A simplified method of computing eigensolution sensitivity derivatives in structural dynamics is developed. It is shown that if the elements of stiffness and mass matrices associated with a design variable are homogeneous functions of that design variable, then eigenvalue derivatives can be computed from element strain and kinetic energies. Furthermore, if cross-mode energies are known, eigensolution derivatives of modified systems can be computed approximately using assume mode reanalysis formulation. A ten bar truss example is used to illustrate the present formulations

Wang, B. P.

English

NASA Technical Reports Server

N87-18870
ON COMPUTING EIGENSOLUTION SENSITIVITY DATA USING FREE
VIBRATION SOLUTIONS
B. P. Wang
Department of Mechanical Engineering
The University of Texas at Arlington
Arlington, Texas
SUMMARY
A simplified method of computing eigensolution sensitivity derivatives in
structural dynamics is developed in this paper. It is shown that if the elements
of stiffness and mass matrices associated with a design variable are homogeneous
functions of that design variable, then eigenvalue derivatives can be computed from
element strain and kinetic energies. Furthermore, if cross-mode energies are known,
eigensolution derivatives of modified systems can be computed approximately using
assume mode reanalysis formulation. A ten bar truss example is used to illustrate
the present formulations.
INTRODUCTION
The usefulness of eigensolution sensitivity derivatives in structural dynamics
research is well known. The sensitivity data can be used for approximate
reanalysis, analytical model improvement, assessment of design trend as well as
structural optimization with eigenvalue constraints. When applied to larger discrete
structural models, these applications typically require long and expensive computer
runs and usually the predominate contributor to the computing time was the
calculation of derivatives. Thus efficient eigensolution sensitivity analysis
procedures would be very useful in structural dynamic research. It is the purpose
of this paper to develop, under certain conditions, efficient eigensolution analysis
procedures using free vibration data.
The equations for computing derivatives of eigenvalues and eigenvectors for free
vibration of undamped structures were known for a long time. Only recently have
these methods been implemented in some general-purpose finite-element programs. In
this paper, a simple method is developed which can be used to compute the eigenvalue
derivatives for a large class of problems by exploiting the similarity between the
equations for eigenvalue derivatives and element strain and kinetic energies.
Furthermore, if the cross-mode element energy data are available, the approximate
eigenvector derivatives can also be computed using a truncated modal expansion
expression. The approximate second derivatives of eigenvalues can then be computed.
Additionally, with the cross-mode strain energy data, the eigenvalue derivatives of a
modified structure can be computed using assumed mode reanalysis formulation.
Numerical examples will be presented to illustrate the various formulations.
233
https://ntrs.nasa.gov/search.jsp?R=19870009437 2020-03-20T11:34:29+00:00Z
EIGENSOLUTIONSENSITIVITYIN STRUCTURALDYNAMICS
The general problem is to compute the rate of change (or derivatives) of
eignevalues and eigenvectors with respect to design variables for the following
generalized eigenvalue problem in structural dynamics.
K¢=AM¢ (1)
Much research has addressed this problem in the past two decades. A
comprehensive survey of literature can be found in a recent paper by Adelman and
Haftka []]. The equations for first order eigenvalue and eigenvector derivatives
as well as second order eigenvalue derivatives are summarized below:
Eigenvalue Derivative:
aA£ T 8K T aM
r r r
(2)
Eigenvector Derivative:
a¢£ n
a-x- = _ A£ij Cj
r j=l
(3)
where for £ _ j
T 8Z£
ARr j = Cj _ ¢_/(_2-_j)
r
(4)
and
ZR = K - h£M
1 T aM
= - i a--f-
r
(5)
(6)
Second Derivative of Eigenvalues:
a2k£ T aZ£ 8¢1 aZ l 8¢i.
8x 8x - Y£ + ¢2 (Sxx 8x + 8x _- )
r s r s s r
(7)
where
.T. a2K a2M aA£ aM aA£ aM
Y£ = _£t_-xS-xx k£ ax ax ax ax ax ax ) CR
r s r s s r r s
(8)
Note that in the above equations, the mode shapes are normalized to unit generalized
mass, i.e.
T
cl'£ M¢£ = 1.0
234
For the eigenvector derivatives, if less than full modes are used, Eq. (2) is an
approximate expression. These will lead to approximate second order derivatives of
eigenvalues.
The above equations have been developed in the literature for some time. For
example, Equations (2) and (3) can be found in Fox and Kapoor [2] and Eq. (7) was
reported by Miura and Schmit [3]. It should be noted in passing that there are some
recently developed algebraic methods [4-5] which can be used to compute eigenvector
derivatives without using modal expansions.
The difficulty of applying the aforementioned equations appears to be the cal-
culation of derivatives of stiffness and mass matrices with respect to design vari-
ables. In the next section it will be shown that under certain assumptions, we can
circumvent the calculation of _K/_x i and _M/_x i in implementing these above equations.
SIMPLIFYING ASSUMPTIONS
In general, the system stiffness and mass matrices in Eq. (I) can be written as
ND
K=K + _ K. (9)
c 1i=l
ND
M = M + Z M. (lO)
c 1i=l
where
K = contribution to stiffness matrix due to structural elements that are to
c
remain constant during the design process.
M = contribution to mass matrix due to the masses of the unchanged elements as
C
well as nonstructural masses.
Ki,M i = contributions to stiffness and mass matrices respectively due to elements
controlled by design variable x..
1
To develop simplified efficient methods for eigensolution sensitivity analysis,
the elements of the matrices K. and M. are assumed to be homogeneous functions of
1 . 1
design variables. That is the matrzces K. and M. have the form
1 1
)_i *K. = K. (11)i (xi i
Yi *
M. = ) Mi (xi i
where K. and M. are constant matrices.
1 1
parameters
(12)
Furthermore, define non-dimensional design
235
xi Pi
i (xi0) (13)
(14)
Then
or
K. = (x i) K. = (x_ 0) (xi0) K.1 1 1
K = u. (15)i i Ki0
Similarly,
M = 8. (16)i i Mi0
where Ki0 and Mi0 are stiffness and mass matrices due to design variable x. at itsi
nominal value xi0.
Based on the above assumptions, the derivatives of stiffness and mass matrices
with respective to design variables can be computed readily:
8K. Be.
8K i i
8x. - 8x. - 8x. Ki0
1 1 1
or
8K. _i
8x. - x. Ki0
1 1
(17)
Similarly, we can derive
8M _i
8x. = _. Hi0
i 1
(18)
It should be noted that at the nominal design, _. = _. = I.
1 1
tions, the eigenvalue derivatives can he computed readily.
With these simplifica-
RATE OF CHANGE OF EIGENVALUES
Using (17) and (18) with a = _i = I, the eigenvalue derivative, eq. (2) becomes
8A£ _r T Yr T
8x - x #_ Kro _ x A_ #_ Mro #_ (19)
r r r
236
Define
1 T
V_r _ = _ #_ Kro 4p_ (20)
1 _£ TT£r £ = _ #£ Mro #_ (21)
Then Eq. (19) can be written as
0k_ _r Yr
0-x- = 2(_- V£r _ x T_r£)
r r r
(22)
Note that from Eqs. (20) and (21), V_r £ and T£r £ can be interpreted as the strain and
kinetic energy respectively of elements associated with design variable x
r"
Thus, given _r and y , the rate of change of eigenvalues can be computed fromr . . .
the energies assoclated with thls design varlable. Since most general-purpose
flnite-element codes provide element strain energy as an output option, one way to
implement (22) is to calculate V£r _ and T_r £ by summing strain energy and kinetic
energy for all elements controlled by design variable x
r"
In the above formulation, we have made use of the form of the stiffness and mass
matrices, Eqs. (II), (12). Not all structural elements can fit into these models but
some important cases do. Some of these are tabulated in Table I.
Using Eqs. (15) to (18), it is possible to derive explicit equations for
eigenvector derivatives as well as second-order derivatives of eigenvalues in terms
of energies associated with various design variables. These are quite tedious and
have not been accomplished so far. In the following, we will discuss the special
case of _i = Yi"
EIGENSOLUTION SENSITIVITIES FOR THE CASE _i = Yi
as:
For this special case, we can use chain rules to rewrite sensitivity derivatives
8A£ OA£ 8a r
Ox O_ Ox
r r r
(23)
Ox O_ Ox
r r r
02A_ 02A_
Ox Ox Ox Ox Ox Ox
r s r s r s
Thus, it remains to find Ok_/O_r, O#Ok£/O_ r and O2h_/OarOas .
Note that
(24)
(25)
237
OK
r
O_ Kro
r
(26)
OM
r
= MrO (27)
r
Using Eqs. (26), (27) and replacing all Xr, xs in Eqs. (2) to (8) by ar and _s' and
making use of the orthogonality properties of normal modes
T
_ M_j = 0 if _ _ j
as well as the linearity assumptions (Eqs. (15) and (16)), we can derive, after
considerable algebraic manipulation, the following results:
aA_
O_ - C_r _ (28)
r
aOl n
as -
r j=l C_rj _J
(29)
where
82A£ aA_
as as - 2[(8-_- C£r£ +
r s s
C£r j = 2(Vir j - Tir j)
OAg_ n _
C£s_) + _ C£r j Clsj]
r j=1
j_
(30)
(31)
1 T
V_r j = _ (_i Kro (_j (32)
:1A£ TT_rj 2 _ Mro Oj (33)
-- C_r J
C£r j = A£ - A.
3
# j (34)
T_r_
C£r£ - A_
(35)
It should be noted that V_r j can be considered as the "cross mode" strain
energy, since it is the work done by the elastic force in jth mode (i.e. Kr_j) moving
through displacement in the _th mode. Similarly, T_r j can be considered as the
"cross mode" kinetic energy. Thus, eigensolution sensitivity derivatives can be
computed readily when these energy terms become available.
238
SENSITIVITYDERIVATIVESFORMODIFIEDSYSTEMS
In iterative analysis, we frequently require the eigensolution derivatives of a
system different from the nominal design. In these situations, assumed mode
reanalysis [6-7] appears to be very efficient. Let AK and AM denote the change to
stiffness and mass matrices, respectively. Then, in term of ei, we have
and
ND
AK = _ (_i- l)Kio (36)
i=l
ND
AM = _ (_i- l)Mio (37)
i=1
Following the development in Ref. 7, the eigensolution of the modified system
can be computed approximately by solving the following reduced eigenvalue problem
q = k M q (38)
where
= sT(K + AK)$
= [AO ] + _ (_i - 1) K'I (39)
= [I] + _ (oi - 1) M'I (40)
where _ is the truncated modal matrix of the original system, and
_. = sT $ (41)
1 KiO
_. = oT _ (42)i Mi0
Once (37) is solved, the eigenvectors of the modified system Oi' in terms of physical
coordinates, can be completed from
_i = _ qi (43)
For modified systems, the eigenvalue derivatives, Eqs. (28) and (30) are still
applicable except V_r j and T£r j are now defined by
1 T
V£rj = 2 q£ Kr q_ (44)
= ! A£ q_ q£T£rj 2 Mr
and the eigenvector derivatives can be computed from
O_g 8q£
r r
(45)
(46)
239
where
aq_ n
a--_-= _ qj
r j=l C_rj (47)
R
C£r j is as defined by (34) or (35) with V_rj, T£r j defined by (44) and (45).
DISCUSSION
In Eq. (7), the second-order derivatives of eigenvalues are shown to be
dependent on eigenvector derivatives. In the present formulation, we can compute
a2A_/a_r a_s Using Eq. (34) without the need to compute eigenvector derivatives
explicitly. Once the derivatives with respective to _'s are known, chain rules can
be used to compute the derivatives with respective to design variables x's (Eqs.
(23) to (25)).
NUMERICAL EXAMPLE
The assumed mode reanalysis sensitivity derivative formulation has been
implemented in a program which post-processes MSC/NASTRAN generated data. The first-
order sensitivity data have been applied to improve analytical model using measured
modal data [8] as well as synthesis of structures with multiple frequency
constraints [9]. Recently the second order derivatives of eigenvalues (Eq. (30))
has also been implemented.
A ten-bar cantilever truss structure, Fig. I, is used to test the program. The
ten members are grouped into 4 design variables as indicated in Figure I. Starting
with a uniform structure with cross sectional area 10 in 2 for all design variables,
the optimal design program described in Ref. 7 is used to mode the first two
natural frequencies from 13.3 and 37.8 Hz to 16 and 39.3 Hz, respectively. This is
accomplished by a sequential linear programming formulation [7,9]. At each
intermediate design, the eigenvalue derivatives are computed using reanalysis
formulations. Table 2 defines the design history. Specifically, the designs at
iteration No. A-O and B-O are analyzed exactly using MSC/NASTRAN. Three iterations
are shown after each exact analysis. The eigensolution at designs A-I to A-3 and
B-I to B-3 are computed using assumed mode reanalysis formulations. Four modes are
used in each case. The first two natural frequencies are tabulated in Table 3.
Also shown in Table 3 are the corresponding exact frequencies. From Table 3, it can
be seen that the accuracy in frequency of assumed mode reanalysis formulation is
very good. Tables 4 to 7 summarize ali/al,ali/ax2,a_/ax I and a_/ax2,
respectively. The results of these tables indicate that the sensitivity derivatives
of modified system can be predicted quite accurately using the assumed-mode
reanalysis formulation.
240
CONCLUDINGREMARKS
General procedures for computing eigensolution sensitivity derivatives for a
class of problems have been proposed in this paper. Detailed formulations have been
carried out for a special case. It is shownthat the eigenvalue derivative with a
design variable can be computedfrom strain energy and kinetic energy for that design
variable. Furthermore, when the cross modeenergy terms are available, assumedmode
method can be used for eigensolution as well as associated sensitivity reanalysis.
This efficient formulation has proved to be very effective in synthesis of structures
with multiple frequency constraints [7,9]. Additionally, the present approach can
be implemented in a post-processor of any finite-element programs without the need
to modify the source code.
Since the current formulation provides an efficient approach for computing
second-order eigenvalue derivatives, it would appear that a second-order method for
structural optimization with frequency constraints could be implemented efficiently.
Finally, in view of the success of the formulation for the special case of _i = Yi'
further development for the general case of _i ¢ Yi seemsto be warranted.
SYMBOLS
K = system stiffness matrix
M = system massmatrix
_ = eigenvector of the £th mode
A_ = eigenvalue of the £th mode
= modal matrix of original system
V£r j = cross-mode strain energy
T_r j = cross-mode kinetic energy
X
r
AK
AM
N
n
ND
= rth design variable
= modification in stiffness matrix
= modification in mass matrix
= eigenvector of the th mode of the modified systems
= number of dof of the system
= number of modes computed, n < N
= number of design variables
241
I °
.
°
.
.
.
.
.
REFERENCES
Adelman, H. M. and Haftka, R. T., "Sensitivity Analysis of Discrete Structural
Systems," AIAA Journal, Vol. 24,No. 5, May 1986, pp. 823-832.
Fox, R. L. and Kapoor, M. P., "Rates of Change of Eigenvalues and Eigenvectors,"
AIAA Journal, Vol. 6, No. 12, December 1968, pp. 2425-2429.
Miura, H. and Schmit, L. A., "Second Order Approximation of Natural Frequency
Constraints in Structural Synthesis," International Journal for Numerical
Methods in Engineering, Vol. 13, 1978, pp. 337-351.
Nelson, R. B., "Simplified Calculation of Eigenvector Derivatives," AIAA
Journal, Vol. 14, No. 9, September 1976, pp. 1201-1205.
Chen, S. Y. and Wei, F.S., "Systematic Approach for Eigensensitivity Analysis,"
Proceedings of AIAA/ASME/ASCE/AHS 26th Structures, Structural Dynamics and
Materials Conference, Part 2, Orlando, Florida, April 15-17, 1985, pp.
178-183.
Wang, B. P., Pilkey, W.D. and Palazzolo, A. B., "Reanalysis, Modal Synthesis and
Dynamic Design," Chapter 8 of State-of -the -Art Survey on Finite Element
Technology, Edited by A. K. Noor and W. D. Pilkey, The American Society of
Mechanical Engineers, 1983, pp. 225-295.
Wang, B. P., "Structural Dynamic Optimization Using Reanalysis Techniques,"
ASME AMD-vol. 76, Reanalysis of Structural Dynamic Models (B. P. Wang, ed.),
1986, pp. 1-19.
Wang, B. P., "Synthesis of Structures with Multiple Frequency Constraints," AIAA
Paper No. 0951-CP, Proceedings of AIAA/ASME/ASCE/AHS 27th Structures, Struc-
tural Dynamics and Materials Conferences, Vol. I, pp. 394-397.
9. Wang, B. P., Chen, T. Y. and Chu, F. H., "Model Refinement Using Test Data,"
Proceedings of 4th International Modal Analysis Conferences, Los Angeles, CA
1986 Vol. II, pp. 1052-1057.
242
TABLEI. - STIFFNESSANDMASSEXPONENTSFOR
SEVERALCOMMONSTRUCTURALE EMENTS
Element Design variable _ y
Truss Cross 1 1
Membrane Thickness 1 1
Plate bending Thickness , 3 1
Beambending Cross-sectional area 2 1
Beambending Section area moment
of inertia* 1 0.5
a,
Circular cross section
TABLE II. - DESIGN HISTORY OF
TEN BAR TRUSS
Iteration
No.
Xl 2 x3 x4
A-0 I0.0 I0.0 I0.0 I0.0
A-I 12.76 8.88 5.0 5.0
A-2 I0.39 8.15 5.0 5.0
A-3 10.44 8.29 5.0 5.0
B-0 10.44 8.29 5.0 5.0
B-I 7.92 7.80 3.44 3.44
B-2 7.47 7.18 2.90 2.90
B-3 7.19 7.0 2.68 2.68
TABLE III. - COMPARISONS OF NATURAL FREQUENCIES
fl(HZ) f2(Hz)
Case Error
No. Approximate Exact (%) Approximate Exact
Error
(%)
A-I 16.88 16.60 1.71 40.16
A-2 15.96 15.78 1.14 39.85
A-3 15.99 15.82 1.10 40.00
B-I 15.90 15.89 0.II 40.23
B-2 15.99 15.98 0.03 39.98
B-3 15.99 15.99 0.00 40.00
39 03
38 99
39 II
39 87
39 44
39 33
2.88
2.19
1.27
0.88
1.37
1.71
TABLE IV. COMPARISON OF 8Al/SX 1 TABLE V. - COMPARISON OF 8_i/8x2
Case 8X/_x 8_/_x
No.
Eq. (28) Exact
Error
(%)
A-I 418.9 454.0
A-2 494.2 509.2
A-3 495.8 505.9
B-I 557.2 578.3
B-2 607.3 609.2
B-3 635.8 620.0
7.7
2.9
1.9
3.7
0.3
2.6
Case 8A/Sx 8_/8x Error
No. (%)
Eq. (28) Exact
A-1 216.7
A-2 208.8
A-3 202.3
B-I 203.8
B-2 203.8
B-3 196.0
294 6
269 9
262 7
205 2
222 9
221 0
26.5
22.5
23.5
0.7
8.6
11.3
243
TABLEVI. - COMPARISONF8k2/SXl TABLEVII. - COMPARISONF8k2/Sx2
Case 8hlNx 8E/_x Error
No. (%)
Eq. (28) Exact
A-1 -812.2
A-2 -747.4
A-3 -764.7
B-I -1232.3
B-2 -1387.7
B-3 -1482.4
-786 9
-793 9
-805 2
-1247 8
-1320 2
-1435 7
3.2
5.8
9.0
1.2
5.1
3.2
Case 8AlSx 8A/Sx Error
No. (%)
Eq. (28) Exact
A-I 3337.9
A-2 3678.3
A-3 3647.5
B-I 3523.5
B-2 3874.9
B-3 4025.3
3652 3
2917 2
2876 9
2928 3
3038 0
3043 1
31.35
26.1
26.8
20.3
27.5
32.3
244
Y2
360 in
2
7 8 9 10
>I
I
6 360 in
4
Modulus of elasticity = 104 ksi
Material density = 0.i ibm/in 3
Design variable x. = 1,2,3,4
1
x2 = 5,6
x3 = member 7,8,9,10
Figure I. Ten Member Cantilever Truss
Yl
245


On computing eigensolution sensitivity data using free vibration solutions

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