- Email: [email protected]
A linearization technique for random vibrations of nonlinear systems
MECHANICS RESEARCH COM3fUNICATIONS Vol. 1 9 (i) , 1-6, 1999. 0093-6413/92 S B . 0 0 + .00 Copyvight (c) 1 9 9 1
l ~ m ~ e d in the U S A Pop~p-rnon Press plc
A LINEARIZATION TECHNIQUE FOR RANDOM VIBRATIONS OF NONLINEAR SYSTEMS
X.X.Lee and J.Q.Chen Institute of Structural Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
(Received 21 June 1990; accepted for print 16 July 1991) Introdution
Stochastically excited linear systems have been studied in great detail and numerous analytic techniques exist for treating both the stationary and nonstationary problems. Unfortunately, many structures of engineering interest can not be considered linear and the techniques for analyzing nonlinear systems are not nearly so well developed. Some systems with simpile nonlinearities can be handled exactly by means of the FPK equation method, but exact solutions are not available for most nonlinear systems. As a consequence, several approximate techniques have been proposed. We know that one of the basic tools for the approximate treatment of nonlinear system is the equivalent linearization method [1]. Recently, the excellent review and works [2-7] show the efficiency and considerable promise for it is not limited by the restriction commonly imposed on the other approaches. Furthermore, this approach can be made quite direct and relatively easy to apply. The usual linearization technique is to minimize the residual beween the actual nonlinear system and the auxiliary linear system. However, this does not certainly lead to the minimization of the residual of responses corresponding to the actual system and the auxiliary system. The pu~ose of the present paper is a cotribution to the study of lincarization approach for treating problem of the stochastic response analysis of nonlinear systems under random excitations. The accuracy of the proposed technique is investigated by means of analytical and numerical examples.
Proposed Linearization Technique
Consider a nonlinear system which can be written in the n-degree-of-freedom state vector form + G(Z) = F(t) (1) where Z is a generalized state vector, G(Z) is a generalized system vector, and F(t) is a stationary i
X.X.
2
LEE and J.Q.
CHEN
Gaussian random vector representing the excitation to the system. As a means of obtaining an approximate solution of (1) let us consider an auxiliary system which is described by a linear differential equation of the form
2 + aZ = F(0 (2) where A is the system matrix which is at this point arbitrary. The measurement of the inequality beween the actual system (1) and the auxiliary system (2) is expressed as = IIG(Y) - A Eli (3) where II ° II is a measuring scaler. F o r usual linearization techniques, this scaler is selected as [1,21 IIG(Y) - A YII = G ( Y ) - A Y (4) Considering the solution of equations (1) and (2), one may see that the minimization o f the residual {$ ~G(Y)dt-$ ~AYdt} gives rise to that o f the response difference. Unfortunately, this residual can not be calculated. As a means of obtaining an approximate formulation of this residual, we choose the measuring scaler as follow = I I G ( Y ) - A Yll =-fn [ G ( Y ) - A Y]df~
(5a)
or as a approximation, the components of e are ki
'; = S~' j-~2 ""I~ ( o , ( r ) -
(Sb)
A,,Y,lay~ay2...dy~,
(i = 1,2,...,n;/" = 1,2,...k~) where Gi(Y) is the component of G(Y) and related to the state variable (Yl Y2 "'" Yra);fl = {Yi[ YiE [0,X], i = 1,2,'",n; X is the stationary solution of (2)} is the definition domain o f Y. The magnitude of this vector t will clearly be function of the auxiliary system matrix A. The approximate solution of the original equation (1) will be generated by selecting A in such a way that some measure of 8 is minimized. As a minimization criterion on ~, it will be required that the mean value o f the scalar product be a minimum. That is T
< 8 ~ > = minimum
(6) where the operator < • > denotes the statistical average of the random variable. The necessary condition for equation (6) are ~
T
8>
~a i]
T ~8
=2<8
--
8a #
> =0
or T
< 8X x 1x ~ . " x , > = 0
(7)
As a consequence o f the fact that F(t) is Gaussian, it may be shown that equation (7) represents a ture minimum of < ere > . Condition (7) can be rewritten as T
2<~aG(Y)df~X xlx2""x ,> -A
2
) > =0
(8)
Hence, if < XXr(xlx2-.,x°)2 > is not singular, the equation (8) has the solution
A = 2 < S n G ( Y ) d I I X r x ~ x 2 . . . x ~> < X X r ( x l x 2 . . . x
2
) >
-1
(9)
which will lead to the minimum for < fie > Now, the original problem has been reduced to the solution of a related linear problem which may be solved by any number of analytic techniques. However, since matrix A depands on the response statistics, an iteration solution procedure is generally required which is also necessary to usual linearization techniques.
RANDOM VIBRATIONS OF NONLINEAR SYSTEMS
3
Accuracy of the technique
The linearization technique proposed in this paper has been tested by studying three different nonlinear systems in which two problems can be solved exactly by F o k k e r - P l a n k equation method and the last problem/s solution will be given by Monte-Carlo simulation. E X A M P L E 1. System with Nonliear Stiffness
Consider a system described by
Y¢ + c ~ + ~ x 3 = R t ) (10) where the excitation f(t) is a zero-mean Gaussian white noise with the power spectral density So. From the nature of the excitation and properties of the stationary reponse of linear system, the auxiliary linear system is readily shown that
(11)
)~ + coYc + k o x = / ( t )
Then, the RMS response of equation (11) can be obtained directly. That is 'gO
1
crx = 1.0588(--)~ (12) c# The RMS response tT~ of the nonlinear system as a function of the excitation level is shown in Fig.1. Results are given for the proposed linearization, the usual linearization, and the exact analysis [3]. By comparison with the exact solution, the proposed linearization gives an even better solution. The error in the proposed solution is 3.27 percent, much less than the error 7.58 percent in the usual solution. As a result, the residual beween the actual system and the usual auxiliary system is 1
3
< ( c # x ~ - k / = x ) 2 > = 6.43(s0#~ / c) ~
(13)
At the same meaning, the residual corresponding to the proposed system is 1
3
< ( c ~ x 3 - k = x ) z > = 8.81(s0/z~ / c) ~
(14)
It is apparent that the system residual caused by proposed technique is larger than that caused by usual technique. E X A M P L E 2. System with Nonlinear Damping
Consider a system described by
~¢ + fl(:c2 + kxZ)]c + kx = ](t) (15) where f(t) is the same excitation as E X A M P L E 1. The approximate RMS response obtained by proposed approach is
o =0.9733( s° )~ (16) x /~k 2 The results given by the proposed iinearization, the usual lincarization and the exact analysis [7] arc plotted in Fig.2, against the excitation level. The solution given by the proposed approach is in good agreement with the exact analysis, The error in the proposed solution is 2.67 percent, much less than the error 5.86 percent in the usual solution. The system residual associated with the usual lincarization is 1
3
< LB(jc z + kxZ)Jc - c'ofc] 2 > -- 5.57(s0/~)i
(17)
The residual associated with the proposed lincarization is 1
3
< [fl(~2 + kx2)]¢ _ c=~]~ > = 7.01(s0~/~)~
(18)
4
X.X. LEE and J.Q. CHEN
The two residuals are different, just as E X A M P L E 1 showns. E X A M P L E 3. Elastoplastic System 3~+~o2J+(1-~) p = k ~ - ylJc I Y I Y I " - '
Consider the nonlinear system described by
2y=/(t) - B~Iyl "
where y represents the hysteretically restoring hysteresis shape; ~ is the ratio of postyield to quency; f(t) is a z e r o - m e a n Gaussian white The auxiliary linear system given by proposed
(19) force; k, y,/~ and n are parameters controling the preyield stiffness; o~ is the preyielding natural frenoise. The response state vector is Z = (x y ~)r. linearization technique is
2
~, = k ~Yc + k a y
(20)
The two coefficients kland k 2 can be approximately evaluated in terms of the second moments of x and y which are given in the Appendix. For example, for the case k = 1, ~ = f l = 0 . 5 , n=~o = 1, and a = 1 / 21, the RMS response a x as a function of the excitation level is shown in Fig.3. The usual linearization solution and the M o n t e - C a r l o simulation [6] are also shown in Fig.. It is observed that the result calculated by the proposed technique compares favourably with what obtained by M o n t e - C a r l o simulation.
Conclusions
An integral-based lincarization technique has been developed for the random vibration analysis of nonlinear systems. Although the two linearization techniques agrec very well thc cxaet analysis or M o n t e - C a r l o simulation and are evidently comparable in accuracy, the proposed linearization techniquc is more efficient procedurc for all nonlinear systcms considered. This proposed technique also has the advantage of being made directly and easy to apply. Consequently, the minimization of the residual between thc actual system and the auxiliary system does not give rise to that of the response difference. Based on examples of three kinds of nonlinear systems, the accuracy of the proposed linearization technique appears to be well within the limits of practical engineering usefulness.
References
1.Lin, Y.K., Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967. 2.Spanos, P.D., Appl Mech Rev, 34(1), 1981. 3.Iwan, W.D., and Yang, I - M , J Appl Mech ASCE, 39(2), 1972. 4.Iwan, W.D., Int J Nonlinear Mech, 8, 1973. 5.Spanos, P.D., and Iwan, W.D., Int J Nonlinear Mech, 13,1978. 6.Wen, Y.K., J Appl mech ASCE, 47(1), 1980. 7.Caughey, T.K., Nonlinear Theory of Random Vibrations in Advances in Applied Mechanics, Academic Press, 1971.
RANDOM VIBRATIONS
OF NONLINEAR SYSTEMS
5
Appendix
The two coefficients k I and k 2 in equation
k, = k - - ( ? J ~ + #J3)o~
(20) are
;
k 2 = --(7J2 + flJ,)%,tr~-' where 2 ~ - F(
)
{(2p + 28p 3
J ' = 37t(n + 1)(1 + 4p 4)
+S:_ sin'+20 N+6 ~ _
2~-F(
J2
=
2p + 36p ~ - 48p s
n + 4
n + 6
~+ )(1 - p 2 ) - q -
• [(1 + 6p 2 - 2 4 p 4 ) s i n 4 0 - (2 + 9p 2 " 26p')sin20+ (1 + 4 p 2 ) ( 1 - p2)]dO )
3n(n + 1)(1 + 4 p ' ) { ( 2 - 16p 2 - 16p' n+ 4
}
,+~ 2 - - 24p 2 + 32p 6 n+ 6 )(1 - p2) -2-
-- ~:_osin'+20 • [(6p - 8p ~ -- 16pS)sin'O - (9p -- 8p 3 - 16pS)sin20 + (3p + 2p3)(1 -- p2)]dO
2'~ r ( ~
)
J3 = 3xf--n(n + 1)(1 + 4 p ' ) • +3
J4
};
{(3 + 12p2)(1 -- p2) _ (3p2 _ 18p')(n + 3) -- 2 p ' ( n + 5)(n + 3)} ;
n+8~
2 ~- r(~-- ; 3xF~-~(n + 1)(1 + 4 p ' ) {(9,0 + 6p3)(I - p2) _ (3p - 6p 3 - 12pS)(n + 3) - (p~ + 2pS)(n + 5)(n + 3)}
a n d w h e r e p is t h e c o r r e l a t i o n
coefficient of random
v a r i a b l e s ~ a n d y.
6
2 1
0
z'oo
3 'oo
tbo
roo
$,Ic/L Fig.1 RMS response of the system with nonlinear stiffness: Proposed Usual iinearization (- - --); Exact analysis (- • -)
linearization
(
),
6
X.X.
it '
;
o
LEE and J.Q.
1
¢'oa
I
CHEN
--
I
I
z'oo 9'oo 5,1¢~(2
~a
~roo
Fig.2 RMS response of the system with nonlinear damping; Legend as in Fig.1
Ioo
•
,
,
,
,,,,,
,
,
,
, , i , , i
7
i
,~
•
,'~
~ fo
~,Of
•
A
. . . .
~il
.f
Fig.3RMS response simulation( ° )
i
,
i
.
,,lJ,
•
,
,
l
of the elastoplastic systcm:
4
.,IL
/0
Legend
as in Fig.l; Monte-Carlo
l ~ m ~ e d in the U S A Pop~p-rnon Press plc
A LINEARIZATION TECHNIQUE FOR RANDOM VIBRATIONS OF NONLINEAR SYSTEMS
X.X.Lee and J.Q.Chen Institute of Structural Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
(Received 21 June 1990; accepted for print 16 July 1991) Introdution
Stochastically excited linear systems have been studied in great detail and numerous analytic techniques exist for treating both the stationary and nonstationary problems. Unfortunately, many structures of engineering interest can not be considered linear and the techniques for analyzing nonlinear systems are not nearly so well developed. Some systems with simpile nonlinearities can be handled exactly by means of the FPK equation method, but exact solutions are not available for most nonlinear systems. As a consequence, several approximate techniques have been proposed. We know that one of the basic tools for the approximate treatment of nonlinear system is the equivalent linearization method [1]. Recently, the excellent review and works [2-7] show the efficiency and considerable promise for it is not limited by the restriction commonly imposed on the other approaches. Furthermore, this approach can be made quite direct and relatively easy to apply. The usual linearization technique is to minimize the residual beween the actual nonlinear system and the auxiliary linear system. However, this does not certainly lead to the minimization of the residual of responses corresponding to the actual system and the auxiliary system. The pu~ose of the present paper is a cotribution to the study of lincarization approach for treating problem of the stochastic response analysis of nonlinear systems under random excitations. The accuracy of the proposed technique is investigated by means of analytical and numerical examples.
Proposed Linearization Technique
Consider a nonlinear system which can be written in the n-degree-of-freedom state vector form + G(Z) = F(t) (1) where Z is a generalized state vector, G(Z) is a generalized system vector, and F(t) is a stationary i
X.X.
2
LEE and J.Q.
CHEN
Gaussian random vector representing the excitation to the system. As a means of obtaining an approximate solution of (1) let us consider an auxiliary system which is described by a linear differential equation of the form
2 + aZ = F(0 (2) where A is the system matrix which is at this point arbitrary. The measurement of the inequality beween the actual system (1) and the auxiliary system (2) is expressed as = IIG(Y) - A Eli (3) where II ° II is a measuring scaler. F o r usual linearization techniques, this scaler is selected as [1,21 IIG(Y) - A YII = G ( Y ) - A Y (4) Considering the solution of equations (1) and (2), one may see that the minimization o f the residual {$ ~G(Y)dt-$ ~AYdt} gives rise to that o f the response difference. Unfortunately, this residual can not be calculated. As a means of obtaining an approximate formulation of this residual, we choose the measuring scaler as follow = I I G ( Y ) - A Yll =-fn [ G ( Y ) - A Y]df~
(5a)
or as a approximation, the components of e are ki
'; = S~' j-~2 ""I~ ( o , ( r ) -
(Sb)
A,,Y,lay~ay2...dy~,
(i = 1,2,...,n;/" = 1,2,...k~) where Gi(Y) is the component of G(Y) and related to the state variable (Yl Y2 "'" Yra);fl = {Yi[ YiE [0,X], i = 1,2,'",n; X is the stationary solution of (2)} is the definition domain o f Y. The magnitude of this vector t will clearly be function of the auxiliary system matrix A. The approximate solution of the original equation (1) will be generated by selecting A in such a way that some measure of 8 is minimized. As a minimization criterion on ~, it will be required that the mean value o f the scalar product be a minimum. That is T
< 8 ~ > = minimum
(6) where the operator < • > denotes the statistical average of the random variable. The necessary condition for equation (6) are ~
T
8>
~a i]
T ~8
=2<8
--
8a #
> =0
or T
< 8X x 1x ~ . " x , > = 0
(7)
As a consequence o f the fact that F(t) is Gaussian, it may be shown that equation (7) represents a ture minimum of < ere > . Condition (7) can be rewritten as T
2<~aG(Y)df~X xlx2""x ,> -A
2
) > =0
(8)
Hence, if < XXr(xlx2-.,x°)2 > is not singular, the equation (8) has the solution
A = 2 < S n G ( Y ) d I I X r x ~ x 2 . . . x ~> < X X r ( x l x 2 . . . x
2
) >
-1
(9)
which will lead to the minimum for < fie > Now, the original problem has been reduced to the solution of a related linear problem which may be solved by any number of analytic techniques. However, since matrix A depands on the response statistics, an iteration solution procedure is generally required which is also necessary to usual linearization techniques.
RANDOM VIBRATIONS OF NONLINEAR SYSTEMS
3
Accuracy of the technique
The linearization technique proposed in this paper has been tested by studying three different nonlinear systems in which two problems can be solved exactly by F o k k e r - P l a n k equation method and the last problem/s solution will be given by Monte-Carlo simulation. E X A M P L E 1. System with Nonliear Stiffness
Consider a system described by
Y¢ + c ~ + ~ x 3 = R t ) (10) where the excitation f(t) is a zero-mean Gaussian white noise with the power spectral density So. From the nature of the excitation and properties of the stationary reponse of linear system, the auxiliary linear system is readily shown that
(11)
)~ + coYc + k o x = / ( t )
Then, the RMS response of equation (11) can be obtained directly. That is 'gO
1
crx = 1.0588(--)~ (12) c# The RMS response tT~ of the nonlinear system as a function of the excitation level is shown in Fig.1. Results are given for the proposed linearization, the usual linearization, and the exact analysis [3]. By comparison with the exact solution, the proposed linearization gives an even better solution. The error in the proposed solution is 3.27 percent, much less than the error 7.58 percent in the usual solution. As a result, the residual beween the actual system and the usual auxiliary system is 1
3
< ( c # x ~ - k / = x ) 2 > = 6.43(s0#~ / c) ~
(13)
At the same meaning, the residual corresponding to the proposed system is 1
3
< ( c ~ x 3 - k = x ) z > = 8.81(s0/z~ / c) ~
(14)
It is apparent that the system residual caused by proposed technique is larger than that caused by usual technique. E X A M P L E 2. System with Nonlinear Damping
Consider a system described by
~¢ + fl(:c2 + kxZ)]c + kx = ](t) (15) where f(t) is the same excitation as E X A M P L E 1. The approximate RMS response obtained by proposed approach is
o =0.9733( s° )~ (16) x /~k 2 The results given by the proposed iinearization, the usual lincarization and the exact analysis [7] arc plotted in Fig.2, against the excitation level. The solution given by the proposed approach is in good agreement with the exact analysis, The error in the proposed solution is 2.67 percent, much less than the error 5.86 percent in the usual solution. The system residual associated with the usual lincarization is 1
3
< LB(jc z + kxZ)Jc - c'ofc] 2 > -- 5.57(s0/~)i
(17)
The residual associated with the proposed lincarization is 1
3
< [fl(~2 + kx2)]¢ _ c=~]~ > = 7.01(s0~/~)~
(18)
4
X.X. LEE and J.Q. CHEN
The two residuals are different, just as E X A M P L E 1 showns. E X A M P L E 3. Elastoplastic System 3~+~o2J+(1-~) p = k ~ - ylJc I Y I Y I " - '
Consider the nonlinear system described by
2y=/(t) - B~Iyl "
where y represents the hysteretically restoring hysteresis shape; ~ is the ratio of postyield to quency; f(t) is a z e r o - m e a n Gaussian white The auxiliary linear system given by proposed
(19) force; k, y,/~ and n are parameters controling the preyield stiffness; o~ is the preyielding natural frenoise. The response state vector is Z = (x y ~)r. linearization technique is
2
~, = k ~Yc + k a y
(20)
The two coefficients kland k 2 can be approximately evaluated in terms of the second moments of x and y which are given in the Appendix. For example, for the case k = 1, ~ = f l = 0 . 5 , n=~o = 1, and a = 1 / 21, the RMS response a x as a function of the excitation level is shown in Fig.3. The usual linearization solution and the M o n t e - C a r l o simulation [6] are also shown in Fig.. It is observed that the result calculated by the proposed technique compares favourably with what obtained by M o n t e - C a r l o simulation.
Conclusions
An integral-based lincarization technique has been developed for the random vibration analysis of nonlinear systems. Although the two linearization techniques agrec very well thc cxaet analysis or M o n t e - C a r l o simulation and are evidently comparable in accuracy, the proposed linearization techniquc is more efficient procedurc for all nonlinear systcms considered. This proposed technique also has the advantage of being made directly and easy to apply. Consequently, the minimization of the residual between thc actual system and the auxiliary system does not give rise to that of the response difference. Based on examples of three kinds of nonlinear systems, the accuracy of the proposed linearization technique appears to be well within the limits of practical engineering usefulness.
References
1.Lin, Y.K., Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967. 2.Spanos, P.D., Appl Mech Rev, 34(1), 1981. 3.Iwan, W.D., and Yang, I - M , J Appl Mech ASCE, 39(2), 1972. 4.Iwan, W.D., Int J Nonlinear Mech, 8, 1973. 5.Spanos, P.D., and Iwan, W.D., Int J Nonlinear Mech, 13,1978. 6.Wen, Y.K., J Appl mech ASCE, 47(1), 1980. 7.Caughey, T.K., Nonlinear Theory of Random Vibrations in Advances in Applied Mechanics, Academic Press, 1971.
RANDOM VIBRATIONS
OF NONLINEAR SYSTEMS
5
Appendix
The two coefficients k I and k 2 in equation
k, = k - - ( ? J ~ + #J3)o~
(20) are
;
k 2 = --(7J2 + flJ,)%,tr~-' where 2 ~ - F(
)
{(2p + 28p 3
J ' = 37t(n + 1)(1 + 4p 4)
+S:_ sin'+20 N+6 ~ _
2~-F(
J2
=
2p + 36p ~ - 48p s
n + 4
n + 6
~+ )(1 - p 2 ) - q -
• [(1 + 6p 2 - 2 4 p 4 ) s i n 4 0 - (2 + 9p 2 " 26p')sin20+ (1 + 4 p 2 ) ( 1 - p2)]dO )
3n(n + 1)(1 + 4 p ' ) { ( 2 - 16p 2 - 16p' n+ 4
}
,+~ 2 - - 24p 2 + 32p 6 n+ 6 )(1 - p2) -2-
-- ~:_osin'+20 • [(6p - 8p ~ -- 16pS)sin'O - (9p -- 8p 3 - 16pS)sin20 + (3p + 2p3)(1 -- p2)]dO
2'~ r ( ~
)
J3 = 3xf--n(n + 1)(1 + 4 p ' ) • +3
J4
};
{(3 + 12p2)(1 -- p2) _ (3p2 _ 18p')(n + 3) -- 2 p ' ( n + 5)(n + 3)} ;
n+8~
2 ~- r(~-- ; 3xF~-~(n + 1)(1 + 4 p ' ) {(9,0 + 6p3)(I - p2) _ (3p - 6p 3 - 12pS)(n + 3) - (p~ + 2pS)(n + 5)(n + 3)}
a n d w h e r e p is t h e c o r r e l a t i o n
coefficient of random
v a r i a b l e s ~ a n d y.
6
2 1
0
z'oo
3 'oo
tbo
roo
$,Ic/L Fig.1 RMS response of the system with nonlinear stiffness: Proposed Usual iinearization (- - --); Exact analysis (- • -)
linearization
(
),
6
X.X.
it '
;
o
LEE and J.Q.
1
¢'oa
I
CHEN
--
I
I
z'oo 9'oo 5,1¢~(2
~a
~roo
Fig.2 RMS response of the system with nonlinear damping; Legend as in Fig.1
Ioo
•
,
,
,
,,,,,
,
,
,
, , i , , i
7
i
,~
•
,'~
~ fo
~,Of
•
A
. . . .
~il
.f
Fig.3RMS response simulation( ° )
i
,
i
.
,,lJ,
•
,
,
l
of the elastoplastic systcm:
4
.,IL
/0
Legend
as in Fig.l; Monte-Carlo