Behavior of frequencies of orthotropic rectangular plate with circular variations in thickness and density

Neeraj Lather1 , Amit Sharma2

1, 2Department of Mathematics, Amity School of Applied Sciences, Gurugram, India

2Corresponding author

Vibroengineering PROCEDIA, Vol. 41, 2022, p. 71-76. https://doi.org/10.21595/vp.2022.22558
Received 1 April 2022; received in revised form 13 April 2022; accepted 19 April 2022; published 21 April 2022

55th International Conference on Vibroengineering, Almaty, Kazakhstan, April 21, 2022

Copyright © 2022 Neeraj Lather, 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.
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Abstract.

In this article, authors dealt with general solution of differential equation of orthotropic rectangular plate with clamped boundary conditions under bi-parabolic temperature variations. Rayleigh Ritz technique is adopted to solve the resultant equation and evaluate the frequencies for first four modes of vibration. The effect of circular thickness and density on frequencies of orthotropic plate are analyzed which is not done yet. The objective of the study is to optimize the frequency modes by choosing the appropriate variation in plate parameters. Th findings of the study complete the objective of the article. All the results are provided in tabular form.

Keywords: orthotropic rectangular plate, thermal gradient, circular tapering, circular density.

1. Introduction

Orthotropic rectangular plate with various plate parameters is widely used in various engineering branches like aerospace, aircraft, automobiles etc. A significant work has been reported by many researchers on orthotropic rectangular plate along with different variations in plate parameters but few work has been reported to study the effect of circular variation in plate parameters (thickness and density).

Effect of nonhomogeneity on vibration of orthotropic rectangular plates of varying thickness resting on Pasternak foundation is discussed in [1]. Free vibration analysis of laminated composite plate with elastic point and line supports by using finite element method was presented by [2]. Buckling analysis of rectangular isotropic plate having simply supported boundary conditions under the influence of non-uniform in-plane loading by using first-order shear deformation theory (FSDT) has been implemented in [3] Free vibration of orthotropic rectangular plate with thickness and temperature variation is studied in [4] and [5] by using Rayleigh Ritz method. Theoretical analysis on time period of vibration of rectangular plate with different plate parameters was proposed by [6] and [7].

New straight forward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method is described in [8]. Natural vibration of skew plate on different set of boundary conditions with temperature gradient is analyzed in [9]. Free vibration analysis of isotropic and laminated composite plate [10] on elastic point supports using finite element method (FEM) is studied. Vibration frequencies of a rectangular plate with linear variation in thickness and circular variation in Poisson’s ratio was provided in [11]. Thermally induced vibrations of shallow functionally graded material arches is considered and analyzed in [12] by using classical theory of curved beams. Fast converging semi analytical method was developed by [13] for assessing the vibration effect on thin orthotropic skew (or parallelogram/oblique) plates. A mathematical model is presented in [14] to analyse the vibration of a tapered isotropic rectangular plate under thermal condition.

In this study, authors examined the behavior of frequencies of orthotropic rectangular plate having circular variations in thickness and density. Authors also discussed the frequency variation when temperature increased on the plate. The present study provided how we can optimize the frequency value by choosing circular variation in plate parameters which will be helpful to structural engineering to get ride from unwanted vibrations.

2. Problem geometry and analysis

Consider a nonhomogeneous orthotropic rectangle plate with sides a and b, thickness l and density ρ as shown in Fig. (1).

Fig. 1. Orthotropic rectangular plate with 2D circular thickness

Orthotropic rectangular plate with 2D circular thickness

The kinetic energy and strain energy for vibration of plate are expressed in the following manner as in [15]:

(1)
Ts=12ω20a0bρlΦ2dψdζ,
(2)
Vs=120a0b[Dx2Φζ22+Dy2Φψ22+2νxDy2Φζ22Φψ2+4Dxy2Φζψ2dψdζ,

where Φ deflection function, ω natural frequency, Dx=Exl3/121-νxνy, Dy=Eyl3/121-νxνy are flexural rigidities in x and y directions respectively and  Dxy=Exl3/121-νxνy is torsional rigidity.

To solve frequency equation, Rayleigh Ritz technique is implemented which requires:

(3)
L=δ(Vs-Ts)=0.

Using Eqs. (1), (2), we get functional equation:

(4)
L=120a0bDx2Φζ22+Dy2Φψ22+2νxDy2Φζ22Φψ2+4Dxy2Φζψ2dψdζ-12ω20a0bρlΦ2dψdζ=0.

Now, introducing non-dimensional variable as:

ζ1=ζa,     ψ1=ψa,

together with two dimensional circular thickness, we get:

(5)
l=l01+β11-1-ζ121+β21-1-a2b2ψ12,

where l0 is thickness at origin and β1, β2 (0β1,β21) are tapering parameters.

For non-homogeneity consideration, authors assumed one dimensional circular variation in density:

(6)
ρ=ρ01+m01-1-ζ12.

Two dimensional parabolic temperature distribution is taken as in [5]:

(7)
τ=τ01-ζ121-a2b2ψ12,

where τ and τ0 denotes the temperature on and at the origin respectively.

For orthotropic materials, modulus of elasticity is given by [17]:

(8)
Ex=E11-γτ,     Ey=E21-γτ,     Gxy=G01-γτ,

where Ex and Ey are the Young’s modulus in x and y directions, Gxy is shear modulus and γ is called slope of variation. Using Eq. (7), Eq. (8) becomes:

(9)
Ex(x)=E11-α1-ζ121-a2b2ψ12,
Ey(x)=E21-α1-ζ121-a2b2ψ12,
Gxy(x)=G01-α1-ζ121-a2b2ψ12,

where α=γτ0 0α<1 is called thermal gradient. Using Eqs. (5), (6), (9) and non dimensional variable, the functional in Eq. (4) become:

(10)
L=D02010ba1-α1-ζ121-a2b2ψ121+β11-1-ζ1231+β2-1-a2b2ψ1232Φζ122+E2E12Φψ122+2νxE2E12Φζ122Φψ12+4G0E11-νxνy2Φζ1ψ12dψ1dζ1-λ2010ba1+m01-1-ζ121+β21-1-ζ121+β21- 1-a2b2ψ12Φ2dψ1dζ1=0,

where:

D0=12E1l03121-νxνy,     λ2=12a4ρω21-νxνyE1h02.

We choose deflection function which satisfies all the edge condition as taken in [16]:

(11)
Φζ,ψ=ζ1eψ1f1-ζ1g1-aψ1bhi=0nΨiζ1ψ11-ζ11-aψ1bi,

where Ψi, i=0,1,2...n are unknowns and the value of e, f, g, h can be 0, 1 and 2, corresponding to free, simply supported and clamped edge conditions respectively. To minimize Eq. (11), we impose the following condition:

(12)
LΨi=0,    i=0,1,...n.

Solving Eq. (11), we have the following frequency equation:

(13)
P-λ2Q=0,

where P=piji,j=0,1,..n and Q=qiji,j=0,1,..n are square matrix of order (n+1).

3. Numerical results and discussion

Vibrational frequency of orthotropic rectangular plate having circular variation for first four modes under thermal effect with circular variation in thickness and density is discussed. Frequency equation is solved by using Rayleigh-Ritz technique with the help of Maple software. Frequency values for first four modes is calculated on different variations of plate parameters (i.e., thermal gradient, tapering parameters and nonhomogeneity). The results have been presented in the form of tables. In through out calculation the aspect ratio i.e., ratio of length to breadth of the plate is considered to be 1.5.

Tables 1 and 2 represents the frequencies for first four modes corresponding to tapering parameters β1, β2 for three different values of thermal gradient, tapering paramter and nonhomogeneity m i.e., α=β2=m=0.2, α=β2=m =0.4, α=β2=m= 0.6. (In Table 1) i.e., α=β1=m= 0.2, α=β1=m=0.4, α=β1=m=0.6 (In Table 2).

Table 1. Tapering parameter β1 vs vibrational frequency λ

β1
α=β2=m=0.2
α=β2=m=0.4
α=β2=m=0.6
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
16.2250
62.5266
140.527
314.224
15.4920
59.2450
132.974
303.002
114.7460
55.8753
125.414
291.617
0.2
16.9440
64.3744
145.547
331.268
16.2045
61.5334
137.954
319.324
15.4520
58.1060
130.414
306.785
0.4
17.7186
67.3744
151.897
347.316
16.9703
63.9622
143.549
334.372
16.2086
60.4747
135.793
322.357
0.6
18.5423
70.0079
156.897
366.204
17.7826
66.5125
149.182
353.426
17.0089
62.9527
141.414
341.070
0.8
19.4086
72.7501
163.162
384.709
18.6352
69.1850
155.236
372.291
17.8470
65.5327
147.325
359.995
1.0
20.3118
75.5910
169.577
405.599
19.5224
71.9365
161.609
391.622
18.7174
68.1935
153.484
380.389

Table 2. Tapering parameter β2 vs vibrational frequency λ

β2
α=β1=m=0.2
α=β1=m=0.4
α=β1=m=0.6
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
16.6318
63.8209
143.168
325.058
16.3257
61.8337
138.677
320.118
16.0163
59.7280
134.016
317.501
0.2
16.9440
64.8730
145.547
331.223
16.6431
62.8843
141.117
326.682
16.3380
60.7803
136.374
325.163
0.4
17.2665
65.9491
147.895
339.639
16.9703
63.9622
143.549
334.372
16.6691
61.8582
138.930
332.343
0.6
17.5987
67.0584
150.517
345.112
17.3068
65.0711
146.010
342.215
17.0089
62.9517
141.463
340.544
0.8
17.9401
68.1874
153.129
352.618
17.6521
66.2017
148.587
349.894
17.3571
64.0756
144.056
348.595
1.0
18.2902
69.3457
155.639
361.626
18.0057
67.3523
151.159
359.151
17.7131
65.2175
146.720
357.118

From Tables 1 and 2, the following facts can be interpreted:

1) Frequency increases in both the Tables 1 and 2 with the increasing in value of tapering parameters β1 and β2.

2) for fixed values of both the tapering parameters β1 and β2, frequency decreases in both the Tables 1 and 2.

3) Rate of decrement of frequencies reported in Table 2 is much slow as compared to Table 1.

Table 3 summarizes the frequency for first four modes corresponding to thermal gradient α for three different value of tapering parameters β1,β2 and non homogeneity m i.e., β1=β2=m=0.2, β1=β2=m=0.4, β1=β2=m=0.6. Table 4 presents frequency for first four modes corresponding to nonhomogeneity m for three different values of thermal gradient α, tapering paramters β1, β2 i.e., β1=β2=α=0.2, β1=β2=α=0.4, β1=β2=α=0.6 respectively.

From Tables 3 and 4, it can be concluded that:

1) Increase in thermal gradient α and nonhomogeneity m results in decrease in frequency in both the tables 3 and 4.

2) Rate of decrement of frequency in nonhomogeneity is slower as compared to thermal gradient.

3) Here, behavior of both the Tables 3 and 4 is quite similar but in Table 4 have higher frequency values as compared to Table 3.

4) Without thermal effect, Table 3 have maximum frequency value then it decreases gradually with the increase in thermal gradient.

Table 3. Thermal gradient α vs vibrational frequency λ

α
β1=β2=m=0.2
β1=β2=m=0.4
β1=β2=m=0.6
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
17.7264
68.0981
152.773
344.414
18.5608
70.5842
158.161
359.868
19.4501
73.1977
163.831
377.364
0.2
16.9440
64.8730
145.547
331.268
17.7860
67.3676
151.009
347.406
18.6797
69.9838
156.690
365.602
0.4
16.1196
61.4568
138.000
317.463
16.9703
63.9628
143.544
334.381
17.8688
66.5801
149.261
353.332
0.6
15.2450
57.8227
129.915
303.204
16.1054
60.3435
135.542
321.048
17.0089
62.9563
141.409
340.651
0.8
14.3085
53.9052
121.350
288.101
15.1795
56.4381
127.131
306.930
16.0884
59.0494
133.108
327.429
1.0
13.2923
49.6287
112.057
272.273
14.1752
52.1774
118.015
292.304
15.0893
54.7852
124.157
313.803

Table 4. Nonhomogeneity m vs vibrational frequency λ

m
α=β1=β2=0.2
α=β1=β2=0.4
α=β1=β2=0.6
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
17.2013
65.9667
148.314
335.442
17.4856
66.1290
148.314
348.144
17.7821
66.1528
149.118
362.262
0.2
16.9440
64.8663
145.634
331.105
17.2222
65.0210
145.943
341.903
17.5128
65.0309
146.362
355.401
0.4
16.6979
63.8335
143.210
323.724
16.9703
63.9628
143.544
334.381
17.2554
63.9650
143.309
348.519
0.6
16.4622
62.8448
140.797
318.704
16.7292
62.9605
141.119
328.422
17.0089
62.9572
141.309
341.897
0.8
16.2363
61.8978
138.611
313.065
16.4980
62.0028
138.814
323.043
16.7728
61.9888
139.107
334.293
1.0
16.0193
60.9914
136.613
306.693
16.2762
61.0890
136.707
316.661
16.5461
61.0639
136.860
329.374

4. Conclusions

Present model provides frequencies of orthotropic rectangular plate with circular variations in thickness and density. from above results and discussion, authors concluded that increasing in tapering parameters β1 and β2, results the decrease in frequency as shown in tables 1 and 2. but increase in thermal gradient α and non homogeneity m, results the increase in frequency is illustrated in tables 3 and 4. the variation in frequency mode (increasing or decreasing) are very slow due to implementation of circular variation in plate parameters i.e., there is no sudden increment or decrement reported in frequencies.

References

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