Vibration of rectangle plate with variable density and Poisson’s ratio
Amit Sharma1 , Vijay Kumar2 , Reeta Bhardwaj3
1, 2, 3Department of Mathematics, Amity University Haryana, Gurgaon, India
Vibroengineering PROCEDIA, Vol. 22, 2019, p. 68-73.
Received 17 January 2019; accepted 28 January 2019; published 15 March 2019
The effect of variation in density and Poisson’s ratio is examined, on the vibration of rectangle plate with linear variable thickness. Bi parabolic temperature effect on the plate is also studied and vibrational modes of rectangular plate is calculated by using Rayleigh Ritz method. Authors also compared the present finding with the published results.
Keywords: vibration, density, Poisson’s ratio, variation.
Isotropic rectangle plate is used in various engineering like civil, mechanical, automobile, aerospace and aircraft engineering. Due to it’s wide applications, it is required to know the vibration characteristics of isotropic rectangular plates. The plates made of non-homogeneous material along with variable thickness has significant impact rather than the homogeneous and uniform plates.
An excellent study related to plate vibration, on different edge conditions is given in [1-7]. A finite layer method  is used to developed three dimensional linear and small deformation solution for the vibration of rectangular plate on different boundary conditions. The mean square bending moment of a rectangular plate has been discussed in . Rayleigh Ritz technique is used to calculate vibrational frequency of non-homogeneous trapezoidal plate  on clamped and simply supported edge condition. The effect of rotary inertia  on vibration of rectangular plate with central cutout is studied, using first order shear deformation theory. Forced vibration of orthotropic circular plate with linear variation in thickness, resting on elastic foundation is discussed in . The effect of axial tension, viscosity coefficients and ratio of length-to-depth is studied on transverse vibration of viscoelastic Timoshenko beam columns . Vibration of square and skew plate [14, 15] is discussed with variation in thickness, density and temperature using Classical plate theory. The effect of circular variation in thickness and Poisson’s ratio on frequency of square plate is presented in . Vibration of circular plate with parabolic thickness, exponential variation in density and Young’s modulus is discussed in  using Ritz method. The effect of various plate parameters [18, 19] on vibration of skew and rectangular plate is presented, on clamped edge, using Rayleigh Ritz method. The temperature effect on vibration of rectangular plate made of non-homogeneous material, with variable thickness, is discussed in [20, 21] using Classical plate theory.
The present paper provides the effect of variation in density as well in Poisson’s ratio on frequency parameter of the plate. Authors also computed the effect of thickness and temperature on frequency. A comparison of frequency with  and , with respect to thermal gradient is also given, to validate the finding of the present study.
Consider a thin rectangular plate with length , breadth , thickness , density and Poisson’s ratio referred to cartesian coordinates (, ).
The maximum kinetic energy of the plate is given by:
where is deflection function and is natural frequency.
The maximum strain energy of the plate is given by:
where , here is Young’s modulus of the plate.
Rayleigh Ritz method requires:
Using Eq. (1) and Eq. (2), we get:
Now introducing non-dimensional variable , together with linear variation in thickness, circular variation in density and exponential variation in Poisson’s ratio:
where , and are the thickness, density and Poisson’s ratio, respectively at origin. Also , and are taper constant and non-homogeneity parameters.
The temperature variation is parabolic on the plate and is given by:
where and are the temperature on the plate at any point and at the origin respectively. The temperature dependent modulus of elasticity is:
where is called slope of variation. Using Eq. (6), Eq. (7) becomes:
where is called temperature gradient. Using non-dimensional variable along with Eq. (5) and Eq. (8), the functional in Eq. (4) become:
Now assuming the deflection function as:
where and represents arbitrary constants and , , . take values 0, 1 and 2 depending upon the support edge condition i.e., take value 0 for free edge, 1 value for simply supported and 2 value for clamped edge.
In order to minimize the functional given in Eq. (9), we require that:
After simplifying we get a homogenous system of equations in whose non-zero solution gives equation of frequency as:
where and are square matrix of order ( + 1) and 0, 1, 2, ..., and 0, 1, 2, ...,
3. Numerical illustration and discussions
The frequency Eq. (12) is solved for first two modes of vibration on clamped edges for various values of temperature gradient , taper constant and non-homogeneity parameters , and presented with the help of figures. The value of aspect ratio is taken as / 1.5.
Fig. 1 shows the effect of non-homogeneity on vibrational frequency for three set of taper constants 0.2, 0.4, 0.6, thermal gradient 0.2, 0.4, 0.6 and non-homogeneity parameter 0.2, 0.4, 0.6. It has been observed that on all set values, frequency parameter increases with the increasing value of non-homogeneity parameter .
Fig. 2 shows the effect of another non-homogeneity parameter on vibration frequency of the plate on three set value of taper constant 0.2, 0.4, 0.6, thermal gradient 0.2, 0.4, 0.6 and non-homogeneity parameter 0.2, 0.4, 0.6. Here, increase in non-homogeneity results the decrease (opposite to the frequency presented in Fig. 1) in vibration frequecy, on all the set values.
Fig. 3 shows the effect of taper constant on vibration frequency of the plate on three set value of thermal gradient 0.2, 0.4, 0.6 and non-homogeneity parameters 0.2, 0.4, 0.6. It is seen that increase in taper constant results the increase in vibrational frequency, on all the three set values. Although the combined increase in mentioned plate parameters results the decrease in vibration frequency.
Fig. 4 depicts the behavior of frequency with respect to increasing value of thermal gradient on three set value of taper constant 0.2, 0.4, 0.6 and non-homogeneity parameters 0.2, 0.4, 0.6. Here, increase in thermal gradient results the decrease in vibrational frequency on all the three set values. Although the combined increase in mentioned plate parameters results the increase in frequency parameter.
Fig. 1. Non-homogeneity vs frequency
Fig. 2. Non-homogeneity vs frequency
Fig. 3. Taper constant vs frequency
Fig. 4. Thermal gradient vs frequency
4. Comparison of results
A comparison of frequency parameters of present paper has been made with  and  in Fig. 5.
Fig. 5 shows the comparison of frequency modes of present study with  and  with respect to temperature gradient for fixed value of non-homogeneity constant i.e., 0.0, for two cases 0.0 and 0.0, 0.6. The frequecy modes of present study is less when compared to frequency modes presented in  and . At 0.0 the frequency modes of present study,  and  are coincides. At 0.0, 0.6 the frequency modes of present study is concides with frequency modes reported in  and less in comparison to the frequency modes reported in .
Fig. 5. Comparison of frequency modes with  and  corresponding to thermal gradient for fixed value of 0.0
From the results discussion and comparison, authors would like to record the followings conclusions:
1) The frequency in case of parabolic temperature variation (present study) is very less in comparison to exponential variation in temperature  and linear variation in temperature  as shown in Fig. 5.
2) Increase in non-homogeneity results the increase in frequency parameter while the increase in non-homogeneity results the decrease in frequency parameter. The rate of increase in frequency is high in comparison to rate of decrease in frequency.
3) The frequency parameters increasing with the increment in tapering parameter of the plate i.e., as thickness of the plate increasing, frequencies are also increasing as shown in Fig. 3.
4) The frequency parameter decreases with the increment in thermal gradient i.e., as the temperature on the plate increasing, frequencies are decreasing as shown in Fig. 4.
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Journal of Vibration and ControlReeta Bhardwaj, Naveen Mani, Amit Sharma