Free vibration of circular annular plate with different boundary conditions

Yash Jaiman1 , Baij Singh2

1Bennett University, Greater Noida, Uttar Pradesh, India

2Indian Institute of Technology (ISM), Dhanbad, India

1Corresponding author

Vibroengineering PROCEDIA, Vol. 29, 2019, p. 82-86. https://doi.org/10.21595/vp.2019.21116
Received 23 October 2019; accepted 4 November 2019; published 28 November 2019

Copyright © 2019 Yash Jaiman, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Creative Commons License
Abstract.

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 (r1/r2) 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 Rayleigh-Ritz approximation method for free vibration of a thin plate to extract natural frequencies. Rajalingham et al. [3] used a Rayleigh-Ritz 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 polynomials-Ritz method for the vibration of circular plates by using three-dimensional elasticity solutions. Zhou et al. [5] used the Chebyshev-Ritz method for three-dimensional vibration and mode shapes of the circular plate. Lim et al. [6] used the state-space 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 Rayleigh-Ritz method to compute the natural frequencies of the thin plate.

2. Basic formulation

Consider a homogeneous, isotropic circular annular plate in cylindrical coordinates r,θ,z 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:

(1)
D 4 W r , θ , t + ρ h w r , θ = 0 ,           4 = 2 2 ,

where Laplacian operator: 2=2r2+1rr+1r22θ2, ρ is the mass density, D=Eh3/[121-ν2] is the flexural rigidity and ν is Poisson’s ratio.

Transverse deflection of natural vibrations for thin circular plate is assumed to be:

(2)
W r , θ , t = w r , θ e i ω t ,

where ω is the natural frequency and wr,θ is natural mode.

Substituting Eqs. (2) in (1), we get:

(3)
4 W - γ 4 W = 0 ,         or         ( 2 + γ 2 ) 2 + γ 2 W = 0 ,

where:

(4)
γ 4 = ρ h ω 2 D .

The general finite element equation for the transverse deflection of thin plate is given by:

(5)
M q ¨ + K q = 0 ,

where M is the mass matrix, K is stiffness matrix, q¨ is the nodal acceleration vector, and q is nodal displacement vector.

The non-dimension natural frequency parameter ϖ are calculated as:

(6)
ϖ = 2 π ω ρ h / D r 1 2 ,

where ω is the natural frequency in Hz.

Fig. 1. Schematic diagram and coordinate system of annular circular plate

Schematic diagram and coordinate system of annular circular plate

3. Results and discussions

In this section, the first six non-dimensional 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 (r1/r2). 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 1-4 shows that the first six non-dimensional 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, r1/r2 = 0.4

Natural modes of a clamped annular plate with a free inner boundary, r1/r2 = 0.4 Natural modes of a clamped annular plate with a free inner boundary, r1/r2 = 0.4 Natural modes of a clamped annular plate with a free inner boundary, r1/r2 = 0.4 Natural modes of a clamped annular plate with a free inner boundary, r1/r2 = 0.4 Natural modes of a clamped annular plate with a free inner boundary, r1/r2 = 0.4 Natural modes of a clamped annular plate with a free inner boundary, r1/r2 = 0.4

Table 5-7 presents the effect of the radii ratio on the non-dimensional frequency parameter of thin plates. It is observed from these tables that as the radii ratio increases non-dimensional frequency parameter increases.

Table 1. Comparison of the non-dimensional natural frequency parameter with Leissa [1] and Zhou [8] for clamped outer and free inner boundary (ν=1/3, r1/r2 =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 non-dimensional natural frequency parameter with Leissa [1] and Zhou [8] for free outer and clamped inner boundary (ν=1/3, r1/r2 =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 non-dimensional natural frequency parameter with Leissa [1] and Zhou [8] for free outer and free inner boundary (ν=1/3, r1/r2 =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 non-dimensional natural frequency parameter with Leissa [1] and Zhou [8] for clamped outer and clamped inner boundary (ν=1/3, r1/r2 =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. Non-dimensional frequency parameter ωr22ρh/D  for the annular circular plate with clamped outer and free inner edge (ν=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. Non-dimensional frequency parameter ωr22ρh/D  for the annular circular plate with free outer and clamped inner edge (ν=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. Non-dimensional frequency parameter ωr22ρh/D  for the annular circular plate with clamped outer and clamped inner edge (ν=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.

References

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