A simplified version of Icerman's problem regarding the design of structures subject to a single harmonic load is discussed. The nature of the restrictive conditions that must be placed on the design space in order to ensure an analytic optimum are discussed in detail. Icerman's problem is then extended to include multiple forcing functions with different driving frequencies. And the conditions that now must be placed upon the design space to ensure an analytic optimum are again discussed. An important finding is that all solutions to the optimality condition (analytic stationary design) are local optima, but the global optimum may well be non-analytic. The more general problem of distributing the fixed mass of a linear elastic structure subject to general periodic loads in order to minimize some measure of the steady state deflection is also considered. This response is explicitly expressed in terms of Green's functional and the abstract operators defining the structure. The optimality criterion is derived by differentiating the response with respect to the design parameters. The theory is applicable to finite element as well as distributed parameter models

Qian, B.

Reiss, Robert

English

NASA Technical Reports Server

OPTIMUM DESIGN OF STRUCTURES SUBJECT TO 
GENERAL PERIODIC LOADS 
R. Reiss and B. Qian 
Large Space Structure Institute 
Howard University 
Department of Mechanical Engineering 
Washington, D.C. 20059 
1265 
https://ntrs.nasa.gov/search.jsp?R=19890015844 2020-03-20T01:48:50+00:00Z
INTRODUCTION 
The optimum design of structures subject to a single harmonic load has been 
considered by Icerman [ I ]  and Plaut , [2].  
single harmonic tip load, and he minimized the amplitude of the tip's steady state 
displacement subject to fixed volume. His derivation of the necessary optimality 
criterion required a steady state extension to the principle of minimum potential 
energy. 
of the optimality condition, provided sufficiently restrictive conditions are 
placed upon the admissible designs. 
Icerman [ l ]  studied a rod acted upon by a 
This new extremum principle can be used to establish the global sufficiency 
Plaut [ 2 ]  generalized Icerman's work by minimizing the amplitude of the steady 
state deflection at any specified location of the structure. 
for several harmonic loads provided all were driven at the same frequency. He 
derived the necessary optimality condition by first extending the principle of 
mutual stationary potential energy to the steady state.He did not address whether 
the optimality condition was also sufficient for the optimal design. 
His problem allowed 
This paper initially addresses a simplified version of Icerman's problem. The 
nature of the restrictive conditions that must be placed on the design space in 
order to ensure an analytic optimum are discussed in detail. 
then extended to include multiple forcing functions with different driving fre- 
quencies. 
sure an analytic optimum are again discussed. An important finding is that all 
solutions to the optimality condition (analytic stationary design) are local optima, 
but the global optimum may well be non-analytic. 
Icerman's problem is 
And the conditions that now must be placed upon the design space to en- 
The more general problem of distributing the fixed mass of a linear elastic 
structure subject to general periodic loads in order to minimize some measure of the 
steady state deflection is also considered. 
in terms of Green's functional and the abstract operators defining the structure. 
The optimality criterion is derived by differentiating the response with respect to 
the design parameters [ 3 ] .  The theory is applicable to finite element as well as 
distributed parameter models. 
This response is explicitly expressed 
AN ELEMENTARY FORCED VIBRATION 
Consider a rod of length L whose cross-sectional area S is piecewise constant. 
In the simplest case S = S1 
The design ratio S1/S2 is denoted by R. 
state response can be represented by U(x) cos wDt, where IU(L)I is the amplitude of 
the tip response. Figures 2 and 3 show, respectively, the natural frequency of the 
design and tip amplitude as a function of the design ratio R. 
R$', the design and driving frequencies coincide and hence the resonance condition .,.. 
shown, Icerman's [ l ]  optimality condition is sufficient provided only designs R >Rn 
are considered. 
for 0 2 x < L/2 and S = S2 for L/2 < x2 L (Fig. 1). 
The tip load is F cos %t. The steady 
For designs with R= 
I I 
I I 
_--- 
s1 
- 
/- 
- L/2 L/2 - 
F cos 
Figure 1. Two-Segment Rod Subject To A Single Driving 
Frequence 
1267 
FREQUENCY AND TIP-DEFLECTION AMPLITUDE VS DESIGN 
0 
Fig. 2. Non-Dimensional Frequency vs. Design. The non-dimensional 
frequency A is defined as wWP/E where P is mass per unit length 
and E is Young's modulus. 
I 
I I 
' I  1 I 
7 
/ 
R* 
Fig. 3. Tip Deflection VS. Design. Rmin is the analytic local optimum. 
Depending upon the value of U(O), Rmin may or may not be globally 
the optimum. 
1268 
TWO ELEMENT ROD SUBJECTED TO TWO DRIVING FREQUENCIES 
The rod shown in Figure 1 is now subjected to a tip load consisting of two 
frequencies - F=F, COS %t + F? COS2 9 t .  While the addition of the second driving 
frequency does not affect the period of the response, it does add a minor complica- 
tion to the calculation of the maximum response. The response can be expressed inthe 
form U (x) cos% t + U~(X) cos 2 w D t .  
setting x=L and maximizing the magnitude of the response over one period. 
which the fundamental frequency equals one of the resonant frequencies!. 
response is depicted in Figure 4 .  
all are minima and one is not analytic. 
forcing frequency, one or both of the resonant designs may not exist. 
The maximum tip response Umax is obtained by 1 
A A 
There are two resonant frequencies and, in general, two designs R and R2 for 
A typical 
Clearly, there are up to three local extrema; 
Analogous to the simpler case of only one 
IUUtlL 
Figure 4 .  Typical Maximum Response as a Function of the Design Parameter. 
R and R are analytic local optimum designs. 1 2 
1269 
~ 
OPTIMIZATION USING GREEN'S FUNCTIONAL 
Consider the following boundary value problem 
(T* ET + a~)u = f in 52 
BYU = g in 
9; * 
B y  E T u  = h in a R 2  
where T, T ... L adjoint operators 9< 2 
E(S) ... Linear stiffness operator 
M(S) ... Linear mass operator 
a ...... A scalar 
y,y 
* ... Trace operators mapping functions defined in 52 onto functions 
defined on ail1 and a52 respectively. 2' 
;t 
B,B ... Boundary operators 
an ,an ..Complementary subsets of 
S ...... Design variable(s) 
1 2  
The following integration by parts formula, due to Oden and Reddy [ 4 ] ,  is postulated 
* 9; * 
= (yu,B y ETv) - (BYu,~ ETv) 
a522 a522 
Also E,M satisfy 
(u,EvIn = (v,EuIn 
( ~ , M V ) ~  = MU)^ for every admissible u and v. 
1270 
STRUCTURAL DYNAMICS 
The equations governing linear structural dynamics can be expressed as 
* 
T E T u + M u = f  in 
Byu = g on anl 
9< * 
B y ET u = h on an2 
If f,g, and h a l l  have the same periodicity, 
N 
1 
f = c fn(x) cos n u,t 
N 
1 
g = c %(x) cos n w t D 
N 
1 
h = C h (x) cos n wDt n 
The solution u ( x , t )  satisfies 
N 
1 
u = c u (XI cos n wDt n 
The Fourier coefficients satisfy 
* 
L (T ET - n2 u2 M) un = fn in n n D 
It can be shown that the solution is 
where Gn is the Green's function corresponding to the operator L n . 
1271 
SOLUTION TO VARIATIONAL EQUATIONS 
Define 
I 6Un = Un(x,S+6S) - un(x,s) 
Then, by using Oden and Reddy's integration by parts formula and taking the first 
variation of the differential equation specifying U it can be shown that n' 
I Consequently, 
I 
2 2  Gu(x,t) = C [n w (6Mvn,Gn)a-(GETU,,TGn)a] cos n w t 
D -D 
I 1 
To determine whether any particular design which causes a stationary response 
also minim'zes the response can be obtained by variational calcu- at fixed x and t 
lus. 
6Un is determined above, and 
5 The second variation 6 u clearly involves the terms G2E, 6Un,G2M. However, 
2 2  6Gn = - (TG n 6E TGn)a + n aD (L~MG~,G,)~ 
2 Consequently, 6 u is completely determined by the variations in the operators E 
and M. 
1272 
APPLICATIONS TO A ROD 
A rod with varying cross-sectional area A(x) is fixed at x=O and acted upon by 
the horizontal periodic force 
C Fn 
at the tip x=L. 
cos n w t D 
The steady state equations are 
- (E A UA) - c E A wD n Un = Fn6(L-x) 
1 -2 2 2  
0 0 
U n (0) = EoA Unl(L)=O 
-2 where Eo is Young's modulus and c = p/E. Clearly 
un(d = F~ G~(L,x) 
Upon making the following identification for the operators: 
$< 
T = - T = d/dx , E=EoA=S(x), M=pA = c-~S(X) 
the solution for 6 Un(L) becomes 
2 L 
{ i  {Un l 2  - n  WD c - ~  Un) 6s dx 1 
n 
6Un(L) = - 
0 F 
Now, let T be selected so that 
By imposing the fixed mass constraint 
L 
1 6s dx = 0, 
0 
the optimality condition for minimizing U 
tained, i.e. 
over all admissible designs S is ob- max 
For rods consisting of piecewise constant cross-sections Si in the domain x 
x < x 
< i-1 
the optimality condition takes the form i' 
1273 
RESULTS FOR A TWO-SEGMENT ROD WITH ONE DRIVING FREQUENCY 
Table 1 shows results for a single driving. The non-dimensional frequency s2 
is wDL/c, the analytic optimum design stiffness ratio R is S /S U 
analytic minimum, Uunif is the amplitude of vibration for R=I and U 
analytic minimum obtained by letting R approach zero. 
is the 
is the non- 2' opt 
0 
It is observed that for low frequencies, the analytic optimum is clearly supe- 
rior to any other design. For Q > 0.854, the awkward non-analytic optimum provides 
a smaller amplitude for the response than does the analytic design. 
larger frequencies SZ > 1.743, even the uniform design is preferable to the analytic 
local optimum. 
For still 
TABLE 1. Comparison of Analytic Optimum to Non-Analytic Optimum 
and Uniform Designs. 
n 
0.1 
0.3 
0.5 
0.7 
0.854 
1.4 
1.743 
1.8 
1.0 , 
R 
1.005 
1.046 
1.130 
1.266 
1.415 
1.597 
2.419 
3.828 
4.176 
'opt"un if 
0.99999 
0.999 
0.996 
0.982 
0.957 
0.911 
0.497 
1.000 
2.824 
Uopt/Uo 
0.010 
0.093 
0.278 
0.604 
1.000 
1.625 
4.066 
13.657 
19.734 
~ 1274 
RESULTS FOR A TWO-SEGMENT ROD WITH TWO DRIVING FREQUENCIES 
Table 2 shows numerical results for two driving frequencies. The symbols R 
and Uo 
1 
o t' 'unif and R are defined the same as in the previous example. are, respectively, the maximum absolute value of the displacemen? over one period 
for a local analytical optimal design, uniform design, and a rod for which S 
to zero. 
frequency R D local optimal designs R 
comparisons based upon &he design R 
2'  parisons based upon R 
Here, U 
tends 
The forcing function is F1 cos w t + F 
is fixed at 0.6 for this example. 
cos 2 w ~ t .  The non-dimensional 
8ote thac there are two analytic 
In the third and fourth Columns in Table 2 are 
D 
and R 2 '  while the sixth and seventh columns are com- 1 
The global optimum is obtained by searching the fourth and seventh columns. 
Since the entries in Column 4 always exceed the corresponding entry in Column 7, it 
follows that R 
R is the global optimum but for F 
always provides a better solution than R . Further, for F1 2 F2, 
< F2, R=O is the gloial optimum. 2 2 
TABLE 2. Comparison of Both Analytic Optimal Designs with Non-Analytic 
Optimal Designs with Non-Analytic Optimum and Uniform Designs. 
All data is for RD = 0.6 
F1/F2 
119 
2/ 8 
317 
416 
5/5 
614 
7/  3 
812 
9 /  1 
R1 
0.168 
0.194 
0.213 
0.234 
0.250 
0.270 
0.288 
0.313 
0.347 
U 
opt 
'unif 
0.81 
0.99 
1.15 
1.29 
1.42 
1.53 
1 . 6 3  
1.69 
1.69 
U 
opt 
U 
-
0 
2.02 
1.88 
1.72 
1.56 
1.41 
1.27 
1.13 
1.00 
0.85 
R2 
1.89  
1.85 
1.79 
1.74 
1.68 
1.61 
1.54 
1 .45  
1.34 
U 
opt 
'un if 
0.80 
0.81 
0.83 
0.85 
0.87 
0.89 
0.92 
0.94 
0.97 
U 
opt u 
0 
1.99  
1.54 
1.24 
1.03 
0.87 
0.74 
0.64 
0.55 
0.48 
1275 
ADDITIONAL RESULTS FOR TWO DRIVING FREQUENCIES 
The identical problem as the preceding one is considered. Here, however, re- 
The driving frequency sults are presented for fixed force amplitudes where F =F 
R is varied in Table 3. 1 2'  D 
Of the analytic solutions, it is clear that R is superior to R for values of 
1 
1 2 R closer to 1 . 0 ,  but R is better for the smaller driving frequencies. However R 
never provides the optimum global design. 
is the global optimum; otherwise the global optimum is not analytic. 
D 2 For forcing frequencies RD < 0.6,  R2 
RD 
0.2 
0 .4  
0.6 
0 . 8  
1 . 0  
TABLE 3 .  
R1 
Comparison of Both Analytic Optimal Designs with 
Non-Analytic Optimum and Uniform Designs. 
for F ~ / F ~  = 1. 
All data is 
Uopt R2 U opt 
U 
opt 
'un i f uO "un i f 
0.025 16.6 1.10 1.05 0.999 
0.105 3.93 1.27 1.24 3.985 
0.251 1.42 1.40 1.68 0.87 
0.482 0.14 1.76 2.67 0.18 
0.824 0.96 2.36 5.09 2.84 
0.07 
0.30 
0.87 
2.31 
7.00 
1276 
TWO SEGMENT ROD SUBJECT TO THREE FREQUENCIES 
The two segment rod is now subjected to the tip force F cos w t + F cos 1 D 2 w t + F cos 3w t. D 3 D designs in addition to the non-analytic optimal design as R+O. Table 4a contains 
results for the lowest driving frequency R = 0.3. In all cases R provides the 
global optimum. However, if RD = 0.6 (Tabye 4b), then R2 is the gjobal optimum. 
There are now three resonant designs and three local optimal 
TABLE 4. Comparison of Three Analytic Local Optimum Designs with Non-Analytic 
Local Optimum Design. = 0.3 for (4a) and QD = 0.6 for (4b), OD 
U 
opt R3 - opt U -F2’F1 F3’F1 R1 R2 
0 
U 
0 
U 
0 
U 
0.50 0.50 0.065 0.82 0.157 0.40 1.22 0.18 
0.75 0.75 0.061 0.89 0.158 0.52 1.25 0.21 
1.00 1.00 0.058 0.94 0.158 0.54 1.27 0.34 
0.50 0.50 0.286 1.07 0.825 0.67 3.12 1.38 
0.75 0.75 0.267 1.15 0.830 0.76 3.24 1.72 
1.00 1.00 0.254 1.20 0.838 0.95 3.32 2.02 
1277 
REFERENCES 
1. Icerman, L.J.: Optimal Structural Design for Given Dynamic Deflection. Int. 
J. Solids and Structures, Vol. 5, 1969, pp. 473-490. 
2. Plaut, R.H.: Optimal Structural Design for Given Deflection under Periodic 
Loading. Q. Appl. Math., July 1971, pp. 315-318. 
3. Reiss, R.: On Singular Cases in the Design Derivative of Green's Functional, 
Symposium: Sensitivity Analysis in Engineering. NASA Conference Publication 
2457, 1986, pp. 319-328. 
4. Oden, J.T. and Reddy, J.N.: Variational Methods in Theoretical Mechanics. 
Springer-Verlag, Berlin, 1976. 
I 1278 


Optimum design of structures subject to general periodic loads

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