's Conference ( 8). Abstracts of these re ports and p

papers are presented below.

A parameter estimation algorithm which employs multi-input excitation

directly has been recently developed and is described fully in Report CA

F ; (9). A paper summarizing this work has been submitted to the AIAA Dynamics 

Specialists Conferer.=. ce (10). With this algorithm it is not necessary to gener1

r a^

Xv
2

ate columns of the FRF matrix prior to entering a modal parameter estimation

algorithm as is done in O e Allemang/Vold procedure,

t

r

F

{t

r^ !

f 9
COMPONENT MODE SYNTHESIS

One of the ultimate goals of refined modal test methods is to produce

math models of structural components which are sufficiently accurate to employ

in synthesizing system models from component models. Component mode synthesis

it

t4 research has been conducted concurrently with the modal test research summarized

L '._ above in order to clarify which component modal parameters are ultimately needed

for synthesizing accurate system models.

Chung developed both Hamiltonian and state vector methods of component

,. mode synthesis. These are described in his Ph.D. dissertation (11). The

Hamiltonian formulation is summarized in References 12 and 13, while the state

vector formulation is summarised in Reference 14. A more simplified state

`, ector component mode synthesis formulation has been developed by Howsman. The

work completed! to date has been submitted as a technical paper to the

AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Materials Conference

(15). Work on this topic is being continued under NASA Contract NAS8-35338.

3
ABSTRACTS OF REPORTS AND TECHNICAL. PAPERS

Modal Analysis Using a Fourier Analyzer, Curve-Fitting, and Modal Tuning

,

tKeT• 01

This report proposes a modal testing procedure including the following

tasks: (1) data acquisition and FFT processing, (2) curve-fitting of single

t

"P

FRF's, (3) modal tuning, (4) mathematical modeling, and/or (5) computercontrolled testing. Steps (1) through (3) are described in the report.

P '

P a

v!

.R

Xj(t ); HyJ(f) v(t) n(t^ 

R.e

A tF .

r 

X1 

t

H(f) v2(t)

x

Figure 1. Two-Input, Single-Output System

The multishaker procedure for acquiring FRF's which was proposed by 

Allemang (4,5) is outlined in the report. For example, for the two-input,

single-output system shown in Fig. 1, the Fourier-transformed response Y(f) is

= a given by

Y(f) H 1 (f) X1 (f) + H 2 (f) X2 (f) + N(f) (1) 

.x

r

4
1

;^' 1 I

L^

1

.A

1

Y/

` • ^^A^^^II a^^ ^ l

PANE i ,

^F P 01IAL.17Y

The least squares estiriates for Hyl and Hy? are giver. by

X2' 21 1

G 1'22 ' 2'21 r Yl ,

-

G..1622

Hyl G11'22 - '12 1 ^ G il 1 - y 1 ?2

1 - ^1 G12

'11'y2 - '12'y1 2

Hy2 '11'22 - 'G

12^ '22 1 - a122

5

(2)

where

Gyl = VX1

G11 = X

1 X 1 etc. (3)

andY12 is the ordinary coherrrcP functicn between inputs X 1 (f) and X2(f).

Figure 2. Dual-Beam with Modified Ends and Modified

Shaker Attachment

C 
6

Dual-shaker uncorrelated random excitation was applied to the dualbeam test structure of Fig. 2. The time histories from two force cells and four

accelerometers were tape recorded and subsequently played bacl through a HP5420A

Fourier analyzer to compute the necessary auto- and cross-spectra. Although 25

records were averaged in computing these auto- and cross-spectra, the computed

FRF's were not of acceptable quality. This appears to be attributable to the

a use of tape-recorded data, which did not permit proper time correlation of the

records input to the two-channel analyzer. Because of the failure of this

attempt to employ multiple shakers, efforts were renewed to secure funding for a

16-channel modal analysis system, and subsequent testing was confined to

2-channel processing of single-shaker FRF's.

The curve-fitting of individual FRF's was performed in order to obtain

M the best analytical fit of the FRF's over limited frequency ranges. The MDOF

' curve-fit algorithm (GE command) in the MODAL PLUS program of Structural Dynamics Research Corporation was employed.

In order to provide a rational procedure for incorporating multiple

FRF's in the determination of system modal parameters, two modal tuning algorithms were employed, the "Asher method" of Ref. 16 and the "minimum coincident

}

response method" of Ref. 17. In the Asher method, approximations to the true

undamped natural frequencies of a system are obtained by computing the roots of

the determinant of the reel part of the FRF matrix.

4

det GC(fc )] = 0 (4)

Its

This method requires a square FRF matrix, i.e. as many exciters as responses.

' *i The minimum coincident response method p permits the number of response DOF s to "

exceed the number of excitation DOF's by selecting frequencies and force

f- distributions which minimize an error function defined as the sum of the squares 4

of the coincident (real) responses, i.e.
7

c Y R Y R X T [Ca T [Cl X (5)

The basic algorithms are presented in Ref. (6), while applications are presented

in Ref. (7).

Modal Vector Estimation for Closel -S aced-Fre uenc Modes Ref. 7

The purpose of this report is to discuss the difficulty of obtaining

accurate mode shapes of systems with closely-spaced Natural frequencies, and to

present examples of the use of modal tuning to obtain modal parameters of such

systems by simultaneously using several FRF's, Table 1 illustrates the fact

that while frequencies of a system are only moderately sensitiv

changes, the mode, shapes are extremely sensitive to small stiffness or mass

changes.

m' fl f2 ^2 

4

1.000 1.0000

1.000 1.0000 1.000

1.0002 -1.0000

1.0000 •1.6192

1.001 0.9994 1.6176

1.0016 1.0000

1.0000 -10.1099

1.010 0.9910 10.0098 1.0011 1.0000

1.0000 -100.1109

1.100 A .9100 91.0100

1.0010 1.0000

"

Table 1. Effect of Subsystem Properties on System Modes

,-^ of a Weakly Coupled 2DOF System 
8

Thus, while modal tuning methods which are based on multiple FRF's can be

expected to produce improved frequency estimates, it may be expected that mode

shapes will be more difficult to determine accurately.

Complete derivations of the Asher method and the minimum coincident

response method are given in the report. Both methods were applied to determine

the frequencies and mode shapes of the weakly-coupled beam structure of Fig. 2.

The effect of analysis bandwidth on modes and frequencies was studied for the

two modes at 118.80 Hz and 119.03 Hz.

The conclusions drawn from this report are (1) that the standard Asher

method and minimum coincident response method are rational procedures for

employing multiple FRF's for identifying modal parameters, and (2) that lightlycoupled systems will always lead to greet difficulty in obtaining "accurate"

mode shapes because of the extreme sensitivity of the mode shapes to system

parameters.

Modal Vector Estimation for Closely-Spaced-Frequency Modes (Ref. 8)

This paper, presented at the GE/CAE International Troubleshooting

Software User's Conference, summarizes the work presented in Refs. 6 and 7 and

abstracted above. 

A Generalized Multiple=Input, Multiple-Ou

ri-'t^ m Re T s. O)

out Modal Parameter Estimation AlgoA new multi-shaker modal analysis algorithm for estimation of modal 

coefficients has been developed in a thesis (Ref: 9) and summarized in a techni-

cal paper (Ref. 10). The algorithm permits multiple inputs (exciter locations)

as well as multiple outputs (accelerometer locations) to be employed. The

excitation may be simultaneous random or simultaneous swept sine. free decay

responses may also be used. The algorithm is an extension of an algorithm
9

published by Coppolino in Ref. 18. Measurement stations are designated as

"independent" or "depend,^nt," wiva; the number of independent static. -.1s being

equal to the number of modes to be identified in a given frequency range, Modal

vectors, which are assumed to be complex, are defined at both independent and

dependent degrees of freedom.

{ The algorithm is based on the theory that a linear structure may be

accurately described over a limited frequency range be a finite number of

degrees of fr-edom. The frequency domain equation relating output acceleration

to input forces is

{()} rf{(^) { F(w )} (6)

It may be rewritten in a form which identifies independent and dependent outputs

(accelerations) and input forces as follows.

{ xi } = [oil rA(w)7 r^i ] T [Did]{f(w)} (7a)

{Xd } = ["hd J r^i 7 `I { xi }

(7b)

Equation (7a) is transformed into an equation for the n independent

coordinates

[M i :l { x i (w)} -+ [ C i ] {x i (w)} + [K1 1 {xi( w)} rgi d7 {fm) (8)

Accelerations are determined at k frequencies (k > n) and assembled into a

4

matrix

[Ki((0)] = [Yw l ) 1 x i ( w2 ) { ( x i ( wk )] (9)

Equation (8) may be rewritten for each of the k frequencies and assembled in

the form
Y

10

Ckil

[Cy CKia .. [D i6 3 ] Cx il (10)

If]

where txi(w)) and (x i (w)) a^e obtained from the measured {X I (w )I by the

equations

{xi (w)} A x i (^)} (11a)

{x i ( ) ^ 2 {ii i (w) (11b)

Equation (10) is solved in a least-squares manner for EC i 1 and CKi1 , which

are used to form an eigenequation whose solution gives the natural frequency and

damping estimates and the Gai l portion of the modal vectors. Equation (7b) is

used to recover Pit,; Rd portion of the modal vectors.

'imu"tIon studies were performed to test the applicability and

accuracy of the algorithm. In one study, randon forces were applied and

accelerations measured at all 8 DOF's of an 8 DOF system having

closely-spaced-frequency modes in the range 5 Hz to 15 Hz. The analysis range

was taken as 0 Hz to 1024 Hz with 256 data points, for a delta frequency of 4

Hz. All modes were identified to extremely high accuracy in spite of the fact

that there were seven natural frequencies between two of the data points. No

noise was introduced into the data used in this study.

More realistic simulation studies were carried out on a 9 DOF system

using 1, 4, 5, and 8 inputs; 5 and 8 independent DOF's; pure random and

random-phased swept sine excitation; and noise ratios of 0%, 10%, and 209.

Table 2 shows the frequencies, damping factors, and mode shapes of the 9 DOF

simulation model. Tables 3 and 4 show the identified modes corresponding to the

5 Hz, 5.5 Hz and 5.55 Hz modes. Eight independent DOF's were employed. Note

that the single exciter in Table 3 was unable to identify the 5 Hz mode because
of its location near a "node line" of the 5 Hz mode, while the 4-exciter results

in Table 4 show acceptable frequencies and made shapes for all three modes.

t
12

Mode No. Frequency (Hz) Damping

1 5.000 0.010

2 5.250 0.010

3 5.500 0.010

4 5.550 0.010

5 5.900 0.010

6 6.200 0.010

7 6.600 0.010

8 15.000 0.010

9 25.000 0.010

DOf

MODE

1 2 3 4 5 6 7 8

1 .010 1.000 1.000 1.000 1.000 1.000 1.000 1.000

2 1.000 1010 1.000 1.000 -1.000 -1.000 -1.000 -1.000

3 1.000 L000 -.010 -1.000 -1.000 -1.000 1.000 1.000

4 1.000 1.000 -1.000 -.010 1.000 1.000 -1.000 -1.000

5 1.000 -1.000 -1.000 1.000 .010 -1.000 -1.000 1.000

6 1.000 -1.000 -1.000 1.000 -1.000 .010 1.000 -1.000

7 1.00t; -1.000 1.000 -1.000 -1.000 1.000 -,010 1.000

8 1.000 -1.000 1.000 -1.000 1.000 -1.000 1.000 -.010

9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Mk

+R

,' t

.t,

Table 2. System Frequencies, Damping Factors, and Mode Shapes

(defined at 8 DOF only

1

4 

1

t

4

^t

p!

f

f1

taw

^...:.

__ 

^ Y
13

No mode corresponding to 5 Hz was identified

Undamped Natural Frequency = .549815E+01

Damping = .922684"E-02

Coordinate Coefficient

1 .100000E+01 0

2 .961693E+00 .154095E-01

3 -.111589E+00 -.243182E+00

4 -.953094E+00 -.189219E-01

5 -.866331E+00 .201760E+00

6 -.874222E+00 .225422E+00

• 7 .852086E+00 -.247594E+00

8 .852671E+00 -.202553E+00

y^

Undamped 'Natural Frequency = .556574E+01

Damping = .125684E-01

Coordinate Coefficient

1 .969798E+00 .718633E-02

2 .100000E+01 0

3 -.601954E+00 .299185E+00

4 -.306331E-01 .115159E+00

5 .663191E+00 -.152995E+00

6 .655221E+00 -.180120E+00

7 -.574999E+00 .258952E+00

` 8 -.641525E+00 .178636E+00

G , ♦ 'V

t

=." Force at DOF No. 1 Noise Ratio = 10%

Table 3. Single-Exciter Results

1

Y

..-,^...^ ------- 

--w_..-..^...",_»... ^,... r-,^.....v... _....-. ...: ^..- mss-.__ .. _ _....
Damping = .979900E-02

Coordinate Coefficient

1 -.995591E+00 .353937E-01

2 -.963411E+00 .140834E-01

3 -..193424E-01 =.112736E-01

4 .9872.21E+00 .125764E-01

5 .952534E+00 -.241061E-01

6 .10000"-7+01 0

7 -.995252E+00 -.490844E-01

a -.960618E+00 -.491432E-01

a

4„

z^

L,

E

r

a

Undamped Natural Frequency = .55d82.5E+01

Damping = .943557E-02

Coordinate Coefficient

1 .993486E+00 -.268232E-01

2 .100000E+01 0

3 -.973952E+00 .261893E-02

4 -.554699E-01 .508959E-02

5 .852456E+00 -.601461E-02

6 .903672E+00 -.198642E-02

7 -.925322E+00 -.291869E-01

8 -.90532,2E+00 -.464260E-01

14

g

t

t

Undamped Natural Frequency = .499588E+01

Damping = .101836E-01

Coordinate Coefficient

1 .180826E-01 .648560E-02

2 .989374E+00 .422064E-03

3 .984854E+00 .001076E-02

4 .100000E+01 0

5 .975333E+00 .364940E-02

6 .987170E+00 -.791027E-02

7 .977551E+00 .291086E-02

8 .975623E+00 -.161876E-03

Undamped Natural Frequency = .549835E+01

Forces at DOF's 1,2,3 and 4 Noise Ratio 10%

Table 4. Four-Exciter Results

r I
9

4

4 8

1 4 8

4%9/6

sj

4 8

.68 %

r

4 8

Freguen Mfrror

0.3 " *-Single Run

•-Three Averages 0 24 ,-

0.18-

F R

0,12.

0.06"

0.00

1 4

r.

Damping Error 25."

201"

15.-

10.-

5.-

0A,

1 4

Modal Vector Error

50.-

40.-

a

OF POOR QUAL" 15

1 4 a 1 -a A

5,9 Hz. 5.5 Hz. 5.55 Hz.

Figure 3. Modal Parameter Errors Versus Number of Shakers

for Single Runs and for 3-Run Average
R

E

16

The results in Tables 3 and 4 were obtained from single "runs."

Averaging may be employed to improve the parameter (frequency, damping, mode

shape) estimates. Figure 3 illustrates the results of single runs with random

excitation and the results of three "runs" averaged.

This algorithm meets the primary objective of this NASA contract,

which was to produce an improved method for multi-shaker modal testing.

Although testing of this algorithm is still in progress, it is believed that it

will prove to be a valuable addition to the multi-input modal identification

software which is currently in great demand in the modal testing industry.

Application and Experimental !Determination of Substr ucture Coupling for Damped

Structural Systems Ref. 11

The principal topic covered in this thesis is the development of a

generalized substructure coupling procedure for a complex structure with general 

viscous damping. A Hamiltonian first-order differential equation formulation is 

employed in order to permit complex substructure modes to be employed easily,

since complex modes are required for systems with non-proportional damping. The

substructure coupling procedure makes use of an incomplete set of complex normal

modes in conjunction with complex residual attachment modes to account for the

contribution of neglected higher-frequency modes. Aiso discussed in the

dissertation are experimental procedures for identifying the substructure data

by modal testing. 

The first-order substructure, or component, equations of motion are

ay + by = f (12)

ra

h
where

O I

a

I c

p

y -

x

OSt1CIINAL P t 

OF POUR QUALITY

-m-' 0

b

0 k

0

f =

f 

17

(13)

where p is the momentum coordinate vector and x is the displacement

coordinate vector. Complex free-interface modes are determined by the

homogeneous equation

4+by=0 (14)

where y is taken to have the form

y 

= *eXt = p eat

-

^x

(15)

Only the case of symmetric damping matrix, c, is considered. Then

orthogonality and normalization of the modes leads to the equations

,YT a Y = I , TT b T _ -A (16)

The x and p-partitions of rigid-body modes are defined by the

equations

kx 0 , km-1p = 0 (17)

r— a :....,...:
18

Convergence is improved by supplementing a truncated set of complex modes with

attachment modes, which are defined by the equation

-1

-m 0 p 0 (18)

0 k

0x a fx

t-

,z

X After extensive manipulation of the attachment modes, the substructure response

is approximated by a truncated set of complex modes, Yn, and a set of residual

attachment modes, gym , by the equation

ya 

= Tnzn + O

m zm

(19)

where ya is the approximated response.

When two or more substructures are combined to form a synthesized

structure, compatability equations must be enforced at substructure interfaces.

These take the form of

E I X = 0 (20)

where X is the union of substructure displacement vectors. Compatibility is

also imposed on the velocity vector X, and the combined compatibility equation

is written in the form

-El M- 1

0 P 0

(21)

0 El X 0

or

EY = 0 (22)

;l
W.

19

Finally, the Y coordinates are expressed in terms of modal coord

nates through

1 x

Y = Ti Z (2

and the Z coordinates expressed in terms of independent system coordinate

through

Z = SU (2

Ti may contain either truncated sets of substructure complex modes or truncate

sets of complex modes plus sets of attachment modes. Details of the derivatio

of the coupling matrix S for these cases are given in the dissertation. I

the former case the final coupled system equation of notion is

(S11TS) ll - ( S TAS) U = -T (2

The Hamiltonian approach outlined above employs the displacemen

vector and momentum vector as coordinates in the first-order equations o

motion. The dissertation also briefly describes a state-vector approach t

substructure coupling, but this is further developed in Ref. (14), which i

abstracted below.

Finally, the dissertation contains a chapter on System Identificatio

which~ describes a procedure for employing experimentally-identified frequencies

damping factors and mode shapes of lightly-damped systems to obtain substructur

equations of motion transformed to modal coordinates, i.e. to form TTaT and

T TbT .

A number of simulation studies are presented in the dissertation

Table 5 presents a portion of the results of one of these studies. It shows th

improvement in convergence resulting from the use of residual attachment modes.

h 

v.
". pl .,~: • ,"e~~ .\ 

Damped Frequency (IDd

) 

Truncation Error Nomal Modes and 

Hade No. Residual Attachment Modes 

(Na=8, NS=6, N

t

=12) % a S t (N =7, N =5, N =12) 

1 0.230647 3.09 0.223737 

2 0.633534 2.70 10.616879 

3 1.227306 1.42 1.210142 

4 2.133363 6.46 2.003981 

5 3.010846 0.30 3.002025 

6 4.457241 5.76 4.215497 

7 5.879491 4.06 5.651209 

8 7.360751 2.19 7.304950 

9 10.17425 11.99 9 .. 103936 

10 12.14582 0.18 12'.19040 

H 

Ii 

*! 11 15.80521 8.08 14.91669 

J~~ 

f! 12 19.15414 7.76 21.25394 

i1 

': 

, 

, 

t 

, 

! 

Table 5. Comparison of Damped Frequencies for Clawped-Clamped Bea

with Non-Proportional Damping - 12 D.O.F. System 
I

21

In summary, the following conclusions were reached in this disserta

tion:

k

k

1. "A generalized first-order Hamiltonian formulation and a generalize

state vector representation have been developed for carrying ou

component mode synthesis for viscously-damped systems. The componen

modes are complex for systems with non-proportional damping. Both th

Hamiltonian approach and the state vector approaO produce the sam

results. However, the state vector representation is more efficien

since it avoids the inversion of the mass matrix, ')he state vecto

formulation is also easier to imp lement when experimental data i

employed."

2. "Complex residual attachment modes, which are based on static ap

proximation of the neglected higher modes, have been defined an

derived. The inclusion of complex residual modes substantiall

improves the approximation to the system eigenvalues and eigenvector

in comparison with a solution based on truncated sets of componen

normal modes only."

3. "The state vector formulation of substructure coupling has bee

employed to identify the system chararteristics of" 

a complex structur

using data from simulated substructure tests. A multiple-degree

of-freedom curve-fit algorithm is used to obtain the poles and res

dues of the substructures, and light damping assumptions are employe

in estimating the complex mode shapes, generalized mass, damping an

stiffness of the substructures. The system results obtained b

substructure coupling show that excellent results can be obtained fo

lightly damped systems."
22

The following recommendations were offered:

"As a result of the work described in this dissertation, further

research on t he identification of the generalized mass, damping and stiffness for damped systems with complex modes is necessary in order tr provide

a more general and accurate system analysis technique via component mode

synthesis. The effect of frequency resolution on the response functions

needs to be investigated. A multiple-reference-multiple-degree-of-freedom

T.

curve fitting algorithm which curve-fits several frequency response

functions simultaneously should be studied in order to improve the est"mation of modal parameters. Finally, the relationship between the residual

attachment modes described in Chapter 3 and frequency response function

<,. residuals obtained in conjunction with multiple-degrees-of.-freedom curve

fitting needs to be elucidated."

A Generalized &,bstructure Coupling Procedure for Dames Systems (Ref. 12)

This paper summarizes the portions of the dissertation, (Ref, 11),

discussed above which deal with substructure coupling using truncated sets of

complex mods . Beam and truss examples are given in the paper. The following

conclusions are reached in the paper:

"A generalized substructure coupling procedure for systems with

damping has been presented. Lagrange multipliers have been employed to

incorporate the substructure interface compatibility constraints into the

first-order system equations of motion. Complex free-interface substructure modes have been employed in two example problems. Although the

E

coupling method is shown to produce the correct system modes if all substructure modes are retained, the results obtained for truncated systems

{ are not very accurate. Since this is most likely a consequence of using

h
23

free-interface modes, further research is needed to explore convergence

when other types of substructure modes are employed in the generalized

substructure coupling procedure."

State Vector Formulation of Substructure Coupling for Damped Systems (Re , '. 14

This paper expands on the brief introduction to state vector formulation in Ref. 11. In the state vector formulation the substructure equation of

motion still has the form of Eq. 12, but a, b, g and f are defined by

} 0 m -m 0 v 0

a ., , b - , y - f

t (26)

m c 0 k x fx

where v is the velocity vector. Free interface complex substructure modes are

again determined by Eq. (14), where y is now taken to have the form

y ^ te xt r eat (27)

4`x

The matrix, ^D m of complex residual attachment modes is defined,

and y is again approximated by y a as in Eq. (19). The procedure for defining complex residual attachment modes requires the computation of all of the

k

modal parameters of the substructure.

141 As in the case of the Hamiltonian formulation described above, compatibility equations are employed to formulate expressions for the coupling

f

matrix, S, of Eq. (24).. These expressions are presented for the case of

coupling using complete or truncated sets of complex normal „odes and for

1 coupling using truncated normal modes plus residual attachment modes. Example:

of these two approaches are pp provided in the paper.. Table 6 illustrates the
24

results, which show the improved convergence of the method employing residu

attachment modes,

Purely truncated modes Truncated modes plus residual

attachment modes

Na=8, N6 =6, N t=12 Na=7, 4=5, Nt=12

e error % wd error % a error % wd error %

-5.31 3.09 0.00 0.00

5.62 2.70 0100 0.00

-0.24 1.42 0.00 0.00

-0.67 6.46 0.00 0.00

-2.85 0.30 0.00 0.00

20.15 5.76 0.05 0.03

-22.87 4.06 -0.11 0.03

20.45 2.19 0.27 0.06

-0.58 11.99 _0:05 0.21

-5.63 0.18 1.23 0.19

54.66 8.08 -5.59 2.01

-31.70 7.76 18.84 19.57

___ --_ --- ._

--- --- ---

Table 6. Frequency and damping errors for a clamped-clamped beam with

non-proportional damping

The conclusions stated in this paper are the following:

"A generalized substructure coupling procedure for systems w

damping has been presented. A state vector formulation leading to 

first-order system equation of motion has been developed. Complex resid

attachment modes, which approximate the contribution of the negle;'

higher modes, are defined. A new method which employs incomplete comp

normal modes in conjunction with the complex residual modes produ

r 0
25

significantly more accurate results than those obtained by using a truncated set of normal modes only. It is observed that the interface generalized coordinates can be eliminated from the final system coordinates by

employing additional constraints on the generalized coordinates of attachment modes."

A Substructure Coupling Procedure Applicable to General Linear Time-Invariant

Dynamic Systems (Ref. 15)

A':though the substructure coupling studies of Refs. 11-14 demonstrated

the use of first-order equations of motion, complex modes, and residual modes,

they were subject to two significant limitations; the damping matrix was

assumed to be symmetric, and all substructure modes were required in forming the

truncated set of complex normal modes and the set of residual attachment modes.

An alternative state vector formulation is derived in this paper, which has been 

submitted to the AIAA/,ASME/ASCE/AHS 25th Structures, Structural Dynamics and

Materials Conference. 

The coupling procedure can be described as a generalized component

mode synthesis technique. The equations of motion for each substructure are

cast in state vector form to facilitate the formation of the substructure 

eigenproblem. If nonsymmetries exist in the mass, damping, or stif

matrices, the eigenproblem will be non-self-adjoint. The low frequency right

hand eigenvectors, along with a set of real attachment modes, are used as the 

Ritz expansion vectors. The concept of interface compatibility is used to

couple the substructure equations of motion together to form the system 

equations of motion. 

The synthesis technique is similar to the method presented in Refer- 

ence 1, but with important differences. The attachment modes, as defined in

Reference 1, are combinations of the high frequency modes not explicitly used as
26

Ritz vectors. This implies that the complete substructure eigenproblem must be

solved - a potentially expensive task if the substructure contains a large

number of degrees of freedom. The method to be presented avoids the problem of

solving for the high frequency component modes by defining the attachment modes

to be the static response of the substructure due to unit loads at the substructure interface. If the contribution of the low frequency component modes is

removed from this static response, the so-called residual attachment modes are

formed. The left-hand (adjoint) eigenvectors of the substructure are required

to form the residual attachment modes.

The coupling procedure has been applied to structural systems containing combinations of rigid body modes, symmetric non-proportional damping, and

non-symmetric damping matrices. Table 1 contains the eigenvalues obtained from

a substructure synthesis compared to the exact values for the following structure,

component A -----.t.---- component B

Figure 4. Structure Used in Skew-Symmetric Damping Study

The damping matrix for each :zubstructure is skew-symmetric, and of the

fo rm

5 1. 1 1

-1 5 1 1

c

-1 -1 5 1

-1 -1 -1 5
7 2/

r-iMC7)Ol m M mmwor-1 1 -4 m0)Ln MwwhmLnr-4w k C) C) r1 r-I o CO m 471 nJ N d• 4t M Ln M m w to m Ln r`% r. W tm 47. a) C)00 o r-Ir--t o a to www -+ r-i r. r. r-I tD w 00m (a +r . • . . . . . • . 0 . . • , . . . . . . .

o i I f 1 mM0)LoLnmr-I

ZA to

f rl

E .ate

rd

Ire a)

r^•• C N a) 0 o Lo Lo CD co N N 00 00 Ln Ln N CV M m N N C0 to N o C) aJ N

44- OCC)OO^rqOO^wrCOI^C^7GNNp0)NOC 2

'V p • • 1 • • • • • • • • • a) 3 tt• d• C) Lo tD C) o

o I o o L L irt' rpl .-i v-•i ^ N

+ra

F•r act

Ln L

.It

r-1 M Ln LC') (a to to to N N N N O o C) C) m w m o w It:l- .i

4- O O N N N N Lo Lo N N ct tt r. n Ln W to tD o w tD C) m L

4 C)COC)o0ommNN 0000MmwwLtdLoetC)0C) + m LL. 4-r

G1 1 1 •

•• • !• ri r4 r 4 r 4 C,,J N O CD 00 C71 Ln N r-I G] ..O 

1 1 r-i 1 lf)

^ 

I ^

I^ M N to

k N

R7 C 

(o

L

a O

C) O O CD r-I r-I r-i r--I .--I rl r•1.. -H r-i r-i -4 ri r-i r-i "I r 4 r-i 00oCDC)OCCDCDoa 000OO C)CDOOOO L 

fj 3

•1• t + t . . . . . t t . . . . + . . + . t oC^WILLJW WwwwW wwLLJLJJwWIJwLLJwwW to O W (1) 4-) m m C) CD m m m M o o CD o I.o w w w N o1 o C:) N O U 

std r. t- mmCDC)wmrnrnrir+Lommm :t Old, nr r— aj m M^ r" r-I r f N N N N m m m Lo tD w w m 0) 0) 0) ny > 4

•

• • • • • • • > 

1 •

I• 1 I• 1 I 1 I L 1 I + C ^J

+j QJ -'

U o •r (Cf •r LY 

Lx

W

Ln

r-1 C7 o O o O C7 C7 O O O C=) CD o r-i r-1 r--1 r-I "H r-I O o r-i U

CD 0 0 o O o o CD o C) C) 0 0 0 o C) 0 CO o o O o o (Cf >, t+ ++ •1• •f- 1 F f f f -h -f f t t t t 1 f t t t LU W W W W W W W

X a

b W W ui LLJ W Li W W W W W W W w W W h m d' ¢t r*^ n ri " d' Ct m w N N d' d' N N m m m 0) Lo 'p

C) m Ln Ln m m Cr t\ r-^ m m N N r-I " N N r-1 r-i m m C) 'C1 aJ w m C"J N M m -- Ln Ln ^ tom• m m r-i r-i r-( r-1 r-i r-1 to to r^ C 4J 

I•1 I•1 I 1 ( t 1 I I•I I 1 1 1 1 1 1 1 1 •1•I• aJ 

QJ N

E

CL

L

M m C D C) 0 o-i r- r- r-1 r-1 v-I r-( r -i r-I Pr-4i r 1 W r-i r-i tD W O 

.d t--1 r-i = C) 0 0 o C) o C7 0 0 o CO C) C) C) C) o O CD CU C) L N

3

1 I. f f... f t f. f f. t ... I+ -f• I I CL •r W W W w w w W W W W W W W W W W W Li ^t W W W W W CL

m m m m o o m m m m CD C7 r7' ei• w w ct ct• w r-i r-1 m m Q cc N

-, t 10 zr it ^ r, M M o O 0) 0) 0) C7) r-i r-1 00 00 00 00 00 tD Ln 

t CV *-I m tM t1 r^ " r-•I CV N N N M M LO Ln tD tD r1 00 00 r-I N 4- Q1 

. . . .

H 1 1 1 I 1 1 I 1 1 i •

O^ 4-

X O U 

O N 

L •r L N d L 4-r 4

G. b N 

r -I o o CO o C) 0 o C) CO o C7 CD C) r--4 r-i r-1 r-i r-1 --4 ri N N p 0. -0 w 

00000Cl000000000OOOOOOOOO 3 E = 

1. t t -h f f.. t t f f f. t.. f f f f f r O Cn 

b W W W W W W W W W W W W W W W W W W W W W LJJ W u^ r

1%,C1CtCf t-I1-%mmmMr-ir-Io01l.fl,mm m Lo } 

C) m m Lo m m Oct -It t-^ t\ Lo m r^ •-i r-1 r-/ o CO r" CD C7 m C) tO m N N m m Cl• Ct Lo Lo (-% n m O) r-1 r-I r-I r-i to ri r-I r-I M

•

. . . . . . . . . . n 

1 1 1 1 1 I 1 1 1 1 1 1 / 1 1 1 1 1 1 1 1 1 II

CU 

IL

~

Q) > 

r- 

> O r-I N M ct Ln tD 1\ co 0) C) r-1 N MCI" In tD 1\ co C71 o r-i N M C Z r-i r-i r-i r-I r-i r1 r-1 r-I .. N N N N uj 

W

st •yy rn ^:.. r .S►u. .. :.r .e.... Vt dvlw \b'MOleu!M^r^
28

As can be seen from Table 7, the method presented is very accurate

even when nonsymmetries exist in the defining matrices, and the method of

defining attachment modes is quite efficient from a computational point of view.
CONCLUSIONS AND RECOMMENDATIONS

Several significant contributions are contained in the reports and

papers prepared under this contract and abstracted above. The principal conelusions of this work are:

1. The modal tuning procedures discussed in Refs. 6-8 provide a systematic

procedure for combining modal data based on single-shaker FRF's to determine a

z

system's undamped modes and frequencies.

2. The generalized multi-input, multi-output modal parameter estimation algo -

rithm presented in Refs. 9 and 10 employs response data from several transdur.,, ers, permits excitation at several points and of several types, and is

capable of accurately identifying modal parameters from moderately "noisy" data.

3. Residual attachment modes significantly improve convergence in substructure

coupling of damped systems. The generalized coupling procedure of Ref. 15

represents the culmination of the effort to develop a procedure capable of

employing complex modes and having reasonable convergence properties.

The following recommendations are made for further work:

1. Development of the generalized multi-input, multi-output modal parameter

estimation algorithm of Refs. 9 and 10 should be continued. An effort should be

made to adapt the algorithm to a minicomputer-based modal analysis system, and

the testing of the algorithm should be extended to modal testing of actual

structures.

2. Development of the generalized substructure coupling procedure of Ref. 15

should be continued. A, special effort should be made to determine appropriate

experimental procedures for determining input data required by the method.

29
SELECTED REFERENCES

1. Vold, H., et al., "A Multi-Input Modal Estimation Algorithm for MiniComputers," SAE Paper No. 820194, International Congress and Exposition,

Detroit, MI, Feb. 1982.

2. Vold, H., and Rocklin, G.J., "The Numerical Implementation of a Multi-Input

` Modal Estimation Algorithm for Mini-Computers," Proceedings First International Modal Analysis Conference, Union College, Schenectady, NY, Nov,

' 1982, pp. 542-548.

3. Vold, H., and Russell, R., "Advanced Analysis Methods Improve Modal Test

r°

Results," J. Sound and Vibration, V. 17, n. 3, Mar. 1983, pp. 36-40. 

0 

4. Allemang, R.J., Investigation of Some Multiple Input/Output Frequency

,. Response Function Experimental Modal Analysis Techniques, Ph.D. Dissertation, University of Cincinnati, 1980.

5. Allemang, R.J. Rost, R.W., and Brown, D.L., "Dual Input Estimation of

Frequency Response Functions for Experimental Modal Analysis of Aircraft

Structures," Proceedings First International Modal Analysis Conferen ce,

Union College, Schenectady, NY, Nov. 1982, pp. 333-340.

f r 

6. Craig, R.R., Jr., and Chung, Y.-T., Modal Analysis Using a Fourier Analyzer, Curve-Fitting, and Modal Tuning, Report No. CAR 81-1, Center for

Aeronautical Research, Bureau of Engineering Research, The University of

Texas at Austin, Austin, TX, Oct. 1981. 

.{ $: 7. Craig, R.R., Jr., Chung, Y.-T., and Blair, M., Modal Vector Estimation for 

Closely-Spaced-Frequency Modes, Report No. CAR 82-1, Center for

Aeronautical Research, Bureau of Engineering Research, The University of

Texas at Austin, Austin, TX., Feb. 1982.

30

` AM1

^;.f^' '- r. r• ^.^ ^ J^rw:..

. ..

_v^d l • "-^Ji: ^ta5tiu I^l- fl.r Y_"iw 
A

I

31

8. Craig, R.R., Jr., Chung, Y.-T., and Blair, M., "Modal Vector Estimation for

Closely-Spaced-Frequency Modes," GE/CAE International Inc. Troubleshooting

,. Software User's Conference, Orlando, FL, March 4-5, 1982.

9. Blair, M.A., and Craig, R.R., Jr., A Generalized Multiple-Input, MultipleOutput Modal Parameter Estimation Algorithm, Report No. CAR 83-1, Center

for Aeronautical Research, bureau of Engineering Research, The University

.' of Texas at Austin, Austin, TX, Oct. 1983.

10. Craig, R.R., j r .. and Blair, M.A., "A Generalized Multiple-Input, MultipleOutput Modal Parameter Estimation Algorithm," Submitted to AIAA Dynamics

Specialists Conference, Palm Springs, CA, May 1984.

11. Chung, Y.-T., Application and Experimental Determination of Substructure

Coupling for Damped Structural Systems, Ph.D. Dissertation, The University

of Texas at Austin, May 1982.

12. Craig, R.R., Jr., and Chung, Y.-T., "A Generalized Substructure Coupling

Procedure for Damped Systems," AIAA Paper No. 81-0560, Proceedings

AIAA/ASME/ASCE/AHS 22nd Structures, Structural Dynamics and Materials

Conference, Atlanta, GA, April 1981, pp. 254-266.

13. Craig, R.R., Jr., and Chung, Y.-T., "Generalized Substructure Coupling

Procedure for Damped Systems," AIAA Journal, V. 70, n. 3, March 1982, pp.

442-444.

14. Chung, Y.-T., and Craig, R.R., Jr., "State Vector Formulation of Substruc

tune Coupling for Damped Systems," Paper No. 83-0965, Proceedings 

AIAA/ASME/ASCE/AHS 24th Structures, Structural Dynamics and Materials

Conference, Lake Tahoe, NV, May 1983, pp. 520-528.
32

15. Howsman, T.G., and Craig, R.R., Jr., "A Substructure Coupling Procedur

Applicable to General Linear Time-Invariant Dynamic Systems," Submitted to

AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Material

Conference, Palm Springs, CA, May 1984.

16. Asher, G.W., "A Method of Normal Mode Excitation Utilizing Admittanc

Measurements," Proc. National Specialists Meeting on Dynamics and Aero

F; elasticity, Ft. Worth, TX, (published by Inst. of Aeronautical Science)

1958, pp. 69-76.

^. 

17. Ensminger, R.R., and Turner, M.J., "Structural Parameter Identification

from Measured Vibration Data," Paper No. 79-0829, AIAA/ASME/ASCE/AHS 20t

Structures, Structural Dynamics and Materials_ Conference, St. Louis, MO,

April 1979, pp. 410-416.

18. Coppo1ino, R.N., "A Simultaneous Frequency Domain Technique for Estimatio

of Modal Parameters from Measured Data," SAE Paper No. 811046, SAE Aero

space Congress and Exposition, Anaheim, CA, Oct. 1981.