Published: 25 November 2023

Vibrational frequency of triangular plate having circular thickness

Neeraj Lather1
Ankit Kumar2
Parvesh Yadav3
Reeta Bhardwaj4
Amit Sharma5
1, 3, 4, 5Department of Mathematics, Amity University Haryana, Gurugram, India
2Chitkara University School of Engineering and Technology, Chitkara University, Himachal Pardesh, India
Corresponding Author:
Amit Sharma
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Abstract

In the current research, modes of frequency of isotropic tapered triangular plate having 1-D (one dimensional) circular thickness and 1-D (one dimensional) linear temperature profile for clamped boundary conditions are discussed. Authors implemented Rayleigh Ritz technique to solve the frequency equation of isotropic triangular plate and computed the first four modes with a distinct combination of plate parameters. Authors have performed the convergence study of modes of frequency of the isotropic triangular plate. Also, conducted comparative analysis of modes of frequency of the current study with available published papers and the results presented in tabular form. The aim of the present study is to show the impact of a one dimensional circular thickness and one dimensional linear temperature on modes of frequency of vibration of an isotropic tapered triangular plate.

1. Introduction

Now a days, study of vibration of nonuniform plates is very essential because vibration plays significantly role in many engineering applications i.e., nuclear reactor, aeronautical field, submarine etc. Study of vibration of triangular plates with variable thickness and temperature has been carried out by many researchers/scientists and has been reported in literature. but till date to the best of the knowledge of the authors, vibration of triangular plate with one dimensional circular thickness has not been considered yet.

Free vibration of cantilevered and completely free isosceles triangular plates based on exact three-dimensional elasticity theory has been investigated in [1] and derived the eigen frequency equation by using Rayleigh Ritz method. Chebyshevs Ritz method is applied in [2] to the free in plane vibration of arbitrary shaped laminated triangular plates with elastic boundary conditions. Time period analysis of isotropic and orthotropic visco skew plate having circular variation in thickness and density at different edge conditions is discussed in [3] and [4]. Two dimensional temperature effect on the vibration is computed in [5] for the first time for a clamped triangular plate with two dimensional thickness by using the Rayleigh Ritz method. A unified formulation was proposed in [6] for the free in-plane vibration of arbitrarily shaped straight-sided quadrilateral and triangular plates with arbitrary boundary conditions by improved Fourier series method (IFSM). Fourier series method is used in [7] for free vibration of arbitrary shaped laminated triangular thin plates. A computationally efficient and accurate numerical model is presented in [8] for the study of free vibration behavior of anisotropic triangular plates with edges elastically restrained against rotation and translation. Free vibration of thick equilateral triangular plates with classical boundary conditions has been investigated in [9] based on a new shear deformation theory. Free vibration of circular and annular three-dimensional graphene foam (3D-GrF) plates under various boundary conditions is discussed in [10].

From the above literature, it is evident that till date to the best of the knowledge of the authors, none of the researchers have worked on triangular plate with one dimensional circular thickness and one dimensional linear temperature environment for clamped boundary conditions. Therefore, in this present study we aim to study the above mentioned problem and investigate the impact on frequency modes of the plate The main purpose of the present study to provide a mathematical model for analyzing the effect of 1-D circular variation in thickness on frequency modes of triangular plate under 1-D linear temperature variation, which had not been investigated earlier. All the numerical results in the form of modes of frequency are presented in tabular form.

2. Problem geometry and analysis

Consider a viscoelastic triangle plate having aspect ratio θ=b/c and μ=c/a and one dimensional thickness l as shown in Fig. 1. Now transform the given triangle into right-angled triangle using the transformation x=aζ+bψ and y=cψ as shown in Fig. 2.

Fig. 1Triangle plate

Triangle plate

Fig. 2Transformed triangle plate

Transformed triangle plate

The kinetic energy and strain energy for vibration of a triangle plate are taken as in [11]:

1
Ts=12ρω20101-ζΦ2acdψdζ,
2
Vs=120101-ζD11a42Φζ22+b2a2c22Φζ2+1c22Φψ2-2bac22Φζψ2+2v1a22Φζ2
+b2a2c22Φζ2+1c22Φψ2-2bac22Φζψ+2(1-ν)-ba2c2Φζ2+1ac22Φζψ2ac dψdζ,

where Φ is the deflection function and D1=E3/121-ν2 is flexural rigidity.

The Rayleigh Ritz method requires:

3
L=δVs-Ts=0.

Using Eqs. (1) and (2), we have:

4
L=120101-ζD11a42Φζ22+b2a2c22Φζ2+1c22Φψ2-2bac22Φζψ2+2v1a22Φζ2
+b2a2c22Φζ2+1c22Φψ2-2bac22Φζψ+2(1-ν)-ba2c2Φζ2+1ac2Φζψ2ac dψdζ
-12ρω20101-ζlΦ2acdψdζ.

Introducing one dimensional circular thickness as:

5
l=l01+β1-1-ζ2,

where l0 are the thickness at origin. Also β is tapering parameter.

One dimensional temperature on the plate is assumed to be linear as:

6
τ=τ01-ζ,

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

The modulus of elasticity is given by:

7
E=E01-γτ,

where E0 is the Young’s modulus at τ=0, and γ is called the slope of variation. Using Eq. (6) and Eq. (7) becomes:

8
E=E01-α1-ζ,

where α=γτ0, (0α<1) is called thermal gradient. Using Eqs. (5) and (8), the functional in Eq. (4) becomes:

9
L=0101-ζ(1-α(1-ζ))1+β1-1-ζ231+θ222Φζ22
+2Φψ2222θ2+1-νμ22Φζψ2+2ν+θ2μ22Φζ22Φψ2
-4θ1+θ2μ2Φζ22Φζψ-4θμ32Φψ22Φζψac dψdζ
-12ρω20101-ζ1+β1-1-ζ2Φ2acdψdζ.

where D0=E0l03/121-v2 and λ2=ρω2l0a2/D0.

The deflection function is taken as:

10
Φζ,ψ=(ζ)e(ψ)f(1-ζ-ψ)gi=1nΨi{(ζ)(ψ)(1-ζ-ψ)}i,

where Ψi, i=0,1,2n are unknowns and the value of e, f, g can be 0, 1 and 2 corresponding to a given edge condition.

To minimize Eq. (9), we have:

11
LΨi=0, i=0,1,n.

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

12
P-λ2Q=0,

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

3. Numerical results and discussion

In the current study, authors evaluated numerical data in the form of modes of frequency (first four modes) for right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate on clamped edge condition for the different value of plate parameters. Throughout the calculation the value of aspect ratio a/b= 1.5, Poisson’s ratio ν= 0.345, E0= 2.80·103 N/M2 and ρ= 2.80·103kg/M3 is taken into consideration. All the results are presented in tabular form (refer Tables 1-3). Table 1 presents the modes of frequency λ for right angled isosceles triangular plate corresponding to tapering parameter β for fixed value of θ= 0, μ= 1.0 and the variable value of thermal gradient α i.e., α= 0.2, 0.6. From the Table 1, it can be seen that the modes of frequency λ decreases with the increasing value of tapering parameter β for all the above mentioned value of thermal gradient α. It is also observed that the value modes of frequency λ decreases with the increasing value of thermal gradient α, while the rate of decrement in modes of frequency λ increases with the increasing value of thermal gradient α.

Table 2 incorporates the modes of frequency λ for right angled scalene triangular plate corresponding to tapering parameter β for fixed value of θ= 0, μ= 1.5 and the variable value of thermal gradient α i.e., α= 0.2, 0.6. In table 2 also, modes of frequency λ decreases with the increasing value of tapering parameter β for all the above mentioned value of thermal gradient α as shown in Table 1. Like in Table 1, it is also observed in Table 2 that the value modes of frequency λ decreases with the increasing value of thermal gradient α, while the rate of decrement in modes of frequency λ increases with the increasing value of thermal gradient α.

Table 1Modes of frequency of right angle isosceles triangle plate corresponding to tapering parameter

θ=0, μ=1.0
α=0.2
α=0.6
β
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
100.130
366.937
831.845
2127.902
84.6067
309.174
702.733
1815.61
0.2
97.9199
360.068
816.224
2063.28
82.5137
302.794
687.724
1753.50
0.4
95.7800
353.380
799.694
2024.049
80.4804
296.555
672.161
1712.71
0.6
93.7147
346.888
785.481
1958.46
78.5112
290.473
658.629
1649.75
0.8
91.7283
340.614
770.259
1916.75
76.6108
284.567
644.601
1606.46
1.0
89.8245
334.544
755.453
1883.44
74.7835
278.878
630.245
1571.98

Table 2Modes of frequency of right angle scalene triangle plate corresponding to tapering parameter

θ=0, μ=1.5
α=0.2
α=0.6
β
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
90.3135
330.939
750.081
1917.22
76.1245
278.106
631.328
1626.59
0.2
88.3716
324.966
736.576
1861.97
74.2734
272.520
618.272
1573.02
0.4
86.4950
319.165
722.302
1829.51
72.4780
267.054
604.933
1538.14
0.6
84.6875
313.532
710.242
1773.00
70.7424
261.745
593.284
1483.65
0.8
82.9526
308.0911
697.482
1737.62
69.0705
256.601
581.116
1447.01
1.0
81.2937
302.856
684.760
1710.59
67.4662
251.626
569.113
1417.35

Table 3Modes of frequency of scalene triangle plate corresponding to tapering parameter

θ=1/3, μ=3/2
α=0.2
α=0.6
β
λ1
λ2
λ3
λ4
λ1
λ2
λ3
λ4
0.0
78.6772
289.064
654.766
1661.231
64.3411
237.763
538.269
1352.55
0.2
77.5677
285.1960
645.948
1632.17
63.2087
233.907
529.352
1321.08
0.4
76.5167
281.485
636.874
1618.99
62.1304
230.217
519.905
1303.43
0.6
75.5254
277.949
629.700
1584.65
61.1079
226.664
512.370
1268.33
0.8
74.5946
274.599
621.852
1569.17
60.1430
223.287
504.310
1248.69
1.0
73.7247
271.443
614.434
1554.99
59.2370
220.074
496.777
1230.15

Table 3 provides the modes of frequency λ for scalene triangular plate corresponding to tapering parameter β for fixed value of θ=1/3, μ=3/2 and the variable value of thermal gradient α i.e., α= 0.2, 0.6. In table 3 also, modes of frequency λ decreases with the increasing value of tapering parameter β for all the above mentioned value of thermal gradient α as shown in Tables 1, 2. Like in Tables 1, 2, it is also reported in Table 3 that the value modes of frequency λ decreases with the increasing value of thermal gradient α, while the rate of decrement in modes of frequency λ increases with the increasing value of thermal gradient α.

4. Convergence study

In this section, authors shows the convergence study done on modes of frequency λ (first two modes) of right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate at clamped edge condition for the plate parameters specified as α=β=0.0, ν= 0.345 and a/b= 1.5. The results are displayed in tabular form (refer Table 4). From the Table 4, one can concluded that modes of frequency for the above mentioned triangular plates converges up to three decimal place in fifth approximation.

Table 4Modes of frequency of scalene triangle plate corresponding to tapering parameter

N
θ=0.0, μ=1.0
θ=0.0, μ=1.5
θ=1/3, μ=3/2
λ1
λ2
λ1
λ2
λ1
λ2
2
107.077
436.191
107.077
436.191
92.7314
377.753
3
107.046
394.917
96.6275
356.480
92.7048
342.008
4
107.045
392.630
96.6272
354.406
92.7045
340.018
5
107.045
392.630
96.6272
354.406
92.7045
340.018

Table 5Comparison of modes of frequency with [12] for right angled isosceles, right angled scalene and scalene triangular plate corresponding to tapering parameter

α=0.0
β
θ=0.0, μ=1.0
θ=0.0, μ=1.5
θ=1/3, μ=3/2
λ1
λ2
λ1
λ2
λ1
λ2
0.0
107.077
436.192
96.665
393.737
84.965
346.119
107.077
436.192
79.2171
322.7010
70.8248
288.5140
0.2
104.798
426.334
94.658
385.287
83.849
341.319
98.813
401.130
74.1212
301.2510
66.7330
271.5330
0.4
102.595
416.894
92.731
377.213
82.792
336.839
91.069
368.577
69.4564
281.9920
63.0304
256.6090
0.6
100.472
407.884
90.877
369.526
81.798
332.682
83.983
339.162
65.3054
265.3940
59.7782
244.1390
0.8
98.432
284.632
89.099
362.233
80.865
328.853
77.713
313.628
61.7481
251.9720
57.0324
234.5460
Bold values are obtained from [12]

5. Results comparison

In this section, authors performed a comparative analysis of modes of frequency λ (first two modes) obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) and modes of frequency λ obtained in [12] at clamped edge condition and presented in tabular form (refer Table 5). In [12], authors assumed the thickness variations in both the direction but in the present study authors taken the thickness in one direction so authors compared the modes of frequency λ of present study with modes of frequency λ obtained in [12] when the value of second tapering parameter β2 is 0.0 in [12]. Table 5 shows the comparison of modes of frequency λ obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) and modes of frequency λ obtained in [12] at clamped edge condition corresponding to tapering parameter β for fixed value of thermal gradient α i.e., α=0.0. From the Table 5, authors conclude that:

1) Modes of frequency λ obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) are higher in comparison to modes of frequency λ obtained in [12].

2) The rate of change in (decrement) in modes of frequency λ obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) are smaller in comparison to modes of frequency λ obtained in [12], at clamped edge condition for all the three above mentioned values of thermal gradient α.

6. Conclusions

The effect of circular thickness on modes of frequency λ of right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate under temperature environment at clamped edge condition is computed. Based on numerical discussions and results comparisons, authors would like to records the following facts:

1) The modes of frequency obtained in present study in case of circular thickness is higher than the modes of frequency obtained in [12] in case of linear thickness. The modes of frequency obtained in present study and modes of frequency obtained in [12] exactly match at β=0.0 (refer Table 5).

2) The variation in modes of frequency obtained in present study in case of circular thickness is less in comparison to modes of frequency obtained in [12] in case of linear variation in thickness (refer Table 5).

3) The modes of frequency obtained for the present study decreases (less rate of decrements) with the increasing value of tapering parameter and thermal gradient. (refer Tables 1-3).

4) As temperature increases on the plate, the modes of frequency decreases but the rate of change (decrement) in modes of frequency increases (refer Tables 1-3).

References

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About this article

Received
28 April 2023
Accepted
25 May 2023
Published
25 November 2023
SUBJECTS
Modal analysis and applications
Keywords
Isotropic triangular plate
temperature
circular thickness
frequency
vibration
Acknowledgements

The authors have not disclosed any funding.

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest

The authors declare that they have no conflict of interest.