Vibration of a circular beam with variable cross sections using differential transformation method

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Vibration of a circular beam with variable cross sections using differential transformation method S.M. Abdelghany a,*, K.M. Ewis a, A.A. Mahmoud b, Mohamed M. Nassar b a b

Engineering Mathematics and Physics Department, Faculty of Engineering, Fayoum University, Fayoum, Egypt Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza, Egypt

A R T I C L E

I N F O

A B S T R A C T

Article history:

In this paper, an application of differential transformation method (DTM) is applied on free

Received 9 December 2014

vibration analysis of Euler-Bernoulli beam. This beam has variable circular cross sections.

Accepted 11 March 2015

Natural frequencies and corresponding mode shapes are obtained for three cases of cross

Available online

section and boundary conditions. MATLAB program is used to solve the differential equation of the beam using DTM. Comparison of the obtained results with the previous solutions

Keywords: Differential transformation method Circular beam

proves the accuracy and versatility of the presented problem. © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Beni-Suef University. This is an open access article under the CC BY-NC-ND license (http://

Variable cross section

creativecommons.org/licenses/by-nc-nd/4.0/).

Vibration

1.

Introduction

Beams with abrupt changes of cross-section are used widely in engineering. It can be easily made in order to save weight or to satisfy various engineering requirements. However, in many theses, beams are treated as volumes, so their cross section is usually rectangular. This paper presents the results of a case study of a beam element with circular cross section. Circular shape of the cross section is chosen because it often occurs in practice. In this paper, the vibration problems of circular Euler-Bernoulli beams have been solved analytically using DTM. The beam has variable cross sections and various end conditions. In order to calculate the fundamental natural fre-

quencies and the corresponding mode shapes, variational techniques were applied in the past such as Rayleigh Ritz, differential quadrature (DQM) and Galerkin methods. Also, some numerical methods were also successfully applied to beam vibration analysis such as finite element method (FEM). The differential transformation method leads to an iterative procedure for obtaining an analytic series solutions of functional equations. Pukhov (1982) have developed a socalled differential transformation method (DTM) for electrical circuits’ problems. In recent years, researchers Ayaz (2004), Moustafa (2008) and Qibo (2012) had applied the method to various linear and nonlinear problems. Comparison between (DTM) and (DQM) were applied by Attarnejad and Shahba (2008), and Rajasekaran (2009).

* Corresponding author. Tel.: +20 1140666453. E-mail address: [email protected] (S.M. Abdelghany). Peer review under the responsibility of Beni-Suef University. http://dx.doi.org/10.1016/j.bjbas.2015.05.006 2314-8535/© 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Beni-Suef University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006

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Mahmoud et al. (2013) used the differential transformation method for the free vibration analysis of rectangular beams with uniform and non-uniform cross sections. Akiji et al. (1982) discussed the importance of the geometrical injection efficiency of a neutral circular beam. Un-damped vibration of beams with variable cross-section was analyzed by Datta and Sil (1996). Au et al. (1999) used C1 modified beam vibration functions to study the vibration and stability of non-uniform beams with abrupt changes of cross-section. Li (2000) presented an exact approach for determining natural frequencies and mode shapes of non-uniform shear beams with arbitrary distribution of mass or stiffness. A three-dimensional method of analysis was presented for determining the free vibration frequencies and mode shapes of thick, tapered rods and beams with circular cross-section Kanga and Leissab (2004). Kisaa and Arif Gurelb (2006) presented a numerical model that combines the finite element and component mode synthesis methods for the modal analysis of beams with circular cross section and containing multiple non-propagating open cracks. Vibration and bending analysis were applied for non-uniform beams, rods and tubes by Mehmet et al. (2007), AL Kaisy et al. (2007), De Rosaa et al. (2008) and Shojaeifard et al. (2012). The beam elements, which are widely used in the absolute nodal coordinate formulation, were treated as iso-parametric elements by Grezegorz (2012). In the present paper, an attempt is made to employ the differential transformation method to solve equations of motion for the free vibration of non-uniform circular beam. Three cases of boundary conditions are considered. Natural frequencies and corresponding mode shapes are obtained.

2.

Basic equations

2.1.

Free vibration of non-uniform circular beam

The governing differential equation for an Euler beam with a circular cross section with variable radius rX as shown in Fig. 1 is given by:

ρA (X)

∂ 2 W ( X, T ) ∂ 2 ⎛ ∂ 2 W ( X, T ) ⎞ + EI ( X ) ⎟=0 ∂T 2 ∂X 2 ⎜⎝ ∂X 2 ⎠

(1)

where ρ is the density of the beam material, A ( X ) is the cross sectional area of the beam, W ( X, T ) is displacement of the

beam, E is young’s modulus of the beam and I ( X ) = inertia of the beam.

2.2.

Boundary conditions

Case a. Simply Supported Beam

W ( 0, T ) = 0;

∂ 2 W ( 0, T ) = 0; ∂X 2

∂ 2 W ( L, T ) =0 ∂X 2

W ( L, T ) = 0;

(2)

Case b. Clamped–Clamped Beam

W ( 0, T ) = 0;

∂W ( 0, T ) = 0; ∂X

W ( L, T ) = 0;

∂W ( L, T ) =0 ∂X

(3)

∂ 2 W ( L, T ) =0 ∂X 2

(4)

Case c. Clamped–Roller Beam

W ( 0, T ) = 0;

∂W ( 0, T ) = 0; ∂X

W ( L, T ) = 0;

where L is the beam length and T is the time.

Table 1 – The first three non-dimensional frequencies (Ω1, Ω2 and Ω3) of Simply Supported uniform Circular Euler beams for different number of terms. No. of terms N 8 9 13 14 15 17 18 23 28 33 34 38 43 44 45

Ω1

Ω2

Ω3

– 9.8902098156 9.8696683979 9.8696683979 9.8696020437 9.8696044699 9.8696044699 9.8696044011 9.8696044011 9.8696044011 9.8696044011 9.8696044011 9.8696044011 9.8696044011 9.8696044011

– 28.2140699048 37.2824413197 37.2824413197 40.0646967922 39.4169139196 39.4169139196 39.4784501712 39.4784176959 39.4784176044 39.4784176044 39.4784176044 39.4784176044 39.4784176044 39.4784176044

– – – – 58.0531514621 – – 87.8912222720 88.8107229014 88.8264496362 88.8264496362 88.8264396509 88.8264396098 88.8264396098 88.8264396098

Table 2 – The first three non-dimensional frequencies (Ω1, Ω2 and Ω3) of Clamped–Clamped uniform Circular Euler beams for different number of terms. No. of terms N

Fig. 1 – A sketch of beam with variable circular cross-section.

π 2 rX is the 4

8 13 17 18 23 24 28 33 36 37 38 39

Ω1

Ω2

Ω3

– 22.3776282691 22.3733025493 22.3733025493 22.3732854786 22.3732854414 22.3732854481 22.3732854481 22.3732854481 22.3732854481 22.3732854481 22.3732854481

– 53.7015697919 60.9657850659 60.9657850659 61.6611232022 61.6779003065 61.6728539182 61.6728228669 61.6728228682 61.6728228679 61.6728228679 61.6728228679

– – – – – 109.8144306817 120.0930730609 120.9036627032 120.9032822584 120.9033934841 120.9033934841 120.9033934841

Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006

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Equation (6) can be conveniently written in terms of dimensionless variables as:

Assume that, the displacement of the beam is given by:

W ( X, T ) = W ( X ) exp ( iω T )

(5)

S (x) where ω is the natural frequency of the beam. Equation (1) can be conveniently written as:

d4 u ( x ) dS ( x ) d3u ( x ) d2S ( x ) d2u ( x ) 13 +2 + = Ω 2 ( S ( x ) ) u ( x ) (7) dx 4 dx dx3 dx2 dx2

EI ( x ) W X A (x) 3 = S ( x ) = (1 − β x ) , u = = S1 3 ( x ) , , x= , EI0 A0 L L r (x) ρ A0 L4 S1 3 ( x ) = = (1 − β x ) , Ω 2 = ω 2 r0 EI0 Ω is the non-dimensional frequency of the beam, r0, A0, I0 are radius, cross sectional area and inertia at the left edge of

where,

− ρ A ( X ) ω 2W ( X ) + E +E

d4 W ( X ) dI ( X ) d3W ( X ) + 2E 4 dW dX dX3

(6)

d2 I ( X ) d2 W ( X ) =0 dX 2 dX 2

Table 3 – Simply supported non-uniform Euler Beams with (Rectangular and Circular) cross sections, for N = 33. EI ( x ) 3 = (1 − β X ) Ω1 Ω2 Ω3 EI ( x ) 0

Rectangular [10, 16] Circular

β=0

β = 0.25

β = 0.5

β=0

β = 0.25

β = 0.5

β=0

β = 0.25

β = 0.5

9.8696 9.8696

8.5772 8.3847

7.1215 6.8020

39.4784 39.4784

34.4062 34.7361

28.9518 31.5322

88.8264 88.8264

77.3794 78.4308

64.9789 71.3759

0.01

0.05

0.005 0

0 Third Mode

-0.005

Third Mode

-0.05

Deflection

Deflection

Second Mode

First Mode

-0.1

Second Mode

-0.01 First Mode

-0.015 -0.02 -0.025 -0.03

-0.15

a

DTM Exact -0.2

0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

-0.04

1

b

DTM Exact

-0.035 0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

0.02 0.01 0 Third Mode

Deflection

-0.01 Second Mode -0.02

Firt Mode

-0.03 -0.04 -0.05 -0.06

c

DTM Exact 0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

Fig. 2 – The first three mode shapes of uniform circular beam (a. Simply Supported, b. Clamped–Clamped and c. Clamped-Roller) boundary. Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006

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the beam, and β is a constant – equals zero for uniform beam-respectively. The non-dimensional boundary conditions are, Case a. Simply Supported Beam

d2u ( 0 ) = 0; dx2

u ( 0 ) = 0;

u ( 1 ) = 0;

d2u ( 1 ) =0 dx2

3.

Differential transformation method

The DTM is a technique that uses Taylor series for the solution of differential equations in the form of a polynomial. The Taylor series method is computationally tedious for high order equations. Following Ayaz (2004) we can obtain the idea of the DTM, The differential transformation of function y(x) is defined as follows;

(8)

Case b. Clamped-Clamped Beam

du ( 0 ) = 0; dx

u ( 0 ) = 0;

u ( 1 ) = 0;

Y (k) =

du ( 1 ) =0 dx

(9)

du ( 0 ) = 0; dx

u ( 1 ) = 0;

(11)

In Equation (1), y(x) is the original function and Y(k) is the transformed function. Differential inverse transform of Y(k) is defined as follows;

Case c. Clamped–Roller Beam

u ( 0 ) = 0;

1 ⎛ ∂k y ( x ) ⎞ k ! ⎜⎝ ∂xk ⎟⎠ x = 0

d2u ( 1 ) =0 dx2

∞

y ( x ) = ∑ xk Y ( k )

(10)

(12)

k=0

0.2

0.5 Beta=0 Beta=0.4 Beta=0.8

0.4

Beta=0 Beta=0.4 Beta=0.8

0.15

0.2

Deflection

Deflection

0.3

0.1

0.1

0.05

0 0

-0.1

-0.2

a 0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

b -0.05

1

0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

0.04 0.02 0

Deflection

-0.02 -0.04 -0.06 -0.08 Beta=0 Beta=0.4 Beta=0.8

-0.1 -0.12

0

0.1

0.2

c 0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

Fig. 3 – Mode shapes of non-uniform Simply Supported circular beam (a. first, b. second and c. third modes) for (β = 0, 0.4 and 0.8). Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006

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from (11) and (12), we obtain

Uk+ 4 =

xk ⎛ ∂ k y ( x ) ⎞ ⎜ ∂xk ⎟ ⎠x =0 x =0 k ! ⎝ ∞

y (x) = ∑

(13)

From the definitions (11) and (12), it is easy to obtain the following mathematical operations; ref. AL Kaisy et al. (2007), Moustafa (2008), Qibo (2012) and Attarnejad and Shahba (2008) 1. If f ( x ) = g ( x ) ± h ( x ) , then F ( k ) = G ( k ) ± H ( k ) . 2. If f ( x ) = cg ( x ) , then F ( k ) = cG ( k ) c is a constant. dn g ( x ) (k + n )! G ( k + n ). 3. If f ( x ) = , then F ( k ) = dxn k! k 4. If f ( x ) = g ( x ) h ( x ) , then F ( k ) = ∑l =0 G ( l ) H ( k − l ). 5. If f ( x ) = xn , then F ( k ) = δ ( k − n ) , δ is the Kronecker delta. m n 7. Cr ( k + n − r ) ! m d g(x) then F ( k ) = a ∑ m If f ( x ) = a ( 1 − bx ) r =0 dxn (k − r )!

( −b )r G ( k + n − r ) , where a and b are constants.

1

{Ω 2Uk − β 2 [( k − 1 )( k ) ( k + 1 ) ( k + 2 ) (k + 1) (k + 2 ) (k + 3) (k + 4 ) + 6k ( k + 1 ) ( k + 2 ) + 6 ( k + 1 ) ( k + 2 )] Uk + 2 + 2 β [k ( k + 1 ) ( k + 2 ) ( k + 3 ) + 3 ( k + 1 ) ( k + 2 ) ( k + 3 )] Uk +3 } (14)

Then apply the DTM to the non-dimensional boundary conditions equations. The solution here will written for simply supported beam only, DT of Equation (8) is written as ∞

u ( 0 ) = ∑ U [k] x k k=0

= U [ 0] x 0 + U [1] x1 + U [2] x2 + U [3] x3 + U [ 4 ] x 4 + U [5] x5 + … = 0 (15.a) which leads to U[0]=0 ∞

u′′ ( 0 ) = ∑ ( k + 1 ) ( k + 2 ) U [k + 2] xk k=0

In solving the problem, governing differential equation must be solved together with applied boundary conditions. Applying the (DTM) to the non-dimensional governing Equation (7) yield,

= ( 1 )( 2 ) U [2] x 0 + ( 2 )( 3 ) U [3] x1 + ( 3 )( 4 ) U [2] x2 + ( 4 )(5) U [3] x3 + (5)( 6 ) U [ 4 ] x 4 + ( 6 )( 7 ) U [5] x5 + … = 0

(15.b)

which leads to U[2]=0

0.2

0.06 Beta=0 Beta=0.3 Beta=0.6

0.15

Beta=0 Beta=0.3 Beta=0.6

0.05 0.04

Deflection

Deflection

0.03 0.1

0.05

0.02 0.01 0 -0.01

0

-0.05

a 0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

b

-0.02 -0.03

1

0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

-3

6

x 10

Beta=0 Beta=0.3 Beta=0.6

4

Deflection

2

0

-2

-4

-6

-8

c 0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

Fig. 4 – Mode shapes of non-uniform Clamped–Clamped circular beam (a. first, b. second and c. third modes) for (β = 0, 0.3 and 0.6). Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006

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Since, C1 and D1 are not zero, for a non-trivial solution to exist the determinant of the matrix must be zero, i.e.

∞

u ( 1 ) = ∑ U [k] x k k=0

= U [ 0] x 0 + U [1] x1 + U [2] x2 + U [3] x3 + U [ 4 ] x 4 + U [5] x5 + … = 0 (15.c)

A11 × D11 − C11 × B11 = 0

(17)

∞

which leads to ∑ U [k] = 0

(15.d)

where, A11, B11 are the coefficients of C1 and D1 in the Equation (15.c) and C11, D11 are the coefficients of C1 and D1 in the Equation (15.e). The root of Equation (17) is the solution for case a of the problem.

(15.e)

4.

k=0

and taking U [1] = C1 ∞

u′′ ( 1 ) = ∑ ( k + 1 ) ( k + 2 ) U [k + 2] xk k=0

= ( 1 )( 2 ) U [2] x 0 + ( 2 )( 3 ) U [3] x1 + ( 3 )( 4 ) U [2] x2 + ( 4 )(5) U [3] x3 + (5)( 6 ) U [ 4 ] x 4 + ( 6 )( 7 ) U [5] x5 + … = 0 which leads to ∑k∞= 0 k ( k − 1 ) U [k] = 0

and taking U [3] = D1

(15.f)

Equations (15.c) and (15.e) may be written as

⎡ A11 ⎢C ⎣ 11

B11 ⎤ ⎡ C1 ⎤ ⎡0 ⎤ = D11 ⎥⎦ ⎢⎣ D1 ⎥⎦ ⎢⎣0 ⎥⎦

(16)

Cases study and discussions

Numerical results are presented in this section for uniform and non-uniform circular beam. Also, a comparison with a rectangular cross section is applied. The boundary conditions are assumed to be simple–simple, clamped-clamped and clampedroller supports. Table 1 compares the accuracy of the first three nondimensional frequencies of simply supported uniform Euler beams – β equals zero – with circular cross section for different number of terms N. The table shows that, when we use

0.3

0.06 0.04

Beta=0 Beta=0.5 Beta=0.75

0.25 0.2

0.02 0

Deflection

Deflection

0.15 0.1 0.05

-0.02 -0.04 -0.06 -0.08

0

-0.1

a

-0.05 -0.1

0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

Beta=0 Beta=0.5 Beta=0.75

-0.12 -0.14

1

0

0.1

0.2

b 0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

0.03 0.02 0.01

Deflection

0 -0.01 -0.02 -0.03 -0.04 -0.05 Beta=0 Beta=0.5 Beta=0.75

-0.06 -0.07

0

0.1

0.2

c 0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

Fig. 5 – Mode shapes of non-uniform Clamped-Roller circular beam (a. first, b. second and c. third modes) for (β = 0, 0.5 and 0.75). Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006

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the differential transformation method, we need 23 terms only to reach exact solution of the first frequency. While, the second and third frequency needs 33 and 43 terms respectively. Table 2 gives results of the first three non-dimensional frequencies of uniform circular clamped–clamped beam. It is observed that the first frequency of the clamped circular beam needs 33 terms, while, the second and the third frequencies needs 39 terms to reach exact solution. From Tables 1 and 2 the reader can observe that the convergence speed increases when the frequency order decreases, i.e. Table 3 gives a comparison of simply supported non-uniform beams with rectangular and circular cross sections for (β = 0, 0.25 and 0.5). It can be demonstrated that the first three frequencies of the simply nonuniform beam decreases with increasing β due to decreasing the beam cross section. Also, Table 3 shows that variation of the beam cross section is effective in the case of circular cross sections. Fig. 2 shows the mode shapes of uniform circular Euler beams i.e β equals zero with three cases of boundary conditions. It can be seen that the first three mode shapes drawn using DTM is highly agreement with the mode shaped drawn using exact methods. Fig. 3 shows, respectively, the first, the second and the third mode shapes of non-uniform simplesimple Euler beams with different circular cross sections (β = 0, 0.4 and 0.8). Fig. 4 shows, respectively, the first, the second and the third mode shapes of non-uniform clamped–clamped Euler beams with different circular cross sections (β = 0, 0.3 and 0.6). Fig. 5 shows, respectively, the first, the second and the third mode shapes of non-uniform clamped-roller Euler beams with different circular cross sections (β = 0, 0.5 and 0.75). From Figs. (3–5), it can be seen that the value of beta β has a significant effect on the mode shapes and the deflection values of circular beams with different boundary conditions. Deflection of non-uniform circular beams increases as cross section decreases (β values increases).

5.

Conclusion

Based on the previous results, it can be demonstrated that the differential transformation method is an efficient method to solve the vibrations problems of the non-uniform circular beams with good accuracy using a few terms. It can be seen that the variation of cross section has a significant effect on the mode shapes and the deflection values of circular beams comparing with rectangular beams. It is also possible to extend this method to the use for other cross section shapes.

7

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Please cite this article in press as: S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni-Suef University Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.bjbas.2015.05.006