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Structural modification with frequency response constraints for undamped mdof systems
Computers & Sfrucrures Printed in Great Britain.
Vol. 36, No. 6. pp. 1013-1018.
STRUCTURAL RESPONSE
1990
CUM57949/w s3.cnl+ 0.00 Pergamon Press plc
MODIFICATION WITH FREQUENCY CONSTRAINTS FOR UNDAMPED MDOF SYSTEMS T. Y. CHEN
Department
of Mechanical Engineering, National Chung-Hsing University, Taichung, Taiwan 40227, Republic of China (Received 8 September 1989)
Abstract-A structural modification method which aims to change the structural design to match the desired frequency response behaviors is introduced. The calculations of frequency response sensitivities and reanalysis of frequency responses of a modified structure are efficiently carried out in a reduced modal space. Either a least square solution or an optimal solution with minimum structural weight subjected to various types of frequency response constraints can be obtained through an iteration loop which repeatedly uses the frequency response sensitivity data and reanalysis technique to achieve the design goal. A tremendous cost saving is expected when this is implemented in practical engineering design.
INTRODUCTION
Finite element models have been widely used to predict the eigenvalues, eigenvectors and structural responses in structural dynamics. For most engineering designs, the initial finite element model must be subjected to many modifications before it satisfies the design requirements, which may include eigenvalue and/or response constraints. For other applications, the accuracy of an existing finite element model is so important so that it must be tested against experimental data. Under either situation, the modification of a finite element model in order to match the design requirements or test data for a large structure is often a time consuming and frustrating job. Sometimes it is even impossible to reach a satisfactory result by limited experience. Sensitivity analyses of some structural data with respect to variations in design parameter values have proved to be a powerful tool leading to a systematic method of solving structural modification problems. The success of structural modification is therefore more or less based on the proper use of sensitivity data as well as the calculations of these data. The satisfaction of all the requirements is often not achieved in a single modification attempt. Several iterations of the modification procedure are necessary. The outcome of each iteration has to be analyzed to see whether the requirements have already been satisfied. The finite element analysis of a large structure is costly. An efficient reanalysis of the modified structure is certainly indispensible. In previous papers, Fox and Kapoor [ 1] derived the sensitivities of eigenvalues and eigenvectors with respect to variation in an arbitrarily designated design parameter. These sensitivity data are a great help to the understanding of the dynamic characteristics of the system. As a result, the structural refine-
ment referring to these data to achieve the design goal becomes much easier. Sharp and Brooks [2] developed a method of calculating the sensitivities of frequency response functions. These data predict the frequency response behavior of a structure when subjected to modifications. Wang et al. [3,4] introduced an efficient re-analysis approach to compute the eigenvalues and eigenvectors for a modified structure. This approach solves the eigenvalue problem in a reduced space and thus avoids the time consuming process of solving the eigenvalue problem in the original global space. Luk and Mitchell [5] solved the reanalysis problem of steady state response by using modal spaces I and II. This method gives the steady state response for the modified system in modal space. Although these previous research outcomes, if properly implemented, will help the structural design engineers to accomplish their work, a comprehenvise and systematic way of attacking the structural modification problems is yet to be established. In the present paper a systematic procedure is created to perform the structural modifications iteratively to match the specified frequency response requirements through efficient computations of frequency response sensitivity data and reanalysis in a reduced space. THEORY
The frequency response function of &undamped MDOF system can be calculated by modal summation as [6]
where H,(n) is the frequency response function at the ith DOF when subjected to a unit input at the jth
1013
1014
T. Y.
DOF; n is the number of modes used in calculating the response function; c$,,represents the value of the ith DOF in the rth mode shape; r#~,~ represents the value of thejth DOF in the rth mode shape; 1, stands for the rth eigenvalue and Q is the excitation frequency in radians. In eqn (1) mode shapes are assumed to be normalized such that {4,}[M]{$,} = 1, where {4r} is the rth mode shape and [M] is the global mass matrix of the structure. The sensitivity of frequency response function with respect to a chosen design parameter is derived as follows:
&EN
Refinement or modification of a structure is seldom accomplished in a single try. Several iterations may be needed. The effect of each modification can only be determined by analyzing the modified structure. The more times the whole modified structure is reanalyzed, the more inefficient the modification process. For this reason, Wang et al. [3] developed an efficient reanalysis in a reduced space called modal space to analyze the eigenvalues and eigenvectors of a modified system. Using this approach, the formulation of the modified eigenvalue problem is much smaller in dimension than the original one. As a result, the saving in the computer time is significant. This efficient reanalysis is repeated here. The eigenvalue equation of the original system is [Kl{&I = wfl~~,N~
(2) where CQis the k th chosen design parameter. To obtain the sensitivity data of the frequency response function, the eigenvalue and eigenvector sensitivities of the original system are required. The computations of these data were introduced by Fox and Kapoor [l]. A brief summary of their results is shown here for convenience. The rth eigenvalue sensitivity is obtained by the following equation:
where [K] is the stiffness matrix of the original design; [M] is mass matrix of the original design; (4,) is the rth mode shape; and I, is the rth eigenvalue. After modification, the modified system has stiffness matrix [K + AK] and mass matrix [M + AM]. The new eigenvalue problem becomes [K + AKl(4:) = bf + A~l{&“}C’,
(8)
where (c#J~}is the modified rth mode shape; ;Lyis the modified rth eigenvalue; [AK] is the amount of change of the stiffness matrix; and [AM] is the amount of change of the mass matrix. Using assumed mode approach, the modified rth mode shape is assumed to be {&Y = [@I{%)>
where {&} is the rth mode shape; [K] is the global stiffness matrix; [M] is the global mass matrix; and i, is the rth eigenvalue. The rth eigenvector sensitivities are expressed as a linear combination of mode shapes.
(7)
(9)
where [@I is the modal matrix of the original system and {qr} is the rth generalized coordinate. Substituting eqn (9) into eqn (8) and premultiplying eqn (8) by [a]’ yields
or (4) where a,,~ are the weighting coefficients for the sensitivity of the rth mode shape. Forr#l
(5) For r = 1
(6)
WI + PlrPKIPI)~qJ
The resultant matrix in parenthesis on the left hand side represents the stiffness matrix of the modified system in a reduced modal space while the matrix in parenthesis on the right hand side stands for the mass matrix of the modified system in modal space. The eigenvalues of the modified system can be obtained directly from eqn (10). The eigenvectors of the modified system can be recovered by eqn (9). Taking advantage of this modal reanalysis, the sensitivity calculation of the frequency response function for a modified system is derived below.
Structural modification with frequency response constraints Employing the equation for frequency response sensitivity of the original system for the modified system, the sensitivity of frequency response for the modified system can be expressed to be
JHT(R)
(11)
ax,
where H;(R) is the frequency response function of the modified system; ~$7 is the value of the ith DOF of the rth mode shape for the modified system; 4:: is the value of the jth DOF of the rth mode shape for the modified system; and J.y is the rth eigenvalue of the modified system. The modified mode shapes and eigenvalue in eqn (11) are obtained from efficient reanalysis, i.e.
where c#+,is the value of the ith DOF of the Ith mode shape for the original system; 4,, is the value of the jth DOF of the ith mode shape for the original system; and q,, is the value of the lth DOF of the rth generalized coordinate. Plugging eqns (12) and (13) into eqn (11) results in the sensitivity of frequency response function of the modified system.
for r = i err = - f{4,)WlT In order to verify the effect of each modification, the reanalysis of the frequency response function of the modified system is necessary. To avoid computing this response function in original global space, the following equation uses the mode shapes and eigenvalues from modal reanalysis to formulate the reanalysis of the frequency response function of the modified system.
Applying these derived equations, an efficient structural modification procedure can be established. For practical application of this proposed modification approach, the modifications of the stiffness and mass matrices are assumed to be the sum of local modifications. [AK] = tll [k,] -t x&Q +
. + a,[k,,J
(20)
and
[AMI= @,b,l+azbzl+
. . + a,h,,l,
(21)
(14) -
(A, -
W)2
.
In eqn (14) the sensitivities of the generalized coordinate {q,} and the eigenvalue of the modified system appear. These sensitivity data are now derived. The associated eigenvalue problem of the modified system has been shown previously by eqn (10). In that equation the stiffness matrix and mass matrix of the modified system in modal space are respectively. Extending Fox and Kapoor’s work for the WI + PIT[AfWI) and (VI + Pl%WPIh sensitivities of modal data to modal space, the sensitivities of 1: and {q,} can be obtained by the following steps.
2
= k,)‘$
(VI + PlPWPlNq,I
- Wq,Y&
WI + PlWflPl){d
(15) The sensitivities of {q,} are assumed to be
(16) for r #i
a,, =
+2
(PI + PlWflPl1d9 k
(17)
1016
T. Y.
where CL, is the design parameter; [AK] is the amount of modification of the stiffness matrix; [AM] is the amount of modification of the mass matrix; [k,] is a constant matrix for modification of the stiffness matrix; and [m,] is a constant matrix for modification of the mass matrix. Employing all the previous efforts in this paper, an iteration loop is created as follows. (1) Initialize all the {q}s. Let the rth entry of {q,} be 1 and all other entries 0. (2) Select constant matrices [k]s and [m]s in eqns (20) and (21). (3) Compute sensitivities of frequency response function with respect to each chosen design parameter using eqns (14)-(18). (4) Solve for the values of x by using the following linear equation:
where Hi(n) is the desired frequency response value and H,(R) is the current frequency response function. Equation (22) may be more than one equation. It depends on the requirements of the modification problem. However, if the number of equality constraint equations (22) is greater than the number of selected design parameters, the least square solution for GIis used. If, on the other hand, the number of equations (22) is less than the number of design parameters, an optimal solution may be obtained by setting up an objective function. (5) Plug the c(s obtained from previous steps into eqns (20) and (21) to compute [AK] and [AM] and then use eqns (IO) and (19) to solve for the frequency response of the modified system H,(R). (6) Convergence check. See if H,,(R) - H;(Q) H;(n)
,-’
(23)
for all the constraint equations. If yes-+stop. Otherwise, go to step (3). The final answers for s(s are expected in a few iterations. Since the dimension of the modal space is generally much smaller than that of the original structure, the approximate solutions from modal space reanalysis result. Therefore the verification of outcomes using updated data in global space is necessary. For each verification run, a new set of modal data is created and the modal space iteration procedure can be applied again. After several verification runs, the best result can be obtained. NUMERICAL
&EN
constraints. The method used to optimize the solution is sequential linear programming (SLP) [A. The subroutine DDLPRS from the IMSL library performs the linear programming. The equality constraint represented by eqn (22) is extended to include inequality and range constraints. The side constraints are also imposed on each design variable. The configurations of these structures are shown in Figs I and 2, respectively. The first example is a IO-bar plane truss. The initial design is a uniform cross-section of 1.0 in* for all members. The constraints are the frequency response of node 2 in the x2 direction less than 0.4 x 10m4in. and the frequency response of node 4 in the x2 direction between 0.55 x 10e4 and 0.65 x 10m4 in. when subjected to a unit harmonic excitation force at node 2 in the x2 direction with an excitation frequency of 10 Hz. The side constraints of lower and upper bounds are 0.5 and 2 in* for all the members. The results are presented in Tables 1 and 2. Table I shows the iteration history. It includes weight, frequency responses, and some lower natural frequencies which have greater influences on the frequency responses. The iteration number 0 represents the initial design of the structure. An asterisk in the weight column indicates infeasible solution. Table 2 consists of the initial and the final properties of the structure. According to the final design, the weight of the structure is reduced by 30.4% while the frequency responses manage to remain in the feasible region. Since the allowed changing range is pretty large for each design variable, the convergence seems a little bit slow but is stable even if the design sometimes strays into the infeasible region. The second example is a transmission tower. This three-dimensional truss structure is composed of 25 elements. Owing to the consideration of design symmetry, these 25 elements are placed into seven groups. Whenever one element changes in a group, it results in the same change for all other elements in the same group. That is, design variable linking is used to maintain the symmetry requirement. The first design variable contains a single element which is element 1 with an area of 0.5 in*. The side
ILLUSTRATIONS
Both examples illustrated in this paper are the optimal solutions of minimum weight design subjected to various types of frequency response
Fig. 1. Ten-bar
truss
Structural modification with frequency response constraints
1017
Fig. 2. Transmission tower. Table 1. Iteration historv of examnle I Iteration No. 0 1 2 3 4 5 6 7 8 9 10 II 12 13
Frequency responses at nodes 2 and 4 (x lo-‘) 4.676 3.344 1.315 2.206 2.429 3.972 3.848 3.787 3.602 3.929 3.893 3.662 3.437 3.956
5.543 7.231 6.155 6.249 6.302 6.378 6.728 6.834 7.204 6.320 6.666 7.031 7.549 6.461
(IL)
(ik)
(IL)
4.50 3.96 3.68 3.82 3.90 3.97 3.92 3.96 3.88 3.96 3.92 3.93 3.85 3.93
12.62 11.57 10.98 11.24 11.28 11.68 11.65 11.63 11.45 11.65 11.55 Il.63 11.21 11.66
13.16 12.11 12.07 12.76 13.46 13.01 12.37 12.92 12.15 12.77 12.42 12.65 12.18 12.50
constraints are 0.375 and 1.0 in’ for this design variable. The second design variable includes elements 2-5 with an area of 1.0 in*. The side constraints are 0.5 and 2.0in2. The third group is composed of elements 6-9 with an area of 3.5 in*. The side constraints are 1.75 and 7.0 in*. The fourth group is of elements 10-13 with an area of 1.0 in2. The side constraints are 0.5 and 2.0 in*. The fifth design Table 2. Initial and final designs of example 1 Cross-sectional area (in*)
AlO CAS 3616-D
Initial design (in*) 1.0000 1.0000 1.OOOo 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Final design (in’) 0.5000 2.0000 0.5000 0.5000 0.5000 0.5000 0.6825 0.7189 0.5000 0.6553
24.01 23.06 20.73 21.08 20.96 20.86 21.37 20.80 21.43 21.86 21.63 20.95 21.57 21.23
Weight (lb) 10.860, 8.898* 8.421 8.092 8.062 7.605 7.482* 7.468* 7.429, 7.667 7.539* 7.405* 7.453+ 7.561
variable comprises elements 14-17 with an area of 1.5 in*. The upper and lower bounds for this design variable are 0.75 and 3.0 in*. The sixth design variable is represented by elements 18-21 with an area of 2.0 in*. The changing limits are 1.0 and 4.0 in*. The last group contains elements 22-25 with an area of 4.0 in*. The side constraints are 2.0 and 8.0 in*, respectively. The frequency response constraints are required as follows. At node 1 in the x2 direction and at node 2 in the x2 direction, the response value must be between 0.35 x 10m4and 0.40 x 10W4in. At node 3 in the direction x2, the frequency response must be less than 0.50 x 10m4in. The unit harmonic excitation force is assumed to be applied at node 3 in the x2 direction with a frequency of 25 Hz. The results of this example are recorded in Tables 3 and 4. As in the previous example, Table 3 shows the iteration history and Table 4 gives the comparison of the initial and the final designs. After 16 iterations, the weight of the final design is 42.7% less than that of the initial design and the frequency responses at specified
T. Y. CHEN
1018
Table 3. Iteration history of example 2 Frequency responses at nodes 1, 2 and 3 ( x 10-z)
Iteration No. 0
4.341 3.797 3.656 3.758 3.852 3.841 3.850 3.849 3.852 3.852 3.854 3.998 3.919 3.981 3.982 3.983 3.983
I
2 3 4 5 6 I 8 9 10 I1 12 13 14 15 16
4.347 3.797 3.656 3.758 3.852 3.841 3.850 3.849 2.852 3.852 3.854 3.998 3.919 3.981 3.982 3.983 3.983
8.833 5.792 5.040 4.970 4.975 4.977 4.919 4.980 4.982 4.983 4.984 4.962 4.973 4.916 4.911 4.977 4.977
(A)
(&
(&
(&
20.53 20.18 19.74 19.55 19.36 19.04 18.76 18.47 18.20 17.93 17.68 17.60 17.40 17.21 17.02 16.91 16.90
23.09
23.39 22.85 22.54 22.42 22.30 22.10 21.92 21.74 21.56 21.38 21.20 21.17 21.10 21.04 20.97 20.93 20.93
30.14 29.29 29.03 28.92 28.84 28.76 28.68 28.61 28.54 28.48 28.42 28.37 28.41 28.44 28.44 28.45 28.45
positive
Table 4. Initial and final designs of example 2 Design variable No.
Initial design (in2)
I
0.5000
2 3 4 5 6
I .oooo
I
3.5000 1.0000
Final design (in*)
1.oooo 0.5000
1.5000
2.9061 0.5000 0.7934
2.0000 4.0000
2.0000
I .oooo
locations are less than those of the initial design as required by the constraints. CONCLUSIONS The efficient calculation of frequency response sensitivity data and reanalysis of the frequency response function for a modified structure have been
established in this research for undamped MDOF systems in a modification process. The merits of the method are that both the computations of the frequency response sensitivity data and the reanalysis of the frequency response function for the modified structure are performed in a much smaller space than the original system. The best result from this modal space computation is selected to be the next starting point for the global iteration. Both of the numerical
22.23 21.89 21.74 21.60 21.39 21.19 20.99 20.79 20.60 20.40 20.37 20.34 20.32 20.31 20.31 20.30
examples method.
give
(&
Weight (lb)
31.15 30.35 30.06 29.94 29.84 29.14 29.64 29.54 29.46 29.37 29.22 29.23 29.23 29.22 29.20 29.19 29.19
31.79 31.13 30.82 30.70 30.57 30.32 30.09 29.87 29.65 29.43 29.29 29.27 29.28 29.30 29.29 29.29 29.31
18.184* 16.807* 16.273* 15.423 14.642 13.829 13.124 12.488 11.916 1I.396 10.922 10.721 10.684 10.640 10.61 10.603 10.597
results
for
proposed
the
REFERENCES 1. R. L. Fox and M. P. Kapoor, Rates of change of eigenvalues and eigenvectors. AIAA Jnl 6, 2426-2429 (1968).
2. R. S. Sharp and P. C. Brooks, Sensitivities of frequency response functions of linear dynamic systems to variations in design parameter values. J. Sound Vibr. 126, 167-172 (1988). 3. B. P. Wang, F. H. Chu and C. Trundle, Reanalysis technique used to improve local uncertainties in modal analysis. In Proceedings of 3rd International Modal Analysis Conference, Orlando, FL (Edited by D. J. DeMichele), pp. 398402. Society for Experimental Mechanics Inc. (1985). 4. B. P. Wang, W. D. Pilkey and A. Palazzola, Reanalysis, modal synthesis and dynamic design. In A Sfate-of-fheArt Survey on Finite Element Technology (Edited by A. K. Noor and W. D. Pilkey), Ch. 8. ASME, New York (1983). 5. Y. W. Luk and L. D. Mitchell, System modeling and modification via modal analysis. In Proceedings ofFirst International Modal Analysis Conference,
Orlando, FL
(Edited by D. J. DeMichele), pp. 423429. Society for Experimental Mechanics Inc. (1982). 6. R. R. Craig Jr, Structural Dynamics. John Wiley, New York (1981). 1. R. L. Fox, Optimizarion Methods for Engineering Design. Addison-Wesley, Reading, MA (1971).
Vol. 36, No. 6. pp. 1013-1018.
STRUCTURAL RESPONSE
1990
CUM57949/w s3.cnl+ 0.00 Pergamon Press plc
MODIFICATION WITH FREQUENCY CONSTRAINTS FOR UNDAMPED MDOF SYSTEMS T. Y. CHEN
Department
of Mechanical Engineering, National Chung-Hsing University, Taichung, Taiwan 40227, Republic of China (Received 8 September 1989)
Abstract-A structural modification method which aims to change the structural design to match the desired frequency response behaviors is introduced. The calculations of frequency response sensitivities and reanalysis of frequency responses of a modified structure are efficiently carried out in a reduced modal space. Either a least square solution or an optimal solution with minimum structural weight subjected to various types of frequency response constraints can be obtained through an iteration loop which repeatedly uses the frequency response sensitivity data and reanalysis technique to achieve the design goal. A tremendous cost saving is expected when this is implemented in practical engineering design.
INTRODUCTION
Finite element models have been widely used to predict the eigenvalues, eigenvectors and structural responses in structural dynamics. For most engineering designs, the initial finite element model must be subjected to many modifications before it satisfies the design requirements, which may include eigenvalue and/or response constraints. For other applications, the accuracy of an existing finite element model is so important so that it must be tested against experimental data. Under either situation, the modification of a finite element model in order to match the design requirements or test data for a large structure is often a time consuming and frustrating job. Sometimes it is even impossible to reach a satisfactory result by limited experience. Sensitivity analyses of some structural data with respect to variations in design parameter values have proved to be a powerful tool leading to a systematic method of solving structural modification problems. The success of structural modification is therefore more or less based on the proper use of sensitivity data as well as the calculations of these data. The satisfaction of all the requirements is often not achieved in a single modification attempt. Several iterations of the modification procedure are necessary. The outcome of each iteration has to be analyzed to see whether the requirements have already been satisfied. The finite element analysis of a large structure is costly. An efficient reanalysis of the modified structure is certainly indispensible. In previous papers, Fox and Kapoor [ 1] derived the sensitivities of eigenvalues and eigenvectors with respect to variation in an arbitrarily designated design parameter. These sensitivity data are a great help to the understanding of the dynamic characteristics of the system. As a result, the structural refine-
ment referring to these data to achieve the design goal becomes much easier. Sharp and Brooks [2] developed a method of calculating the sensitivities of frequency response functions. These data predict the frequency response behavior of a structure when subjected to modifications. Wang et al. [3,4] introduced an efficient re-analysis approach to compute the eigenvalues and eigenvectors for a modified structure. This approach solves the eigenvalue problem in a reduced space and thus avoids the time consuming process of solving the eigenvalue problem in the original global space. Luk and Mitchell [5] solved the reanalysis problem of steady state response by using modal spaces I and II. This method gives the steady state response for the modified system in modal space. Although these previous research outcomes, if properly implemented, will help the structural design engineers to accomplish their work, a comprehenvise and systematic way of attacking the structural modification problems is yet to be established. In the present paper a systematic procedure is created to perform the structural modifications iteratively to match the specified frequency response requirements through efficient computations of frequency response sensitivity data and reanalysis in a reduced space. THEORY
The frequency response function of &undamped MDOF system can be calculated by modal summation as [6]
where H,(n) is the frequency response function at the ith DOF when subjected to a unit input at the jth
1013
1014
T. Y.
DOF; n is the number of modes used in calculating the response function; c$,,represents the value of the ith DOF in the rth mode shape; r#~,~ represents the value of thejth DOF in the rth mode shape; 1, stands for the rth eigenvalue and Q is the excitation frequency in radians. In eqn (1) mode shapes are assumed to be normalized such that {4,}[M]{$,} = 1, where {4r} is the rth mode shape and [M] is the global mass matrix of the structure. The sensitivity of frequency response function with respect to a chosen design parameter is derived as follows:
&EN
Refinement or modification of a structure is seldom accomplished in a single try. Several iterations may be needed. The effect of each modification can only be determined by analyzing the modified structure. The more times the whole modified structure is reanalyzed, the more inefficient the modification process. For this reason, Wang et al. [3] developed an efficient reanalysis in a reduced space called modal space to analyze the eigenvalues and eigenvectors of a modified system. Using this approach, the formulation of the modified eigenvalue problem is much smaller in dimension than the original one. As a result, the saving in the computer time is significant. This efficient reanalysis is repeated here. The eigenvalue equation of the original system is [Kl{&I = wfl~~,N~
(2) where CQis the k th chosen design parameter. To obtain the sensitivity data of the frequency response function, the eigenvalue and eigenvector sensitivities of the original system are required. The computations of these data were introduced by Fox and Kapoor [l]. A brief summary of their results is shown here for convenience. The rth eigenvalue sensitivity is obtained by the following equation:
where [K] is the stiffness matrix of the original design; [M] is mass matrix of the original design; (4,) is the rth mode shape; and I, is the rth eigenvalue. After modification, the modified system has stiffness matrix [K + AK] and mass matrix [M + AM]. The new eigenvalue problem becomes [K + AKl(4:) = bf + A~l{&“}C’,
(8)
where (c#J~}is the modified rth mode shape; ;Lyis the modified rth eigenvalue; [AK] is the amount of change of the stiffness matrix; and [AM] is the amount of change of the mass matrix. Using assumed mode approach, the modified rth mode shape is assumed to be {&Y = [@I{%)>
where {&} is the rth mode shape; [K] is the global stiffness matrix; [M] is the global mass matrix; and i, is the rth eigenvalue. The rth eigenvector sensitivities are expressed as a linear combination of mode shapes.
(7)
(9)
where [@I is the modal matrix of the original system and {qr} is the rth generalized coordinate. Substituting eqn (9) into eqn (8) and premultiplying eqn (8) by [a]’ yields
or (4) where a,,~ are the weighting coefficients for the sensitivity of the rth mode shape. Forr#l
(5) For r = 1
(6)
WI + PlrPKIPI)~qJ
The resultant matrix in parenthesis on the left hand side represents the stiffness matrix of the modified system in a reduced modal space while the matrix in parenthesis on the right hand side stands for the mass matrix of the modified system in modal space. The eigenvalues of the modified system can be obtained directly from eqn (10). The eigenvectors of the modified system can be recovered by eqn (9). Taking advantage of this modal reanalysis, the sensitivity calculation of the frequency response function for a modified system is derived below.
Structural modification with frequency response constraints Employing the equation for frequency response sensitivity of the original system for the modified system, the sensitivity of frequency response for the modified system can be expressed to be
JHT(R)
(11)
ax,
where H;(R) is the frequency response function of the modified system; ~$7 is the value of the ith DOF of the rth mode shape for the modified system; 4:: is the value of the jth DOF of the rth mode shape for the modified system; and J.y is the rth eigenvalue of the modified system. The modified mode shapes and eigenvalue in eqn (11) are obtained from efficient reanalysis, i.e.
where c#+,is the value of the ith DOF of the Ith mode shape for the original system; 4,, is the value of the jth DOF of the ith mode shape for the original system; and q,, is the value of the lth DOF of the rth generalized coordinate. Plugging eqns (12) and (13) into eqn (11) results in the sensitivity of frequency response function of the modified system.
for r = i err = - f{4,)WlT In order to verify the effect of each modification, the reanalysis of the frequency response function of the modified system is necessary. To avoid computing this response function in original global space, the following equation uses the mode shapes and eigenvalues from modal reanalysis to formulate the reanalysis of the frequency response function of the modified system.
Applying these derived equations, an efficient structural modification procedure can be established. For practical application of this proposed modification approach, the modifications of the stiffness and mass matrices are assumed to be the sum of local modifications. [AK] = tll [k,] -t x&Q +
. + a,[k,,J
(20)
and
[AMI= @,b,l+azbzl+
. . + a,h,,l,
(21)
(14) -
(A, -
W)2
.
In eqn (14) the sensitivities of the generalized coordinate {q,} and the eigenvalue of the modified system appear. These sensitivity data are now derived. The associated eigenvalue problem of the modified system has been shown previously by eqn (10). In that equation the stiffness matrix and mass matrix of the modified system in modal space are respectively. Extending Fox and Kapoor’s work for the WI + PIT[AfWI) and (VI + Pl%WPIh sensitivities of modal data to modal space, the sensitivities of 1: and {q,} can be obtained by the following steps.
2
= k,)‘$
(VI + PlPWPlNq,I
- Wq,Y&
WI + PlWflPl){d
(15) The sensitivities of {q,} are assumed to be
(16) for r #i
a,, =
+2
(PI + PlWflPl1d9 k
(17)
1016
T. Y.
where CL, is the design parameter; [AK] is the amount of modification of the stiffness matrix; [AM] is the amount of modification of the mass matrix; [k,] is a constant matrix for modification of the stiffness matrix; and [m,] is a constant matrix for modification of the mass matrix. Employing all the previous efforts in this paper, an iteration loop is created as follows. (1) Initialize all the {q}s. Let the rth entry of {q,} be 1 and all other entries 0. (2) Select constant matrices [k]s and [m]s in eqns (20) and (21). (3) Compute sensitivities of frequency response function with respect to each chosen design parameter using eqns (14)-(18). (4) Solve for the values of x by using the following linear equation:
where Hi(n) is the desired frequency response value and H,(R) is the current frequency response function. Equation (22) may be more than one equation. It depends on the requirements of the modification problem. However, if the number of equality constraint equations (22) is greater than the number of selected design parameters, the least square solution for GIis used. If, on the other hand, the number of equations (22) is less than the number of design parameters, an optimal solution may be obtained by setting up an objective function. (5) Plug the c(s obtained from previous steps into eqns (20) and (21) to compute [AK] and [AM] and then use eqns (IO) and (19) to solve for the frequency response of the modified system H,(R). (6) Convergence check. See if H,,(R) - H;(Q) H;(n)
,-’
(23)
for all the constraint equations. If yes-+stop. Otherwise, go to step (3). The final answers for s(s are expected in a few iterations. Since the dimension of the modal space is generally much smaller than that of the original structure, the approximate solutions from modal space reanalysis result. Therefore the verification of outcomes using updated data in global space is necessary. For each verification run, a new set of modal data is created and the modal space iteration procedure can be applied again. After several verification runs, the best result can be obtained. NUMERICAL
&EN
constraints. The method used to optimize the solution is sequential linear programming (SLP) [A. The subroutine DDLPRS from the IMSL library performs the linear programming. The equality constraint represented by eqn (22) is extended to include inequality and range constraints. The side constraints are also imposed on each design variable. The configurations of these structures are shown in Figs I and 2, respectively. The first example is a IO-bar plane truss. The initial design is a uniform cross-section of 1.0 in* for all members. The constraints are the frequency response of node 2 in the x2 direction less than 0.4 x 10m4in. and the frequency response of node 4 in the x2 direction between 0.55 x 10e4 and 0.65 x 10m4 in. when subjected to a unit harmonic excitation force at node 2 in the x2 direction with an excitation frequency of 10 Hz. The side constraints of lower and upper bounds are 0.5 and 2 in* for all the members. The results are presented in Tables 1 and 2. Table I shows the iteration history. It includes weight, frequency responses, and some lower natural frequencies which have greater influences on the frequency responses. The iteration number 0 represents the initial design of the structure. An asterisk in the weight column indicates infeasible solution. Table 2 consists of the initial and the final properties of the structure. According to the final design, the weight of the structure is reduced by 30.4% while the frequency responses manage to remain in the feasible region. Since the allowed changing range is pretty large for each design variable, the convergence seems a little bit slow but is stable even if the design sometimes strays into the infeasible region. The second example is a transmission tower. This three-dimensional truss structure is composed of 25 elements. Owing to the consideration of design symmetry, these 25 elements are placed into seven groups. Whenever one element changes in a group, it results in the same change for all other elements in the same group. That is, design variable linking is used to maintain the symmetry requirement. The first design variable contains a single element which is element 1 with an area of 0.5 in*. The side
ILLUSTRATIONS
Both examples illustrated in this paper are the optimal solutions of minimum weight design subjected to various types of frequency response
Fig. 1. Ten-bar
truss
Structural modification with frequency response constraints
1017
Fig. 2. Transmission tower. Table 1. Iteration historv of examnle I Iteration No. 0 1 2 3 4 5 6 7 8 9 10 II 12 13
Frequency responses at nodes 2 and 4 (x lo-‘) 4.676 3.344 1.315 2.206 2.429 3.972 3.848 3.787 3.602 3.929 3.893 3.662 3.437 3.956
5.543 7.231 6.155 6.249 6.302 6.378 6.728 6.834 7.204 6.320 6.666 7.031 7.549 6.461
(IL)
(ik)
(IL)
4.50 3.96 3.68 3.82 3.90 3.97 3.92 3.96 3.88 3.96 3.92 3.93 3.85 3.93
12.62 11.57 10.98 11.24 11.28 11.68 11.65 11.63 11.45 11.65 11.55 Il.63 11.21 11.66
13.16 12.11 12.07 12.76 13.46 13.01 12.37 12.92 12.15 12.77 12.42 12.65 12.18 12.50
constraints are 0.375 and 1.0 in’ for this design variable. The second design variable includes elements 2-5 with an area of 1.0 in*. The side constraints are 0.5 and 2.0in2. The third group is composed of elements 6-9 with an area of 3.5 in*. The side constraints are 1.75 and 7.0 in*. The fourth group is of elements 10-13 with an area of 1.0 in2. The side constraints are 0.5 and 2.0 in*. The fifth design Table 2. Initial and final designs of example 1 Cross-sectional area (in*)
AlO CAS 3616-D
Initial design (in*) 1.0000 1.0000 1.OOOo 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Final design (in’) 0.5000 2.0000 0.5000 0.5000 0.5000 0.5000 0.6825 0.7189 0.5000 0.6553
24.01 23.06 20.73 21.08 20.96 20.86 21.37 20.80 21.43 21.86 21.63 20.95 21.57 21.23
Weight (lb) 10.860, 8.898* 8.421 8.092 8.062 7.605 7.482* 7.468* 7.429, 7.667 7.539* 7.405* 7.453+ 7.561
variable comprises elements 14-17 with an area of 1.5 in*. The upper and lower bounds for this design variable are 0.75 and 3.0 in*. The sixth design variable is represented by elements 18-21 with an area of 2.0 in*. The changing limits are 1.0 and 4.0 in*. The last group contains elements 22-25 with an area of 4.0 in*. The side constraints are 2.0 and 8.0 in*, respectively. The frequency response constraints are required as follows. At node 1 in the x2 direction and at node 2 in the x2 direction, the response value must be between 0.35 x 10m4and 0.40 x 10W4in. At node 3 in the direction x2, the frequency response must be less than 0.50 x 10m4in. The unit harmonic excitation force is assumed to be applied at node 3 in the x2 direction with a frequency of 25 Hz. The results of this example are recorded in Tables 3 and 4. As in the previous example, Table 3 shows the iteration history and Table 4 gives the comparison of the initial and the final designs. After 16 iterations, the weight of the final design is 42.7% less than that of the initial design and the frequency responses at specified
T. Y. CHEN
1018
Table 3. Iteration history of example 2 Frequency responses at nodes 1, 2 and 3 ( x 10-z)
Iteration No. 0
4.341 3.797 3.656 3.758 3.852 3.841 3.850 3.849 3.852 3.852 3.854 3.998 3.919 3.981 3.982 3.983 3.983
I
2 3 4 5 6 I 8 9 10 I1 12 13 14 15 16
4.347 3.797 3.656 3.758 3.852 3.841 3.850 3.849 2.852 3.852 3.854 3.998 3.919 3.981 3.982 3.983 3.983
8.833 5.792 5.040 4.970 4.975 4.977 4.919 4.980 4.982 4.983 4.984 4.962 4.973 4.916 4.911 4.977 4.977
(A)
(&
(&
(&
20.53 20.18 19.74 19.55 19.36 19.04 18.76 18.47 18.20 17.93 17.68 17.60 17.40 17.21 17.02 16.91 16.90
23.09
23.39 22.85 22.54 22.42 22.30 22.10 21.92 21.74 21.56 21.38 21.20 21.17 21.10 21.04 20.97 20.93 20.93
30.14 29.29 29.03 28.92 28.84 28.76 28.68 28.61 28.54 28.48 28.42 28.37 28.41 28.44 28.44 28.45 28.45
positive
Table 4. Initial and final designs of example 2 Design variable No.
Initial design (in2)
I
0.5000
2 3 4 5 6
I .oooo
I
3.5000 1.0000
Final design (in*)
1.oooo 0.5000
1.5000
2.9061 0.5000 0.7934
2.0000 4.0000
2.0000
I .oooo
locations are less than those of the initial design as required by the constraints. CONCLUSIONS The efficient calculation of frequency response sensitivity data and reanalysis of the frequency response function for a modified structure have been
established in this research for undamped MDOF systems in a modification process. The merits of the method are that both the computations of the frequency response sensitivity data and the reanalysis of the frequency response function for the modified structure are performed in a much smaller space than the original system. The best result from this modal space computation is selected to be the next starting point for the global iteration. Both of the numerical
22.23 21.89 21.74 21.60 21.39 21.19 20.99 20.79 20.60 20.40 20.37 20.34 20.32 20.31 20.31 20.30
examples method.
give
(&
Weight (lb)
31.15 30.35 30.06 29.94 29.84 29.14 29.64 29.54 29.46 29.37 29.22 29.23 29.23 29.22 29.20 29.19 29.19
31.79 31.13 30.82 30.70 30.57 30.32 30.09 29.87 29.65 29.43 29.29 29.27 29.28 29.30 29.29 29.29 29.31
18.184* 16.807* 16.273* 15.423 14.642 13.829 13.124 12.488 11.916 1I.396 10.922 10.721 10.684 10.640 10.61 10.603 10.597
results
for
proposed
the
REFERENCES 1. R. L. Fox and M. P. Kapoor, Rates of change of eigenvalues and eigenvectors. AIAA Jnl 6, 2426-2429 (1968).
2. R. S. Sharp and P. C. Brooks, Sensitivities of frequency response functions of linear dynamic systems to variations in design parameter values. J. Sound Vibr. 126, 167-172 (1988). 3. B. P. Wang, F. H. Chu and C. Trundle, Reanalysis technique used to improve local uncertainties in modal analysis. In Proceedings of 3rd International Modal Analysis Conference, Orlando, FL (Edited by D. J. DeMichele), pp. 398402. Society for Experimental Mechanics Inc. (1985). 4. B. P. Wang, W. D. Pilkey and A. Palazzola, Reanalysis, modal synthesis and dynamic design. In A Sfate-of-fheArt Survey on Finite Element Technology (Edited by A. K. Noor and W. D. Pilkey), Ch. 8. ASME, New York (1983). 5. Y. W. Luk and L. D. Mitchell, System modeling and modification via modal analysis. In Proceedings ofFirst International Modal Analysis Conference,
Orlando, FL
(Edited by D. J. DeMichele), pp. 423429. Society for Experimental Mechanics Inc. (1982). 6. R. R. Craig Jr, Structural Dynamics. John Wiley, New York (1981). 1. R. L. Fox, Optimizarion Methods for Engineering Design. Addison-Wesley, Reading, MA (1971).