free vibration analysis of timoshenko beams by dsc method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERINGInt. J. Numer. Meth. Biomed. Engng. 2010; 26:1890–1898Published online 8 June 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1279COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

Free vibration analysis of Timoshenko beams by DSC method

Omer Civalek∗,† and Okyay Kiracioglu

Faculty of Engineering, Division of Mechanics, Civil Engineering Department, Akdeniz University, Antalya, Turkey

SUMMARY

Free vibration analysis of Timoshenko beams has been presented. Discrete singular convolution methodis used for numerical solution of equation of motion of Timoshenko beam. Clamped, pinned andsliding boundary conditions and their combinations are taken into account. Typical results are presentedfor different parameters and boundary conditions. Numerical results are presented and comparedwith that available in the literature. It is shown that very good results are obtained. This method isvery effective for the study of vibration problems of Timoshenko beam. Copyright q 2009 John Wiley& Sons, Ltd.

Received 22 December 2008; Revised 20 April 2009; Accepted 21 April 2009

KEY WORDS: shear deformable beam; Timoshenko beam; free vibration; discrete singular convolution

1. INTRODUCTION

Plates and beams are widely used as structural component in various engineering applications.Therefore, free vibration analysis of such structures is a most important task for engineer in thedesign stage of civil, mechanical, aerospace and railroad applications. The shear deformation theorywas first demonstrated by Timoshenko [1] for elastic beams. After this, various shear deformationtheories were purposed for elastic beams. There are many studies in the literature on theory andanalysis of shear deformable beams and plates. Vibrations of shear deformable beams have beenof interest to the researchers for many years since they are found in wide application of variousproblems in mechanical, aeronautical and structural engineering. The majority of the availablepublications are based on the analytical and numerical solution of shear deformable beams [2–12].In the past 10 years, the methods of differential quadrature (DQ), discrete singular convolution(DSC) and meshless methods have become increasingly popular in the numerical solution ofinitial and boundary value problems [13–22]. These methods can yield accurate solutions withrelatively much fewer grid points. It has been also successfully employed for different solid andfluid mechanic problems [21–34]. The main objective of this study is to give a numerical solutionof free vibration analysis of Timoshenko beams. To the author knowledge, it is the first time theDSC method has been successfully applied to Timoshenko beam for the numerical analysis ofvibration.

∗Correspondence to: Omer Civalek, Faculty of Engineering, Division of Mechanics, Civil Engineering Department,Akdeniz University, Antalya, Turkey.

†E-mail: [email protected]

Contract/grant sponsor: Scientific Research Projects Unit of Akdeniz University

Copyright q 2009 John Wiley & Sons, Ltd.

FREE VIBRATION ANALYSIS OF TIMOSHENKO BEAMS BY DSC METHOD 1891

2. DISCRETE SINGULAR CONVOLUTION (DSC)

The DSC method is a relatively new numerical technique in applied mechanics. The method ofDSC was proposed to solve linear and nonlinear differential equations by Wei [35, 36], and laterit was introduced to solid and fluid [37–46]. It has also been successfully employed for differentvibration problems of structural members such as plates and shells [47–50]. It is shown from thesestudies that the method of DSC has high level of accuracy and reliability. It is also emphasized thatDSC method yields more efficient and accurate approximation compared with the other numericalmethods for higher-order frequencies [51–60].

The method of DSC is an effective and simple approach for the numerical verification of singularconvolutions, which occur commonly in mathematical physics and engineering. The DSC methodhas been extensively used in scientific computations in past 10 years. For more details of themathematical background and application of the DSC method in solving problems in engineering,the readers may refer to some recently published References [35–40]. In the context of distributiontheory, a singular convolution can be defined by [61]

F(t)=(T ∗�)(t)=∫ ∞

−∞T (t−x)�(x)dx (1)

where T is a kind of singular kernel such as Hilbert, Abel and delta type, and �(t) is an elementof the space of the given test functions. In the method of DSC, numerical approximations of afunction and its derivatives can be treated as convolutions with some kernels. According to DSCmethod, the r th derivative of a function F(x) can be approximated as [62]

F (r)(x)≈M∑

k=−M�(r)�,�(xi −xk) f (xk) (r =0,1,2, . . . , ) (2)

where � is the grid spacing, xk are the set of discrete grid points which are centered around x and2M+1 is the effective kernel, or computational bandwidth. It is also known that the regularizedShannon kernel (RSK) delivers very small truncation errors when it uses the above convolutionalgorithm. For this reason we use RSK in this study.

3. FUNDAMENTAL EQUATIONS

The governing equations for free vibration of Timoshenko beam can be written as [1]

kGAd2W

dx2−kGA

d�

dx+�A�2W = 0 (3)

E Id2�

dx2+kGA

dW

dx−kGA�+�I�2� = 0 (4)

By using DSC discretization, Equation (1) take the form

kGAN∑j=1

�(2)�/�,�(�x)W (xi )−kGA

N∑j=1

�(1)�/�,�(�x)�(xi ) = −�A�2Wi (5)

E IN∑j=1

�(2)�/�,�(�x)�(xi )+kGA

N∑j=1

�(1)�/�,�(�x)W (xi ) = −�I�2�i (6)

Three-types of boundary conditions are considered. These are:

Clamped (C)

�=0 and W =0 (7)

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2010; 26:1890–1898DOI: 10.1002/cnm

1892 O. CIVALEK AND O. KIRACIOGLU

Pinned (P)

M=0 and W =0 (8)

Sliding (S)

�=0 and V =0 (9)

Implementation of boundary conditions is an important task on numerical methods especiallyfree edges. In the method of DQ, for examples, a few different approaches have been proposedin the past 10 years for the implementation of boundary conditions. For free edge, this is animportant stage during the numerical modeling. In the method of DSC iteratively matched boundarymethod is an efficient and effective method for free boundary conditions. Furthermore, some newmethods such as local adaptive DQ methods [61], matched interface and boundary method [63] andfictitious-domain approach [60] have also been proposed. There is no any difficulty to implementof clamped, pinned and sliding boundary and all other combinations. For free boundary, the IMBmethod [60] is an efficient way for implementation. This is explained for vibration problem ofEuler beam [60] and eigenvalue problems in mechanics [62].

In this study, however, we cannot obtain the efficient results using the IMB method for freeboundary. Hence, the free boundary condition had not been used. We used the standard approachproposed by Wei et al. [41–43]. In these equations V and M are the shear and moment resultantsand are given by

V = kGh

(�W�x

−�

)(10)

M = E I��

�x(11)

After implementation of the given boundary conditions, Equations (5) and (6) can be expressed by

[R]{U}=�2{U} (12)

whereU is the displacements vector,R is the stiffness matrix. The frequency values for Timoshenkobeam are given by the following non-dimensional form:

�2=�L2

√�A

E I(13)

where � is the mass density, A the cross-sectional area, I the second moment of area of cross-section, E the Young modulus, L is the length of the beam and � is the circular frequency.

4. NUMERICAL EXAMPLES

In the past 50 years some numerical and analytical methods such as finite elements method, spectralmethods, DQ method and some analytical methods have widely used for modeling and analysis ofengineering problems [64–74]. The method of DSC is a new and effective method. In this section,the method of DSC is applied to study the vibration behavior of shear deformable beams. ThePoisson ratio is chosen as 0.3 throughout study. For the definition of the boundary conditions ofthe beam, for example, the symbols C-P identifies a beam having clamped support at x=0 andpinned at x= L .

To validate the accuracy of the present DSC method, the results are compared with theresults given by Lee and Schultz [11], Ferreira and Fasshauer [31], Simsek and Kocaturk [17],Eisenberger [8] for shear deformable beams. These results are listed in Tables I–V. The natural



Table I. Comparison of frequency parameters of P-P Timoshenko beam for h/L=0.02 (k= 56 ; =0.3).

DSCCBT Lee and Schultz [11] Ferreira and Fasshauer [32]Mode [11] (N =35) (N =35) N =13 N =15 N =21

1 3.1415 3.14053 3.1405 3.1405 3.1405 3.14052 6.2831 6.27471 6.2747 6.2747 6.2747 6.27473 9.4247 9.39632 9.3963 9.3965 9.3963 9.39634 12.5664 12.4994 12.4994 12.4996 12.4995 12.49955 15.7080 15.5784 15.5784 15.5786 15.5784 15.57846 18.8496 18.6282 18.6282 18.6885 18.6885 18.6884

Table II. Comparison of fundamental frequency of C-C Timoshenko beam (k= 56 ; =0.3).

DSCLee and Schultz Ferreira and Fasshauer Simsek and Kocaturkh/L [11] (N =35) [32] (N =35) [17] N =13 N =15 N =21

0.002 4.7299 4.7308 4.7299 4.7302 4.7302 4.73020.01 4.7284 4.7287 4.7284 4.7286 4.7286 4.72860.02 4.7235 4.7236 4.7235 4.7238 4.7236 4.72360.05 4.6899 4.6899 4.6902 4.6902 4.6899 4.6899

Table III. First six frequency parameters of P-P Timoshenko beam (k= 56 ; =0.3; N =13).

Mode h/L=0.002 h/L=0.01 h/L=0.02 h/L=0.1 h/L=0.2

1 3.1415 3.1413 3.1405 3.1156 3.04532 6.2831 6.2811 6.2747 6.2313 5.67153 9.4245 9.4176 9.3963 9.2553 7.83954 12.5657 12.5494 12.4995 12.1813 9.65715 15.7066 15.6750 15.5784 14.9928 11.22216 18.8474 18.7927 18.6885 17.6811 12.6023

Table IV. First six frequency parameters of C-C Timoshenko beam (k= 56 ; =0.3; N =13).

Mode h/L=0.002 h/L=0.01 h/L=0.02 h/L=0.1 h/L=0.2

1 4.7299 4.7284 4.7235 4.5796 4.24012 7.8530 7.8469 7.8282 7.3312 6.41793 10.9950 10.9801 10.9341 9.8561 8.28534 14.1361 14.1062 14.0155 12.1454 9.90385 17.2767 17.2247 17.0680 14.2324 11.34886 20.4168 20.3341 20.0869 16.1487 12.6403

Table V. First six frequency parameters of S-S Timoshenko beam (k= 56 ; =0.3; N =13).

Mode h/L=0.002 h/L=0.01 h/L=0.02 h/L=0.1 h/L=0.2

1 3.1416 3.1413 3.1405 3.1157 3.04532 6.2831 6.2810 6.2747 6.2314 5.67163 9.4246 9.4176 9.3963 9.2553 7.83954 12.5658 12.5494 12.4995 12.1813 9.65715 15.7066 15.6749 15.5785 14.9927 11.22206 18.8475 18.7946 18.6282 17.6810 12.6023



frequencies of P-P Timoshenko beam are obtained to illustrate the convergence of the solutions andthe accuracy of the DSC method. Table I shows the effects of the grid numbers for different modesof beams. Comparison of fundamental frequency of C-C Timoshenko beam for different h/Lratio is listed in Table II. For a validation, the present results are compared with other publishedresults by using pseudospectral method [11], the radial basis function-based spectral method [32],the analytical solution using third-order shear deformation theory and the classical beam theory(CBT). Tables I and II show that a good convergence and accuracy of the solutions are obtainedby increasing the grid numbers for all cases. It is seen that good results are obtained for beam byusing N =15 and M=16. Figure 1 shows the convergence of first five frequencies for differentgrid numbers. C-P boundary conditions are considered for the results given in this figure. It is seenfrom this figure that when the grid point numbers reach N =7 the present method gives accuratepredictions for the first and second vibration modes. For other modes of vibration, however, theaccurate results are obtained for N =15.

Some parametric results are also presented. Variation of frequency with h/L ratios of C-CTimoshenko beam for different mode numbers is shown in Figure 2. It is shown from this figurethat the increasing value of h/L always decreases the frequency parameter for third and fourthmodes. However, the effect of the h/L ratio on the frequency parameter is nearly insignificant forfirst mode. In other words, the results show that the thickness becomes more influence as modenumbers bigger. The effects of boundary conditions on the frequencies of the Timoshenko beamare presented in Figure 3 for different modes. It is seen that the frequency parameter for all thefour boundary conditions increases as mode number increases. This figure indicates that for agiven mode number of a Timoshenko beam, the maximum frequency values are obtained for C-Csupport condition. Furthermore, the lowest frequency values are obtained for P-S support condition.Figure 4 describes the manner of variation of frequency parameter with respect to mode numberfor different h/L ratios. As expected, the variation is linear with the mode numbers for smallvalue of h/L . Some further results have also been presented for different boundary conditions.Non-dimensional frequencies of P-P, C-C and S-S Timoshenko beam for different geometricparameters are given in Table III–V. In general, the frequencies decrease with the increasing h/Lratios.

3

8

13

18

3

N

Ω

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5

5 7 9 11 13 15 17 19

Figure 1. Convergence of frequency parameter for first five modes with gridnumbers of C-P beam (h/L=0.05).



3

7

11

15

0.03

(h/L)

Freq

uenc

y

Mode 1

Mode 2

Mode 3

Mode 4

0.05 0.08 0.1 0.13 0.15 0.18 0.2 0.23 0.25

Figure 2. Variation of frequency with h/L ratios of C-C Timoshenko beam for different mode numbers.

0

6

12

18

1

Mode Number

Freq

uenc

y

C-C P-P

P-S C-P

3 4 5 62

Figure 3. Variation of frequency with mode numbers for different boundary conditions (h/L=0.1).

5. CONCLUSIONS

In this study, using the DSC method, a numerical approach for the free vibration analysis of sheardeformable beam is presented. Several examples were worked to demonstrate the convergenceof the method. Excellent convergence behavior and accuracy in comparison with exact resultsor results obtained by other numerical methods were obtained. Although not provided here, the



0

6

12

18

24

1

Mode Number

Freq

uenc

y

h/L=0.002

h/L=0.05

h/L=0.2

2 3 4 5 6

Figure 4. Variation of frequency with mode numbers of C-C beam for different h/L ratios.

method is also useful in providing vibration solutions of Euler beam. DSC provides a controllablenumerical accuracy by using the suitable bandwidth. This is important for large-scale computations.The present study is being further developed to overcome the convergence problems encounteredin the nonlinear vibration analysis of beams. Present work also indicates that the method of DSCis promising and potential approach for computational mechanics.

ACKNOWLEDGEMENTS

The financial support of the Scientific Research Projects Unit of Akdeniz University is gratefullyacknowledged.

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free vibration analysis of timoshenko beams by dsc method

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