An algorithm is described for synthesizing the mass and stiffness matrices from experimentally derived modal data in a way that preserves the physical significance of the individual mass and stiffness elements. The mass and stiffness matrices are derived for a rollup solar array example, and are then used to define the modal response of a modified array

Ross, R. G., Jr.

English

NASA Technical Reports Server

JPL Quarterly Technical Review Volume 2, Number 3 October 1972
Index: mathematical sciences, mechan-
ics, power sources, solid-state physics,
structural engineering
An Algorithm for Synthesizing Mass and
Stiffness Matrices From Experimental
Vibration Modes
R. G. Ross, Jr.
Engineering Mechanics Division
It is sometimes desirable to derive a dynamic model of highly complex structures
from experimental vibration data. This article presents an algorithm for synthesizing
the mass and stiffness matrices from experimentally derived modal data in a way that
preserves the physical significance of the individual mass and stiffness elements. The
mass and stiffness matrices are derived for a rollup solar array example, and are then
used to define the modal response of a modified array.
Introduction
Structural design often requires an estimate of the response of a conceived
structure to dynamic excitation before the design is finalized. The predicted
response is generally obtained using a finite element discretization of the
governing differential equations to calculate the matrices K and M that
characterize the structure's stiffness and mass properties. The solution to the
algebraic eigenvalue problem Kx = o>2Mx defines the predicted natural
frequencies &> and vibration mode shapes of the structure.
To verify the analytical model, the low-order natural frequencies and
mode shapes may also be determined experimentally in a modal test. These
tests are usually run on the final structure or on a structural simulation of the
final structure, often at a time when it is difficult to incorporate anomalies
into the design cycle. For this reason, there is a growing desire to
incorporate modal test results of early prototypes or skeletal systems into
the analytical models of these systems so that extrapolation to final designs
can be made with greater confidence.
In addition to' space applications as described in this article, one area of
current interest is the modal testing of multistory building structures that is
conducted after the completion of the primary structure but before the
addition of interior walls, trim, and furnishings. It is desired to use the results
12
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of preliminary modal tests to improve the analytical model of the primary
structure so that extrapolation to the dynamic earthquake response of the
finished building can be made by adding the elemental stiffness and mass
matrices describing the structural modifications directly to the synthesized
matrices.
Several methods for synthesizing the governing mass and stiffness matrices
from experimental vibration modes have been recently described (Refer-
ence 1). This article describes an improved matrix synthesis technique that is
shown to allow elemental stiffness and mass matrices describing structural
modifications to be added directly to the synthesized matrices.
Definition of Problem
Following the rationale outlined in Reference 1, the problem is the
following: with a set of p eigenvalues A,- (Af = to,-2) and corresponding
eigenvectors <£,- (mode shapes) of the unknown structural system, we wish to
determine the nth order mass and stiffness matrices so that the eigensystem
Kx = XMx (1)
will possess eigenvalues and eigenvectors that are either equal to or as close
as possible to the measured ones. Since the most significant characteristics of
the mass and stiffness matrices are their representation of the system's
kinetic and strain energies (Reference 1), M and K are further required to
accurately represent known kinetic and strain energy characteristics of the
system. This is necessary if M and K are to be compatible with the analytical
model of the structure.
To determine the requirements for M and K, we note that if M and K are
required to be symmetric and to define the measured eigensolution, then the
eigenvectors must be orthogonal with respect to them; i.e., if 4> is the modal
matrix of eigenvectors, then
3>TM3> = D and 4>rK4> = AD (2)
where D is an arbitrary diagonal matrix and A is a diagonal matrix of the A(.
If we temporarily assume that n eigenvectors are available so that O is
nonsingular, then Equation 2 requires
and
13
However, because the number of coordinates used is assumed to be much
larger than the number p of natural modes determined, only the first p out of
n eigenvectors are available, and the inverse of the modal matrix is not
defined. To alleviate the problem, we consider the inverse matrices
(4)
K"1 =<J>A"1D"1<I)'r
which are singular, rank p matrices when only p modes are available. Since
ultimately M and K are required, the inversion of these rank deficient
matrices will be necessary. However, first consider the problem of
determining the diagonal normalization matrix D so as to preserve as much
as possible the known kinetic and strain energy characteristics.
Determination of the Normalization Matrix D
In most cases, knowledge of the system energies is limited to that
represented by the analytical mass and stiffness matrices. Allowing for other
known energy characteristics, M is defined as the approximate mass matrix
that represents knowledge of the system kinetic energy, and similarly K is
defined as the approximate stiffness matrix that represents knowledge of the
system strain energy.
Recalling that the strain and kinetic energies associated with an arbitrary
displacement qt or velocity distribution v( are given by l/2qiTKq( and
\/2v^Av{, respectively, the normalization D can be defined by requiring
that
v.TMv. =v.TMv. (5)
and
qftol^ qfKq, (6)
for a total of p arbitrary vectors q( and «,-. If more than p important energy
relationships exist, D can be alternatively defined by requiring that
Equations 5 and 6 be satisfied in the least squares sense.
To preserve the important energies defined by M and K, the arbitrary
vectors should be chosen as meaningful displacement and velocity distribu-
tions for the structural system under examination. Though the optimum
vectors will depend on the specific characteristics of K and M, a logical
choice for most problems can be made by noting that M will approximate
14
the kinetic energies of the low-order modes reasonably well; i.e., <£lrM<£>i is
well described by fyTMfyj for low-order <J>(. For such problems, the measured
</>, are therefore a logical choice for the arbitrary vectors. With this
selection, and the use of Equation 3 for M, the p dt are defined by
d. - <S>. M0. \i = 1. p] (7)l l l * - ' r j x '
Similarly, if a measured mode <£, is used as a qh the ith diagonal element is
defined by
d. = (0TK0.)/X. (8)
Inversion of the Rank Deficient Inverse Matrices
If the number p of vibration modes determined is equal to the number of
measured coordinates n, then Equations 4, and 7 and/or 8 define nonsingular
inverse mass and stiffness matrices that, when inverted, define the measured
frequencies and mode shapes exactly. When p is less than n, the inverse
matrices are singular, rank p, and Equations 7 and 8 generally only
approximate the equalities defined by Equations 5 and 6.
In Reference 1 it was suggested that Equations 4 could be made
invertable by filling out the modal matrix with n-p arbitrary linearly
independent vectors. Though these vectors will be eigenvectors of the
resulting eigensystem, choosing larger A,- for them will make them higher
order modes out of the range of interest. The advantage of this approach is
that the resulting mass and stiffness matrices define the measured frequen-
cies and mode shapes exactly, and Equations 5 and 6 are satisfied exactly for
a total of n vectors. The disadvantage of this technique is the difficulty
sometimes encountered in choosing linearly independent vectors to fill out <I>
that are not nearly dependent on the measured mode shapes, and that do not
lead to ill-conditioned mass and stiffness matrices.
An alternate inversion approach, which is considered here, is a modified
spectral inversion of the rank difficient inverse mass and stiffness matrices.
Recall that these symmetric matrices can be described in terms of their
spectral decompositions (eigenvalues and eigenvectors) as follows
(9)
1=1
15
and
K-'=f;«.?.f.T = ZAZT (10)
1=1
where
|3. = zth eigenvalue of M~' (ordered smallest to largest)
£. = iih eigenvector of M"l (normalized I,-7?,- = 1)
a, = zth eigenvalue of K"1 (ordered smallest to largest)
f, = zth eigenvector of K"1 (normalized f/7"^. = 1)
B = diagonal matrix of (S.
X .= modal matrix of %.
A = diagonal matrix of a.
Z = modal matrix of f.
Since the eigenvectors of symmetric matrices are orthonormal, Xr = X"1, ZT
= Z"1 and thus
M = XB-1XT (11)
and
K=ZA-1ZT (12)
However, because K"1 and M"1 are rank p, a,- = 0 and /8( = 0 [i = 1,
n - p] and A"1 and B"1 are undefined. This is overcome by requiring
StTWst = t i r W i ( i= l ,n - P ]
and
?.TK?.=r.TKf. [ /=! ,„ -p]
Substituting into Equations 11 and 12 gives
a-^f^KfJi^l .n-p] (13)
/
16
(14)
Using a,"1 = l/af and /Jf1 = 1//3,- for i = n - p+ 1, n in conjunction with
Equations 13 and 14 completely defines A-1 and B"1.
The complete algorithm for synthesizing the mass and stiffness matrices
from experimental modal data is defined by Equations 4, 7 and/or 8, 11, 12,
13, and 14.
Example—Application to Roll up Solar Array
As a demonstration, the matrix synthesis technique just described is used
to synthesize the mass and stiffness matrices for the rollup solar array shown
in Figure 1. The array consists of two solar-cell blankets tensioned between
the base and a relatively stiff leading edge beam that is attached to the tip of
the deployed boom (Reference 2). For the purposes of this example, the out-
of-plane dynamic behavior of the array is described by the seven-degree-of-
freedom model in Figure 2. The parameter values associated with the array
are given in Table 1. Note that the actual array is assumed to have unequal
blanket tensions on the two sides and an uneven mass distribution. The first
few eigenvalues (natural frequencies squared) and mode shapes defined by
this model will be used as the "experimentally determined modal data" and
are given in Table 2.
The idealized equal tension uniform mass values in Table 1 are assumed in
the approximate analytical model that defines K and M. For comparison, the
eigenvalues and mode shapes defined by K and M are given in Table 2. Note
that this "approximate" model leads to a pair of equal eigenvalue pure
blanket resonances in addition to the pure symmetric and antisymmetric
BOOM
BLANKETS
SUPPORT
STRUCTURE
SUNLIGHT
BEAM
Figure 1. Deployed array
17
Table 1. Solar array parameter values
Parameter
Length
Width
Boom mass
Boom bending stiffness ( El )
Boom torsional stiffness (GJ)
Leading edge beam mass
Boom tip mass
Blanket mass:
Outer half left side
Inner half left side
Outer half right side
Inner half right side
Blanket tension:
Left side
Right side
Units
m
m
kg
Nm2
Nm2
kg
kg
kg
kg
kg
kg
N
N
Actual
array
1.22
0.61
0.56
18.6
0.
0.41
0.04
0.95
0.86
0.95
0.86
9.3
7.1
Approximate
model
1.22
0.61
0.54
- 16.5
0.
0.45
0.0
0,91
0.91
0.91
0.91
8.9
8.9
Figure 2. Finite element model of array
18
Table 2. Eigensolutions for example array
"Measured" eigensolution of actual array
0.787 + 01
1
0.504
0.002
0.002
-0.462
0.003
0.001
0.730
0.811 + 01
1
-0.479
-0.000
-0.000
0.479
-0.000
-0.000
-0.735
0.787+01
1
0.504
0.002
0.002
-0.462
0.003
0.001
0.730
0.137 + 02
2
0.414
0.455
0.388
0.423
O'.SOS
0.192
-0.045
0.121+02
2
-0.407
-0.407
-0.407
-0.407
-0.542
-0.208
0.000
0.137+02
2
0.414
0.455
0.388
0.423
0.503
0.192
-0.045
0.392 + 02
3
0.589
-0.797
0.045
0.123
0.015
0.006
-0.026
Eigensolution
0.483 + 02
3
0.130
0.094
-0.801
0.577
0.000
0.000
-0.000
Eigensolution
0.392 + 02
3
-0.589
0.797
-0.045
-0.123
-0.015
-0.006
0.026
0.493 + 02
4
-0.228
-0.084
0.785
-0.567
0.029
0.011
0.050
0.758+02
5
0.660
0.237
0.408
-0.319
-0.236
-0.088
-0.421
0.924 + 02
6
-0.157
0.184
0.272
0.777
-0.240
-0.089
0.445
0.801 + 04
7
0.018
0.018
0.018
0.018
-0.075
-0.996
0.000
from approximate model
0.483 + 02
4
-0.587
0.789
-0.021
-0.182
-0.000
-0.000
-0.000
0.878+02
5
0.455
0.455
0.455
0.455
-0.389
-0.145
0.000
0.885+02
6
-0.603
-0.000
-0.000
0.603
0.000
0.000
0.523
0.720+04
7
0.018
0.018
0.018
0.018
-0.072
-0.997
-0.000
from synthesized model
0.493 + 02
4
0.228
0.084
-0.785
0.567
-0.029
-0.011
-0.050
0.537 + 02
5
0.506
0.331
-0.013
-0.344
-0.387
-0.212
-0.567
0.881 + 03
6
0.189
0.055
-0.352
-0.590
0.457
0.158
-0.505
0.121+04
7
0.012
0.006
-0.010
-0.020
-0.350
0.936
-0.022
aThe eigenvalues are listed horizontally using scientific notation and the eigenvectors
are listed vertically beneath them.
structural modes. These are different from the modes of the "actual" array
as given in Table 2.
The problem is to synthesize new mass and stiffness matrices that better
approximate the assumed "measured" modal data, and that remain
compatible with the finite element analysis used to define the approximate
model. Using the previously derived algorithm in conjunction with the first
four measured modes (p — 4) gives new matrices for the structure. The
eigenvalues and mode shapes defined by these synthesized mass and stiffness
19
matrices are given in Table 2, and compare quite favorably with the
"measured" modal data.
To demonstrate the compatibility with the approximate finite element
model, the effect of assuming a 1.36 Nm/rad (1 ft-lb/rad) torsional stiffness
for the boom and adding a 0.45 kg (1 Ib) lumped mass to the tip of boom is
considered. Adding the appropriate mass and stiffness terms to the "actual,"
approximate, and synthesized matrices leads to the modified eigenvalues and
mode shapes given in Table 3. From these results, it is clear that the
Table 3. Eigensolutions for modified array
"Measured" eigensolution of modified array
0.116+02
1
0.443
0.435
0.383
0.377
0.534
0.204
0.009
0.240 + 02
2
0.707
-0.049
-0.033
-0.548
-0.029
-0.011
0.442
0.392 + 02
3
-0.552
0.820
-0.052
-0.138
-0.017
-0.006
0.029
Eigensolution from
0.101 + 02
1
0.397
0.397
0.397
0.397
0.568
0.218
-0.000
0.246 + 02
2
0.619
0.000
-0.000
-0.619
0.000
0.000
0.483
0.483+02
3
0.500
-0.500
-0.500
0.500
0.000
0.000
-0.000
Eigensolution from
0.116 + 02
1
-0.411
-0.426
-0.401
-0.415
-0.525
-0.201
-0.005
0.237 + 02
2
0.733
0.048
-0.092
-0.446
-0.099
-0.056
0.491
0.392 + 02
3
-0.557
0.818
-0.055
-0.126
-0.024
-0.011
0.012
0.495+02
4
-0.199
-0.071
0.757
-0.616
0.025
0.009
0.045
0.702 + 02
5
0.533
0.336
0.636
0.264
-0.294
-0.110
-0.172
0.117 + 03
6
-0.400
0.022
0.029
0.591
-0.036
-0.013
0.699
0.782+04
7
0.013
0.013
0.013
0.013
-0.054
-0.998
0.000
modified approximate model
0.483 + 02
4
-0.316
0.632
-0.632
0.316
0.000
0.000
-0.000
modified
0.752+02
5
0.474
0.474
0.474
0.474
-0.297
-0.111
-0.000
0.117 + 03
6
-0.524
-0.000
-0.000
0.524
0.000
0.000
0.671
0.704+04
7
0.013
0.013
0.013
0.013
-0.052
-0.998
0.000
synthesized model
0.495 + 02 0.627 + 02
4
0.188
0.055
-0.778
0.596
-0.001
0.004
-0.018
5
0.406
0.228
-0.120
0.146
-0.281
-0.153
-0.803
0.705+03
6
0.174
0.060
-0.398
-0.583
0.374
-0.192
-0.539
0.113+04
7
0.076
0.023
-0.110
-0.206
-0.120
0.944
-0.186
aThe eigenvalues are listed horizontally using scientific notation and the eigenvectors
are listed vertically beneath them.
20
modified synthesized model provides a much better description of the
modified structure than does the modified approximate model. Though the
agreement between the modified synthesized and modified exact models is
not perfect, it is considered good in light of the size of the error in the
starting approximate model and the size of modification that was made.
References
1. Ross, R. G., "Synthesis of Stiffness and Mass Matrices from Experimen-
tal Vibration Modes," Paper No. 710787, presented at the Society of
Automotive Engineers 1971 National Aeronautics and Space Engi-
neering and Manufacturing Meeting, Los Angeles, California, Sept.
28-30, 1971. To be published in SAE Transactions, Vol. 80, Society of
Automotive Engineers, Inc., New York.
2. Final Report, Design and Development of a 30 Watt per Pound, 250
Square Foot Rollup Solar Array, Document 705D4286. Prepared for
the Jet Propulsion Laboratory by Missiles and Space Division, General
Electric Co., Philadelphia, Pennsylvania, Dec. 1,1970.
21


An Algorithm for Synthesizing Mass and Stiffness Matrices from Experimental Vibration Modes

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