A novel variational numerical method for analyzing the free vibration of composite conical shells

Abstract

This paper proposes an efficient numerical method in the context of variational formulation and on the basis of Rayleigh-Ritz technique to address the free vibration problem of laminated composite conical shells. To this end, the energy functional of Hamilton's principle is written in a quadratic form using matrix relations first. Displacements are then approximated via a linear combination of base functions, by which the number of final unknowns reduces. Afterthat, the strain tensor is discretized by means of matrix differential quadrature (DQ) operators. In the next step, using Taylor series and DQ rules, a matrix integral operator is constructed which is embedded into the stiffness matrix so as to discretize the quadratic representation of energy functional. Finally, the reduced form of mass and stiffness matrices are readily obtained from the aforementioned discretized functional. To obtain the natural frequencies of conical shell, hybrid harmonic-beam base functions are employed as modal displacement functions. The accuracy of the present numerical method is examined by comparing its results with those from the published literature. It is revealed that the method is capable of accurately solving the problem with a little computational effort and ease of implementation.

Free vibration analysis of embedded functionally graded carbon nanotube-reinforced composite conical/cylindrical shells and annular plates using a numerical approach

“…In this regard, applying numerical differential operators, the discretized form of the strain vector can be presented as where ε=, U and E are the discretized form of ε, U and E , respectively. According to the mathematical approach explained by Ansari et al. (2014b), equation (15) can be given as where S=diag(S∼) and S∼ is an accurate numerical integral operator.…”

“…It should be pointed out that M=diag(V) returns a square matrix M of order n , when V is a vector of n components and V=diag(M) returns the main diagonal of M as a vector. Furthermore, sign ⊗ introduces the Kronecker product (Ansari et al., 2014b). Substituting equation (19) into (20), the strain energy can be expressed as …”

“…Performing the variation and taking the integration by parts scheme into account in the time domain, one can write (Ansari et al., 2014b) …”

“…Taking the variation and considering the integration by parts scheme in the time domain results in (Ansari et al., 2014b) where the mass matrix M and stiffness matrix K are given as follows in which the two-dimension-integral operator S, can be written as where Sθ and Sx are the integral operators in the circumferential and axial directions, respectively. Moreover, the discretized matrix operators E, ℍ and G are given as follows where in equations (51)–(53) and …”

Free Vibration Analysis of Composite Conical Shells with Variable Thickness

“…Similarly, the generalized differential quadrature method was used by Bacciocchi et al [4] to analyse the vibration of plates of variable thickness and shells. A novel vibrational numerical method was used by Ansari et al [5] to investigate the free vibration of composite conical shells. 2D-FGM truncated conical shell resting on Winkler-Pasternak foundations was studied by Asanjarani et al [6] using the differential quadrature method for different boundary conditions.…”