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
JVE Conferences
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.
References
 Leissa A. W. Vibration of Plates. Office of Technology Utilization, Washington, 1969. [CrossRef]
 Kim C. S., Dickinson S. M. On the free, transverse vibration of annular and circular, thin, sectorial plates subject to certain complicating effects. Journal of Sound and Vibration, Vol. 134, Issue 3, 1989, p. 407421. [Publisher]
 Rajalingham C., Bhat R. B., Xistris G. D. Vibration of rectangular plates using plate characteristic functions as shape functions in the RayleighRitz method. Journal of Sound and Vibration, Vol. 193, Issue 2, 1996, p. 497509. [Publisher]
 Liew K. M., Yang B. Threedimensional elasticity solutions for free vibrations of circular plates: a polynomialsRitz analysis. Computer Methods in Applied Mechanics and Engineering, Vol. 175, Issues 12, 1999, p. 189201. [Publisher]
 Zhou D., Au F. T. K., Cheung Y. K., Lo S. H. Threedimensional vibration analysis of circular and annular plates via the ChebyshevRitz method. International Journal of Solids and Structures, Vol. 40, Issue 12, 2003, p. 30893105. [Publisher]
 Lim C. W., Li Z. R., Xiang Y., Wei G. W., Wang C. M. On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates. Advances in Vibration Engineering, Vol. 4, Issue 3, 2005, p. 221248. [CrossRef]
 Zhou Z. H., Wong K. W., Xu X. S., Leung A. Y. T. Natural vibration of circular and annular thin plates by Hamiltonian approach. Journal of Sound and Vibration, Vol. 330, Issue 5, 2011, p. 10051017. [Publisher]
 Kumar S., Vinayak Ranjan, Jana P. Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Composite Structures, Vol. 197, 2018, p. 3953. [Publisher]
 Pratap Singh P., Azam M. S., Vinayak Ranjan Vibration analysis of a thin functionally graded plate having an out of plane material inhomogeneity resting on WinklerPasternak foundation under different combinations of boundary conditions. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2018. [CrossRef]