Vibration analysis of a cantilevered trapezoidal moderately thick plate with variable thickness

Abstract

This paper presents a numerical solution for vibration analysis of cantilevered non-uniform trapezoidal thick plates. Based on the first shear deformation theory, kinetic and strain energies of the plate are derived and using Hamilton's principle, governing equations and boundary conditions are derived. A transformation of coordinates is used to convert the equations and boundary conditions from the original coordinates into a new computational coordinates. Using Differential quadrature method (DQM), natural frequencies and corresponding modes are derived numerically. Convergence and accuracy of the proposed solution are confirmed using results presented by other authors and also results obtained based on the finite element method using ANSYS software. Finally, as the case studies, two cases for variation of thickness are considered and the effects of angles, aspect ratio and thickness of the plate on the natural frequencies are studied. It is concluded that two angles of the trapezoid have opposite effect on the natural frequencies. Also, it is shown that all frequencies rise as value of thickness increases or value of the aspect ratio of the plate decreases. The most advantage of the proposed solution is its applicability for plates with variable thickness.

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“…Sönüm elemanlarının ve kütlelerin artırılmasının doğal frekanslar değerlerini yükselttiği ifade edilmiştir [8]. Bir başka çalışmada ise trapez kesitinin et kalınlığı ile doğal frekanslar arsındaki ilişki ANSYS ile incelenmiştir, et kalınlığının artırılması ile doğal frekans değerlerinin yükseldiği gösterilmiştir [9]. Bu yöntemler en genel titreşim azaltma yöntemleri olup, ağırlık ve maliyet artışına neden olmanın yanı sıra imalat süresinin uzamasına neden olabildiğinden her zaman uygulanması mümkün olmamaktadır.…”

Vibration and flutter analyses of cantilever trapezoidal honeycomb sandwich plates

“…There is a weak possibility to find an exact solution for the set of governing equations (equation (20)) and they can be numerically solved. Vibration and flutter analyses of skew and trapezoidal plates were numerically solved by the FEM [29–31], Ritz method [32–36], and DQM [26,28,37–40]. In the current research, the DQM is employed to solve the set of governing equations and boundary conditions.…”

Optimization for flutter boundaries of cantilevered trapezoidal thick plates

“…Thus, in this paper, these equations are solved numerically using DQM (see Appendix 2). DQM is a strong numerical method which was introduced by Bellman [26] and was used by many authors to investigate static and dynamic analyses of structures [19,20,[27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. Some authors mixed this method with other exact or numerical approaches and studied structural problems [42][43][44][45][46][47][48].…”

“…Despite of its high similarity to geometry of wing and tail fin of aircrafts, cantilever trapezoidal plates are not investigated as much as rectangular ones. However, some authors presented numerical and semi-analytical solutions for vibration analysis of trapezoidal plates [14][15][16][17][18][19][20]. The critical velocity of a trapezoidal plate is affected by many parameters, such as length, width, thickness, variation of thickness, and angles of the leading and training edges of the plate.…”

“…1), and (ζ 4 , η 4 ) = (1, 1) using the following relations: in which Substituting Eq (20). into Eq.…”