application of differential transform in free vibration

Reza Attarnejad, Ahmad Shahba and Shabnam Jandaghi Semnani

APPLICATION OF DIFFERENTIAL TRANSFORM IN FREE VIBRATION ANALYSIS OF TIMOSHENKO BEAMS RESTING ON TWO-PARAMETER ELASTIC FOUNDATION Reza Attarnejad, Ahmad Shahba* and Shabnam Jandaghi Semnani School of Civil Engineering, University College of Engineering University of Tehran, Tehran, Iran

1. INTRODUCTION Non-prismatic beams have received great attention from engineers due to their capability in optimizing the strength and weight of the structure. In recent years, many researchers have worked on engineering problems related to static and dynamic analysis of either Euler–Bernoulli [1–3] or Timoshenko [4,5] beams. For short thick beams and rotating machineries, the Timoshenko beam theory presents a more realistic model in comparison with the Euler– Bernoulli beam theory due to both the shear deformation and rotary inertia. When encountering problems such as beams on different types of elastic foundations, including buried pipelines, shallow foundations, and piles, understanding the static and dynamic response of Timoshenko beams on elastic foundations seems to be of great significance. To achieve this, a perception of the interaction between the soil and structural elements is required. Some researchers have investigated the effect of different soils on structural members. Mahmood and Ahmed [6] evaluated the sensitivity of concrete-reinforced beam structures to different behaviors of the soil and the interface layer when influenced by an earthquake excitement. Due to the complex behavior of different types of soils, it is difficult to obtain analytical solutions; therefore, simplified mechanical models have been proposed by several researchers, among which are one-parameter, two-parameter ,and threeparameter elastic foundations. First, Winkler [7] proposed a simple model with only one parameter, that is, the stiffness of linearly elastic and mutually independent vertical springs. Yankelevesky and Eisenberger [8] performed an exact analytical solution for a finite element beam-column resting on Winkler foundation leading to derivation of exact static stiffness matrix. Later, Yankelevsky et al. [9] presented an iterative method for beams on nonlinear Winkler foundation by approximating the nonlinear characteristics of the foundation by piecewise linear curves. Deriving the translational and rotational dynamic stiffness matrcies, Ruge and Birk [10] compared infinite Euler– Bernoulli and Timoshenko beams on Winkler foundation in frequency and time-domain. Although readily applied to a wide range of mechanical problems, the Winkler model merely takes into account the resistance against vertical deformation. Thus, some two-parameter models were developed in order to include the effect of continuity and cohesion of the soil; among them are those suggested by Pasternak [11], Vlasov [12], and Filonenko–Borodich [13]. Despite adding a second parameter, two-parameter models remain sufficiently simple. Accordingly, they have been vastly used in the literature. Using Vlasov assumption, Kim et al. [14] derived an exact dynamic stiffness matrix for shear deformable thin-walled beams with non-symmetric cross-section by applying a power series expansion of the displacement components. Moreover, Kim et al. [15] expanded the method for non-symmetric thin-walled curved beams. Ayvaz and Ozgan [16] applied a modified Vlasov model for the free vibration analysis of beams resting on an elastic foundation. Kerr [17] offered a third parameter to be included, providing more realistic results.

___________________ *Corresponding Author: E–mail: [email protected] ________________________________________________________________________________________________________ Classification: civil engineering, Timoshenko beam, free vibration, natural frequency, two-parameter elastic foundation, Differential Transform Method (DTM) ________________________________________________________________________________________________________ Paper Received April 11, 2009: Paper Revised August 8, 2009; Paper Accepted October 12, 2009

October 2010

The Arabian Journal for Science and Engineering, Volume 35, Number 2B

125


The differential equations governing the transverse free vibration of non-prismatic Timoshenko beams resting on a two-parameter elastic foundation are two coupled differential equations in terms of transverse displacement and bending rotation angle with variable coefficients. Except for some special cases, there exists no closed-form solution in the literature; hence, approximate methods have played a notable role in the solution of this problem. Employing Chebyshev polynomials, Ruta [18] solved the free vibration and stability problems of non-prismatic Timoshenko beams resting on a two-parameter elastic foundation, providing that the moment of inertia and cross-sectional area can vary as general functions along the beam length. He introduced the effects of distributed normal, tangential, axial, and moment loads in the governing differential equations. De Rosa [19] used a geometrical approach to obtain the differential equations of motion for Timoshenko beams. He proposed two models for the free vibration analysis of Timoshenko beams resting on a Pasternak foundation, which is characterized by two parameters, namely Winkler spring and shear layer constants. In one of the models, the shear layer constant is interpreted as the proportion between bending moment and bending rotation, while in the other one, it is considered as the proportion between bending moment and total rotation. Deriving approximate shape functions, Klasztorny [20] studied the vibration problem of non-prismatic Euler–Bernoulli and Timoshenko beams. Considering the presence of subtangential follower force, Auciello and De Rosa [21] carried out the dynamic analysis of beams on two–parameter elastic soil via application of the differential quadrature method (DQM) and the Rayleigh-Ritz method. Chen [22] studied the vibration of prismatic beams on one-parameter elastic foundation using the differential quadrature element method (DQEM). Later, Chen [23] employed DQEM for free vibration analysis of non-prismatic shear deformable beams resting on elastic foundations. On the basis of the exact solution of governing differential equations, Avramidis and Morfidis [24] studied prismatic Timoshenko beams resting on Kerr-type three-parameter foundations, estimating the first two parameters on the basis of a modified Vlasov model and the third one through parametric investigation. Later, Morfidis [25] used a similar procedure for derivation of structural matrices of a prismatic Timoshenko beam resting on a Kerr-type elastic foundation. Arboleda-Monsalve et al. [26] derived a dynamic stiffness matrix and a load vector of a Timoshenko beam-column with generalized boundary conditions resting on a two-parameter elastic foundation. Eisenberger [27] also derived a dynamic stiffness matrix for variable cross-section members. Zhou [28] was the first to introduce the concept of the differential transform method (DTM) in the solution of differential equations in the analysis of circuits. Being simple and widely applicable, DTM has been used in civil/mechanical problems concerning the system of differential equations. Ho and Chen [29] applied DTM to solve the free and forced vibration problems of non-uniform beams on a non-homogenous elastic foundation. Catal [30] solved the free vibration equations of an axially loaded beam on a Winkler foundation using DTM. DTM has been recently used in the determination of natural frequencies of Timoshenko beams. Ozgumus and Kaya [31] applied DTM for free vibration analysis of double-tapered rotating Timoshenko beams. Balkaya et al. [32] used DTM to obtain natural frequencies of prismatic Euler–Bernoulli and Timoshenko beams resting on Winkler or Pasternak elastic foundations. In this paper, DTM is used to study the free vibration analysis of non-prismatic Timoshenko beams resting on two-parameter elastic soil. The method poses no restriction on either the type of cross-section or variation of crosssectional area and moment of inertia along beam axis. Numerical examples are carried out to verify the present procedure. 2. DIFFERENTIAL TRANSFORM METHOD DTM is an iterative method for obtaining the solution of differential equations in the form of Taylor series. It is different from higher order Taylor series expansions which require the computation of derivatives of the data functions. DTM constructs an analytical solution in the form of polynomials and involves less computational effort in comparison with Taylor series solution in solving higher order problems. Applying DTM in engineering problems involves two transformations, namely, differential transform (DT) and inverse differential transform (IDT), which are respectively defined as

F (k ) =

1 ⎛dk f ⎜ k! ⎜⎝ dx k

⎞ ⎟⎟ ⎠ x = x0

(1)

∞

f ( x) = ∑ ( x − x 0 ) k F (k )

(2)

k =0

In practice, the function f ( x ) is presented by a finite series; hence, IDT is modified as

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m

f ( x) = ∑ ( x − x 0 ) k F (k )

(3)

k =0

∞

where m is chosen such that

∑ (x − x )

k

0

k = m +1

F (k ) is negligibly small. The general theorems referring to application

of DTM are given in Table 1. Table 1. Fundamental Theorems of DTM Original Functions

Transformed Functions

f ( x) = g ( x) ± h( x)

F ( k ) = G (k ) ± H ( k )

f ( x) = cg ( x) ; c = cons.

F (k ) = cG(k )

f ( x) = g ( x)h( x)

F ( k ) = ∑ G ( k − i ) H (i )

k

i=0

f ( x) =

d n g ( x) dx n

F ( k ) = (k + 1)( k + 2)...( k + n)G (k + n)

3. STRUCTURAL FORMULATION AND APPLICATION OF DTM

Assume a general Timoshenko beam made of linear elastic material with modulus of elasticity E and shear modulus G subjected to external transverse load q ( x, t ) as shown in Figure 1. Applying the basic equilibrium equations, one obtains

∂V ∂ ∂w ∂2w − kw ( x) w + (kφ ( x) ) − ρ A( x) 2 = q( x, t ) ∂x ∂x ∂x ∂t

V−

∂M ∂ 2ϕ − ρI ( x) 2 = 0 ∂x ∂t

(4)

(5)

From Timoshenko beam theory, we have

V ( x, t ) = κGA( x)(

∂w − ϕ) ∂x

M ( x, t ) = − EI ( x)

∂ϕ ∂x

(6)

q(x) V mi

M fi

fs

M+dM

V+dV ms

Figure 1: Forces and moments on differential Timoshenko beam element

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Considering free vibration conditions q( x, t ) = 0 and assuming a sinusoidal variation of W ( x, t ) and ϕ ( x, t ) with circular natural frequency ω , the governing equations of motion are obtained as

(κGA(ξ ) + k

ϕ (ξ ) )

d 2W ⎛ dA dk ϕ + ⎜⎜ κG + 2 dξ dξ dξ ⎝

⎞ dW dϕ dA 2 ⎟ ⎟ dξ − L k w (ζ )W (ξ ) − LκGA(ξ ) dξ − LκG dξ ϕ (ξ ) ⎠

+ L2ω 2 ρA(ξ )W (ξ ) = 0 LκGA(ξ )

(7)

d 2ϕ dI dϕ dW + EI (ξ ) 2 + E − L2κGA(ξ )ϕ (ξ ) + L2ω 2 ρI (ξ )ϕ (ξ ) = 0 dξ dξ dξ dξ

(8)

in which ξ = x / L . Applying DTM in Equations (7) and (8) besides the theorems presented in Table 1, the following two recurrence relations are obtained: k

k

i =0

i =0

∑ κG A(k − i)(i + 1)(i + 2)W (i + 2) +∑ kϕ (k − i)(i + 1)(i + 2)W (i + 2) + k

k

k

∑ κG(k − i + 1) A(k − i + 1)(i + 1)W (i + 1) + ∑ (k − i + 1)kϕ (k − i + 1)(i + 1)W (i + 1) − L2 ∑ k w (k − i)W (i) i =0

i =0

i=0

k

k

k

i =0

i=0

i =0

− L ∑ κG A( k − i )(i + 1)ϕ (i + 1) − L∑ κG (k − i + 1) A(k − i + 1)ϕ (i) + L2ω 2 ρ ∑ A( k − i )W (i ) = 0 k

k

k

i =0

i =0

i =0

(9)

L∑ κG A(k − i)(i + 1)W (i + 1) + ∑ E I (k − i)(i + 1)(i + 2)ϕ (i + 2) + ∑ (k − i + 1) E I (k − i + 1)(i + 1)ϕ (i + 1) k

k

i =0

i =0

− L2 ∑ κG A(k − i )ϕ (i) + L2ω 2 ρ ∑ I (k − i)ϕ (i ) = 0

(10)

Using Equations (9) and (10), it is possible to express all DT terms of transverse displacement and bending angle of rotation, W (k ) and ϕ (k ) , respectively, in terms of four terms, namely W (0) , W (1) , ϕ (0) and ϕ (1) . Once , obtaining the IDTs of W and ϕ , the IDTs of bending moment and shear force are computed using Equation (6) via the theorems presented in Table 1. In this paper, two different types of boundary conditions, i.e., clamped-free and simply supported beams, are considered. Each one can be described as

W ξ = 0 = 0 ϕ ξ =0 = 0 V ξ =1 = 0 M and W =0 M ξ =0

ξ =0

ξ =1

=0

for cantilevered beam

= 0 W ξ =1 = 0 M ξ =1 = 0

for simply supported beam

(11) (12)

Imposing the boundary conditions, four simultaneous equations are obtained, which can be presented in matrix form as

⎧W (0) ⎫ ⎪ ⎪ ⎪W (1) ⎪ m Cij (ω ) 4×4 ⎨ ⎬=0 ⎪ ϕ ( 0) ⎪ ⎪ ϕ (1) ⎪ ⎩ ⎭

[

]

, (i, j = 1,2,3,4)

(13)

where the superscript, m , implies the number of DT terms taken into account. Setting the determinant of [C ] to zero, the characteristic equation of the system is obtained from which the natural frequencies of the structure are computed. Let Ω i denote the ith estimated non-dimensional natural frequency corresponding to m

Ωi

m −1

associated with desirable accuracy :

128

m terms and

m − 1 terms. The following criterion helps one decide the sufficient value for m to hold a


October 2010


Ω im − Ω im−1 ≤ ε where

ε is

a small value. Considering a beam, whose cross-sectional area and moment of inertia vary as

A(ξ ) = A0 (1 − 0.2ξ ) and I (ξ ) = I 0 (1 − 0.2ξ ) , Figure 2 shows Ω1 − Ω1 m

m −1

for different values of

m . It is

observed that as a greater number of terms are considered, the above inequality converges to zero, in a way that for

m = 20 the inequality is satisfied for ε = 10−6 .

Figure 2: Convergence of the method with respect to m

4. NUMERICAL RESULTS AND DISCUSSIONS

In what follows, three different problem configurations are studied for which the natural frequencies are determined. 4.1. Example 1. Prismatic Beam

Consider a prismatic beam with modulus of elasticity E = 2.1× 1011 Pa , shear modulus G = 3E / 8 , beam length L = 0.4m , cross-sectional shape factor κ = 2 / 3 , mass density ρ = 7850kg / m3 , and rectangular cross-section with 0.08m height and 0.02m width. The first nine natural frequencies of the beam are determined and compared with Ruta [18], as reported in Table 2. Table 2. Natural Frequencies of Beam in Example 1 Clamped-Free Simply Supported

Present

Ruta [18]

Present

Ruta [18]

ω1

2529.4926

2529.4927

6838.8333

6838.8336

ω2

13279.9048

13279.905

23190.8264

23190.827

ω3

31044.7902

31044.791

43443.4922

43443.493

ω4

50825.8335

50825.834

64939.1839

64939.1850

ω5

71565.0461

71565.047

86710.8977

86710.889

ω6

91994.8230

91994.824

108431.3438

108431.34

ω7

110975.9806

110975.98

111981.2934

111981.29

ω8

119244.5763

119244.57

120647.2391

ω9

131606.5217

131606.52

130003.6125

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120647.23 130003.61


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4.2. Example 2. Non-Prismatic Beam

Assume a simply supported non-prismatic Timoshenko beam, whose cross-sectional area and moment of inertia vary as 3 A(ξ ) = A0 (1 − αξ ) , I (ξ ) = I 0 (1 − αξ )

(14)

The dimensionless parameter r = I 0 / A0 is introduced to configure the problem. It is assumed that κ = 5 / 6 ,

E / G = 2.6 and r = 0.0707 . The dimensionless natural frequencies are obtained and compared with Auciello and Ercolano [33] as tabulated in Table 3. It is observed that the results are highly compatible with those in the literature. In order to investigate the convergence of the method, different number of terms, m , is used in IDTs and the relative errors in determination of the first natural frequency for different values of taper ratio are calculated and plotted in Figure 3. The results in Eisenberger [27] are considered as exact natural frequencies and used as the basis in relative error calculations. It is observed that the relative error increases with taper ratio. As expected, a more accurate modelling is obtained as a greater number of terms are considered, as shown in Figure 3. Table 3. Dimensionless Natural Frequencies Ω = ω L2 ρ A0 of Beam in Example 2 i i EI 0

α

0.5

0

-1

Present

Ref. [33]

Present

Ref. [33]

Present

Ref. [33]

Ω1

6.754

6.765

9.022

9.023

11.893

11.901

Ω2

24.353

24.462

29.901

29.914

36.403

36.427

Ω3

47.281

47.371

55.173

55.201

62.798

63.004

Ω4

72.629

72.674

81.754

81.817

67.976

68.143

Ω5

98.968

99.231

108.602

108.856

89.613

89.984

4.3. Example 3. Non-Prismatic Beam Resting on Two-Parameter Foundation

Consider a clamped-free non-prismatic Timoshenko beam with properties similar to those of Example 2 resting on a foundation with dimensionless parameters α = K w L4 / EI = 1 and sϕ = Kϕ L2 / π 2 EI = 1 . The first four dimensionless natural frequencies of the beam are tabulated for different values of taper ratio in Table 4. 8 7 6 5

Taper 0.6

4

Taper 0.5

3

Taper 0.4 Taper 0.66

2 1 0 -1

10

15

20

25

30

35

40

45

50

Figure 3: Relative error in determination of natural frequencies (Example 2)

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Table 4. Dimensionless Natural Frequencies of the Beam in Example 3

α

First Mode

Second mode

Third mode

Fourth mode

0.0

12.5571

40.5664

71.4005

105.7895

0.3

12.3580

39.6259

69.7041

102.4202

0.5

12.3434

39.1951

68.7980

100.9933

0.7

12.6228

39.2083

68.4403

100.3003

0.8

13.0199

39.6273

68.7900

100.5918

5. CONCLUSIONS

The practical problem of non-prismatic Timoshenko beams resting on two-parameter foundations were studied by using the well-established numerical-analytical technique, namely DTM. The provided numerical examples validate the proposed method for determining the natural frequencies. It was observed that the number of terms used in IDT of both transverse displacement and bending angle of rotation can directly affect the accuracy of the results, particularly for higher modes. REFERENCES [1]

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