[American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC

American Institute of Aeronautics and Astronautics

1

MULTIMODE LARGE AMPLITUDE FREE VIBRATION OF SHALLOW SHELLS CONSIDERING INPLANE INERTIA

Adam Przekop*, M. Salim Azzouz*, Xinyun Guo*, Chuh Mei�

Old Dominion University, Norfolk, VA 23529-0247

Lahcen Azrar� Universite Abdelmalek Essaadi, Tanger, BP 416, Morocco

A finite element modal formulation for large amplitude free vibration of arbitrary laminated composite shallow shells is presented. The system equations of motion are formulated first in the physical structural-node degrees of freedom. Then the system is transformed into general Duffing-type modal equations with modal amplitudes of coupled linear bending and inplane modes. Multiple modes, inplane inertia, and the first-order transverse shear deformation for composites are considered in the formulation. A shallow shell finite element is developed as an extension from the triangular Mindlin (MIN3) plate element with the improved shear correction factor by Tessler. Time numerical integration is employed to determine the nonlinear periodic frequency characteristics. An iterative procedure to determine the judicious initial conditions for periodic panel response is developed and presented. The general Duffing modal equations in functions of modal amplitudes of linear bending modes only, by dropping the inplane inertia, are also formulated and presented. The inaccuracy in characterizing a shallow shell behavior with bending modes only is demonstrated and discussed. Nonlinear vibrations of antisymmetrically laminated composite cylindrical shell panels are also investigated.

Introduction

Shallow shells are common structural components in many fields of engineering. Various theories of shells were described and outlined in many monographs, for example references.1-3 A review of vibration of shallow shells covering the advances since 1970s was given by Liew et al.4 Marguerre curved plate theory was used by Cummings5 to study large amplitude vibration of a freely supported cylindrical shell segment. Perturbation and exact elliptic integral methods were employed for the panel frequency. Leissa and Kadi6 derived the nonlinear equations of motion for doubly curved shallow shells and studied curvature effects on period of free vibration. They employed the general elliptic equation and the Galerkin method for shells of rectangular boundary supported by shear diaphragms. Donnell’s shell theory was applied by Hui7 for simply supported cylindrical panels with geometric imperfections. Using the Galerkin procedure, the nonlinear vibration frequency was obtained from the Duffing equation with perturbation method. Fu and

*Graduate Research Assistant, Department of Aerospace Engineering, Student Member AIAA. �Professor, Department of Aerospace Engineering, Associate Fellow AIAA. �Professor, Faculty of Sciences and Techniques at Tanger, Department of Mathematics.

Chia8 presented a multi-mode solution for nonlinear free vibration of anti-symmetric angle-ply shallow cylindrical panels with edges elastically supported against rotation. Effects of transverse shear deformation and geometric imperfection were included in their analysis. The harmonic balance method was employed in determining nonlinear frequency of vibration. The Donnell-Mushtari-Vlasov shell theory was used by Raouf and Palazotto9 to model curved orthotropic cylindrical panels with simply supported edges. The spatial domain was discretized using the Galerkin procedure, and the perturbation method was used to evaluate nonlinear natural frequency. Kobayashi and Leissa10 derived governing equations for nonlinear vibration of doubly curved shallow shells based on the first order shear deformation theory. Applying Galerkin procedure, the governing equations were reduced to an elliptic ordinary differential equation in time. Period of vibration for shells with rectangular boundary supported by shear diaphragms was obtained using the Gauss-Lagrange integration method. Shin11 studied the large amplitude vibration of symmetrically laminated moderately thick doubly curved shallow open shells with simply supported edges. By applying a Galerkin approximation, five governing equations of motion were reduced to a single nonlinear time differential equation. The Runge-Kutta time integration scheme was then employed to obtain the nonlinear frequency. Abe et al.12 investigated nonlinear vibration of clamped

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia

AIAA 2003-1772

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

laminated shallow shells by considering first two modes (1st symmetrical and 1st antisymmetrical) and applying the Galerkin’s procedure to the equations of motion. The authors did not treat internal resonance between the first and the second modes, thus the second mode was neglected in determining nonlinear free vibration frequency for the first mode. The influence of the first mode on nonlinear vibration of the second mode was investigated, and the shooting method13 was employed for initial conditions. Pillai and Rao14, and Bhimaraddi15 were concerned about the softening effect in flat plates due to antisymmetrical lamination. Alhazza and Nayfeh16 studied forced vibration of shells and found that solution characteristics may severely change in function of number of modes retained in the analysis. Their results, however, utilizing the multiple scale method is limited to relatively small nonlinearities. All the aforementioned studies have shown both hard- and/or soft-spring behaviors for shallow shells with different geometries, materials and boundary conditions. Based on an exhaust literature search, it is interesting to note that the classical analyses of large amplitude free vibration of shallow shells5-12 have all neglected the inplane inertia terms due to mathematical difficulties. Also problems with obtaining the initial conditions for the steady periodic response resulted in the prevailing number of investigations using a single mode approximation.5-7,9-11,14,15 Moreover, classical solutions were usually obtained for geometries based on rectangular plan-form, isotropic or orthotropic materials, and fully simply supported or fully clamped boundary conditions. By neglecting the inplane inertia terms and using classic analytical method for large amplitude free vibration of shallow shells, it leads to the case that the linear inplane modes are also dropped out from the analysis. The nonlinear Duffing modal equations are thus in functions of linear bending modal amplitudes only. For shallow shell structures, however, the linear bending and inplane modes are inherently physically coupled due to curvature, and to characterize their nonlinear large amplitude behavior with linear bending modes only may yield inaccurate results. The much versatile finite element methods, on the other hand, have been dealt with this coupled linear bending and inplane modes for flat unsymmetrically laminated composite plates (due to laminate stiffness � � 0�B ). The nonlinear general Duffing equations in functions of modal amplitudes of coupled linear bending-inplane modes were reported by Shi et al.17 Abdel-Motaglay et al.18 studied panel flutter with inplane inertia neglected and expressed the nonlinear Duffing equations in functions of linear bending modal amplitudes. This is one of the objectives of the present paper to investigate the inaccuracy in predicting

nonlinear frequency of shallow shells by neglecting inplane inertia terms using finite element method. This paper presents a finite element modal formulation for large amplitude free vibration of arbitrary laminated composite shallow shells. The system equations of motion are formulated first in the physical structural-node degrees of freedom (DOF). Then, the system is transformed into general Duffing-type modal equations with modal amplitudes of coupled linear bending-inplane modes. This linear bending-inplane coupling is due to the shell curvature as well as the unsymmetric lamination ( � � 0�B ). Multiple modes, inplane inertia, and the first-order transverse shear deformation for composites are considered in the formulation. A triangular shallow shell finite element is developed from an extension of the triangular Mindlin (MIN3) element with the improved shear correction factor by Hughes19, and Tessler.20,21 Time numerical integration is employed to determine nonlinear frequency of vibration with judicious initial conditions. An iterative procedure to determine the initial conditions for periodic panel response is developed and presented. By dropping the inplane inertia effect, the general Duffing modal equations in functions of modal amplitudes of linear bending modes only are also formulated and presented, and it is used for comparison of results with existing classic analytical methods. The inaccuracy in characterizing a shallow shell behavior with bending modes alone is demonstrated and discussed.

Formulation Equations of Motion in Structure Node DOF The inplane strain, change of curvature, and shear strain vectors based on the von Karman large deflection and the first-order shear deformation theory for a doubly curved shallow shell are given by

� � � � � � � �

��

�

��

�

�

��

�

��

�

�

��

��

�

��

��

�

��

��

�

��

��

�

�

��

0221

,,

2,

2,

,,

,

,

00

000

y

x

yx

y

x

xy

y

x

bm

Rw

Rw

wwww

vuvu

��

� ��

��

�

��

��

�

�

yyxx

yx

xy

,,

,

,

��

�

�

� (1)

� ��

��

�

��

��

�

��

��

��

��

y

x

y

x

x

ys

Rv

Ru

ww

�

��

,

,

where u, v and w are the inplane and transverse displacements, respectively, and y� and x� are the


3

rotations of the normal to the mid-surface about y and x axes, respectively, and Rx and Ry are the radii describing the shallow shell geometry, as presented in Figure 1. The constitutive equations for a laminated composite shallow shell panel are

��

��

�

��

��

�

��

��

�

��

��

�

��

��

�

ssADBBA

QMN

�

�

�0

0000

(2)

where [A], [B], [D] and [As] are the laminate inplane, inplane-bending coupling, bending, and shear stiffness matrices, respectively. Applying the Hamilton’s principle and standard finite element assembly process, the system equations of motion for a composite shallow shell under large amplitude free vibration can be expressed as

��

��

��

�

��

��

�

��

��

��

��

��

�

��

��

��

��

��

��

�

01110

0

000

00

mb

bmbsRm

sRmb

sRbm

Rmb

Rbm

Rb

sb

mmb

bmb

m

b

m

b

KKK

KKK

KKK

KKKKK

WW

MM

��

��

��

��

��

��

��

��

��

�

��

��

��

��

��

�

��

��

�

��

��

00

0002

0001

0001

0001

0001

m

bbRb

Nb

Nb

Nb

WWKK

KKK Rmb

(3a)

or

� �� 021 �� WKKKWM L�� (3b)

where [M] is mass matrix, [KL] is linear stiffness matrix, and [K1] and [K2] are first- and second-order nonlinear stiffness matrices, respectively, such that they depend linearly and quadratically on the unknown system displacement vector � � � � � �� T

mT

b WWW � , where

� � � � � � � �� Ty

Tx

Tt

Tb WW �� and � � � � � �� TTT

m VUW �.

Indices s and R are corresponding to shear and curvature, respectively, b and m to bending and membrane displacements, respectively, and � � � �� BNb � , � � � �� 0

mm AN �� and � � � �� 0RR AN �� .

Solving the system of Eq. (3) turns out to be computationally costly because 1) at each time step, the element nonlinear stiffness matrices are evaluated and the system nonlinear stiffness matrices are assembled and updated, 2) an iterative-incremental procedure is needed for numerical solution, 3) the number of structure node DOF of � �W is usually very large, and 4) the time step of integration should be extremely small. An efficient solution procedure is to transform Eq. (3) into the modal coordinates17,18 with a modal reduction. This is presented as follows.

Figure 1. A triangular shallow shell finite element Equations of Motion in Modal Coordinates Express the panel deflection as a linear combination of some known functions as

� � � �� qtqWn

r

rr�

�

��

1�� (4)

where the number of retained linear modes is much smaller then the number of structure node DOF. The normal mode � �� r

� , which is the coupled bending-inplane mode normalized with the maximum component to unity, and the linear natural frequency

r� are obtained from the solution of the linear vibration problem (neglecting nonlinear terms in Eq. (3)). The modal mass matrix and the modal linear stiffness matrix are � � � �� L

T KMKM ,, � (5) Nonlinear stiffness matrices in modal coordinates17,18 are defined as

� � � ��

�

n

r

rrq KqK

11 (6)

� � � ��

�

n

r

rss

n

srqq KqqK

1 12 (7)

Now, the equation of motion in the truncated modal DOF has a form � � � � � � � � � �� 01

�� qKKKMq qqq�� (8)

The iterative procedure to determine the initial conditions providing the periodic solution is presented in the Appendix A. The values of mode contributions are defined as

� �

� ��

� n

ss

r

q

qionParticipatModal

1

max

max

max (9a)


4

and � �

� ��

� n

ss

r

q

qionParticipatModal

1

min

min

min (9b)

Validation of the Formulation No work has been reported so far for large amplitude free vibration of shallow shells considering effects of inplane inertia, characterizing behavior with coupled linear bending-inplane modes and multiple-mode solutions. The validation of the developed model was conducted by neglecting the inplane inertia and characterizing the nonlinear vibration behavior with linear bending modes only. The finite element formulation without inplane inertia is presented in Appendix B, so that the finite element vibration results could be compared with classical analytical results. Firstly, the single-mode solutions for isotropic shells supported by shear diaphragms were compared with results obtained by Kobayashi and Leissa10. Figures 2 to 4 present the comparison of frequency ratios

LNL �� in function of nondimensional maximum deflection Wmax/h for various shallow shell geometries. Secondly, for a flat cross-ply plate, the nonlinear terms’ coefficients of Duffing single-mode equation were compared with Pillai and Rao14 in Table 1. Both factors, the curvature of the panel (Figures 2-4) and the non-symmetrical lamination sequence (Table 1), contributing to the quadratic term in the Duffing equation were investigated. Table 1. Duffing equation coefficients for a flat, simply supported cross-ply (0/90)3 rectangular plate a=20 mm, b=10 mm, h=0.6 mm, E11 = 5000 kg/mm2, E22 = 500 kg/mm2, �12 = 0.25, G12 = 250 kg/mm2

Table 2. Natural frequencies for isotropic square doubly curved shallow shell supported by shear diaphragms a=b=0.10m, h=1mm, Rx=Ry=1m

Results and Discussions

For all cases studied in this paper, the symmetrical initial conditions are assumed. Subsequently the

response consists only of symmetrical modes. It allows for the quarter of shallow shell is being studied for refined discretization. The mesh size used is 14 by 14 or 392 triangular shallow shell elements. First, a square isotropic doubly curved shell used for FE model validation, rx = ry = 10, b/a = 1, H = 0.01, � = 0.3, where rx=Rx/a, ry=Ry/a, and H = h/a, is studied to investigate the effect of inplane inertia neglected/not-neglected and the discrepancy between single- and multi-mode solutions. Subsequently, two graphite-epoxy simply supported cylindrical panels are studied to determine the influence of the varying sequence of lamination on the response. Inplane Inertia Effect Analytical methods have neglected inplane inertia effects on the response of shallow shells as described earlier. The FE formulation presented here does not neglect this term (in the validation part inplane inertia was not included only for the sake of comparison, Figures 2-4). It is found, as presented in Figure 5, that the vibrating shallow shell exhibits considerably less pronounced softening effect when membrane effects are included. In some cases inplane inertia effect can change not only quantitatively, but also qualitatively the response of the shallow shell. For shallow shells that exhibit weak softening effects when the inplane inertia is not included, it is very likely to have purely hardening characteristic when this effect is accounted for, which is presented in Figure 5. Finite element formulation where inplane inertia is neglected is presented in Appendix B. Multimode Solution In this section the square doubly curved shell used in the model validation and inplane inertia effect investigation is studied further with multimode solutions. The two-mode and three-mode solutions are determined and compared with single-mode solution. The first three natural frequencies are given in Table 2. A typical set of the 3-mode results consisting of the time response and the phase plots is shown in Figures 6a and 6b, respectively. It is seen that multimode solution departure significantly from the single mode solution, especially for the inbound part of oscillation (negative deflection – shallow shell in compression). For this reason it was thought reasonable to include both values, namely Wmax/h and Wmin/h, instead of only positive one, as it is traditionally done for isotropic or symmetrically laminated flat plates. Sample modal participation values for the case shown in Figure 6 are given in Table 3. Mode (1,3)-(3,1) is not shown in Figure 6 since it does not contribute to total Wmax/h nor Wmin/h. The overall effect of accounting in the solution for more than one mode is presented in Figure 7.

� � 0322�� qqqq ��

�

� Exact14 0.8896 5.9153

FE (15x15) quarter plate 0.8759 5.8965 Difference, % -1.54 -0.32

Mode (1,1) (1,3)+(3,1) (1,3)-(3,1)

(3,3)

Frequency, Hz 944.46 2550.6 4427.6


5

Figure 2. Comparison of analytical and FE frequency ratio for various curvatures ratios rx = 10, h/a = 0.01, b/a = 1, � = 0.3

Figure 3. Comparison of analytical and FE frequency ratio for various curvatures h/a = 0.01, b/a = 1, � = 0.3

Figure 4. Comparison of analytical and FE frequency ratio for various thickness ratios, H=h/a, rx = ry = 10, b/a=1, � = 0.3 Lamination Sequence Responses of cylindrical rectangular simply supported composite panels with different antisymmetrical lamination sequences, namely (0/90)

Table 3. Modal participations for isotropic doubly curved shallow shell

and (90/0), are investigated. For the (0/90) cylindrical panel, the (0) layer is closer to the center of cylinder. The composite shallow shell of the same plane-form

Figure 5. Effect of inplane inertia on the square, isotropic, doubly curved shell response, ( 3.0�� ,

10�� yx rr , 01.0�H )

Modal Participation max, % Modal Participation min, %

L�

� hwmax

hwmin

q11

q13 + q31

q13 - q31

q33

1.0029 0.0950 -0.1013

98.50 98.54

1.36 1.36

0.00 0.00

0.14 0.10

1.0042 0.2596 -0.3115

95.79 95.93

3.73 3.88

0.00 0.00

0.48 0.19

1.0069 0.3317 -0.4205

94.60 94.77

4.74 5.05

0.00 0.00

0.66 0.18

1.0117 0.3987 -0.5320

93.56 93.71

5.60 6.14

0.00 0.00

0.84 0.15

1.0265 0.5234 -0.7621

92.13 91.81

6.66 8.16

0.00 0.00

1.21 0.02

1.0376 0.5847 -0.8807

91.82 90.83

6.79 9.11

0.00 0.00

1.40 0.06

1.0511 0.6479 -1.0020

91.80 89.82

6.62 10.04

0.00 0.00

1.58 0.15

1.0671 0.7149 -1.1260

92.05 88.81

6.17 10.96

0.00 0.00

1.78 0.24

1.0858 0.7870 -1.2527

92.54 87.81

5.46 11.86

0.00 0.00

2.01 0.33

1.1064 0.8655 -1.3817

93.18 86.85

4.56 12.73

0.00 0.00

2.26 0.42

1.1302 0.9505 -1.5123

93.98 85.96

3.48 13.55

0.00 0.00

2.54 0.49

1.1563 1.0529 -1.6505

95.64 84.82

1.49 14.62

0.00 0.00

2.87 0.55


6

Figure 6a. 3-mode solution - time response (mode (1,1), (1,3)+(3,1), (3,3), and 3-mode solution compared with single mode)

Figure 6b. 3-mode solution – phase plots (mode (1,1), (1,3)+(3,1), (3,3), and 3-mode solution compared with single mode)


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Figure 7. Multiple mode solutions vs. single mode solution for square doubly curved simply supported aluminum panel

Figure 8a. Multimode solution for cylindrical rectangular panel with (0/90) lamination

Figure 8b. Multimode solution for cylindrical rectangular panel with (90/0) lamination

Figure 9. Lamination sequence influence on the cylindrical rectangular panel response dimensions 10x15”, curvatures Rx = 100”, Ry = �, and thickness h=0.050” is studied. It is found that substantial bending and inplane coupling occurs. This coupling is generated due to two factors, namely panel’s curvature and antisymmetrical lamination of the panel ([B]�0). Certain sequence of antisymmetrical lamination can be used to influence characteristics of the abovementioned coupling, which has an impact on the dynamic response. Table 4 presents first three natural frequencies of the panels. Nonlinear multiple mode solutions are presented in Figures 8a and 8b, and they indicate that single mode approach will not give accurate results. As the sequence of lamination is reversed, the major differences are found with respect to natural frequencies, mode shapes and nonlinear response characteristics. Lamination (0/90) results in the fundamental frequency being higher by 17.6% then for lamination (90/0). For (0/90) case, the sequence of mode shapes is (1,1) followed by (3,1) and (1,3) while for (90/0) lamination sequence case modes’ ordering is (1,1), (1,3) and (3,1) in increasing of natural frequency. As for the non-linear behavior, lamination sequence (0/90) introduces softening characteristics, while lamination (90/0) gives pure hardening response. Table 4. Natural frequencies for graphite-epoxy rectangular cylindrical simply supported panel with antisymmetrical lamination

Frequency, Hz Lamination Mode

(1,1) Mode (1,3)

Mode (3,1)

(0/90) 381.66 437.87 414.49 (90/0) 324.54 431.60 584.04


8

More examples of the influence of lamination sequence on the simply supported cylindrical rectangular (10x15x0.050”, rx=10) panel response are illustrated with single-mode solutions in Figure 9. A vibrating flat plate always remains in tensile strain. For the shallow shell, the outbound part of oscillation is also associated with positive strain (panel is in tension), but for the inbound part of oscillation the strain becomes negative (panel is in compression). For that reason the time when shallow shell remains below the undeflected position is longer than half of the period and the negative deflection has larger absolute value than positive deflection. It is seen that even for the moderately large deflection of order of one thickness, higher modes contribution could approach 15%. It is also observed that the modal participations based on the outbound (maximum) and inbound (minimum) deflections differ. The difference increases as the deflection increases.

Conclusions The inplane inertia effects play important role in the large amplitude response of the shallow shell panels. Analytical solutions fail to include this factor. At this point, the authors do not intend to settle the question whether inplane inertia dropped or kept in the formulation would yield the accurate results. However the answer that a model with less simplifying assumptions is always more accurate seems to be very natural and intuitive. From the examples studied, it is concluded that inplane inertia may give completely different characteristics (hard- or soft-spring), while multiple modes will improve the accuracy of the nonlinear frequency. Flexibility of enforcing complicated boundary conditions and non-rectangular geometries of the panel promote FE approach with the transformation into the modal degrees of freedom to be the essential tool for variety of shallow shell panel response problems, including flutter, sonic fatigue22 and post-buckling behavior. Since analytical methods fail to address inplane inertia, further numerical solutions are needed and verified with experiments.

Acknowledgent

Dr. L. Azrar wishes to acknowledge the assistance of the Fellowship Grant from Fulbright and the Morocco-American commissions during his stay visit at Aerospace Engineering Department at ODU.

Reference

1 Leissa, A. W., “Vibration of Shells,” NASA SP-288, Washington DC, 1973. 2 Kraus, H., “Thin Elastic Shells,” John Wiley & Sons, Inc., New York, 1967. 3 Libai, A., and Simmonds, J. G., “The Nonlinear Theory of Elastic Shells,” Cambridge University Press, Cambridge, UK, 1998, 2nd Edition. 4 Liew, K. M., Lim, C. W., and Kitipornchai, S., “Vibration of shallow shells: A review with bibliography,” Applied Mechanics Review, Vol. 50, No. 8, 1997, pp. 431-444. 5 Cummings, E. A., “Large amplitude vibration and response of curved panels,” AIAA Journal, Vol. 2, 1964, pp. 709-716. 6 Leissa, A. W., and Kadi, A. S., “Curvature effects on shallow shell vibrations,” Journal of Sound and Vibration, Vol. 16, 1971, pp. 173-187. 7 Hui, D., “Influence of geometric imperfections and in-plane constraints on nonlinear vibrations of simply supported cylindrical panels,” Journal of Applied Mechanics, Vol. 51, June 1984, pp. 383-390. 8 Fu, Y. M., and Chia, C. Y., “Multi-mode non-linear vibration and postbuckling of anti-symmetric imperfect angle-ply cylindrical thick panels,” International Journal of Non-Linear Mechanics, Vol. 24, No. 5, 1989, pp. 365-381. 9 Raouf, R. A., and Palazotto, A. N., “On the non-linear free vibration of curved orthotropic panels,” International Journal of Non-Linear Mechanics, Vol. 29, No. 4, 1994, pp. 507-514. 10 Kobayashi, Y., and Leissa, A. W., “Large amplitude free vibration of thick shallow shells supported by shear diaphragms,” Journal of Non-Linear Mechanics, Vol. 30, No. 1, 1995, pp. 57-66. 11 Shin, D. K., “Large amplitude free vibration behavior of doubly curved shallow open shells with simply-supported edges,” Computers and Structures, Vol. 62, No. 1, 1997, pp. 35-49. 12 Abe, A., Kobayashi, Y., and Yamada, G., “Non-linear vibration characteristics of clamped laminated shallow shells,” Journal of Sound and Vibration, Vol. 234, No. 3, 2000, pp. 405-426. 13 Tamura, H., and Matsuzaki, K., “Numerical scheme and program for the solution and stability analysis of a steady periodic vibration problem,” Japanese Society of Mechanical Engineering International Journal, Vol. 39, No. 3, 1996, pp. 456-463. 14 Pillai, S. R. R., and Rao, B. N., “Reinvestigation of non-linear vibrations of simply supported rectangular cross-ply plates,” Journal of Sound and Vibration, Vol. 160, No. 1, 1993, pp. 1-6. 15 Bhimaraddi, A., “Large amplitude vibrations of imperfect antisymmetric angle-ply laminated plates,”


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Journal of Sound and Vibration, Vol. 162, No. 3, 1993, pp. 457-470. 16 Alhazza, K. A., and Nayfeh, A. H., “Nonlinear vibrations of doubly-curved cross-ply shallow shells,” AIAA-2001-1661, Proceedings 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference and Exhibit, Seattle, WA, April 2001. 17 Shi, Y., Lee, R. Y. Y., and Mei, C., “Finite element method for non-linear free vibration of composite plates,” AIAA Journal, Vol.35, No.1, 1995, pp. 159-166. 18 Abdel-Motaglay, K., Chen, R., and Mei, C., “Nonlinear flutter of composite panels under yawed supersonic flow using finite elements,” AIAA Journal, Vol. 37, No. 9, 1999, pp. 1025-1032. 19 Tessler, A., Hughes, T., Jr., “A three-node Mindlin plate element with improved transverse shear,” Computer Methods in Applied Mechanics and Engineering, Vol. 50, 1985, pp. 71-101. 20 Tessler, A., “A priori identification of shear locking and stiffening in triangular Mindlin elements,” Computer Methods in Applied Mechanics and Engineering, Vol. 53, 1985, pp. 183-200. 21 Tessler, A., “A C0-anisoparametric three-node shallow shell element,” Computer Methods in Applied Mechanics and Engineering, Vol. 78, 1990, pp. 89-103. 22 Przekop, A., Guo, X., Azzouz, M. S., Mei, C., “Nonlinear response and fatigue of shallow shells to acoustic excitation using finite element”, AIAA-2003-1710, 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Norfolk, VA, April 2003.

Appendix A – Iterative Procedure To Determine Initial Conditions For Periodic Response

Modal approach, where the solution is sought in the form of Eq. (4) results in the general system of Duffing-type modal equations in reduced DOF of Eq. (8). The second order system of n ODE’s can be transformed into 2n order system of first order ODE’s

� � � ��

��

��

�

�

��

��

��

qp

KKKI

pq

qqq00

�

� (A1)

Let us denote � � � � � �� TqqX �,� . We have to solve the following differential equation

� � � �� XFX �� (A2)

with a judicious initial conditions � �� 0X that lead to periodic solutions. The initial conditions are

� ��

Tii

Tii qqqqX 00 ,0,00 �� (A3a)

� � � �n221 ...,,, �� (A3b)

For the periodic solution with period T it follows that

� ��

� �� tXtqtq

TtqTtqTtXT

ii

Tii

�

��

�

�

,

, (A4)

or setting t = 0

� �� 0XTX � (A5)

Recalling Eq. (4) for t = 0 results in

� �� rn

rr yxqyxW ,0,,

10 ��

�

� (A6)

When vector � �� is known along with the period T one can obtain the relationship between the amplitude and the non-linear period or non-linear frequency. Let us denote by T the period of the non-linear system corresponding to the initial conditions � �� prescribed by Eq. (A3). We assume that

� � � � � ��

��

��

�� 0

0 TTT (A7)

where T0 and � �0� are initial approximations, and �T and � �� are corrections that need to be computed. An approximation of T0 may be obtained by solving single-mode problem. Initial conditions for multimode approach are assumed to be

� � � �Tq 00010 �� (A8)

where � �0101 qq � is a given value. Now, using Eqs. (A3), (A5) and (A7) one can write

� ��

� ��

��

��

��

00

00

,0,,

XTTXTX (A9)

Taking Taylor series in the neighborhood of � �00 ,�T and neglecting the non-linear terms of the expansion yields

� � � �

� ��

� �

� ��

� �

� ��

�

��

��

��

��

�

��

��

0000 ,,00

0

,TT

XTtXTX

(A10)

Eq. (A10) can be rearranged into form

� ��

� � � ��

� �

� � � �� 000

,,

,0000

��

��

��

TX

TtXIX

TT

�

��

��

��

�

�

��

�

� (A11)

One needs to know � �� 00 ,�TX , � �

� �00 ,�TtX�

� and

� �� 00 ,�� T

X�

� in order to solve Eq. (A11). � �� 00 ,�TX can be

found by solving the system of Duffing’s equations with the initial guess (for the first iteration) or the previous solution (for the subsequent iterations) of the


10

initial conditions. In order to estimate � �

� �00 ,�TtX�

� one can

use Eqs. (A2) and (A3) to notice that

� �

� �

� �� 0000,

,0,00

��

�

FXFTXFtX

T

��

� (A12)

The Jacobian � ��

� �00 ,�� T

X�

� can be evaluated using the

forward scheme � ��

� � � ��

�

��

��

0000

,

,,

00

TXeTXXX iji

j

i

T

��

��

�

��

�

(A13)

where � is a small parameter, � �je is the unit thj

vector and i, j = 1, 2, …2n. Denoting � ��

��

X Eq.

(A11) can be rewritten in the form

� � � ��

� � � �� 000

0

,��

��

TXFTI

�

�� (A14)

Eq. (A14) represents system of 2n equations. Since the amplitude of the first mode is arbitrary prescribed by Eq. (A8) and is to remain constant during the iteration process the number of unknowns is equal to the number of equations available. Ruling out 01�� , the unknowns can be expressed as a vector

� � � �TT

nxn TT �� 210202 � (A15) It follows that the system to be solved is

� � � � � �� 000 ,~��

�TX

T��

��

��

(A16)

where

� �

� �

� �

� ��

�

�

��

�

�

�

�

�

�

�

�

�

��

�

�

�

�

��

�

�

��

�

�

�

�

�

�

�

�

�

�

�

�

�

��

�

�

�

�

�

�

n

nnn

n

n

n

nnn

n

n

XXF

XXF

XXF

XXt

X

XXt

X

XXt

X

2

2

2

202

2

2

2

202

2

1

2

101

2

2

2

22

2

2

2

22

2

1

2

11

~

��

��

��

��

��

��

�

��

�

�

�

��

�

�

(A17)

Finally,

� � � � � �� 0001

,~��

�TX

T��

��

��

� (A18)

Presented process of determining the initial conditions is of the iterative nature. The corrected period and deflections become the initial conditions for the subsequent iteration. One needs to set satisfactory rate of convergence where the iterative process is assumed to result in desired accuracy. The choice of parameter� has an effect on the accuracy of the results. The Runge-Kutta 4th order numerical integration scheme is used to solve the equations giving � �00 ,�TX i and � �ji eTX �� 00 , . The system to be solved is conditionally stable. The stability check requires eigenvalues of the Jacobian matrix of Eq. (A13) called monodromy matrix. Appendix B – Finite Element Formulation Without Inplane Inertia When the inplane inertia is not neglected, the system of Eq. (3) is transformed into modal coordinates of Eq. (8). In order to neglect inplane inertia, the formulation is modified as follows: second equation of Eq. (3a) is solved for � �mW under the assumption, that

� �� 0�mm WM �� (B1) This results in an expression for inplane displacement being

� � � � � �� bmb

sRmb

Rmbmb

sRmmm

WKKKK

KKW

1

1

��

��

�

(B2)

while the first (corresponding to bending and rotation) equation becomes � ��

� � � � � � � � � ��

� � � � � � � �� 01

21111

1

��

��

��

mbmsRbm

Rbmbm

bbNb

Rb

Nb

Nb

bRb

sbbbb

WKKKK

WKKKKK

KKKKWMRmb

��

(B3)

Substitution of Eq. (B2) into Eq. (B3) results in the formulation when the inplane inertia is neglected. It is observed that the cross product of the underlined portion of Eq. (B3) by Eq. (B2) will result in 16 new terms. It is further observed that among these 16 terms, nine are linear and seven nonlinear. Then, the modal solution is sought in a form expressed in bending and rotational degrees of freedom only, as

� � � �� qtqWn

rb

rbrb �

�

��

1�� (B4)


11

Table B1. Comparison of coefficients of second order nonlinear terms with and without neglecting inplane inertia

In particular it is seen, that this approach generates two additional second order nonlinear terms being products of first order nonlinear terms. First of the two second order nonlinear terms is generated by simple multiplication of two first order nonlinear terms and it becomes � � � � � �� mb

sRmmbm KKKK 11 1�

�� . The second term originates from the first order nonlinear matrix � �mN

bK1 . Since this term is expressed in function of

� �mW (see Eq. (B2)), two first order nonlinearities are crossed resulting in a second order nonlinear term of � �mN

bK 2 . As a result, the second order nonlinear term when the inplane inertia is neglected reads as � � � � � � � �� mN

bmbsRmmbmb KKKKKK 2112 1

��

� (B5)

where the underlined portion constitutes the difference between formulations with and without inplane inertia.

Certainly, expressions for linear and first order nonlinear terms are also differ as a result of assumption Eq. (B1), but numerical results showed that this differences are minor (of order not greater then 1-2%), while differences in the second order nonlinear term of Eq. (B5) are substantial. Second order nonlinear stiffness of the formulation involving assumption of Eq. (B1) is always smaller, what results in more pronounced softening characteristics of the shallow shell response. The comparison based on the doubly curved isotropic panel is shown in Table B1.

Coefficient x10 9 W/ Inplane Inertia

W/o Inplane Inertia

Second Order

Nonlinear Term

1st Eq.

2nd Eq.

3rd Eq.

311q 0.0171

0.0040 -0.0059 0.0065

-0.0104 -0.0022

3113211 �

qq -0.0089 0.0099

0.1692 0.0448

0.0211 0.0094

33211qq -0.0304

-0.0063 0.0408 0.0183

0.2460 0.0374

2311311 �

qq 0.0852 0.0254

0.1221 0.0838

0.0961 0.0372

33311311 qqq�

0.0411 0.0184

0.3727 0.1444

0.0227 0.0235

23311qq 0.2402

0.0365 0.0220 0.0228

0.0014 -0.0405

33113�q 0.0205

0.0141 0.2830 0.1097

0.0593 0.0341

332

3113 qq�

0.0938 0.0364

0.3450 0.1982

0.7842 0.2859

2333113 qq

� 0.0111

0.0115 1.5205 0.5542

0.0208 0.0120

333q 0.0005

-0.0132 0.0135 0.0078

1.3554 0.4794

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[American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Norfolk, Virginia ()] 44th AIAA/ASME/ASCE/AHS/ASC