Free Vibration Analysis of Fgm Sandwich Beam Resting

FREE VIBRATION ANALYSIS OF FGM SANDWICH BEAM RESTING ON THE WINKLER FOUNDATION USING

DIFFERENTIAL TRANSFORMATION METHOD (DTM)

Presented by: Parimal Priyadarshi (09AE6001)

Guided BY:Prof. S.C.Pradhan Department of Aerospace

Engineering, I.I.T Kharagpur

Contents

• Introduction• Literature Survey• Formulation• Results and Discussion• Conclusion• References

INTRODUCTION

• Understanding of soil-structure interaction. • The well known and widely used mechanical model is the one

devised by Winkler. According to the Winkler model, the beam-supporting soil is modeled as a series of closely spaced, mutually independent, linear elastic vertical springs which provide resistance in direct proportion to the deflection of the beam.

• In this work we have taken FGM sandwich beam resting on winkler foundation and Euler Bernoulli theory has been applied for the governing equation.

FUNCTIONALLY GRADED MATERIALS

• FGM may be characterized by the variation in composition and structure gradually over volume, resulting in corresponding changes in the properties of the material.

• The concept of FGM was originated in Japan in 1984 during a space plane project.

• FGM material have the best of both materials.

Most of the FGMs are being used in high-temperature environment and their material properties are temperature dependent. A typical material property Pi can be expressed as a function of the environment temperature T

where P0, P−1, P1, P2 and P3 are temperature coefficients and are constants for a specific FGM constituent material.

Young's modulus of FGM beam made up of two different materials are expressed as

and from temp. variation it can be shown as

LITERATURE SURVEY• Zhou [1] and Eisenberger [2] studied a general

solution to vibrations of beams on a variable Winkler elastic foundation.

• Eisenberger and Clastornik [3] examined the vibrations and buckling of a beam on a variable Winkler elastic foundation.

. • Also, some researchers [4-5] studied the analysis

of elastic foundations with Winkler-Pasternak models. In addition to differential transform method for structures on elastic foundation.

• Zhou[6] was the first to use the DTM to solve the initial boundary value problem. In the electric circuit analysis.

• Ozdemir and kaya [7] calculated natural frequencies for the nonprismatic beams whose crosssectional area and moment of inertia vary.

• Ozgumus and kaya [8] determined the first Natural frequencies of a cantilevered double tapered rotating Euler-Bernoulli beam.

• Pradhan et al., [9] found out the nonlocal critical buckling load of a beam embeded in an elastic system using DTM.

• Praveen and Reddy [10] carried out thermo-elastic analysis of FG plates. They investigated the static and dynamic response of the FGM plates by varying the volume fraction of the ceramic and metallic constituents using the simple power-law distribution.

• Reddy and Cheng [11] studied the three-dimensional distribution of displacement and stresses of smart FG plates.

• Librescu et al. [12] studied the behavior of thin-walled beams made of FGM operating at high temperatures, which included vibration and instability analysis with effects of volume fraction, temperature gradients, etc.

• Li et al. [13] reported free vibration response of FGSW rectangular plates based on the three-dimensional elasticity theory. Their work included sandwich structures with FGM face sheet as well as with FGM core.

Objective

• To find out the natural frequencies of FGM Sandwich Beam based on Euler Bernoulli theory.

• Applying different Boundary Conditions to both ends and find out the changes in the frequencies as we change the value of Winkler’s Elasticity, Temperature and Rn.

Formulation

The differential transform method (DTM) is a transformation technique based on the Taylor series expansion and is a useful tool to obtain analytical solutions of the differential equations. In this method, certain transformation rules are applied and the governing differential equations and the boundary conditions of the system are transformed into a set of algebraic equations in terms of the differential transforms of the original functions and the solution of these algebraic equations gives the desired solution of the problem.

DTM Theorems used for the equation of motion and B.C’s [9]

Mathematical FormulationConsider Euler-Bernoulli beam resting on Winkler foundation. The equation of motion for this problem is given as follows.

Where k is spring constant; w is deflection (m); is the mass density (kg/m3); A is the cross sectional area (m2); E is the Young’s Modulus (Pa) and I is the area moment of inertia about the neutral axis (m4). Here x is the horizontal space coordinate measured along the length of the beam and t is any particular instant of time.

4 2

4 2( ) ( ) 0

w wEI k x A x

x t

Assuming the displacement function as follows, where ω is the natural frequency;

So we can again write the eq of motion as

The elastic foundation is represented by a set of linear springs in Winkler modelling. Winkler elastic foundation can vary linearly or parabolically or even constantly through the length of the beam. In this study it is being taken as constant k.

( , ) ( ) i tw x t w x e

42

4( ) 0

wEI A w k x w

x

Now we can nondimensionalize this eq using these Non- dimesional parameters

Using these non-dimensional parameters again the equation can be written as

Now using the DTM table it can be written as

Where

42

4(1 ) 0

d WW

d

( 1)( 2)( 3)( 4) ( 4) ( )k k k k W k rW k

2(1 )r

max0,1,2.....k N

Application of Boundary Conditions

• For C-C BeamAs we know

Using DTM Table We can write above boundary Condition as

W(0)=0, w(1)=0 and

0 0, 0 0, 0, 0w( )= w'( )= w(L)= w'(L)=

0

0k=

W(k)=

0

0k=

k W(k)=

• For S-S BeamNow again using DTM we can write these B.C’s

as

W(0)=0 and w(2)=0And

0 0, '' 0 0, 0, '' 0w( )= w ( )= w(L)= w (L)=

0

0k=

W(k)=

0

1 0k=

k (k )W(k)=

• For C-F BeamNow again using DTM we can write these B.C’s

as

W(0)=0 and w(1)=oAnd

0 0, ' 0 0, 0, '' 0w( )= w ( )= w(L)= w (L)=

0

0k=

W(k)=

0

1 0k=

k (k )W(k)=

Validation(For L=1,EI=1,K=1,ρ=1)

Results Fig1: Natural Frequency Vs RN for Different Boundary Conditions (At k=0, T= -300 k)

Fig 2: : Natural Frequency Vs RN for Different Boundary Conditions (At k=500, T= -300 k)

Fig3: Natural Frequency Vs RN for Different Boundary Conditions (At k=1000, T= -300 k)

Fig4: Natural Frequency Vs RN for Different Boundary Conditions (At k=2000, T= -300 k)

Fig5: Natural Frequency Vs RN At different T and k=0 for the Clamped-clamped Boundary Condition

Fig6: Natural Frequency Vs K for the Clamped-Free Boundary Condition at different RN (For T= 300 k)

Fig7: Natural Frequency Vs K for the Different Boundary Conditions (For T= 300 k, Rn= 0)

Fig8: Natural Frequency Vs K for the Different Boundary Conditions (For T= 300 k, Rn= 10)

Fig9: E Vs Rn for Alumina And stainless steel FGM Beam

Fig10: E Vs Rn for alumina and steel FGM beam fir the different temperature variations

C-C Beam At T=3000 c, k=0

C-C Beam at T=3000 c, k=1000

C-C Beam at T=3000 c, k=2000

Conclusion

• DTM gave the accurate result up to four decimal places.

• Natural Frequency decreases as Temperature and Rn increases.

• Natural frequency doesn’t much change with the value of Winkler elasticity.

References1. D. Zhou, A General solution to vibrations of beams on variable Winkler elastic

foundation. Computers & Structures 47 (1993), 83-90.2. M. Eisenberger, Vibration frequencies for beams on variable one- and two-

paramter elastic foundations. Journal of Sound and Vibrations 176(5) (1994), 577-584.

3. M. Eisenberger, J. Clastornik, Vibrations and buckling of a beam on a variable Winkler elastic foundation. Journal of Sound and Vibration 115 (1987), 233-241.

4. X. Ma, J.W. Butterworth, G.C. Clifton, Static analysis of an infinite beam resting on a tensionless Pasternak foundation. European Journal of Mechanics A/Solids 28 (2009), 697-703.

5. O. Civalek, Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods. Applied Mathematical Modeling 31 (2007), 606-624.

6. Zhou, J.K., Differential transformation and its application for electrical circuits. Huazhong University Press, Wuhan, China, 1986.

Continued….7. Ö. Özdemir, M.O. Kaya, Flapwise bending vibration analysis of a rotating tapered

cantilever Bernoulli–Euler beam by differential transform method. Journal of Sound and Vibration 289 (2006), 413–420.

8. Ö. Özdemir, M.O. Kaya, Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method. Meccanica 41(6) (2006), 661–670.

9. S.C.Pradhan, G.K.Reddy, Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM. Computational material sciences 50(5)(2011), 1052-1056.

10. G.N. Praveen and J.N. Reddy, Nonlinear transient thermo-elastic analysis of functionally graded ceramic–metal plates, International Journal of Solids and Structures 35 (1998), pp. 4457–4476.

11. J.N. Reddy and Z.Q. Cheng, Three-dimensional solutions of smart functionally graded plates, Journal of Applied Mechanics 68 (2) (2001), pp. 234–241.

Continued…..12. L. Librescu, S.Y. Oh and O. Song, Thin-walled beams made of functionally

graded materials and operating in a high temperature environment: vibration and stability, Journal of Thermal Stresses 28 (2005), pp. 649–712.

13. Q. Li, V.P. Iu and K.P. Kou, Three-dimensional vibration analysis of functionally graded material sandwich plates, Journal of Sound and Vibration 311 (1–2) (2008), pp. 498–515.

14. Chen CN. Vibration of prismatic beam on an elastic foundation by the differential quadrature element method. Computers and Structures 2000; 77:1–9.

Thank You