Vibration of Curvilinearly Stiffened Plates Using Ritz Method With Orthogonal Jacobi Polynomials

Abstract

This paper presents a general approach for the free vibration analysis of curvilinearly stiffened rectangular and quadrilateral plates using the Ritz method by employing classical orthogonal Jacobi polynomials. Both the plate and stiffeners are modeled using first-order shear deformation theory (FSDT). The displacement and rotations of the plate and stiffeners are approximated by separate sets of Jacobi polynomials. The ease of modification of the Jacobi polynomials enables the Jacobi weight function to satisfy geometric boundary conditions without loss of orthogonality. The distinctive advantage of Jacobi polynomials, over other polynomial-based trial functions, lies in that their use eliminates the well-known ill-conditioning issues when a high number of terms are used in the Ritz method, e.g., to obtain higher modes required for vibro-acoustic analysis. In this paper, numerous case studies are undertaken by considering various sets of boundary conditions. The results are verified both with the detailed finite element analysis (FEA) using commercial software msc.nastran and with those available in the open literature. New formulation and results include: (i) exact boundary condition enforcement through Jacobi weight function for FSDT, (ii) formulation of quadrilateral plates with curvilinear stiffeners, and (iii) use of higher order Gauss quadrature scheme for required integral evaluations to obtain higher modes. It is demonstrated that the presented method provides good numerical stability and highly accurate results. The given new numerical results and convergence studies may serve as benchmark solutions for validating the new computational techniques.

Free vibration analysis of a laminated honeycomb sandwich panel: a suggested analytical solution and a numerical validation

“…Maji and Mahato (2020) reviewed the developments and applications of shear deformation theories for the analysis of laminated composite plates. Ritz method was incorporated by B Alanbay (2020) for the vibration analysis of sandwich panels. Praveen A et al (2020) used higher order theory to study the vibration response of honeycomb panels.…”

Skin-Stringer Assembly Using Radial Basis Functions for Curvilinearly Stiffened Panels

“…When modeling the stiffeners as beam elements, the glue contact may influence the accuracy of the analysis results because it is very sensitive to the offset between the stiffener's centroid and the plate's median plane [10]. Additionally, both semi-analytical and mesh-less methods were developed for buckling and vibration analysis of curvilinearly stiffened plates by Tamijani and Kapania [11][12][13], Vescovini et al [14,15] and Alanbay et al [16]. For practical aircraft panels, which have both complex geometry and complex boundary conditions, it is not efficient to use these semi-analytical methods as one needs to find admissible trial functions to satisfy those complex essential boundary conditions.…”