Free vibration of circular annular plate with different boundary conditions
Yash Jaiman^{1} , Baij Singh^{2}
^{1}Bennett University, Greater Noida, Uttar Pradesh, India
^{2}Indian Institute of Technology (ISM), Dhanbad, India
^{1}Corresponding author
Vibroengineering PROCEDIA, Vol. 29, 2019, p. 8286.
https://doi.org/10.21595/vp.2019.21116
Received 23 October 2019; accepted 4 November 2019; published 28 November 2019
43rd International Conference on Vibroengineering in Greater Noida (Delhi), India, November 2830, 2019
This paper deals with the numerical simulation of free vibration analysis of a thin circular annular plate for various boundary conditions at the outer edge and inner edge. Classical plate theory is used to derive the governing differential equation for the transverse deflection of the thin isotropic plate. The finite element method is used to evaluate the first six natural frequencies and mode shapes of the thin uniform circular annular plate with radius ratios $({r}_{1}/{r}_{2})$ for different boundary conditions. These natural frequencies results are compared with those available in the literature. The results are verified with classical plate theory with our Abaqus results and checked with the previous research literature on the topic.
Keywords: circular annular plate, free vibration, numerical simulation.
1. Introduction
Plates are widely used as a structural element and have vast practical applications in many engineering fields such as aerospace, mechanical, civil, nuclear, electronic, automotive, marine and heavy machinery, etc. Various researchers have analyzed the free vibration behavior of circular annular plates of different shapes, sizes, thickness for different boundary conditions. Leissa [1] used the Ritz method to estimate the natural frequencies of the isotropic plate for different boundary conditions. Kim and Dickinson [2] used the RayleighRitz approximation method for free vibration of a thin plate to extract natural frequencies. Rajalingham et al. [3] used a RayleighRitz method to analyze the plate characteristics parameter as shape functions and continued his work to formulate a variational reduction expression to analyze frequencies and mode shapes. Liew et al. [4] used the polynomialsRitz method for the vibration of circular plates by using threedimensional elasticity solutions. Zhou et al. [5] used the ChebyshevRitz method for threedimensional vibration and mode shapes of the circular plate. Lim et al. [6] used the statespace method to analyze transverse vibration and mode of a thick circular plate. Zhou et al. [7] used the Hamiltonian principle to solve governing equations for free vibration analysis by using the variational principle of mixed energy method. Kumar et al. [8] use a dynamic stiffness method to extract the natural frequency and mode shapes of a thin plate. Piyush et al. [9] used the RayleighRitz method to compute the natural frequencies of the thin plate.
2. Basic formulation
Consider a homogeneous, isotropic circular annular plate in cylindrical coordinates $\left(r,\theta ,z\right)$ with uniform thickness $h$ as shown in Fig. 1.
Classical plate theory is used to derive the governing differential equation for transverse vibration in the polar coordinate system is defined as:
where Laplacian operator: ${\nabla}^{2}=\frac{{\partial}^{2}}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial}^{2}}{{\partial \theta}^{2}}$, $\rho $ is the mass density, $D=E{h}^{3}/\left[12\left(1{\nu}^{2}\right)\right]$ is the flexural rigidity and $\nu $ is Poisson’s ratio.
Transverse deflection of natural vibrations for thin circular plate is assumed to be:
where $\omega $ is the natural frequency and $w\left(r,\theta \right)$ is natural mode.
Substituting Eqs. (2) in (1), we get:
where:
The general finite element equation for the transverse deflection of thin plate is given by:
where $\left[M\right]$ is the mass matrix, $\left[K\right]$ is stiffness matrix, $\left\{\ddot{q}\right\}$ is the nodal acceleration vector, and $\left\{q\right\}$ is nodal displacement vector.
The nondimension natural frequency parameter $\varpi $ are calculated as:
where $\omega $ is the natural frequency in Hz.
Fig. 1. Schematic diagram and coordinate system of annular circular plate
3. Results and discussions
In this section, the first six nondimensional natural frequencies and mode shapes of the circular annular plate are estimated by using the finite element method. Here, we calculated different eigenvalues for different boundary conditions with different radii ratio $({r}_{1}/{r}_{2})$. Different combinations of boundary conditions are applied to compute the natural frequencies and mode shapes of the circular annular plate. 2820 elements and 5922 nodes are used to estimate the natural frequencies and mode shape function of thin circular annular plates after convergence study. The present natural frequencies results are compared with those available in the literature.
Tables 14 shows that the first six nondimensional natural frequencies values for the circular annular plate. These present results are nearly the same as Leissa [1] and Zhou [8] under different boundary conditions.
Fig. 2. Natural modes of a clamped annular plate with a free inner boundary, ${r}_{1}/{r}_{2}$= 0.4
Table 57 presents the effect of the radii ratio on the nondimensional frequency parameter of thin plates. It is observed from these tables that as the radii ratio increases nondimensional frequency parameter increases.
Table 1. Comparison of the nondimensional natural frequency parameter with Leissa [1] and Zhou [8] for clamped outer and free inner boundary ($\nu =1/3$, ${r}_{1}/{r}_{2}=\text{0.4}$)
Results  Mode number  
1  2  3  4  5  6  
Leissa [1]  13.54  19.80  31.34  –  –  – 
Zhou [8]  13.500  19.389  31.338  46.855  65.984  66.924 
Present  12.871  18.497  29.901  44.70  62.982  63.935 
Table 2. Comparison of the nondimensional natural frequency parameter with Leissa [1] and Zhou [8] for free outer and clamped inner boundary ($\nu =1/3$, ${r}_{1}/{r}_{2}=0.4$)
Results  Mode number  
1  2  3  4  5  6  
Leissa [1]  9.096  10.37  –  –  –  – 
Zhou [8]  9.0719  9.1294  10.366  14.726  22.530  33.455 
Present  9.0257  9.0868  10.3267  14.6805  22.5682  33.4769 
Table 3. Comparison of the nondimensional natural frequency parameter with Leissa [1] and Zhou [8] for free outer and free inner boundary ($\nu =1/3$, ${r}_{1}/{r}_{2}=0.4$)
Results  Mode number  
1  2  3  4  5  6  
Leissa [1]  –  4.567  –  –  –  – 
Zhou [8]  4.5325  8.5510  11.765  17.043  21.262  31.356 
Present  4.5197  8.5070  11.7368  16.9765  21.233  31.2565 
Table 4. Comparison of the nondimensional natural frequency parameter with Leissa [1] and Zhou [8] for clamped outer and clamped inner boundary ($\nu =1/3$, ${r}_{1}/{r}_{2}=0.4$)
Results  Mode number  
1  2  3  4  5  6  
Leissa [1]  62.33  62.92  66.406  –  –  – 
Zhou [8]  61.872  62.966  66.672  73.630  84.594  99.904 
Present  62.0056  63.1121  66.7350  73.6148  84.5044  99.7988 
Table 5. Nondimensional frequency parameter $\omega {r}_{2}^{2}\sqrt{\rho h/D}$ for the annular circular plate with clamped outer and free inner edge ($\nu =1/3$)
$n$  ${r}_{1}/{r}_{2}$  
0.2  0.4  0.6  0.8  
0  10.2922  12.871  25.3966  92.3617 
1  20.4306  18.497  28.3519  94.2334 
2  33.6865  29.001  36.2745  98.3097 
3  50.4963  44.70  47.9049  105.6264 
4  69.6342  62.982  62.8458  115.7173 
5  91.0597  83.751  79.4797  128.4994 
Table 6. Nondimensional frequency parameter $\omega {r}_{2}^{2}\sqrt{\rho h/D}$ for the annular circular plate with free outer and clamped inner edge ($\nu =1/3$)
$n$  ${r}_{1}/{r}_{2}$  
0.2  0.4  0.6  0.8  
0  4.795  9.0257  20.5213  84.5418 
1  5.190  9.0868  20.8923  85.1491 
2  6.322  10.3267  22.3356  87.0250 
3  12.367  14.6805  25.6329  90.3110 
4  21.502  22.5682  31.5306  95.2026 
5  32.438  33.4769  40.3467  101.941 
Table 7. Nondimensional frequency parameter $\omega {r}_{2}^{2}\sqrt{\rho h/D}$ for the annular circular plate with clamped outer and clamped inner edge ($\nu =1/3$)
$n$  ${r}_{1}/{r}_{2}$  
0.2  0.4  0.6  0.8  
0  4.795  9.0257  141.4272  593.2696 
1  5.190  9.0868  144.8754  593.8935 
2  6.322  10.3267  149.467  595.8484 
3  12.367  14.6805  156.3015  599.0929 
4  21.502  22.5682  165.7894  603.7515 
5  32.438  33.4769  178.2596  609.866 
4. Conclusions
In this paper, numerical analysis for free vibration analysis of a thin annular solid plate is carried out using the finite element method for different boundary conditions at the inner and outer radius. It is found that those natural frequency results are quite close to those reported in previous works of literature. The novelty of this paper is the effect of the radii ratio on natural frequency is discussed and found that with increasing radii ratio, natural frequency increases and another novelty is by using shell element modeling as per Abaqus convention the dimensionless frequency parameter as found in the literature are completely validated.
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