Nonlinear Symmetric Free Vibration Analysis of Super Elliptical Isotropic Thin Plates

Da-Guang Zhang and Hao-Miao Zhou

1 Introduction

Super elliptical plates which are defined by shapes between an ellipse and a rectangle have a wide range of use in engineering applications,and it is more difficult to analyze nonlinear behaviors of super elliptical plates than rectangular,circular and elliptical plates.

Some studies for linear behaviors of super elliptical plates are available in the literature,for example,Wang et al.(1994)presented accurate frequency and buckling factors for super elliptical plates with simply supported and clamped edges by using Rayleigh-Ritz method.Lim(1998)investigated free vibration of doubly connected super-elliptical laminated composite plates.Then Chen et al.(1999)reported a free vibration analysis of laminated thick super elliptical plates.Liew and Feng(2001)studied three-dimensional free vibration analysis of perforated super elliptical plates.Zhou(2004)analysed three-dimensional free vibration of super elliptical plates based on linear elasticity theory using Chebyshev-Ritz method.Altekin(2008)gave out free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports.Altekin and Altay(2008)calculated static analysis of point-supported super-elliptical plates,then Altekin(2009;2010)discussed free vibration and bending of orthotropic super elliptical plates on intermediate supports.?eriba?s?et al.(2008)gave out static linear analysis of super elliptical clamped plates based on the classical plate theory by Galerkin’s method.?eriba?s?and Altay(2009)investigated free vibration of super elliptical plates with constant and variable thickness by Ritz method,then ?eriba?s?(2012)investigated static and dynamic linear analyses of thin uniformly loaded super elliptical clamped functionally graded plates.Jazi and Farhatnia(2012)discussed buckling of functionally graded super elliptical plate based on the classical plate theory using Pb-2 Ritz method.Tang et al.(2012)presented upper and lower bounds of the solution for the superelliptical plates problem using genetic algorithms.

Many studies for nonlinear vibration of rectangular and circular plates are available in the literature.For example,Ostiguy and Sassi(1992)investigated the influence of initial geometric imperfections on the dynamic behavior of simply supported rectangularplates subjected to the action ofperiodic in-plane forces.Sanieiand Luo(2001)presented the natural frequency and responses for the nonlinear free vibration of heated rotating disks when non-uniform temperature distributions pertaining to the laminar and turbulent airflow induced by disk rotation were considered.Haterbouch and Benamar(2003;2004)examined the effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates,then Haterbouch and Benamar(2005)investigated nonlinear free axisymmetric vibration of simply supported isotropic circular plates by using the energy method and a multimode approach.Allahverdizadeh et al.(2008)studied vibration amplitude and thermal effects on the nonlinear behavior of thin circular functionally graded plates.Bakhtiari-Nejad and Nazari(2009)calculated nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate.Shooshtari and Razavi(2010)presented a closed form solution for linear and nonlinear free vibrations of composite and fiber metal laminated rectangular plates.Alijani et al.(2011)investigated geometrically nonlinear vibrations of FGM rectangular plates in thermal environments via multi-modal energy approach.Ma et al.(2012)reported nonlinear dynamic response of a stiffened plate with four edges clamped under primary resonance excitation.Xie and Xu(2013)applied a simple proper orthogonal decomposition method to compute the nonlinear oscillations of a degenerate two-dimensional fluttering plate undergoing supersonic flow.

A literature review of works on the nonlinear vibration of rectangular and circular plates is given by Chia(1980),Sathyamoorthy(1987)and Chia(1988),while investigations on nonlinear vibration of super elliptical plates haven’t been reported.Zhang(2013)first investigated non-linear bending of super elliptical thin plates based on classical plate theory,and presented approximate solutions and conver-gence studies by Ritz method.The present paper extends the previous works[Zhang(2013)]to the case of nonlinear vibration analysis for super elliptical thin plates,and approximate solutions are also obtained by Ritz method.

2 Basic formulations of thin plates based on classical plate theory

Figure 1:Geometry and coordinates of a super elliptical plate.

Consider a super elliptical plate of major axis 2a,minor axis 2band thicknessh,and the coordinate system is illustrated in Fig.1.The boundary shape equation of the super elliptical plates can be represented by

kis the power of the super ellipse,and ifk=1,the shape becomes an ellipse,ifk=∞,the shape becomes a rectangle.According to classical plate theory,the displacement fields are

in whichu,vandware total displacements,u0andv0are mid-plane displacements in thexandydirections,respectively.Considering nonlinear von Kármán straindisplacement relationships,the strains can be expressed by

According to Hooke’s law,the stresses can be determined as

The constitutive equations can be deduced by proper integration.

In Eq.(8)AijandDijare the plate stiffnesses,defined by

In the following analysis,all edges of plate are assumed to be simply supported and clamped with no in-plane displacements,i.e.prevented from moving in thex-andy-directions.

wherenrefers to the normal directions of the plate boundary.

3 Ritz method for approximate solutions of nonlinear vibrations of super elliptical thin plates

Ritz method is adopted in this section to obtain approximate solutions of super elliptical plates.The key issue is first to assume the deflection and mid-plane displacements of the plate

whereMis total number of series.For symmetrical problems about the plate withCase1,it can be assumed that

For symmetrical problems about the plate withCase2,it can be assumed that

in whichj=2m-i-2 in Eqs.(12-13).Note thataij,dijandeijare undetermined coefficients,and Eqs.(12-13)satisfy displacement boundary conditions.In addition,Eqs.(12-13)are adapt to analysis of symmetrical nonlinear fundamental vibration modes,but not adapt to other modes,so other modes are not discussed in this paper.

Nonlinear algebraic equations aboutaij,dijandeijcan be obtained by substitutingw,u0andv0into the following expression.

in which Π=K-U,and the strain energy is

The kinetic energy may be expressed by

where ? denotes domain of plates and ?0denotes mid-plane of plates.

As for plates with given nonlinear fundamental frequencyωNLand other known coefficients,aij,dijandeijcan be solved by Newton-Raphson method or other equivalent methods.For the sake of brevity,nonlinear algebraic equations and the solving process are omitted.Substituting these coefficients back into Eqs.(12-13),w,u0andv0may then be completely determined.In addition,linear fundamental frequencyωLcan be easily obtained by making solutions of coefficientsaij,dijandeijapproach to zero.

4 Results and discussion

4.1　Comparison studies

The presentpaperextends convergence studiesofthe previousworks[Zhang(2013)]to nonlinear vibrations analysis of super elliptical plates,andM=5 is used in all the following calculations in consideration of both simplicity and convergence.To ensure the accuracy and effectiveness of the present method,two examples are solved for nonlinear vibration analysis of isotropic circular and rectangular plates.Example 1.The nonlinear-to-linear fundamental frequency ratio(ωNL/ωL)for the isotropic circular plates withCase2 is calculated and compared in Table 1 with results of Haterbouch and Benamar(2003)and Allahverdizadeh et al.(2008).In this example,the plates havea/h=136.59 andν=0.3.

Example 2.The nonlinear-to-linear fundamental frequency ratio(ωNL/ωL)for the isotropic rectangular thin plates withCase1 is calculated and compared in Fig.2 with results of Shooshtari and Razavi(2010).In this example,the super elliptical plate withk=10 represents a shape similar to a rectangular plate,a/h=50 andν=0.3.

Figure 2:Comparison of nonlinear-to-linear frequency ratio(ωNL/ωL)for isotropic rectangular plates with Case 1.

These two comparisons show that the present results agree well with existing results,and thus the validity can be confirmed.

4.2　Parametric studies

The relations of nonlinear-to-linear frequency ratioωNL/ωLand non-dimensional vibration amplitudeswmax/hfor the plate are calculated in Table 2-9.It can be observed that the present results agree well with existing results of Wang et al.(1994)for linear vibration of isotropic super elliptical thin plates.It can be concluded that nonlinear vibration frequencies increase significantly with increasing the value of vibration amplitudes and ratio of major to minor axisa/b.It can be also observed that the characteristics of nonlinear vibration are significantly influenced by different boundary conditions and the power of the super ellipsek.

Table 2:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 1 and k=1.

Table 3:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 1 and k=2.

Table 4:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 1 and k=3.

Table 5:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 1 and k=4.

Table 6:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 2 and k=1.

Table 7:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 2 and k=2.

Table 8:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 2 and k=3.

Table 9:Nonlinear-to-linear frequency ratio(ωNL/ωL)for super elliptical thin plates with Case 2 and k=4.

5 Conclusions

In this paper,nonlinear vibration analyses are first presented for super elliptical platesbased on classicalplate theory.Ritz method isemployed to analyze nonlinear vibration behaviors.Numerical results confirm that the characteristics of nonlinear vibration behaviors are significantly influenced by different boundary conditions,vibration amplitudes,the power of the super ellipsek,as well as ratio of major to minor axis.

Acknowledgement:This research was supported by a grant of the Fund of the National Natural Science Foundation of China(Nos.11172285,10802082)and the Natural Science Foundation of Zhejiang Province(Nos.LR13A020002).The authors would like to express their sincere appreciation to these supports.

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