Rayleigh-Ritz Technique to Analysis Natural Frequency of Plate with Quadratic Thermal Distribution

IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 08, 2015 | ISSN (online): 2321-0613

Rayleigh-Ritz Technique to Analysis Natural Frequency of Plate with Quadratic Thermal Distribution 1

Preeti Prashar1 Sofia2 Department of Mathematics Engineering 2Department of Applied Science 1,2 Pacific Univ., Raj., India

Abstract—This paper is to investigate the vibration problem of square plate with thickness and quadratic temperature distribution. Rapid advancements made in various materials science and manufacturing technologies that has lead to the application of composite materials for different areas of engineering like mechanical structures, automobiles structures, aerospace industries, power plant systems etc.

Rayleigh-Ritz technique is used to obtain solution of frequency equation with a two-term deflection function.

Results of frequencies are calculated for various value of taper constant and thermal gradient for Ist and 2nd the modes of vibration.

Key words: Thickness, Square Plate, Taper Parameter, Thermal Gradient, Vibration I. INTRODUCTION The technological development in mechanical, civil and aerospace engineering provides visco-elastic square plate structures gained popularity in the 20th century. On the one hand, vibrations in various cases are of great helpful, while on the other, there are many cases where vibrations show devastating effects of excessive vibration on engineering or automobiles structures. When a body is disturbed after applying external force from its initial position, then it tries to reach its initial position by the elastic property of the material of the body. In general, we may feel or see that every nearly things vibrates in nature.

Many researchers of engineering field have been motivated by the various engineering applications of vibration, such as the mechanical design of machines, automobiles structures and many mechanical or civil control systems.

The various application of plate’s vibration with varying thickness is widely used in modern technology. In making of complex structural design in aeronautical engineering, study of frequencies with non- homogeneity, temperature and variable varying thickness of plates has an important role. In the modern world, it is desirable to make such large machines with optimum vibration and smooth working.

Recently, Leissa [1, 2] discussed for variable varying thickness in rectangular plate. An orthotropic rectangular plate with non-homogeneity which is parabolically varying in both directions with thermal effect is studied by Gupta, Johri and Vats [3]. Gupta and Khanna [4] discussed free vibration on visco-elastic rectangular plate having linearly varying thickness in x - y directions. The transverse vibration of rectangular plates is studied by Singh and Saxena [5] having bi-directional thickness variation.

Sobotka [6, 7] has investigated the vibrational analysis of orthotropic rectangular visco-elastic plates. A. K. Gupta, Amit Kumar and Dhram Veer Gupta [8] found two mode of vibration for orthotropic parallelogram plate with parabolically varying thickness. Thermal deflection effect is discussed by Warade and Deshmukh [9] in the thin clamped

circular plate where heat is partially distributive. Gupta and Kumar [10] analyzed vibration analysis with nonhomogeneity in visco-elastic rectangular plates with linearly thickness. The consequence of varying poisson ratio is analyzed by Khanna A., Kaur N. and Sharma A. K. [11] on non-homogeneous rectangular plate having temperature variation. Sharma S. K. and Sharma A. K. [12] has discussed the mechanical vibration with 2 dimensional thickness variation and temperature effect of rectangular plate. Square plate with variable varying thickness, temperature effect is solved by Khanna A. and Sharma A. K. [13]. Sinusoidal thickness variation has discussed by Kumar Sharma A., and Sharma, S. K. [14] on visco-elastic plate with two dimensional thermal effects. Computational investigation on vibration is given by Khanna A., Kumar A., and Bhatia M. [15] on 2-dimensional thermal variation with variable thickness in visco-elastic square plate. Khanna A. and Sharma A. K. [16] calculated regular vibration of viscoelastic plate with temperature effect and variable varying thickness. Kumar Sharma A. and Sharma S. K. [17] discussed free vibrational analysis of orthotropic viscoelastic rectangular plate with parabolic thermal effect and linear thickness variation in x and y directions. Bi-parabolic thermal and thickness variation is studied by Sharma S. K. and Sharma A. K. [18] on vibration of orthotropic viscoelastic rectangular plate. Poisson ratio variation with thermal effect of non-homogeneous rectangular plate is solved by Khanna A., Kaur N., and Sharma A. K. [19]. Analysis of free vibration using Rayleigh-Ritz method on 2D varying thickness and thermal effect of orthotopic rectangular plate is studied by Sharma Subodh Kumar and Sharma Ashish Kumar [20]. Also Sharma Subodh Kumar and Sharma Ashish Kumar [21] solved vibration problems for viscoelastic parallelogram plate.

The main aim of the text is to provide some authentic mathematical models on natural frequency of isotropic square plates of variable thickness along with thermal effect which are extensively used in modern technology to help mechanical engineers, technocrats, researchers, practitioners etc. All the results are shown in graphical forms for mode of vibration.

II. FORMULATION AND ITS SOLUTION The equation of motion square plate in Cartesian coordinates, as [2]  4W D1   3 W D1   3 W 4W 4W  3W  3W  D1  2 2 2    2  2   4 x y y 4  x  x 3 xy 2  y  y3 x 2 y   x 

 2 D1   2 W  2 D1  2 W  2 W   2 D1   2 W 2W  ν ν  ρhp 2 W  0      2(1  ν) x 2  x 2 y 2  y 2  y 2 x 2  xy xy

(1) Here, we have considered linearly temperature distribution in x-direction i.e.    0 ( 1  x / a) (2) Where signifies the excess of temperature at any point on plate higher than the reference temperature and

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Rayleigh-Ritz Technique to Analysis Natural Frequency of Plate with Quadratic Thermal Distribution (IJSRD/Vol. 3/Issue 08/2015/198)

denotes the temperature at any point on the boundary of plate and “a” is the length square plate. For engineering materials, the temperature dependence of the modulus of elasticity is given as (3) E  E0 1- γτ  E0 is the value of the young's modulus of plate at reference temp. On using (2) in (3) modulus of elasticity is E  E0 [1   (1  x / a)(1  y / a)] (4) where,    0 (0    1) , a parameter.

Consider, thickness of plate varies quadratic along x-axis or direction as (5) h  h0 (1  1 x / a  2 x2 / a 2 ) Here, the length of square plate is denoted by a , 1 and 2 are taper parameters in x and y directions of plate at h=h0 at x=y=0.

Deflection function of vibration for the plate can be written as (6) w( x, y, t )  W ( x, y )T (t ) Here, D1 is the flexural rigidity of plate i.e.

(7) D1  Eh3 / 12(1  v 2 ) Using the E & h from equation (4) & (5) in the expression of D1, one obtain

D1  [ E0[1   (1  x / a)(1  y / a)]h03 (1  1 x / a  2 x2 / a2 )3 ] /12(1  v2 )

(8) Here, Rayleigh-Ritz method is applied for solving the frequency equation. In which, maximum strain energy and kinetic energy must be equal as

 (V *  T * )  0

(9) It’s necessary for the plate that for arbitrary variations of W satisfying relevant geometrical boundary conditions.

Here, satisfying relevant geometrical boundary conditions and the plate is clamped at all the four edges, then the boundary conditions are (10) The two term deflection function for plate as (11) *( ) ( ) (

Here

)(

)+

and

* 〈( ) ( ) ( )( )〉 + Assuming, non-dimensional variables as ̅ ̅

(12) The expressions for kinetic energy T* and strain energy V* are [2] 1 1 (13) T *  (1 / 2)  p 2 h0 a 5 [(1  1 X   2 X 2 )W 2 ]dYdX

 0

0

and V *  Q

1

0

1

 [1   (1  X )(1  Y )](1   X   1

0

2

X 2 )3{(W , XX ) 2  (W ,YY ) 2

(14)

2vW , XX W ,YY 2(1  v)(W , XY ) }dYdX 2

Where, Q  E0 h03a3 / 24(1  v2 ) Using equations (10) & (11) in equation (6), one get (V **   2 p 2T ** )  0 (15) Where V    [1   (1  X )(1  Y )](1   X   X ) {(W , )  (W , )  2vW , W , 2(1  v)(W , ) }dYdX (16) **

1 1

0 0

2 3

1

2

2

XX

2

YY

and T **  

1

0

1

 [(1   X   0

1

2

X 2 )W 2 ]dYdX

(17)

Here, And  2  12 (1  v2 )a2 / E0h02 is a frequency parameter.

Equation (11) containing two unknown constants i.e. A1 & A2 and determined as follows (18) (V **   2 p2T ** ) / An  0 , n  1, 2 On simplification (18), we get

Qn1 A1  Qn2 A2  0 , n  1, 2

(19)

Here, include parametric constant and the frequency parameter. For having solution to be non-trivial, the determinant of the coefficient matrix should be zero. So we get equation of frequency as Q11 Q12  0 (20) Q21 Q 22 On solving equation (18), we obtained quadratic equation in λ2. The solution value of λ2 represent the frequency vibration of two modes i.e. λ1 (Ist Mode) & λ2 (IInd Mode) for clamped plate with different values of thermal gradient and taper constant.

III. RESULT AND DISCUSSION Frequency equation (18) is quadratic in λ2, which gives two roots. The frequency is finding for the I st and 2nd modes of vibration for plate with linearly varying thickness in both the directions and value of Poisson ratio ν has been taken 0.345.

Here, results are present in tabular as well as graphical form from1 to 4 for Ist and 2nd modes of vibration for plate.

In Figure 1: It is clearly seen that value of frequency of plate seems to be continuously decreases as value of thermal gradient increases (α =0.0 to 1.0) for different values of taper parameters i.e. β1=β2 =0.0, 0.4 and 0.6 for both the modes of vibrations.

Actually, Frequency and Young’s modulus (E) are directly proportional to each other (Frequency ∝√ E). So, that on increasing the value of Young’s modulus (E), frequency will also increase. Using the value of E from equation (8), above relation becomes: Frequency ∝√ E0 {1(1-x2/a2) (1-y2/a2)} which simply shows that if thermal gradient increases, frequency decreases. So, the results in numerical form for frequency of plate in Table 1 are authentic and acceptable.

In Figure 2: Also, it is evident that value of frequency continuously of plate seems to be continuously increases as value of taper parameter increases from 0.0 to 1.0 for different values thermal gradient and β2=0.2 i.e. α =0.3, 0.6 and 0.9 for both the modes of vibrations.

From equation (19), it’s clearly seen that Kinetic energy of plate material directly depend on taper parameter (β1) which shows that as β1 increases, Kinetic energy will also increase.

Due to this increment in Kinetic energy, molecules of materials get extra energy so that their velocity increases.

From this, their rate of oscillation or vibration i.e. frequency increases. So, one can conclude that as taper parameter increase then frequency also increases. So, the numeric results of Table 2 for plate are also authentic and acceptable.

2

XX

YY

XY

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Rayleigh-Ritz Technique to Analysis Natural Frequency of Plate with Quadratic Thermal Distribution (IJSRD/Vol. 3/Issue 08/2015/198) 

Mode1 For  180

150

Mode2 For 

120

Mode3 For  

Frequency  90

60

30

0 0.0

0.2

0.4

0.6

0.8

1.0



Fig. 1: Frequency vs. Thermal gradient

Fig. 2: Frequency vs. Taper constant IV. CONCLUSION Authors have for accuracy compare the results of present paper with [11] the values of frequency in Table 1 and Table 2. On comparing the present results with [11], authors concluded that values of frequencies in present paper decrease for Ist and 2nd modes of vibrations for the corresponding values of taper parameter i.e. β1=β2 =0.0, 0.4 and 0.6as thermal gradient increases. In Table 1 and 2, it is also interesting to note that difference in Ist mode is quite more as compared to 2nd mode. Hence, our main aim for our work is to develop a theoretical model for engineers and scientists so that they use of it with a practical approach, for the welfare of the human beings and advancement of technology.

REFERENCES [1] Leissa A. W.: Vibration of Plate. NASA SP-160 (1969).

[2] Leissa A. W.: Recent studies in plate vibration 19811985 part II, complicating effects. Shock and Vibration Digest, Vol. 19, p. 10-24 (1987).

[3] Gupta A. K., Johri T., Vats R. P.: Thermal effect on vibration of non-homogeneous orthotropic rectangular plate having bi-directional parabolically varying thickness. Proceeding of International Conference in World Congress on Engineering and Computer Science, San-Francisco, USA, p. 784-787 (2007).

[4] Gupta A. K., Khanna A.: Vibration of visco-elastic rectangular plate with linearly thickness variations in both directions. Journal of Sound and Vibration, Vol. 301, p. 450-457 (2007).

[5] Singh B., Saxena V.: Transverse vibration of rectangular plate with bi-directional thickness variation.

Journal of Sound and Vibration, Vol. 198, p. 51-65 (1996).

[6] Sobotka Z. Vibration of rectangular orthotropic viscoelastic plate. Acta Techanica CSAV, p. 511-514 (1969).

[7] Sobotka Z.: Free vibration of visco-elastic orthotropic rectangular plates. Acta Techanica CSAV, p. 678-705 (1978).

[8] Gupta A. K., Kumar Amit, Gupta Dhram Veer.: Vibration of visco-elastic orthotropic parallelogram plate with parabolically thickness variation.

International Journal of Engineering, Vol. 8, Issue 2, p.

61-70 (2012).

[9] Warade R. W., Deshmukh K. C.: Thermal deflection of a thin clamped circular plate due to a partially distributive heat supply. Ganita, Vol. 55, p. 179-186 (2004).

[10] Gupta A. K., Kumar L.: Thermal effect on vibration of non-homogenous visco-elastic rectangular plate of linear varying thickness. Meccanica, Vol. 43, p. 47-54 (2008).

[11] Khanna, A., & Sharma, A. K.: Effect of Thermal Gradient on Vibration of Visco-Elastic Plate with Thickness Variation.

American Journal of Computational and Applied Mathematics, 2(1), 34-36 (2012).

[12] Sharma S. K., Sharma A. K. Mechanical vibration of orthotropic rectangular plate with 2D linearly varying thickness and thermal effect. International Journal of Research in Advent Technology, Vol. 2, Issue 6, p. 184190 (2014).

[13] Khanna A., Sharma A. K.: Vibration analysis of viscoelastic square plate of variable thickness with thermal gradient. International Journal of Engineering and Applied Sciences, Vol. 3, Issue 4, pp. 1-6.

[14] Kumar Sharma A., Sharma S. K.: Vibration computational of visco-elastic plate with sinusoidal thickness variation and linearly thermal effect in 2D. Journal of Advanced Research in Applied Mechanics and Computational Fluid Dynamics, Vol. 1, Issue 1, p.

46-54 (2014).

[15] Khanna A., Kumar A., Bhatia M.: A computational prediction on two dimensional thermal effects on vibration of visco-elastic square plate of variable thickness. Proceeding of CONIAPS XIII, Deharadun, (2011).

[16] Khanna A., Sharma A. K.: Natural vibration of viscoelastic plate of varying thickness with thermal effect.

Journal of Applied Science and Engineering, Vol. 16, Issue 2, pp. 135-140 (2013).

[17] Kumar Sharma A., Sharma S. K.: Free vibration analysis of visco-elastic orthotropic rectangular plate with bi-parabolic thermal effect and bi-linear thickness variation. Journal of Advanced Research in Applied Mechanics and Computational Fluid Dynamics, Vol. 1, Issue 1, p. 10-23 (2014).

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Rayleigh-Ritz Technique to Analysis Natural Frequency of Plate with Quadratic Thermal Distribution (IJSRD/Vol. 3/Issue 08/2015/198)

[18] Sharma S. K., Sharma A. K.: Effect of bi-parabolic thermal and thickness variation on vibration of viscoelastic orthotropic rectangular plate. Journal of Advanced Research in Manufacturing, Material Science and Metallurgical Engineering, Vol. 1, Issue 2, p. 26-38 (2014).

[19] Khanna A., Kaur N., Sharma A. K.: Effect of varying Poisson ratio on thermally induced vibrations of nonhomogeneous rectangular plate. Indian Journal of Science and Technology, Vol. 5, Issue 9, p. 3263-3267 (2012).

[20] Sharma S. K., Sharma A. K.: Rayleigh‑Ritz method for analyzing free vibration of orthotropic rectangular plate with 2D thickness and temperature variation. Journal of Vibroengineering, Vol. 17, Issue 4, p. 1989‑2000 (2015).

[21] Sharma S. K., Sharma A. K.: Mathematically study on vibration of visco‑elastic parallelogram plate.

Mathematical Models in Engineering, Vol. 1, Issue 1, p.

12‑19 (2015).

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