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Transverse Vibration of Thin Rectangular Orthotropic Plates on Translational and Rotational Elastic Edge Supports: A Semi-analytical Approach

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

The free vibration analysis of thin orthotropic plates with elastically constrained edges is presented using the energy-based Rayleigh–Ritz (RR) method. Various edge conditions are modeled with rotational and translational linear springs. The complete set of admissible functions, which is a combination of (1) trigonometric functions, (2) the unit function, (3) the linear function, and (4) the lowest order polynomial, has been used in the RR method. It has been demonstrated that the use of a combination of the lowest order polynomial and the cosine series results in a rapid convergence of the solution, without any ill-conditioning of the admissible functions upon expansion. In particular, this work proposes a simple guideline for determining a set of trial functions that can be universally utilized for the vibration analysis of plates with non-classical boundary conditions. The convergence and exactness of this approach have been demonstrated through several examples. The results indicate that the elastically restrained stiffness and the plate aspect ratio impact the frequency parameters considerably.

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Abbreviations

\(x, y, z\) :

Axis of the reference system

\(a\) :

Length of the plate (m)

\(b\) :

Breadth of the plate (m)

\(h\) :

Thickness of the plate (m)

\(A\) :

Cross-sectional area (m2)

\(E_{1}\) :

Young’s modulus of the material in bending for the \(x\) direction (N/m2)

\(E_{2}\) :

Young’s modulus of the material in bending for the \(y\) direction (N/m2)

\(G_{12}\) :

Shear modulus of the material in bending in \(xy\) plane (N/m2)

\(D_{ij}\) :

Standard bending rigidities

\(\upsilon_{12}\) :

Major Poisson’s ratio (–)

\(\upsilon_{21}\) :

Minor Poisson’s ratio (–)

\(\rho\) :

Mass density of the specified material (kg/m3)

\(\omega\) :

Dimensional plate natural frequency (rad/s)

\(\Omega\) :

Non-dimensional plate natural circular frequency (–)

\(M_{x} ,M_{y}\) :

Bending moments

\(Q_{x} ,Q_{x}\) :

Shear forces

\(M_{xy}\) :

Twisting moments

K :

Stiffness matrix

M :

Mass matrix

\(A_{ij}\) :

Weight coefficients

\(T\) :

Kinetic energy of the plate (J)

\(U\) :

Strain energy of the plate (J)

\(K_{\text{T}}\) :

Non-dimensional translational spring constant (–)

\(K_{\text{R}}\) :

Non-dimensional rotational spring constant (–)

\(\alpha\) :

Aspect ratio (–)

\(W\left( {x, y} \right)\) :

Transverse displacement of the plate (m)

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Authors and Affiliations

  1. Indian Institute of Technology, Kharagpur, India

    R. K. Praharaj, N. Datta, M. R. Sunny & Y. Verma

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  1. R. K. Praharaj
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  2. N. Datta
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  3. M. R. Sunny
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  4. Y. Verma
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Correspondence to R. K. Praharaj.

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Praharaj, R.K., Datta, N., Sunny, M.R. et al. Transverse Vibration of Thin Rectangular Orthotropic Plates on Translational and Rotational Elastic Edge Supports: A Semi-analytical Approach. Iran J Sci Technol Trans Mech Eng 45, 863–878 (2021). https://doi.org/10.1007/s40997-019-00337-5

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