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Design sensitivity analysis of nonlinear mode spectrum for large amplitude forced vibrations of beams
Pergamon
0045-7949(94)E0264-3
Compurers & Bructures Vol. 53, No. 2. pp. 363-366, 1994 Copyright tC 1994 Elwier Science Ltd Printed in Great Britain. All rights reserved 0045.7949/94 17.00 + 0.00
DESIGN SENSITIVITY ANALYSIS OF NONLINEAR MODE SPECTRUM FOR LARGE AMPLITUDE FORCED VIBRATIONS OF BEAMS Suhuan Chen, Aijun Ma and Wanzhi Han Department of Mechanics, Jilin University of Technology, Changchun 130022, People’s Republic of China (Received 17 Muy 1993) Abstract-A computational method for design sensitivity analysis of a nonlinear mode spectrum for forced vibrations of beams is presented. The large deflection leads to geometric nonlinearity. Longitudinal displacement and inertia are considered in the nonlinear vibration formulation. A numerical example is given to illustrate the validity of the proposed method.
INTRODUCIION
Design sensitivity analysis of eigenvalues and eigenvectors plays an important role in structural optimization, because this analysis will reveal how the change in some design parameters in the system affects the dynamic characteristics of the structure. The designer can use this information directly in the interactive computer-aided design procedures as a valuable guide. There have been a number of works on the design sensitivity analysis of eigenproblems published during the past two decades [l-3], but most of them are associated with linear vibration. Recently, some authors discussed the design sensitivity analysis of eigenvalues and eigenvectors of nonlinear vibrations. Rao and Narayanaswami [4] used an optimality criterion approach to extract the optimum configuration of beam and plate structures executing large amplitude vibrations subject to a frequency constraint. They presented a simple relation between the nonlinear geometrical matrix and the design variables. The design variables are the thickness of the beam and the thickness of the plate. With this given relation, the eigenvalue derivative can be obtained in the same fashion as the one obtained for the linear vibration problem. Hou and Yuan [5] investigated a beam with immovable ends under large amplitude oscillations which was subjected to an amplitudedependent stretching force. The geometrical nonlinearity results from the interaction between the axial deformation and lateral deflection. They derived the analytical equations of design sensitivity analysis and developed a finite element method to calculate the derivatives of the eigenvalue and eigenvector. But in that paper the longitudinal inertia was not considered in the nonlinear vibration formulation, so that the governing differential equation for the
nonlinear vibration of a beam could be described in terms of the lateral deflection alone. In 1990, Hou et al. [6] proceeded to study the design sensitivity analysis of beams under nonlinear forced vibrations. The in-plane inertia and displacement are considered in the nonlinear vibration formula. This paper presents a finite element perturbation method to compute the design sensitivity of a nonlinear mode spectrum for large amplitude forced vibrations of beams subjected to harmonic loading. The effect of longitudinal deformation and longitudinal inertia are both included in the formulation. First, we use the direct differential approach to derive the design sensitivity formula of the nonlinear mode spectrum, and then transform the differential approach into a finite element perturbation. Finally, the validity of the present approach is illustrated by a numerical example. The results show that the proposed method has a high accuracy and proves to be effective. THEORY
The finite element equation of motion for a beam under influence of inertia, elastic, large deflection and harmonic excitation forces may be stated as MU + (KL + K&U = P,
(1)
where the symmetric matrices K,_. KNL, and M are the global linear stiffness matrix, the nonlinear stiffness matrix and the mass matrix, respectively; U is the displacement vector and P is the external force vector. Considering the effect of large deflection, P will change as the beam vibrates. Consider the two-dimensional beam element, shown in Fig. 1, in which each node has three degrees of freedom, two translational and one rotational.
363
364
Suhuan Chen et al.
eqn (6) becomes
A,.jJ/rM*,=II/T(KL,j+K,,,-‘iM,i)*i.
L3fe-----
As we assume the system is linear in every integral step, so the eigenvector satisfies
Fig. 1. Beam element. Referring to [7], we can find the element linear stiffness matrix Ki, nonlinear stiffness matrix K& and consistent mass matrix M’. Using the standard assembly procedure, the global linear stiffness matrix KL, the nonlinear geometric &.,, and the mass matrix M can be assembled. Equation (1) is nonlinear. An incremental form of the finite element equations of motion can be derived as follows: M, Ats, + (K& + K,,,) AU, = AP, ,
(KL+ &L M = aM$,
The sensitivity of the eigenvector, expressed as the following series
J/i.i, can
where n is the number of the eigenvector, and C,, is a coefficient. Premultiplying eqn (5) by tj: leads to
eqn (10) into eqn (1 1), we have
Substituting
I,$:(KL,~+ KNL.~-
AiM,, - &,M)d’i
(4)
Let il,, and $ti.; denote the derivatives of the eigenvalue Izi and the eigenvector (l// with respect to the Using eqn (8) and letting s = k, we obtain design variables b, (j = 1,2, . . . , m); and &, KNL.1 and M,, denote the derivative of the linear stiffness, nonlinear stiffness and mass matrix with respect to bi, respectively. Upon differentiation with respect to bj, eqn (4) leads to ,..., n. i# k, i,k=1,2
+ K
be
(3)
where 1 is the eigenvalue, and $ is the corresponding mode shape. By using iterative perturbation [8], eqn (3) can be solved. Let $, denote the eigenvector corresponding to the eigenvalue i,, then they satisfy (K, + KNL - &M)I)Q = 0.
where 6, is the Kronecker delta. Thus, from eqn (7), we get the design sensitivity of the eigenvalue
(21
where the index t denotes current time. The mass and linear stiffness matrices are constant and the nonlinear stiffness matrix, KtNL, can be treated as constant within the time increment. That is to say, in every integral step, we can assume the system is linear. Thus the approximate linearized eigenvalue equation of the beam can be obtained:
Premultiplying
(7)
I
+ KNL- djMM,., = 0. (5)
(13)
The coefficient Ciji in eqn (10) can be determined by the normalized condition
eqn (5) by Jf,‘, we obtain Upon differentiation
this equation with respect to bi,
we have +~r(K,+K,,-~iM)~i,,=O.
Considering
(6)
the following relation
#ziMtLi= -‘/2+fM,,rlt,. Substituting
eqn (10) into eqn (14), we arrive at
$:(KL + KNL- 4MM,,i = ‘k:, x (KL + KNL- liM)l(li = 0,
i
C,j&fW = -
ww,,*,.
(14)
Design sensitivity anaiysis of a nonlinear If s = i
in the above equation, we obtain C,=
-l/Z$TM,,$,.
(15)
FINlTR ELEMENT P~~~A~~~ OF DESIGN SENSITIVITY ANALYSIS
Although the direct differential method is quite straightforward, it needs to construct the derivative matrices of stiffness and mass. So the above direct differential method is inconvenient and it is necessary to transform the differential approach into finite element perturbation. Let AK,, AKNL and AM denote the increment of the linear stiffness, the nonlinear stiffness and the mass matrices resulting from an incremental change of the design variable, Abj; and A& and A$i denote the co~sponding increment of the eigenvalue and the eigenvector. The direct differential method of design sensitivity analysis of vibrational modes can now be converted into perturbation form, approximately I, z AlzifAbj = $;(AlY,
+
AKN, - ~iAM)~~iAb~ (16)
365
mode spectrum
where ij is the eth element eigenvector. It is important to observe in eqns (21) and (22) that the calculations are on the element basis, therefore, the calculations are greatly simplified. Using the foIIowing shorthand notations
n;j
= i
@(AK;
+
AKC,, - li AM’)+,
(23)
J
eqns (21) and (22) can be written as: 1, = c n;J e
(25)
where JTaand I& are the design ~nsitivity of the eth element for the eigenvalue 1, and the eigenvector +i, respectively.
NUMERICAL EXAMPLE
-
lAbj* l/‘X$TAM@i)$i
(17)
1
In finite element analysis, AKL, AK,, and AM are known to be the sum of the element increment AKt, AKeNLand AM” AK, = c AK;.
(18)
e AKNL = c A&_ e
AM = c AM=, e
(20)
The theory developed in this paper will be applied to the design sensitivity analysis of a nonlinear forced vibration of a beam. Consider a IO-element cantilever beam with a concentrated mass, MC (20 kg), attached to the right-hand end, as shown in Fig. 2. The beam has 30 degrees of freedom. The length L and radius of the cross-section D are 2 and 0.025 m, respectively. The Young’s modulus of elasticity E is 7.0 x 10’0Nm-2. The mass density p is 2.7 x 10’ kgmm3. The initial axial loading P,, and lateral loading P., are defined by 19.6 cos(55t)N and 49cos(55t)iV, respectively. Consider the effect of the large deflection, 9, and cy will change as the beam vibrates. Assuming the rotational angle of the node 11 is 8, P, and Py can be formulated as:
where summation is over the elements. Substituting eqns (18)-(20) into eqns (16) and (17), we get
P_,= 19.6 cos(55t)cos 0 - 49 cos(55r)sin 6, T,, = 19.6 cos(55r)sin t? + 49 cos(55t )cos 8.
(21) As for the design sensitivity analysis, the radius (D) of the cross-section of the uniform beam is considered
and
x [$J(AKC, + AK:,.
- A, AMe)&)+,
_
2m _ Fig. 2. Ten-element
cantilever beam.
Suhuan Chen et al.
366
Table 1. Design sensitivity analysis of eigenvalues associated with various perturbations of design variables for a cantilever beam Percentage change AD(%) - 10.0 - 6.0 -3.0 + 3.0 + 6.0 + 10.0
Actual change AI
Prediction &,n x AD -8.0114 -5.1171 - 2.6807 2.9407 6.1574 10.9043
- 8.3567 - 5.3282 - 2.7861 3.0418 6.3518 11.2053
Error (%)t 4.1 4.0 3.8 3.3 3.1 2.7
t Error = Id,,, x AD l/An x 100%. Table 2. Design sensitivity analysis of eigenvalues at different times {percentage change AD is 1%) t/At
Actual value
Sensitivity
20 40 60 80 100 120
0.252279:OE + 0.24854806E + 0.24364575E + 0.24083665E + 0.24235875E + 0.24697225E +
20 40 60 80 100 120
A_ O.J55893;2E f 05 0.1556122lE + 05 0.15524328E + 05 O.l5503188E+05 0.155146438+05 0.15549361E +05
k 02 02 02 02 02 02
Prediction?
Error(%)1
04 04 04 04 04 04
A, 0.25222920E + 02 0.24836295E + 02 0.24327721E + 02 0.240487~E + 02 0.24226076s + 02 0.24712415E +02
0.02 0.07 0.15 0.14 0.04 0.06
AZ.0 O.l2731052E$O? 0.12731211E+07 0.12730687E + 07 0.1272940JE + 07 0.12728184E + 07 0.12728175E+07
I‘2 0.15594964E + 05 0.15565875E + 05 O.l55276iOE+O5 0.15506620E + 05 0.1551996lE +05 0.155565528 + 05
0.04 0.03 0.02 0.02 0.03 0.05
AI,, 0.38931235E + 0.386422768 + 0.38252781E + 0.380263598 + 0.38144256E + 0.38510173E +
t Prediction = I,, x AD + I,, f Error = II -&l/i, x 100%.
as the design variable. For the changes of beam radius of the cross-section ranging from - 10% to + lo%, as shown in Table 1, the predicted changes of the first mode eigenvalue are in good agreement with the actual changes calculated by the finite difference. In Table 2, the results of the design sensitivity analysis of the first and the second mode eigenvalue (2,. 2,) are shown at different times when the percentage change (AD) is 1%. The actual eigenvalues, 1,, and ILc2,are calculated by the iterative perturbation [S]. The error is very small. These results demonstrate that the presented algorithm for calculating design derivatives of the nonlinear mode spectrum is valid.
CONCLUSION
An efficient numerical method, the finite element ~rturbation method, is developed to calculate derivatives of eigenvalues and eigenvectors for nonlinear vibrations of beams subjected to harmonic excitation. By using the sum of finite element matrices, elements.
calculations can be based on individual Thus, this method offers a very efficient and
convenient way to carry out the sensitivity analysis of nonlinear mode spectrum for large amplitude forced vibrations of structures.
~cknowle~ge~e~f-Thus work is supported Naturai Science Foundation of China.
by National
REFERENCES
1. W. Y. Tseng and J. Dugundji, Nonlinear vibrations of a beam under harmonic excitation. J. uppi. Me&. 38, 467-476 (1971). 2. C. Mei. Comments on ‘Lagrange-type formulation for finite element analysis of nonlinear beam vibrations’. /. Sound Vibr. 94, 445-447
(1984).
3. C. Mei and K. Decha-Umpkai, A finite element method for nonlinear forced vibrations of beams. J. Sound Vibr. 102, 369-380
(198.5).
4. G. V. Rao and R. Narayanaswami, Optimization of simply supported beams on large amplitude vibration subject to a frequency. J. Sound Vibr. 74, 1399142 (1981). 5. J. W. Hou and J. Z. Yuan, Calculation of eigenvalue and eigenvector derivatives for nonlinear beam vibrations. AIAA JI 26, 872-880 (1988). 6. Jean W. Hou, Chuh Mei and Yong X. Xue, Design sensitivity analysis of beams under nonlinear forced vibrations. AZAA JI 28, 1067-1068 (1990). 7. J. S. Przemieniecki, Theory of‘Murri.x Structural Analysis. McGraw-Hill, New York (1968). 8. Suhuan Chen. Tao Xu and Zhongsheng Liu, Nonlinear frequency spectrum in nonlinear structural analysis. Coals?. Srruct. 45, 5533556 (1992). 9. Suhuan Chen and H. H. Pan, Design sensitivity analysis of vibration modes by finite element perturbation. Proc. Ih IMAC, pp. 38-43 (1986).
0045-7949(94)E0264-3
Compurers & Bructures Vol. 53, No. 2. pp. 363-366, 1994 Copyright tC 1994 Elwier Science Ltd Printed in Great Britain. All rights reserved 0045.7949/94 17.00 + 0.00
DESIGN SENSITIVITY ANALYSIS OF NONLINEAR MODE SPECTRUM FOR LARGE AMPLITUDE FORCED VIBRATIONS OF BEAMS Suhuan Chen, Aijun Ma and Wanzhi Han Department of Mechanics, Jilin University of Technology, Changchun 130022, People’s Republic of China (Received 17 Muy 1993) Abstract-A computational method for design sensitivity analysis of a nonlinear mode spectrum for forced vibrations of beams is presented. The large deflection leads to geometric nonlinearity. Longitudinal displacement and inertia are considered in the nonlinear vibration formulation. A numerical example is given to illustrate the validity of the proposed method.
INTRODUCIION
Design sensitivity analysis of eigenvalues and eigenvectors plays an important role in structural optimization, because this analysis will reveal how the change in some design parameters in the system affects the dynamic characteristics of the structure. The designer can use this information directly in the interactive computer-aided design procedures as a valuable guide. There have been a number of works on the design sensitivity analysis of eigenproblems published during the past two decades [l-3], but most of them are associated with linear vibration. Recently, some authors discussed the design sensitivity analysis of eigenvalues and eigenvectors of nonlinear vibrations. Rao and Narayanaswami [4] used an optimality criterion approach to extract the optimum configuration of beam and plate structures executing large amplitude vibrations subject to a frequency constraint. They presented a simple relation between the nonlinear geometrical matrix and the design variables. The design variables are the thickness of the beam and the thickness of the plate. With this given relation, the eigenvalue derivative can be obtained in the same fashion as the one obtained for the linear vibration problem. Hou and Yuan [5] investigated a beam with immovable ends under large amplitude oscillations which was subjected to an amplitudedependent stretching force. The geometrical nonlinearity results from the interaction between the axial deformation and lateral deflection. They derived the analytical equations of design sensitivity analysis and developed a finite element method to calculate the derivatives of the eigenvalue and eigenvector. But in that paper the longitudinal inertia was not considered in the nonlinear vibration formulation, so that the governing differential equation for the
nonlinear vibration of a beam could be described in terms of the lateral deflection alone. In 1990, Hou et al. [6] proceeded to study the design sensitivity analysis of beams under nonlinear forced vibrations. The in-plane inertia and displacement are considered in the nonlinear vibration formula. This paper presents a finite element perturbation method to compute the design sensitivity of a nonlinear mode spectrum for large amplitude forced vibrations of beams subjected to harmonic loading. The effect of longitudinal deformation and longitudinal inertia are both included in the formulation. First, we use the direct differential approach to derive the design sensitivity formula of the nonlinear mode spectrum, and then transform the differential approach into a finite element perturbation. Finally, the validity of the present approach is illustrated by a numerical example. The results show that the proposed method has a high accuracy and proves to be effective. THEORY
The finite element equation of motion for a beam under influence of inertia, elastic, large deflection and harmonic excitation forces may be stated as MU + (KL + K&U = P,
(1)
where the symmetric matrices K,_. KNL, and M are the global linear stiffness matrix, the nonlinear stiffness matrix and the mass matrix, respectively; U is the displacement vector and P is the external force vector. Considering the effect of large deflection, P will change as the beam vibrates. Consider the two-dimensional beam element, shown in Fig. 1, in which each node has three degrees of freedom, two translational and one rotational.
363
364
Suhuan Chen et al.
eqn (6) becomes
A,.jJ/rM*,=II/T(KL,j+K,,,-‘iM,i)*i.
L3fe-----
As we assume the system is linear in every integral step, so the eigenvector satisfies
Fig. 1. Beam element. Referring to [7], we can find the element linear stiffness matrix Ki, nonlinear stiffness matrix K& and consistent mass matrix M’. Using the standard assembly procedure, the global linear stiffness matrix KL, the nonlinear geometric &.,, and the mass matrix M can be assembled. Equation (1) is nonlinear. An incremental form of the finite element equations of motion can be derived as follows: M, Ats, + (K& + K,,,) AU, = AP, ,
(KL+ &L M = aM$,
The sensitivity of the eigenvector, expressed as the following series
J/i.i, can
where n is the number of the eigenvector, and C,, is a coefficient. Premultiplying eqn (5) by tj: leads to
eqn (10) into eqn (1 1), we have
Substituting
I,$:(KL,~+ KNL.~-
AiM,, - &,M)d’i
(4)
Let il,, and $ti.; denote the derivatives of the eigenvalue Izi and the eigenvector (l// with respect to the Using eqn (8) and letting s = k, we obtain design variables b, (j = 1,2, . . . , m); and &, KNL.1 and M,, denote the derivative of the linear stiffness, nonlinear stiffness and mass matrix with respect to bi, respectively. Upon differentiation with respect to bj, eqn (4) leads to ,..., n. i# k, i,k=1,2
+ K
be
(3)
where 1 is the eigenvalue, and $ is the corresponding mode shape. By using iterative perturbation [8], eqn (3) can be solved. Let $, denote the eigenvector corresponding to the eigenvalue i,, then they satisfy (K, + KNL - &M)I)Q = 0.
where 6, is the Kronecker delta. Thus, from eqn (7), we get the design sensitivity of the eigenvalue
(21
where the index t denotes current time. The mass and linear stiffness matrices are constant and the nonlinear stiffness matrix, KtNL, can be treated as constant within the time increment. That is to say, in every integral step, we can assume the system is linear. Thus the approximate linearized eigenvalue equation of the beam can be obtained:
Premultiplying
(7)
I
+ KNL- djMM,., = 0. (5)
(13)
The coefficient Ciji in eqn (10) can be determined by the normalized condition
eqn (5) by Jf,‘, we obtain Upon differentiation
this equation with respect to bi,
we have +~r(K,+K,,-~iM)~i,,=O.
Considering
(6)
the following relation
#ziMtLi= -‘/2+fM,,rlt,. Substituting
eqn (10) into eqn (14), we arrive at
$:(KL + KNL- 4MM,,i = ‘k:, x (KL + KNL- liM)l(li = 0,
i
C,j&fW = -
ww,,*,.
(14)
Design sensitivity anaiysis of a nonlinear If s = i
in the above equation, we obtain C,=
-l/Z$TM,,$,.
(15)
FINlTR ELEMENT P~~~A~~~ OF DESIGN SENSITIVITY ANALYSIS
Although the direct differential method is quite straightforward, it needs to construct the derivative matrices of stiffness and mass. So the above direct differential method is inconvenient and it is necessary to transform the differential approach into finite element perturbation. Let AK,, AKNL and AM denote the increment of the linear stiffness, the nonlinear stiffness and the mass matrices resulting from an incremental change of the design variable, Abj; and A& and A$i denote the co~sponding increment of the eigenvalue and the eigenvector. The direct differential method of design sensitivity analysis of vibrational modes can now be converted into perturbation form, approximately I, z AlzifAbj = $;(AlY,
+
AKN, - ~iAM)~~iAb~ (16)
365
mode spectrum
where ij is the eth element eigenvector. It is important to observe in eqns (21) and (22) that the calculations are on the element basis, therefore, the calculations are greatly simplified. Using the foIIowing shorthand notations
n;j
= i
@(AK;
+
AKC,, - li AM’)+,
(23)
J
eqns (21) and (22) can be written as: 1, = c n;J e
(25)
where JTaand I& are the design ~nsitivity of the eth element for the eigenvalue 1, and the eigenvector +i, respectively.
NUMERICAL EXAMPLE
-
lAbj* l/‘X$TAM@i)$i
(17)
1
In finite element analysis, AKL, AK,, and AM are known to be the sum of the element increment AKt, AKeNLand AM” AK, = c AK;.
(18)
e AKNL = c A&_ e
AM = c AM=, e
(20)
The theory developed in this paper will be applied to the design sensitivity analysis of a nonlinear forced vibration of a beam. Consider a IO-element cantilever beam with a concentrated mass, MC (20 kg), attached to the right-hand end, as shown in Fig. 2. The beam has 30 degrees of freedom. The length L and radius of the cross-section D are 2 and 0.025 m, respectively. The Young’s modulus of elasticity E is 7.0 x 10’0Nm-2. The mass density p is 2.7 x 10’ kgmm3. The initial axial loading P,, and lateral loading P., are defined by 19.6 cos(55t)N and 49cos(55t)iV, respectively. Consider the effect of the large deflection, 9, and cy will change as the beam vibrates. Assuming the rotational angle of the node 11 is 8, P, and Py can be formulated as:
where summation is over the elements. Substituting eqns (18)-(20) into eqns (16) and (17), we get
P_,= 19.6 cos(55t)cos 0 - 49 cos(55r)sin 6, T,, = 19.6 cos(55r)sin t? + 49 cos(55t )cos 8.
(21) As for the design sensitivity analysis, the radius (D) of the cross-section of the uniform beam is considered
and
x [$J(AKC, + AK:,.
- A, AMe)&)+,
_
2m _ Fig. 2. Ten-element
cantilever beam.
Suhuan Chen et al.
366
Table 1. Design sensitivity analysis of eigenvalues associated with various perturbations of design variables for a cantilever beam Percentage change AD(%) - 10.0 - 6.0 -3.0 + 3.0 + 6.0 + 10.0
Actual change AI
Prediction &,n x AD -8.0114 -5.1171 - 2.6807 2.9407 6.1574 10.9043
- 8.3567 - 5.3282 - 2.7861 3.0418 6.3518 11.2053
Error (%)t 4.1 4.0 3.8 3.3 3.1 2.7
t Error = Id,,, x AD l/An x 100%. Table 2. Design sensitivity analysis of eigenvalues at different times {percentage change AD is 1%) t/At
Actual value
Sensitivity
20 40 60 80 100 120
0.252279:OE + 0.24854806E + 0.24364575E + 0.24083665E + 0.24235875E + 0.24697225E +
20 40 60 80 100 120
A_ O.J55893;2E f 05 0.1556122lE + 05 0.15524328E + 05 O.l5503188E+05 0.155146438+05 0.15549361E +05
k 02 02 02 02 02 02
Prediction?
Error(%)1
04 04 04 04 04 04
A, 0.25222920E + 02 0.24836295E + 02 0.24327721E + 02 0.240487~E + 02 0.24226076s + 02 0.24712415E +02
0.02 0.07 0.15 0.14 0.04 0.06
AZ.0 O.l2731052E$O? 0.12731211E+07 0.12730687E + 07 0.1272940JE + 07 0.12728184E + 07 0.12728175E+07
I‘2 0.15594964E + 05 0.15565875E + 05 O.l55276iOE+O5 0.15506620E + 05 0.1551996lE +05 0.155565528 + 05
0.04 0.03 0.02 0.02 0.03 0.05
AI,, 0.38931235E + 0.386422768 + 0.38252781E + 0.380263598 + 0.38144256E + 0.38510173E +
t Prediction = I,, x AD + I,, f Error = II -&l/i, x 100%.
as the design variable. For the changes of beam radius of the cross-section ranging from - 10% to + lo%, as shown in Table 1, the predicted changes of the first mode eigenvalue are in good agreement with the actual changes calculated by the finite difference. In Table 2, the results of the design sensitivity analysis of the first and the second mode eigenvalue (2,. 2,) are shown at different times when the percentage change (AD) is 1%. The actual eigenvalues, 1,, and ILc2,are calculated by the iterative perturbation [S]. The error is very small. These results demonstrate that the presented algorithm for calculating design derivatives of the nonlinear mode spectrum is valid.
CONCLUSION
An efficient numerical method, the finite element ~rturbation method, is developed to calculate derivatives of eigenvalues and eigenvectors for nonlinear vibrations of beams subjected to harmonic excitation. By using the sum of finite element matrices, elements.
calculations can be based on individual Thus, this method offers a very efficient and
convenient way to carry out the sensitivity analysis of nonlinear mode spectrum for large amplitude forced vibrations of structures.
~cknowle~ge~e~f-Thus work is supported Naturai Science Foundation of China.
by National
REFERENCES
1. W. Y. Tseng and J. Dugundji, Nonlinear vibrations of a beam under harmonic excitation. J. uppi. Me&. 38, 467-476 (1971). 2. C. Mei. Comments on ‘Lagrange-type formulation for finite element analysis of nonlinear beam vibrations’. /. Sound Vibr. 94, 445-447
(1984).
3. C. Mei and K. Decha-Umpkai, A finite element method for nonlinear forced vibrations of beams. J. Sound Vibr. 102, 369-380
(198.5).
4. G. V. Rao and R. Narayanaswami, Optimization of simply supported beams on large amplitude vibration subject to a frequency. J. Sound Vibr. 74, 1399142 (1981). 5. J. W. Hou and J. Z. Yuan, Calculation of eigenvalue and eigenvector derivatives for nonlinear beam vibrations. AIAA JI 26, 872-880 (1988). 6. Jean W. Hou, Chuh Mei and Yong X. Xue, Design sensitivity analysis of beams under nonlinear forced vibrations. AZAA JI 28, 1067-1068 (1990). 7. J. S. Przemieniecki, Theory of‘Murri.x Structural Analysis. McGraw-Hill, New York (1968). 8. Suhuan Chen. Tao Xu and Zhongsheng Liu, Nonlinear frequency spectrum in nonlinear structural analysis. Coals?. Srruct. 45, 5533556 (1992). 9. Suhuan Chen and H. H. Pan, Design sensitivity analysis of vibration modes by finite element perturbation. Proc. Ih IMAC, pp. 38-43 (1986).