# Vibrational characteristics of annular plates and rings of radially varying thinkness

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## Abstract

In this paper, annular plates having thickness variation are studied by deriving the equations of motion on the basis of the Mindlin plate theory. The Chebyshev collocation method is employed to solve the differential equation governing the transverse motion of such plates. The dimensionless frequencies are evaluated for different values of taper constant (α), thickness ratio (*h* _{u}). radii ratio (ε) and power (*n*). The results of an experimental investigation are also presented, and the agreement between these findings and the predicted values in theory is remarkably good. As a result of this study, it is found that the effects of rotatory inertia and transverse shear deformation reduce the natural frequencies for all boundary conditions and for all values of*n. h* _{o}, ∈, a and*s* (mode number). This study also showed that the natural frequencies of annular plates with thickness expressed by the nth power function are higher than those by the (*n*−1)th power function for positive values of α, and vice versa for negative values of*e* for all three boundary conditions. Moreover, there is a proof that the natural frequencies of annular plates tend to be higher as the taper constant decrease and/or as the radii ratio increase for all three boundary conditions and for all values of*n, s* and*h* _{o}.

## Key Words

Forced Vibration Rofor Natural Frequency Annular Plate Inertia Shear Deformation Boundary Condition## Nomenclature

*D*Flexurai rigidity of plate *−

*Eh*^{3}/12(1*ν*^{2}))*E*Young’s modutus of annular plate

*G*Shear modulus of annular plate

*H*Dimensionless variable (−

*h/a*)*K*_{i}Unknown constants

*M*_{r}, M_{θ}Bending moments per unit length

*M*_{rθ}Twisting moment per unit length

*Q*_{r}, Q_{θ}Radial and tangential shearing forces per unit length

*T*Chebyshes polynomials

*T*^{o}Chebyshes polynomials with superseript meaning integration with respect

*y**a*Outer radius

*b*Inner radius

*h*Thickness of plate defined by Eq. (5)

*h*_{o}Thickness ratio

*m*Number of Chebyshev collocation point

*n*Power

*q*External force per unit area

*s*Mode number

*t*Time

*x*Dimensionless variable (−

*r/a*)*y*Chebyshev constant defined by Eq. (7

*b*)

## Greek Letters

- α
Taper constant

- ε
Ratio of the inner and outer radin (

*b/a*)*k*Averaging shear coefficient (−

*π*^{2}/12)- ν
Poisson’s ratio

- ϱ
Density (mass per unit velume)

- ω
Circular frequency

*Ω*_{s}Dimensionless frequency parameter including the effects of rotatory inertia

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