An improved Fourier series method for vibration analysis of moderately thick annular and circular sector plates subjected to elastic boundary conditions

Fazl e Ahad1 , Dongyan Shi2 , Zarnab Hina3 , Syed M. Aftab4 , Hafiz M. Waqas5

1, 2, 4, 5College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, China

3College of Information and Communication Engineering, Harbin Engineering University, Harbin, China

1Corresponding author

Journal of Vibroengineering, Vol. 18, Issue 6, 2016, p. 3841-3857. https://doi.org/10.21595/jve.2016.16765
Received 23 December 2015; received in revised form 3 April 2016; accepted 20 May 2016; published 30 September 2016

Copyright © 2016 JVE International Ltd. 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.
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Abstract.

In this paper, an improved Fourier series method is presented for vibration analysis of moderately thick annular and circular sector plates subjected to general elastic boundary conditions along its edges. In literature, annular and circular sector plates subjected to classical boundary conditions have been studied in detail however in practical engineering applications the boundary conditions are not always classical in nature. Therefore, study of vibration response of these plates subjected to general elastic boundary conditions is far needed. In the method presented, artificial boundary spring technique has been employed to simulate the general elastic boundary conditions and first order shear deformation theory has been employed to formulate the theoretical model. Irrespective of the boundary conditions, each of the displacement function is expressed as a new form of trigonometric expansion with accelerated convergence. Rayleigh-Ritz method has been employed to determine the expansion coefficients. Unlike most of the studies on vibration analysis of moderately thick annular sector plates, the present method can be universally applied to a wide range of vibration problems involving different boundary conditions, varying material and geometric properties without modifying the solution algorithms and procedure. The effectiveness, reliability and accuracy of the present method is fully demonstrated and verified by several numerical examples. Bench mark solutions for moderately thick annular sector and circular plates under general elastic boundary conditions are also presented for future computational methods.

Keywords: vibrations, circular plates, annular plates, sector plates, natural frequency, mode shapes, arbitrary boundary conditions.

1. Introduction

Annular, circular and their sector parts are key structural components that are used in various engineering fields like civil, marine, aerospace and mechanical engineering. Due to different geometrical shapes of these structures, they have been analyzed separately using different solution techniques. Most of the initial research on these plates was done using Classical plate theory (CPT) in which the shear deformation and rotary inertia was neglected which in turn limited its application on moderately thick and thick plates. Later a lot of theories were proposed incorporating the shear deformation and rotary inertia which resulted in an increase in accuracy of the results for moderately thick and thick annular and circular plates. These theories’ have been well explained in Leissa’s book on vibration of plates.

Different methods have been employed by various researchers to study the vibration characteristics of annular and circular plates subjected to different boundary conditions. However, a few prominent studies related to these plates are highlighted here in this manuscript. Employing the Mindlin plate theory on thick sector plates, Guruswamy et al. [1] studied the dynamic response of these plates by proposing a sector finite element. Fully clamped boundary conditions were employed on all edges. Taking the effect of shear deformation in thickness direction, another study was performed by Soni et al. [2] on axisymmetric non uniform circular disks. In their research they employed Chebyshev collocation technique to study the vibration characteristics of these plates. In another study Rayleigh-Ritz method was employed by Liew et al. on circular plates with multiple internal ring supports. Later in another study they studied the characteristics of these plates subjected to in-plane pressure [3-4].

Using three dimensional finite strip model, thick and thin sector plates subjected to various combinations of classical boundary condition were analyzed by Cheung et al. [5]. The integral equation technique and finite strip method was employed by Sirinivasan et al. and Misuzawa et al. respectively to study the vibration characteristics of Mindlin annular sector plates [6-7]. Employing the Mindlin plate theory and differential quadrature method, Liu et al. studied the effect of sector angle and thickness to radius ratio on the vibration characteristics of moderately thick sector plates [8]. Later they extended the same differential quadrature method to annular sector plates having shear deformation subjected to different combinations of classical boundary conditions [9]. In another study, the same differential quadrature method was employed on solid circular plates with variable thickness in radial direction and subjected to elastic boundary conditions by Wu et al. [10]. Similarly, Xiang et al. employed domain decomposition technique and studied the vibration response of circular plates with stepped thickness variation [11].

Using Rayleigh-Ritz method So et al. [12] studied the vibration characteristics of annular and circular plates by employing three dimensional elasticity theory. A similar three dimensional study was performed by Hashemi et al. on annular sector plates resting on elastic foundations. He employed polynomial-Ritz approach and studied the vibration characteristics for different sets of classical boundary conditions [13]. In another study similar polynomial-Ritz model was presented by Liew et al. to investigate the effect of boundary conditions and thickness on the vibration characteristics of annular plates [14]. In another prominent three dimensional study Chebyshev-Ritz technique was employed on circular and annular plates by Zhou et al. to study the vibration characteristics of these plates [15]. In plane vibrations of circular plates subjected to different boundary conditions were investigated by various researchers using different solution techniques [16-17]

From the studies mentioned above it can be seen that most of the previous studies on annular and circular plates were limited to classical boundary conditions which includes free, simply supported, clamped or combination of these. However, in practical engineering applications the boundary conditions are not always classical in nature. Therefore, the development of an analytical method universally dealing with arbitrary boundary conditions was much needed. An improved Fourier series method was developed for vibration analysis of beams and plates by Li [18-22]. Later Xianjie Shi et al. extended this method to thin annular plates to study its vibration characteristics [23-26]. The main objective of this study is to realize and extend the same generalized Fourier series method to study the vibration analysis of Mindlin annular sector and circular sector plates under various boundary conditions including the general elastic restraints.

2. Theoretical formulation

2.1. Model description

Consider an annular sector plate of constant thickness h, inner radius a, outer radius b and width R in radial direction as shown in Fig. 1. The plate geometry and dimensions are defined in the cylindrical coordinate system (r,ϕ,z). A local coordinate system (s,ϕ,z) is also shown in Fig. 1. The radial and thickness coordinates s and z are measured normally from the inner edge and mid plane of the annular sector plate respectively whereas ϕ is the circumferential angle. Four sets of distributed springs (one translational and two rotational) of arbitrary stiffness values are attached at each edge to simulate arbitrary boundary conditions. All the classical sets of boundary conditions can easily be achieved by varying the stiffness value of each spring from zero to an infinitely large number i.e. 1014.

The domain of the annular sector plate can be defined as:

(1)
0 s R ,           R = b - a ,           - h 2 z h 2 ,           0 ϕ < 2 π ,

The relationship between local and global coordinate system can be expressed as:

(2)
s = r - a .

The material of the plate is assumed to be isotropic with material density ρ, young’s modulus E and Poisson ratio ν. It should be noted that a circular sector plate can be defined as a special case of an annular sector plate if the inner radius ‘a’ is set either equal to 0 or to a very small number say 0.00001.

The displacement field of Mindlin annular sector plate in cylindrical coordinates is given by:

(3)
u r r , ϕ , z , t = u r r , ϕ , z + z θ r r , ϕ , t ,
u ϕ r , ϕ , z , t = u ϕ r , ϕ , z + z θ ϕ r , ϕ , t ,
w r , ϕ , z , t = w o r , ϕ , t ,

where z is the thickness coordinate, ur and uϕ are displacements of the mid plane in r and ϕ directions, respectively, w is the transverse displacement. θr and θϕ are the rotation functions of the middle surface and t is the time. Assuming the plain stress distribution in accordance with Hooks law, the stress resultants are obtained for Mindlin annular plate by integrating the stresses as shown below:

(4)
M r r = - h / 2 h / 2 σ r r z d z = - h / 2 h / 2 E 1 - ν 2 ε r r + ν ε ϕ ϕ z d z = D θ r r + ν r θ r + θ ϕ ϕ ,
M ϕ ϕ = - h / 2 h / 2 σ ϕ ϕ z d z = - h / 2 h / 2 E 1 - ν 2 ε ϕ ϕ + ν ε r r z d z = D 1 r θ r + θ ϕ ϕ + ν θ r r ,
M r ϕ = - h / 2 h / 2 τ r ϕ z d z = - h / 2 h / 2 G γ r ϕ z d z = D 1 - ν 2 1 r θ r ϕ - θ ϕ + θ ϕ r ,
Q r r = K 2 - h / 2 h / 2 τ r z d z = K 2 G h θ r + w o r ,
Q ϕ ϕ = K 2 - h / 2 h / 2 τ ϕ z d z = K 2 G h θ ϕ + 1 r w o ϕ ,

where Mrr, Mϕϕ and Mrϕ are the bending moments per unit length of the plate, Qrr and Qϕϕ are the transverse shear forces per unit length of the plate, σrr, σϕϕ are the normal stresses, and τrϕ, τrz and τrz are the shear stresses, h is the plate thickness, E is the modulus of elasticity, G is the shear modulus, ν is the Poisson ratio, D=Eh3/121-ν2 is the flexural rigidity and K =π2/12 is the shear correction factor to compensate for the error in assuming the constant shear stress throughout the plate thickness.

The equation of motion of Mindlin plates in (r,ϕ,z) is given by:

(5)
M r r r + 1 r M r ϕ ϕ + 1 r M r r - M ϕ ϕ - Q r r = ρ h 3 12 2 θ r t 2 ,
M r ϕ r + 1 r M ϕ ϕ + 2 r M r ϕ - Q ϕ ϕ = ρ h 3 12 2 θ ϕ t 2 ,
Q r r r + 1 r Q ϕ ϕ ϕ + Q r r r = ρ h 2 w o t 2 .

Fig. 1. Mindlin sector plate geometry

 Mindlin sector plate geometry

2.2. Solution scheme

2.2.1. Selection of admissible displacement function

Assume that the displacement field of Mindlin annular sector plate in local coordinate system (s,ϕ,z) is defined by the following series:

(6)
θ s ( s , ϕ ) = m = n = - 2 A m n φ m s φ n ϕ ,         r = s + a ,
θ ϕ ( s , ϕ ) = m = n = - 2 B m n φ m s φ n ϕ ,
w o ( s , ϕ ) = m = n = - 2 C m n φ m s φ n ϕ ,

where:

φ m s = c o s λ m s ,       m 0 s i n λ m s ,         m < 0 ,       φ n ϕ = c o s λ n ϕ ,       n 0 s i n λ n ϕ ,         n < 0 ,

and λm=mπ/R,λn=nπ/α and Amn,Bmn,Cmn denotes the Fourier series expansion coefficients. The sine terms in the equations Eq. (6) are introduced to overcome the potential discontinuities (convergence problem) of the displacement function, along the edges of the plate, when it is periodically extended and sought in the form of trigonometric series expansion. The addition of these auxiliary functions in the admissible functions plays an important role in the convergence and accuracy of the present method or in other words the elimination of potential discontinuities at the ends or elimination of Gibbs effect.

In order to illustrate this, take a beam problem for example. The governing equations for free vibration of a general supported Euler beam is obtained as:

(7)
D 4 w x x 4 - ρ A ω 2 w x = 0 ,

where D, ρ and A are, respectively, the flexural rigidity, the mass density and the cross sectional area of the beam, and ω is frequency in radian. From Eq. (7) it can be observed that the displacement solution wx on a beam of length L is required to have up to the fourth derivatives, that is, wxC3. In general, the displacement function wx defined over a domain [0, L] can be expanded into a Fourier series inside the domain excluding the boundary points:

(8)
w x = m = 0 A m c o s m π x L ,

where Am are the expansion coefficients. From the Eq. (8), we can see that the displacement function wx can be viewed as a part of an even function defined over -L,L, as shown in Fig. 2.

Fig. 2. An illustration of the possible discontinuities of the displacement at the end points

 An illustration of the possible discontinuities  of the displacement at the end points

Fig. 3. An illustration of removal of possible discontinuities (convergence problem) at ends

 An illustration of removal of possible discontinuities (convergence problem) at ends

Thus, the Fourier cosine series is able to correctly converge to wx at any point over [0, L]. However, its first-derivative w'x is an odd function over -L,L leading to a jump at end locations. The corresponding Fourier expansion of w'x continue on [0, L] and can be differentiated term-by-term only if w0=wL=0. Thus, its Fourier series expansion (sine series) will accordingly have a convergence problem due to the discontinuity at end points (Gibbs phenomena) when w'x is required to have up to the first-derivative continuity.

To overcome this problem, this Improved Fourier Series technique was proposed by Li [18, 19]. In this technique a new function Px is considered in the displacement function:

(9)
w - x = W x + P x = m = 0 A m c o s m π x L + P x ,

where the auxiliary function Px in equation above represents an arbitrary continuous function that, regardless of boundary conditions, is always chosen to satisfy the following equations:

(10)
P ' = w ' 0 ,           P ' L = w ' L ,             P ' ' ' 0 = w ' ' ' 0 ,             P ' ' ' L = w ' ' ' L .

The actual values of the first and third derivatives (a sine series) at the boundaries need to be determined from the given boundary conditions. Essentially, w-x represents a residual beam function which is continuous over [0, L] and has zero slopes at the both ends, as shown in Fig. 3. Apparently, the cosine series representation of w-x is able to converge correctly to the function itself and its first derivative at every point on the beam.

Thus, based on the above analysis, Px can be understood as a continuous function that satisfies Eq. (9), and its form is not a concern but must be a closed-form and sufficiently smooth over a domain [0, L] of the beam in order to meet the requirements provided by the continuity conditions and boundary constraints. Furthermore, it is noticeable that the auxiliary function Px can improve the convergent properties of Fourier series.

2.2.2. Determining the expansion coefficients

Once the proper admissible function for the displacement field is selected Eq. (6), the next step is to find the expansion coefficients in the assumed displacement field. In order to do so Rayleigh-Ritz method is employed which is an energy-based method. To employ this method, it is necessary to state the potential and kinetic energies first in terms of displacement fields. The expression for the potential energy of the sector plate in local coordinates (s,ϕ,z) is derived from the constitutive laws and strain-displacement relations. According to the Mindlin plate theory. The strain energy of the annular sector plates can be expressed as:

(11)
U p = 1 2 0 α 0 R D θ s s 2 + 2 ν s + a θ s s θ ϕ ϕ + θ s + 1 ( s + a ) 2 θ ϕ ϕ + θ s 2 + 1 - υ 2 1 ( s + a ) 2 θ ϕ - s + a θ ϕ s - θ s ϕ 2 + K 2 G h w o s + θ s 2 + 1 ( s + a ) 2 w o ϕ + s + a θ ϕ 2 s + a d s d ϕ ,

the kinetic energy expression for annular sector plate is expressed as:

(12)
T p = 1 2 ω 2 0 α a b ρ h w o 2 + ρ h 3 12 θ r 2 + θ ϕ 2 r d r d ϕ .

The potential energy stored in the boundary springs is given by:

(13)
U s p = 1 2 0 α a k a   w o 2 + K a r θ s 2 + K a t θ ϕ 2     s = 0 + b k b     w o 2 + K b r θ s 2 + K b t θ ϕ 2 s = R d ϕ
              + 1 2 0 R k 0   w o 2 + K 0 r θ s 2 + K 0 t θ ϕ 2 ϕ = 0 + k α   w o 2 + K α r θ s 2 + K α t θ ϕ 2 ϕ = α d s ,

where ka , kb  (k0  and kα ) are linear spring constants, Kar, Kbr (K0r and Kαr) are rotational spring constants in radial direction, Kat, Kbt (K0t and Kαt) are rotational spring constants in tangential direction at edges s= 0 and s=R and ϕ= 0 and ϕ=α respectively. All the classical homogeneous boundary conditions can be simply considered as special cases when the spring constants are either extremely large or substantially small. The units for the translational and rotational springs are N/m and Nm/rad, respectively.

After the potential and kinetic energies are expressed, then all the assumed displacement functions are inserted in the potential and kinetic energy equations and these equations are then further minimized with respect to the expansion coefficients in the displacement field. Mathematically, the Lagrangian for the annular sector plate can be generally expressed as:

(14)
L = U p + U s p - T p ,

where Up is strain energy of the plate, Usp is strain energy stored in the boundary springs and Tp is the kinetic energy of the plate. Substituting Eq. (6) in Eqs. (11)-(13) and then minimizing Lagrangian Eq. (14) against all the unknown series expansion coefficients that is:

(15)
L Θ = 0 ,       where     Θ =   A m n ,   B m n ,   C m n ,

we can obtain a series of linear algebraic expressions in a matrix form as:

(16)
( K - ω 2 M ) E = 0 ,

where E is a vector which contains all the unknown series expansion coefficients and K and M are the stiffness and mass matrices, respectively. E, K and M can be expressed as:

(17)
E = A - 2 , - 2 , A - 2 , - 1 , A - 2,0 , . . . , A m ' , - 2 , A m ' , - 1 , . . . A m ' , n ' , . . . , A M N B - 2 , - 2 , B - 2 , - 1 , B - 2,0 , . . . , B m ' , - 2 , B m ' , - 1 , . . . B m ' , n ' , . . . , B M N C - 2 , - 2 , C - 2 , - 1 , C - 2,0 , . . . , C m ' , - 2 , C m ' , - 1 , . . . C m ' , n ' , . . . , C M N   T ,
(18)
K = K i i p K i j p K i k p K j i p K j j p K j k p K k i p K k j p K k k p + K i i s p K i j s p K i k s p K j i s p K j j s p K j k s p K k i s p K k j s p K k k s p ,
(19)
M = M i i 0 0 0 M j j 0 0 0 M k k ,

where the subscripts i, j and k represents w, θs and θϕ and the superscripts p and sp represents plate and boundary springs respectively. For conciseness, the detailed expressions for the stiffness and mass matrices are not shown here.

2.2.3. Determining the eigen values and eigen vectors

Once Eq. (16) is established, the eigenvalues (or natural frequencies) and eigenvectors of Mindlin annular sector plates can now be easily and directly determined from solving a standard matrix eigenvalue problem i.e. Eq. (16) using MATLAB. For a given natural frequency, the corresponding eigenvector actually contains the series expansion coefficients which can be used to construct the physical mode shape based on Eq. (6). Although this investigation is focused on the free vibration of Mindlin annular sector plate, the response of the annular sector plate to an applied load can be easily considered by simply including the work done by this load in the Lagrangian, eventually leading to a force term on the right side of Eq. (16).

3. Results and discussion

To check the accuracy and usefulness of the proposed technique, several numerical examples are presented in this section. It is important to mention here that the accuracy of the proposed method is greatly controlled by the number of truncation terms i.e. M=N, the more number of truncation terms we use we get more accurate results however the computational cost and time will increase with increasing number of truncation terms. Theoretically, there are infinite terms in the assumed displacement functions. However, the series is numerically truncated and finite terms are counted in actual calculations which will be further explained in the text to follow. Moreover, in identifying the boundary conditions in this section, letters C, S, and F have been used to indicate the clamped, simply supported and free boundary condition along an edge, respectively. Therefore, the boundary conditions for a plate are fully specified by using four alphabets with the first one indicating the B.C. along the first edge, r=a. The remaining (the second to the fourth) edges are ordered in the counterclockwise direction.

First of all, in order to check the accuracy and usefulness we first consider a fully clamped Mindlin annular sector plate. Fully clamped (CCCC) boundary conditions can easily be achieved by setting the stiffnesses of the restraining springs to an infinitely large number (1014) in the numerical calculations. The first six non-dimensional frequency parameter, Ω=ωb2ρh/D1/2 are tabulated in Table 1 along with the reference results from [9] and [27].

Table 1. First six non dimensional frequency parameter Ω=ωb2ρh/D1/2 for fully clamped (CCCC) Mindlin Annular sector plates (ϕ=2π/3, a/b= 0.25, h/b= 0.2)

M = N
Mode sequence
1
2
3
4
5
6
2
31.481
42.907
62.894
66.199
73.800
94.776
4
31.084
41.872
56.108
62.435
72.906
75.775
6
31.059
41.823
55.972
62.411
71.269
72.862
8
31.054
41.813
55.951
62.406
71.148
72.852
10
31.053
41.810
55.946
62.405
71.125
72.849
12
31.053
41.809
55.943
62.404
71.118
72.847
14
31.053
41.809
55.942
62.404
71.116
72.847
Ref. [9]
31.056
41.814
55.951
62.420
71.127
72.862
Ref. [27]
31.057
41.814
55.951
62.420
71.127
72.862

Similarly, in Table 2, first six non-dimensional frequency parameter for Mindlin annular sector plate having simply supported radial edges and clamped circumferential edges (CSCS) boundary conditions has been given along with the reference results from [9, 27]. A good agreement in the present values and reference values can be observed.

Table 2. First six non dimensional frequency parameter Ω=ωb2ρh/D1/2 for Mindlin Annular sector plates (ϕ=π/3, a/b= 0.5, h/b= 0.1) having simply supported radial edges and clamped circumferential edges (CSCS)

M = N
Mode sequence
1
2
3
4
5
6
2
77.625
104.054
161.254
170.932
194.112
243.145
4
76.567
102.955
149.795
166.952
190.467
215.603
6
76.476
102.765
149.410
166.772
190.093
206.703
8
76.449
102.696
149.323
166.726
189.961
206.315
10
76.439
102.667
149.290
166.710
189.908
206.228
12
76.435
102.654
149.275
166.703
189.883
206.196
14
76.432
102.647
149.267
166.699
189.871
206.182
Ref. [9]
76.902
103.682
150.413
167.327
191.593
207.276
Ref. [27]
76.902
103.682
150.413
167.327
191.593
207.276

Next we consider annular sector plate simply supported at radial edges and having different combination of boundary conditions (free-clamped, free-simply supported, simply supported-simply supported, simply supported-free and clamped-free) at the circumferential edges. The simply supported condition is simply produced by setting the stiffnesses of the translational and rotational springs to and 0, respectively, and the free edge condition by setting both stiffnesses to zero. The fundamental frequency parameters with different boundary conditions are shown in Table 3. The current results agree well with those taken from references [28, 29].

Next to illustrate the convergence and numerical stability of the current solution procedure, several sets of results for fully clamped Mindlin annular sector plates having different sector angles and using different truncation numbers (M=N= 2, 4, 6, 8, 10, 12, 14) are presented in Tables 4-7. Furthermore, the fast convergence pattern can also be observed in Fig. 4.

Table 3. Fundamental frequency parameter Ω=ωb2ρh/D1/2 for Mindlin Annular sector plates having simply supported radial edges and different boundary conditions at circumferential edges (a/b= 0.5)

Sector angle (ϕ)
Thickness to radius ratio (h/b)
Method
Boundary conditions at circumferential edges
F-C
F-S
S-S
S-F
C-F
195
0.1
Present
19.998
10.224
38.365
4.560
12.696
Ref. [28]
19.999
10.227
38.636
4.675
12.680
Ref. [29]
20.097
10.239
38.764
0.2
Present
17.503
9.130
32.508
4.005
11.413
Ref. [28]
17.582
9.366
32.871
4.542
11.427
Ref. [29]
17.764
9.396
33.190
210
0.1
Present
19.620
9.685
38.222
4.507
12.678
Ref. [28]
19.610
9.664
38.455
4.584
12.659
Ref. [29]
19.706
9.675
38.582
0.2
Present
17.235
8.681
32.419
3.997
11.417
Ref. [28]
17.294
8.877
32.734
4.458
11.425
Ref. [29]
17.294
8.904
33.050
270
0.1
Present
18.654
8.213
37.868
4.392
12.639
Ref. [28]
18.622
8.130
38.010
4.372
12.615
Ref. [29]
18.715
8.139
38.134
0..2
Present
16.548
7.450
32.200
3.999
11.433
Ref. [28]
16.566
7.546
32.394
4.263
11.430
Ref. [29]
16.739
7.567
32.704

A fast convergence pattern can be observed in the tabulated results as well as Fig. 4, therefore it can be concluded that sufficiently accurate results can be obtained with a small number of terms in the series expansion and the solution is consistently refined as more and more terms are included in the series expansion.

Table 4. First five non-dimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)

M = N
Sector angle
Mode sequence
1
2
3
4
5
2
ϕ = π 6
145.584
240.451
251.723
331.721
391.843
4
144.104
237.420
249.040
328.660
350.364
6
144.032
237.301
248.941
328.487
349.816
8
144.020
237.280
248.924
328.452
349.739
10
144.017
237.274
248.920
328.442
349.720
12
144.016
237.272
248.918
328.438
349.713
14
144.015
237.271
248.917
328.436
349.711

Table 5. First five non-dimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)

M = N
Sector angle
Mode number
1
2
3
4
5
2
ϕ = π 2
104.250
116.563
163.387
223.325
232.424
4
102.977
112.649
130.467
173.504
220.564
6
102.911
112.460
129.978
155.396
187.319
8
102.900
112.420
129.896
154.985
185.972
10
102.897
112.408
129.871
154.895
185.781
12
102.896
112.403
129.862
154.865
185.726
14
102.895
112.401
129.857
154.852
185.705

Table 6. First five non-dimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)

M = N
Sector angle
Mode number
1
2
3
4
5
2
ϕ = 2 π 3
102.797
109.453
139.708
221.999
227.027
4
101.566
106.400
115.457
143.396
194.321
6
101.503
106.239
115.077
128.766
147.504
8
101.492
106.203
115.006
128.447
146.310
10
101.489
106.192
114.984
128.368
146.140
12
101.488
106.188
114.975
128.339
146.089
14
101.488
106.186
114.970
128.327
146.068

Table 7. First five non-dimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)

M = N
Sector angle
Mode sequence
1
2
3
4
5
2
ϕ = 7 π 6
101.717
103.774
115.254
220.887
222.496
4
100.537
101.885
104.287
114.756
138.302
6
100.480
101.779
104.088
107.716
113.103
8
100.470
101.753
104.042
107.560
112.462
10
100.467
101.744
104.026
107.512
112.360
12
100.467
101.741
104.019
107.492
112.325
14
100.466
101.739
104.015
107.482
112.308

From Tables 4-7 it can be seen that when the truncated numbers change from M×N= 10×10 to 12×12, the maximum difference of the frequency parameters does not exceed 0.003 for the worst case, which is acceptable. Furthermore, in modal analysis the natural frequencies for higher order modes tend to converge slower as compared to the lower order modes which can easily be observed in Fig. 4 that the 9th mode frequency converges slowly as compared to the 6th and 3rd mode. Thus a suitable truncation number should be used to achieve the accuracy of the largest desired natural frequency. In view of above and excellent numerical behavior of the current solution, the truncation number for all subsequent calculations in the present method is taken as M=N= 12.

Next we study the effect of sector angle and thickness-radius ratio and cutout ratio on non-dimensional frequency parameter. The effect has been graphically represented in Figs. 5-7, respectively.

Fig. 4. Convergence pattern of frequency parameters with no. of terms (M=N)

 Convergence pattern of frequency  parameters with no. of terms (M=N)

Fig. 5. Effect of sector angle on non-dimensional frequency parameter ‘Ω

 Effect of sector angle on  non-dimensional frequency parameter ‘Ω’

Fig. 6. Effect of thickness to radius ratio (h/b) on the frequency parameter

 Effect of thickness to radius ratio (h/b)  on the frequency parameter

Fig. 7. Effect of cutout ratio (a/b) on the frequency parameters

Effect of cutout ratio (a/b)  on the frequency parameters

It can be seen in Fig. 5 that for smaller sector angles i.e. ϕ2π/3, the decrease in the frequency parameters is more as compared to the sector angles greater than 2π/3. Similarly, in figure 6, 1st, 3rd and 5th mode frequency parameters have been plotted against different thickness to radius ratios (h/b) for a fully clamped (CCCC) Mindlin annular sector plate having sector angle =2π/3 and cutout ratio =a/b= 0.6. It can be observed that with the increase in thickness to radius ratio the frequency parameter always decreases. Similarly, the effect of cutout ratio i.e. inner radius to outer radius (a/b) on the frequency parameters for a FSFS mindlin annular sector plate having sector angle =π/3, and h/b= 0.2 can be seen in Fig. 7.

As mentioned previously, using this method, a Mindlin circular sector plate can also be analyzed easily just by equating the inner radius of Mindlin annular sector plate to zero without modifying the equations or the solution algorithm. Table 8 shows first six non-dimensional frequency parameter along with reference results for Mindlin circular sector plates having inner radius =a= 0.0001, different thickness to radius ratio and sector angles and subjected to simply supported radial edges and clamped circular edge (SCS) boundary condition respectively. A close agreement can be observed in the present values and the reference results.

Table 8. First six non-dimensional frequency parameters for SCS Mindlin Circular sector plates (a/b= 0.0001)

Sector angle (ϕ)
Thickness to radius ratio (h/b)
Method
Mode sequence
1
2
3
4
5
6
π 6
0.1
Present
91.419
148.357
205.117
208.046
278.356
281.097
Ref. [8]
93.450
152.630
206.900
213.080
274.650
283.190
0.2
Present
66.272
98.992
131.454
131.981
161.648
163.749
Ref. [8]
67.933
102.560
132.860
135.610
165.680
167.820
π 2
0.1
Present
31.657
60.112
71.009
92.956
110.151
118.760
Ref. [8]
32.205
60.637
72.221
93.450
111.320
120.450
0.2
Present
26.298
46.323
53.227
67.436
77.370
82.512
Ref. [8]
26.993
46.906
54.466
67.933
78.582
83.903
π
0.1
Present
20.778
31.969
45.586
54.582
60.614
71.698
Ref. [8]
20.223
32.205
45.773
53.859
60.637
72.221
0..2
Present
17.752
26.630
36.470
42.169
46.726
53.813
Ref. [8]
17.773
26.993
36.758
42.393
46.906
54.466
7 π 6
0.1
Present
19.915
28.398
39.635
51.986
52.898
65.557
Ref. [8]
19.489
28.599
39.786
51.998
52.416
65.091
0.2
Present
16.871
23.958
32.269
40.788
40.916
49.836
Ref. [8]
17.092
24.293
32.519
41.067
41.195
49.859

All the examples mentioned above are limited to different combinations of classical boundary conditions which are viewed as special case of elastically restrained edges. After verifying the convergence, accuracy and effectiveness of the proposed method for different combinations of classical boundary conditions, the method is further employed here to study the vibration characteristics of Mindlin annular sector and circular sector plates subjected to general elastic boundary conditions.

In order to simulate the elastic boundary conditions, it is important to study the effect of restraining springs first on the frequency parameters so that proper value to the restraining springs could be assigned. Fig. 8 shows the effect of restraining springs stiffness on the frequency parameter for Mindlin annular sector plate (a/b= 0.6, h/b= 0.2 and ϕ= 120).

Fig. 8 shows 1st, 5th and 10th mode frequency parameters plotted against the spring stiffnesses by varying the stiffnesses of one group of boundary spring from 0 to 1016 while keeping the stiffnesses of the other group equal to infinite i.e 1016. It can be seen in Fig. 8(a) that the frequency parameter almost remains at a level when the stiffness of the translational spring in z direction is less than 108 and greater than 1012 where as other than this range the frequency parameter increases with increasing stiffness values. Similar phenomena can be observed in case of rotational spring stiffness however a slight change in frequency parameter can be observed with in the stiffness range from 108 to 1010. Based on the analysis it can be concluded that stable frequency parameter can be obtained when the stiffnesses for all the restraining springs is more than 1012 or less than 108 and also it is suitable and valid to use the stiffness value 1014 to simulate the infinite stiffness in the numerical calculations since the frequency parameter remain at the same level for values greater than equal to 1014.It can also be concluded that the elastic stiffness range for translational spring is more than the two rotational springs.

Fig. 8. a) Effect of translational spring stiffness (k) on Ω, and effect of rotational spring stiffness b) in tangential direction (Kt) on Ω, c) in radial direction (Kr) on Ω

a) Effect of translational spring stiffness (k) on Ω, and effect of rotational spring stiffness  b) in tangential direction (Kt) on Ω, c) in radial direction (Kr) on Ω

a)

a) Effect of translational spring stiffness (k) on Ω, and effect of rotational spring stiffness  b) in tangential direction (Kt) on Ω, c) in radial direction (Kr) on Ω

b)

a) Effect of translational spring stiffness (k) on Ω, and effect of rotational spring stiffness  b) in tangential direction (Kt) on Ω, c) in radial direction (Kr) on Ω

c)

From Fig. 8, an elastic boundary condition can easily be defined with any stiffness value between 108 to 1012. To the author’s best knowledge, no reported results are available in literature for vibration analysis of Mindlin annular sector plates under general elastic boundary conditions. As mentioned earlier the present method can be used to obtain natural frequency parameters for Mindlin annular sector plates under general elastic boundary condition regardless of modifying solution algorithm and procedure.

In order to achieve valuable results for annular sector plates subjected to elastic boundary conditions, we define an elastic restraint ‘E1having corresponding translational and rotational spring stiffness values k= 1e8, Kr= 0 and Kt= 0. Tables 9-11 shows first five natural frequency parameters for Mindlin annular sector plates having different sector angles and thickness-radius ratio (h/b) subjected to E1E1E1E1, FE1 FE1 and CE1CE1 boundary conditions where E1E1E1E1, FE1 FE1 and CE1CE1 represents the combination of classic and elastic boundary conditions at edges a and b as well as ϕ= 0 and α.

Due to unavailability of results in previous literature for these types of boundary conditions (E1E1E1E1, FE1 FE1 and CE1CE1), the comparison has been made with results obtained from ABAQUS. However, we know that the core algorithm of the ABAQUS software is based on the finite element method. Furthermore, the FEM computational accuracy strongly depends on the size of the mesh and the type of element selection. For more accuracy in the higher frequency region and for complex geometries, a highly refined mesh and a higher order finite element is needed. We know that the smaller the mesh size, the greater the number of elements we get for analysis which further requires more computer memory and subsequently a high computational cost. Since the geometry under investigation in this manuscript is a simpler geometry therefore a simple free meshing technique with mesh size 0.005 and S4R (Shell 4 node Reduced Integration) element type has been used which is computationally inexpensive and is considered suitable for this type of geometry and modal analysis. Keeping the mesh size 0.005, the number of elements used in the analysis for annular sector plate having sector angle π/3, π/2, 2π/3 and π are 13040, 20080, 29727 and 40240 respectively.

A very good agreement can be observed in Tables 9-11 between the calculated results and the one obtained from ABAQUS. This shows that the present method can be easily applied to classical and elastic type boundary conditions as well as their combination without modifying the solution algorithm and procedure. The results tabulated in Tables 9-11 can be used as a bench mark for future computational methods.

Table 9. First five natural frequency parameters for Mindlin Annular sector plates subjected to E1E1E1E1 type elastic boundary conditions (a/b= 0.6)

Sector angle (ϕ)
Thickness to radius ratio (h/b)
Method
Natural frequency modes
1
2
3
4
5
π 3
0.1
Present
337.934
435.140
506.867
1520.326
1966.903
ABAQUS
337.690
437.350
507.130
1522.500
1969.600
0.2
Present
241.050
302.391
330.646
2587.184
3074.622
ABAQUS
240.970
303.960
330.420
2590.100
3080.200
π 2
0.1
Present
314.276
389.282
479.851
870.662
1294.086
ABAQUS
313.770
390.700
480.060
872.380
1295.400
0.2
Present
225.655
274.476
318.276
1411.322
1991.271
ABAQUS
225.400
275.570
318.090
1412.500
1994.500
2 π 3
0.1
Present
300.548
361.101
440.420
703.848
884.878
ABAQUS
301.310
364.010
443.800
707.710
901.740
0..2
Present
216.209
257.039
302.020
1032.654
1342.753
ABAQUS
216.710
259.290
303.580
1037.500
1374.000
π
0.1
Present
288.597
325.207
378.969
516.005
622.710
ABAQUS
288.110
325.230
380.110
517.160
622.670
0.2
Present
206.659
235.054
270.651
650.104
811.858
ABAQUS
206.290
235.350
271.530
650.900
811.980

Table 10. First five natural frequency parameters for Mindlin Annular sector plates subjected to FE1FE1 type elastic boundary conditions (a/b= 0.4)

Sector angle (ϕ)
Thickness to radius ratio (h/b)
Method
Natural frequency
1
2
3
4
5
π 3
0.1
Present
119.832
264.818
319.661
1483.220
1916.028
ABAQUS
119.940
264.970
319.570
1484.100
1917.400
0.2
Present
83.397
180.756
222.669
2577.428
3060.712
ABAQUS
83.395
180.750
222.710
2580.000
3066.200
π 2
0.1
Present
57.023
251.087
267.380
786.925
1222.649
ABAQUS
56.935
251.210
267.440
787.520
1223.700
0.2
Present
41.696
179.512
185.090
1389.419
1971.191
ABAQUS
40.802
179.550
184.680
1390.300
1974.300
2 π 3
0.1
Present
24.833
201.973
271.669
561.948
806.253
ABAQUS
24.039
202.070
271.430
562.430
806.550
0.2
Present
18.952
149.812
189.712
995.529
1318.058
ABAQUS
17.523
149.840
189.290
995.930
1319.700
π
0.1
Present
3.867
120.269
233.439
411.437
434.108
ABAQUS
3.856
120.260
233.370
411.390
434.060
0.2
Present
3.869
105.112
171.054
610.370
758.047
ABAQUS
0.000
105.100
170.900
610.820
758.160

4. Conclusion

An improved Fourier series method has been presented for vibration analysis of moderately thick annular and circular sector plates with classical and general elastic restraints along its edges. Regardless of the boundary conditions, the displacement function is invariantly expressed as an improved trigonometric series which converges uniformly at an accelerated rate. The efficiency, accuracy and reliability of the present method have been fully demonstrated by various numerical examples for moderately thick annular sector plates having different cutout ratios and sector angles.

Table 11. First five natural frequency parameters for Mindlin Annular sector plates subjected to CE1CE1 type elastic boundary conditions (a/b= 0.4)

Sector angle (ϕ)
Thickness to radius ratio (h/b)
Method
Natural frequency modes
1
2
3
4
5
π 3
0.1
Present
5161.242
5329.372
6024.050
7500.647
9771.648
ABAQUS
5182.600
5349.600
6044.000
7521.400
8046.600
0.2
Present
6628.586
6823.656
7744.270
9710.654
12411.918
ABAQUS
6665.600
6859.200
7778.100
9746.900
10565.000
π 2
0.1
Present
5167.620
5242.262
5540.684
6140.785
7122.635
ABAQUS
5188.700
5261.800
5560.000
6158.100
7141.200
0.2
Present
6633.347
6724.016
7105.490
7936.324
9269.663
ABAQUS
6673.500
6762.800
7142.300
7970.700
9306.600
2 π 3
0.1
Present
5171.158
5212.374
5378.060
5694.631
6205.673
ABAQUS
5191.900
5231.300
5397.000
5711.500
6222.000
0..2
Present
6636.080
6687.398
6895.770
7327.756
8047.005
ABAQUS
6676.100
6726.100
6933.100
7362.200
8080.800
π
0.1
Present
5175.030
5192.529
5265.762
5397.220
5599.831
ABAQUS
5195.000
5210.600
5284.500
5414.600
5616.400
0.2
Present
6639.126
6661.568
6752.481
6926.255
7207.997
ABAQUS
6678.800
6699.900
6790.300
6961.700
7241.800

The effect of sector angle, thickness to radius ratio and restraining springs on the frequency parameters has been discussed. The present method is also employed to study the vibration analysis of moderately thick circular sector plates without modifying the solution procedure. Results for moderately thick annular sector plates under general elastic boundary conditions for various thicknesses to radius ratio and sector angle are presented which can serve as a bench mark for future computational methods. The accuracy of the results has been verified by comparing it with those available in literature and with ABAQUS. An excellent agreement is observed between the results obtained using the present method and with those available in literature. Keeping in view the accuracy and fast convergence behavior this method can easily be further extended to study the vibration analysis of various built up structures without modifying the solution algorithm and procedure.

Acknowledgements

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. U1430236) and Natural Science Foundation of Heilongjiang Province of China (No. E2016024).

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