Strong SaS formulation for free and forced vibrations of laminated composite plates

Abstract The application of the sampling surfaces (SaS) method to the three-dimensional (3D) stress analysis of laminated composite plates is presented in a companion paper (Kulikov GM, Plotnikova SV. Strong sampling surfaces formulation for laminated composite plates. Compos Struct 2017;172:73–82). In this paper, we extend the strong SaS formulation to the 3D vibration analysis of laminated plates. The SaS formulation is based on choosing the arbitrary number of SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate unknowns. Such choice of unknowns with the use of the Lagrange polynomials in spatial approximations of displacements through the layer thicknesses leads to a robust laminated plate formulation. The feature of the proposed approach is that all SaS are located inside the layers at Chebyshev polynomial nodes. The use of outer surfaces and interfaces is avoided that makes it possible to minimize uniformly the error due to the Lagrange interpolation. Therefore, the strong SaS formulation based on direct integration of equations of motion of the elastic body can be applied efficiently for the 3D free and forced vibration analyses of laminated composite plates.

[1]  Chih‐Ping Wu,et al.  Dynamic responses of functionally graded magneto-electro-elastic shells with open-circuit surface conditions , 2008 .

[2]  A. Rao,et al.  Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates , 1970 .

[3]  Chih‐Ping Wu,et al.  An asymptotic theory for dynamic response of doubly curved laminated shells , 1996 .

[4]  Tiangui Ye,et al.  Elasticity solution for vibration of 2‐D curved beams with variable curvatures using a spectral‐sampling surface method , 2017 .

[5]  Erasmo Carrera,et al.  Finite deformation higher-order shell models and rigid-body motions , 2008 .

[6]  Seyed Rasoul Atashipour,et al.  ON THE EXACT IN-PLANE AND OUT-OF-PLANE FREE VIBRATION ANALYSIS OF THICK FUNCTIONALLY GRADED RECTANGULAR PLATES: EXPLICIT 3-D ELASTICITY SOLUTIONS , 2013 .

[7]  J. Tarn.  An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated cylindrical shells , 1994 .

[8]  G. Kulikov,et al.  Solution of three-dimensional problems for thick elastic shells by the method of reference surfaces , 2014 .

[9]  G. Kulikov,et al.  On the Use of a New Concept of Sampling Surfaces in Shell Theory , 2011 .

[10]  Romesh C. Batra,et al.  The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators , 1997 .

[11]  Ernian Pan,et al.  Modal analysis of magneto-electro-elastic plates using the state-vector approach , 2007 .

[12]  Wang Yung-Ming,et al.  An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated plates , 1994 .

[13]  Romesh C. Batra,et al.  Analytical solution for the cylindrical bending vibration of piezoelectric composite plates , 2004 .

[14]  Jean-François Deü,et al.  Free-vibration analysis of laminated plates with embedded shear-mode piezoceramic layers , 2005 .

[15]  E. Carrera,et al.  Stress, vibration and buckling analyses of FGM plates—A state-of-the-art review , 2015 .

[16]  S. Plotnikova,et al.  Benchmark solutions for the free vibration of layered piezoelectric plates based on a variational formulation , 2017 .

[17]  G. Kulikov,et al.  On the use of sampling surfaces method for solution of 3D elasticity problems for thick shells , 2012 .

[18]  G. Kulikov.  Refined Global Approximation Theory of Multilayered Plates and Shells , 2001 .

[19]  S. Srinivas,et al.  An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates , 1970 .

[20]  Senthil S. Vel,et al.  Exact solution for the vibration and active damping of composite plates with piezoelectric shear actuators , 2005 .

[21]  G. Kulikov,et al.  Strong sampling surfaces formulation for laminated composite plates , 2017 .

[22]  Chih‐Ping Wu,et al.  A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells , 2016 .

[23]  Y. M. Ghugal,et al.  On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results , 2015 .

[24]  Jianqiao Ye,et al.  A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges , 1997 .

[25]  Jiaying Wang,et al.  THREE DIMENSIONAL ANALYSIS FOR FREE VIBRATION OF RECTANGULAR COMPOSITE LAMINATES WITH PIEZOELECTRIC LAYERS , 1998 .

[26]  Jyh-Yeuan Lo,et al.  An asymptotic theory for dynamic response of laminated piezoelectric shells , 2006 .

[27]  S. Plotnikova,et al.  An analytical approach to three-dimensional coupled thermoelectroelastic analysis of functionally graded piezoelectric plates , 2017 .

[28]  Tiangui Ye,et al.  Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method , 2016 .

[29]  R. Batra,et al.  Missing frequencies in previous exact solutions of free vibrations of simply supported rectangular plates , 2003 .

[30]  Dimitris A. Saravanos,et al.  Exact free‐vibration analysis of laminated plates with embedded piezoelectric layers , 1995 .

[31]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[32]  Tiangui Ye,et al.  A spectral-sampling surface method for the vibration of 2-D laminated curved beams with variable curvatures and general restraints , 2016 .

[33]  M. Levinson.  Free vibrations of a simply supported, rectangular plate: An exact elasticity solution , 1985 .

[34]  Ernian Pan,et al.  Wave propagation in magneto-electro-elastic multilayered plates , 2007 .

[35]  Francesco Ubertini,et al.  Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey , 2015 .

[36]  Fan Jiarang,et al.  An exact solution for the statics and dynamics of laminated thick plates with orthotropic layers , 1990 .

[37]  Paul R. Heyliger,et al.  Free vibration of piezoelectric laminates in cylindrical bending , 1995 .

[38]  Jiann-Quo Tarn,et al.  An asymptotic variational formulation for dynamic analysis of multilayered anisotropic plates , 1996 .

[39]  Jianqiao Ye,et al.  Three-dimensional vibration of laminated composite plates and cylindrical panels with arbitrarily located lateral surfaces point supports , 1996 .

[40]  Z. Zhong,et al.  Vibration of a simply supported functionally graded piezoelectric rectangular plate , 2006 .

[41]  G. Kulikov,et al.  Three-Dimensional Solution of the Free Vibration Problem for Metal-Ceramic Shells Using the Method of Sampling Surfaces , 2017, Mechanics of Composite Materials.

[42]  J. N. Reddy,et al.  Frequency of Functionally Graded Plates with Three-Dimensional Asymptotic Approach , 2003 .

[43]  Gennady M. Kulikov,et al.  Exact 3D stress analysis of laminated composite plates by sampling surfaces method , 2012 .

[44]  Ernian Pan,et al.  FREE VIBRATIONS OF SIMPLY SUPPORTED AND MULTILAYERED MAGNETO-ELECTRO-ELASTIC PLATES , 2002 .

[45]  Weiqiu Chen,et al.  On free vibration of a functionally graded piezoelectric rectangular plate , 2002 .

[46]  Chih‐Ping Wu,et al.  A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells , 2008 .

[47]  N. Bakhvalov.  Numerical methods : analysis, algebra, ordinary differential equations , 1977 .

[48]  Romesh C. Batra,et al.  The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators , 1996 .

[49]  M. G. Kulikov,et al.  Three-dimensional vibration analysis of layered and functionally graded plates through sampling surfaces formulation , 2016 .

[50]  Gennady M. Kulikov,et al.  Exact 3D Thermoelectroelastic Analysis of Piezoelectric Plates through a Sampling Surfaces Method , 2015 .

[51]  S. Vel,et al.  Three-dimensional exact solution for the vibration of functionally graded rectangular plates , 2004 .

[52]  Rongqiao Xu,et al.  ON FREE VIBRATION OF A PIEZOELECTRIC COMPOSITE RECTANGULAR PLATE , 1998 .

[53]  Weiqiu Chen,et al.  On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates , 2005 .

[54]  M. Levinson.  The simply supported rectangular plate: An exact, three dimensional, linear elasticity solution , 1985 .