# Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia

• Kanak Kalita
• Salil Haldar
Original Contribution

## Abstract

A nine-node isoparametric plate element, in conjunction with first-order shear deformation theory, was used for free vibration analysis of rectangular plates. Both thick and thin plate problems were solved for various aspect ratios and boundary conditions. In this work, the primary focus is on the effect of rotary inertia on the natural frequencies of rectangular plates. It is found that rotary inertia significantly affects thick plates, while it can be ignored for thin plates. The numerical convergence is very rapid and based on a comparison with data from the literature; it is proposed that the present formulation can yield highly accurate results. Finally, some numerical solutions are provided here, which may serve as benchmarks for future research on similar problems.

## Keywords

Finite element method (FEM) FSDT Rectangular plate Rotary inertia Natural frequency

## List of Symbols

[B]

Strain displacement matrix

[D]

Rigidity matrix

[K]

Global stiffness matrix

[N]

Shape function

[N0]

Null matrix

[M]

Consistent mass matrix

|J|

Jacobian matrix

[Nr]

Interpolation function of the rth point

[K0]

Overall stiffness matrix

[M0]

Overall Mass matrix

w

Transverse displacement

θxθy

Total rotations in bending

E

Modulus of elasticity

G

Modulus of rigidity

ν

Poisson’s ratio

h

Thickness of plate

a, b

Plate dimensions

D

Flexural rigidity

ω

Natural frequency

ϕxϕy

Average shear rotation

θxθy

Total rotation in bending

{σ}

Stress vector

{ε}

Strain vector

Mx, My

Bending moments in x and y direction

Mxy

Twisting moment

QxQy

Transverse shear forces

$$\xi , \eta$$

Natural coordinates

ρ

Density

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