Paper id 26201436

International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637

Mechanical Vibration of Orthotropic Rectangular Plate with 2D Linearly Varying Thickness and Thermal Effect Subodh Kumar Sharma1 and Ashish Kumar Sharma2 Associate Prof. and Head of Dept., Govt. P.G College, Ambala Cantt. Haryana, India1 Research Scholar Dept. of Mathematics, Pacific University, Udaipur, Raj. India2 Email: subodhamb@rediffmail.com 1, ashishk482@gmail.com 2

Abstract- A mathematical model is constructed with an aim to assist the design engineers for the making of various structure used in the satellite and aeronautical engineering. In this paper effect of Bi-Linear variation in temperature is premeditated on an orthotropic rectangular plate as:  x   y  T = T0  1  1 a   b  

and whose thickness varies linearly in 2D as  x  y h=h 0  1+β1   1+β 2  a  b 

Rayleigh Ritz approach is applied for the solution of the problem. Fundamental frequencies, Logarithmic decrement, Time period and Deflection functions are calculated for first two modes of vibration of an orthotropic rectangular plate with diverse values of temperature gradient and taper constants.

Index Terms- Frequency, Logarithmic decrement, Time period, Deflection, Taper constants 1. INTRODUCTION Orthotropic plates of various geometries are commonly used as structural elements in various fields of engineering such as civil, naval and mechanical. In particular, rectangular plates are widely used in ocean structures and aerospace industry. Plates with varying thickness possess a number of attractive features such as material saving, weight reduction, stiffness enhancing, high strength and also meet the desirability of economy.

During the past four decades, vibration of plates has become an important subject in engineering applications. So, a good comprehension of the free vibration behavior of plate structures is necessary to design mechanical systems. An abundant number of plates with different shapes, thickness variations, and boundary conditions have been the subject of numerous research works. Many studies have been devoted to transverse vibration of plates by Leissa (1969).

Usually, an analytical or computer model is needed to analyze the vibration in an engineering system. Models are also useful in the process of design and development of an engineering system for good performance with respect to vibrations.

Vibration monitoring, testing, and experimentation are important as well in the design, implementation, maintenance, and repair of engineering systems.

Vibration is the study of oscillatory motions of mechanical system. It is both useful and harmful for engineering systems.

Sometimes, it is profitably used in musical instruments, propagation of sound etc. On the other hand, excessive vibration causes damage to buildings and rapid wear of machine parts such as gear & bearing. At the stage of resonance, it may even lead to the shutdown of the turbine units which directly disturbed the economical system, time management, employment system of a nation. Resonance is the stage where the natural frequency of the system is equal to the external excitation frequency. Due to this, the amplitude of vibration is excessive at resonance.

So, determination of first few modes of natural frequency of a system is a must for design point of view. So, it is essentially required to have the knowledge of vibration (natural frequency) for a designer.

Time period, Deflection and Logarithmic decrement at different points for the first two modes of vibration are calculated for various values of thermal gradients, aspect ratio and taper constants.

Also results are illustrated with graphs. Here it is important to note that all the numerical calculations have been made using the material constants of ‘Duralium’ an alloy of Aluminium, Copper and traces of Magnesium and Manganese.

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International Journal of Research in Advent Technology, Vol.2, No.6, No. June 2014 E-ISSN: 2321-9637 is called flexural rigidities of the plate in y-direction, y

2. GEOMETRY OF PLATE Consider a visco-elastically elastically supported plate with side lengths a, b and thickness h subjected to a concentrated force as shown in Fig I. Translational stiffness and damping coefficients were assigned equally along supported edges. The elastic symmetry axes of the plate material coincide with the OX and OY axes. Therefore the plate is especially orthotropic.

D

xy

=

G

xy

h

3

12

is called is torsional rigidity, D 1 = ν x D y ( = ν y D x ) , D% is Rhelogical operator and Ex and Ey are the modules of elasticity in xx and ydirection, vx and vy are the poisson ratios and Gxy is the shear modulus.

Let the plate under consideration is subjected to a steady two dimensional temperature distribution T linearly along the length, i.e., x-- axis and y-axis as

T = T

0

 x   y  1−  1−  a   b  

(3)

where, T is the temperature excess above the reference temperature at a distance x/a and y/b . T0 is the temperature excess above the reference temperature at the end of the plate i.e. at x=a, y=b.

Temperature dependence of the modulus of elasticity for most of engineering materials can be expressed in this form

E x (T ) = E1 [1- α T ]

Figure I :- Visco-elastically elastically supported orthotropic

E y (T ) = E2 [1- α T ]

rectangular plate

Gxy (T ) = G0 [1- α T ]

2.1. Equation of Motion The governing differential equation of transverse motion of an orthotropic rectangular plate of variable thickness in Cartesian coordinate is [2],

∂ 4w ∂4w ∂2w ∂2w + D + 2 H y ∂x 4 ∂y 4 ∂x 2 ∂y 2 ∂H y ∂ 2 w ∂w ∂H x ∂w ∂ 2w +2 + 2 ∂x ∂x ∂y 2 ∂y ∂x 2 ∂y ∂D y ∂ 3w ∂D x ∂ 3w ∂ 2Dx ∂ 2w +2 + 2 + ∂x ∂x3 ∂y ∂y 3 ∂x 2 ∂x 2 Dx

+

∂2D y ∂2w ∂ 2 D1 ∂ 2 w ∂ 2 D1 ∂ 2 w + + 2 2 2 2 ∂y ∂y ∂x ∂y ∂y 2 ∂x 2

∂ 2 D xy ∂ 2 w ∂ 2w (1) +4 + ρh = 0 2 ∂x∂y ∂x∂y ∂t and && 2 DT=0 % T+p (2) Eqs. (1) and (2) are the differential equation of motion for orthotropic rectangular plate and time function for visco-elastic elastic plate of free vibration having variable two dimensional linearly varying thickness and here p2 is a constant.

H = D 1 + 2 D xy , E xh 3 Dx = 1 2 (1 -ν x ν y ) is called flexural rigidities of the plate in x-direction, x

D

y

=

(4)

E yh

3

1 2 (1 -ν x ν y )

Here E1 and E2 are values of the Young’s moduli respectively along the x and y axis at the reference temperature, i.e., at τ = 0 and γ is the slope of the variation of modulus of elasticity with τ. Thus modulus variation become   x  y   E x ( T ) = E 1  1 -α   1 − 1 −     a  b       x  y   E y ( T ) = E 2  1 -α   1 − 1 −     a  b       x  y    (5) G x y ( T ) = G 0  1 -α   1 − 1 −     a  b     where is thermal gradient parameter

α = γ T 0 (0 ≤ α < 1)

The expression for the strain energy V and Kinetic energy T are respectively expressed as, 2   ∂2W 2   ∂2W  D + D +   x y 2  2  ab 1   ∂x    ∂y  V = ∫∫  dydx (6) 2 2 2 2 200 2D ×  ∂ W  ∂ W  + 4D  ∂ W   xy    1  ∂x2  ∂y2      ∂x∂y   

and

T =

1 2 p ρ 2

a b

∫ ∫ hW

2

d y d x (7)

0 0

where ρ is the mass density.

Assuming that thickness h of the plate is assumed to be varying linearly in both directions, i.e.

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 Here limit of X is 0 to 1 and Y is 0 to b/a.

On substituting the values of T & V from equations (13) and (14) in equation (9), one obtains

x  y   h=h 0 1+β1  1+β 2  (8) a  b   where β1 and β2 are taper constants.

(V

1

− λ 2 p 2 T1 ) = 0

(16)

2.2. Solution and Frequency Equation

where

An approximate solution to the current problem is given by the application of Rayleigh – Ritz method. In order to apply their procedure, maximum Strain energy must be equal to maximum Kinetic energy.

Therefore it is desired that following equation must be satisfied: (9) δ V −T = 0

 ∂2W2 E* ∂2W2    2  + 2*  2  +      a      ∂x  E1  ∂y   b 1−α(1−X) 1−Y ×  * 2  1 a   b   E2 ∂ W∂2W    V1 =∫∫ × 2vx *  2  2 + dydx 3    E1  ∂x  ∂y   0 0 a  (1+β1X)31+β2Y    2   2  b     G0  ∂ W (17)   4 * (1−vv ) x y     ∂x∂y   E1 

(

)

for arbitrary variations of W satisfying relevant geometrical boundary conditions which are

W

= W

x

= 0

at

x = 0,a

W

= W

y

= 0

at

y = 0,b

(10)

and the corresponding two term deflection function is taken as [1]

 x  y  x  y   2    1- 1-      a  b  a  b    W(x,y)=    x  y  x  y      1- 1-     A1 +A 2  a  b  a  b    

A2

Where A1 and

(11)

are constants to be evaluated.

The non-dimensional variables are

X = x / a, Y = y / a, W = W / a, h = h / a (12)

E =E 2 /(1-v x v y ) E * =v x E *2 =v y E 1* By using equations (5), (8) and (12) in (6) and (7), we get   ∂2W 2 E*  ∂2W 2   (1− X )    2  + 2*  2       ∂x  E1  ∂y   b 1−α  1 a  1−Y a  ×   E*  ∂2W  ∂2W      V = Q∫∫ b   ×+2vx 2*  2  2  dydx      E1  ∂x  ∂y   0 0 3 2  (1+ β X )3 1+ β Y a    2 (13)   G   ∂ W     1 2 0  b   +4 * (1−vxvy )     E x y ∂ ∂     1 and b a 1  a 2  T = ρp2 h0a5 ∫∫ (1+ β1X ) 1+ β2Y W  dydx (14) 2 b   0 0 1

(

 a  2  T = ∫ ∫  (1 + β1 X )  1 + β 2Y  W  dydx b   0 0  where

λ = 2

(18)

12 a 4 ρ (1 − v x v y ) E1 h02

Eq. (16) involves the unknown A1 & A2 arising due to the substitution of W from eq. (16). These two constants are to be determined from Eq. (18), as follows: ∂ (V1 − λ 2 p 2T1 ) / ∂ An = 0,

q1

* 2

Q =

b 1 a

(19)

n=1,2.

On simplifying equation (21), we get following form, (20) c A + c A = 0

and E 1* =E 1 /(1-v x v y )

where

and

)

3 1 E 1* h 0 / 1 2 a 2

1

q 2

2

where q= 1,2 here cq1 and cq2 involves the parametric constants and the frequency parameter.

For a non - zero solution, determinant of coefficients of equation (20) must vanish. In this way frequency equation comes out to be:

c11 c12 c21 c22

= 0

(21)

On solving (21) one gets a quadratic equation in p2, so it will give two roots. On substituting the value of A1=1, by choice, in equation (11) one get A2=-c11/c12 and hence W becomes: 2 a a     c11  XY (1 − X) 1+  −  XY    c b b    12  W(x,y)=  (22)  a    1-Y  a       (1 − X ) 1-Y  b     b   



(15)

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 2.3. Time function Time functions of free vibrations of visco-elastic plates are defined by the general ordinary differential equation (2). Their form depends on visco-elastic operator D% and which for Kelvin’s model, one can be taken as: % = 1 +  η   d   (23) D     G   dt   Where η is visco-elastic constant and G is shear modulus. Taking temperature dependence of viscoelastic constant η and shear modulus G is the same form as that of Young’s moduli, we have

G(T)= G0 (1−γ1 T), η(T) =η0 (1−γ2 T)

(24)

Where G0 is shear modulus and η0 is visco-elastic constant at some reference temperature, i.e., T=0, γ1 and γ2 are slope variation of τ from equation (3) and using equation (12) in equation (24), one gets:

(25)

(35)

3. NUMERICAL EVALUATIONS

Here α1 and α2 are thermal constants.

After using equation (23) in equation (2), one gets

T&& + p2 kT& + p2T = 0

(26)

a  η0 α1(1− X)(1− Y ) b where k = η =  a G  G0 α1(1− X)(1− Y ) b 

(27)

Equation (26) is a differential equation of second order for time function T. On solving equation (26), one gets (28) T (t) = ea t [ C1 cos b1t + C2 sin b1t ] 1

2 p2k and b1 = p 1 −  pk  2  2 

(29)

and C1, C2 are constants which can be determined easily from initial conditions of the plate.

Let us take initial conditions as T=1 and ܶሶ=0 at t=0 (30) On using equation (29) in equation (28), one gets

C1 = 1 & C2 = −

w2 ) w1

where w1 is the deflection at any point on the plate at time period K = K1 and w2 is the deflection at same point at the time period succeeding K1.

a η=η0(1−α2(1−X)(1−Y )) , b where α2=γ2T0, 0 ≤ α2 ≤ 1

a1 = −

by using equation (32) and equation (22).

Time period of the vibration of the plate is given by 2π (34) K = p where p is the frequency given by equation (21).

Logarithmic decrement of the vibrations given by the standard formula

∧ = log e (

a G= G0 (1−α1(1− X)(1− Y )) , b where α1=γ1T0 , 0 ≤ α1 ≤ 1 and

where

Thus, deflection of vibrating mode w(x,y,t), which is equal to W(x,y)T(t), may be expressed as 2   a a    XY (1 − X)  1-Y    b b       a a    (33)    c11   w =  1+  −  XY (1 − X )  1-Y    b b       c12     x e a1t  cos b t + ( − a1 )sin b t     1 1  b1    

a1 b1

(31)

one has

  a T (t) = ea1t cos b1t + (− 1 ) sin b1t  b1   (32) after using equation (30) in equation (28).

For calculations, the material of ‘Duralium’ which is an alloy of Aluminium, Copper, Magnesium and Manganese have been taken. Computations have been made for calculating the values of logarithmic decrement (^), time period (K) and deflection (w) for a isotropic visco-elastic rectangular plate for different values of taper constants β1 & β2 and aspect ratio a/b at different points for first two modes of vibrations. In calculations, the following material parameters are used: E=7.08 x 1010N/M2 G = 2:632 x 1010 N/M2, η =14:612 x 105 Ns/M2, ρ= 2:80 x 103 kg/M3, v =0:345.

These values have been reported [9] for ‘Duralium’.

The thickness of the plate at the center is taken as h0 = 0.01 m.

4. RESULTS AND DISCUSSION All numerical results for a visco-elastic orthotropic rectangular plate of linearly varying thickness in both directions have been computed with accuracy by using latest computer technology. Computations have been made for calculating time period K and deflection w for different values of taper constants β1 and β2, thermal constants (α, α1, α5), and aspect ratio a/b for first two modes of vibration. All these results are presented in Figure 1 to Figure 7.

Figure1:- The results of time period K for a fixed value of aspect ratio a/b (=1.5) and for different

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International Journal of Research in Advent Technology, Vol.2, No.6, No. June 2014 E-ISSN: 2321-9637 thermal gradient α ranging from 0.0 to 1.0 with a difference of 0.2 but for all the values of X, Y and for all α1, α2 and for two combinations of β1 and β2 for the first two modes of vibration as follows: β1

β2

0.0 0.4

0.0 0.2

It can be clearly seen that as thermal gradient α increases, time period K increases continuously for both the modes of vibration.

Figure 2:- The values of time period K for different values of aspect ratio a/b ranging from 0.5 to 2.5 with a difference of 0.5., and all values of X, Y and for all α1, α2 for both the modes of vibration having the following cases: α 0.0 0.3

β1 0.0 0.4

β2 0.0 0.2

decreases till zero as X increases for different values of Y.. It can also be observed that deflection w for the second mode of vibration, for Y = 0.2, 0.2 first increases then decreases and then increases and finally becomes zero but for Y = 0.6,, deflection w first increases and then decreases till zero as X increases.

Figure 7:- The variation of deflection w for different values of aspect ratio a/b,, and for fixed values valu of X (= 0.2) and Y (= 0.2) at initial time 0.K and at time 5.K for following case: Figure 7

α 0.0

β1 0.0

β2 0.0

α1 0.0

α2 0.0

In Figure 7, it can be concluded that deflection w for the first mode of vibration continuously increases at initial time 0.K and first increases and then slightly decreases at the time 5.K but an increase followed by decrease is observed for second mode of vibration, with increase in aspect ratio a/b, at initial time 0.K and at time 5.K.

In both the cases, it can be seen that the value of time period K decreases as aspect ratio a/b increases for both the modes of vibration. It is clearly shown in Fig.2 that time period K continuously decreased as aspect ratio a/b increased.

Figure 3:- Shows the results of time period K, for the first two modes of vibration, for fixed aspect ratio a/b and for fixed thermal gradient α (=0.3) and for fixed value of taper constant β2(=0.2) for different values of taper constant β1 ranging from 0.0 to 1.0 with a difference of 0.2 but for all the values of X, Y and α1, α2. It can be noted that time period K decreases as taper constant β2 increase for both the modes of vibration. Fig. 3 supports the result of a steady decrease in time period K with increase in taper constants β1. The results of time period K, for the first two modes of vibration for fixed aspect ratio a/b, fixed thermal gradient α and for fixed value of taper constant β1 but for different values of taper constant β2 ranging from 0.0 to 1.0 with a difference of 0.2 and for all X, Y and α1, α2 are displayed in Fig.4. For both the modes of vibration, time period K decreases with increase in taper constant β2.

Figure 1:- Variation of Time period K with Thermal Gradient radient and Aspect ratio

Figures 5 and 6:- In both figures 5 an 6, the variation of deflection w for different X for fixed aspect ratio a/b (=1.5) for the first two modes of vibration at initial time 0.K and time 5.K for different values of Y for the following cases: Figures α β1 β2 α1 α2 0.0 0.0 0.0 0.2 0.3 5 0.2 0.3 0.4 0.2 0.3 6 gures 5 and 6 show that deflection w for the Both the figures first mode of vibration first increases and then

Figure 2:- Variation of Time period K with different Aspect ratio

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International Journal of Research in Advent Technology, Vol.2, No.6, No. June 2014 E-ISSN: 2321-9637

Figure 3:- Variation of Time period K with Taper constant β1 and Aspect ratio Figure 6:- Deflection w vs X with constant Aspect ratio and different X and Y

Figure 4:- Variation of Time period K with Taper constant β2and Aspect ratio

Figure 7:-Variation Variation of Deflection w for different Aspect Ratio and X=Y=0.2 5. CONCLUSION

Figure 5:- Deflection w vs X with Aspect Ratio and for different X and Y

It is concluded that as thermal gradient increases, time period increases continuously and hence frequency decreases for both the modes of vibration, therefore thermal effects can never be neglected. If thermal stresses are removed in the above case, the result match with the unheated plate in which temperature effect was not taken into account. After comparing authors conclude that as temperature effect introduced, time period and deflection increase gradually in comparison to unheated plate of varying thickness. Therefore engineers can see and develop the plates in the manner so that they can fulfill the requirements.

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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 REFERENCES [1] Leissa, A.W. (1969): Vibration of plates, NASA SP-160, U.S. Govt. Printing office.

[2] Khanna A, Ashish Kumar Sharma, (2013): Natural Frequency of Viso-Elastic Square Plate with Thickness Variation”, Elixir Mech. Engg., 56, pp. 13264-13267.

[3] A.K. Gupta and Anupam Khanna, (2010): Thermal Effect on Vibrations of Parallelogram Plate of Linearly Varying Thickness, Advanced Studies of Theoretical Physics, 4 (17), pp. 817826.

[4] Gupta A.K., Johri T., Vats R.P., (2010): Study of thermal gradient effect on vibrations of a nonhomogeneous orthotropic rectangular plate having bi-direction linearly thickness variations, Meccanica, 45(3), pp. 393-400.

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[6] Gupta A.K., Kaur H., (2008): Study of the effect of thermal gradient on free vibration of clamped visco-elastic rectangular plates with linearly thickness variation in both directions, Meccanica, 43(4), pp. 449-458.

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