What question did this study set out to answer?

The research aims to model the effects of parabolic non-homogeneity on the vibration frequencies of orthotropic trapezoidal plates.

February 12, 2026Open Access

Effect of Parabolically Varying Non-homogeneity on Thermally Induced Vibration of Orthotropic Trapezoidal Plate with Thickness Varies Parabolically in Both Directions

Key Points

The research aims to model the effects of parabolic non-homogeneity on the vibration frequencies of orthotropic trapezoidal plates.
Applied Rayleigh-Ritz method to solve governing differential equations.
Defined two-term deflection function for clamped-simply supported boundary conditions.
Investigated frequency effects with varying structural parameters such as taper constant and thermal gradient.
Utilized Mathematica software for computations and graphical representations.
Found vibration frequency variations for first and second modes depending on structural parameters.
Results matched well with preexisting literature, confirming model accuracy.
Demonstrated that parabolic variation improves sustainability and reduces environmental hazards.

Abstract

The aim of present paper is to find the frequencies using the mathematical model: Effect of parabolically varying non-homogeneity on thermally induced vibration of orthotropic trapezoidal plate with thickness varies parabolically in both directions. In the above model both thickness and density varies parabolically. Rayleigh-Ritz method is used to solve the governing differential equation for maximum strain energy and maximum kinetic energy for orthotropic trapezoidal plate. Two term deflection function corresponding to clamped-simply supported clamped-simply supported (C-S-C-S) boundary condition is defined by the product of the equation of the prescribed continuous piecewise boundary shape. The effect of frequencies for first and second mode investigated with the variations in structural parameters such as taper constant, non-homogeneity constant, aspect ratio and thermal gradient respectively. Differential equations for maximum strain energy and maximum kinetic energy for orthotropic trapezoidal plate are solved using the mathematica software. All the results are calculated with great accuracy and are displayed graphically. To validate the model all the results are also compared with the preexisting literature and fit well. So by developing such type of model we increase sustainability and also reduce the environmental hazards.

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