A numerical method for free vibration analysis of beams

^IUniversity of Novi Sad, Faculty of Civil engineering Subotica, Kozaracka 2a, Subotica, Serbia

^IIUniversity of Novi Sad, Faculty of Civil engineering Subotica, Kozaracka 2a, Subotica, Serbia. [e-mail]

^IIIUniversity of Novi Sad, Faculty of Civil engineering Subotica, Kozaracka 2a, Subotica, Serbia. [e-mail]

ABSTRACT

In this paper, a numerical method for solution of the free vibration of beams governed by a set of second-order ordinary differential equations of variable coefficients, with arbitrary boundary conditions, is presented. The method is based on numerical integration rather than the numerical differentiation since the highest derivatives of governing functions are chosen as the basic unknown quantities. The kernelsof integral equations turn out to be Green's function of corresponding equation with homogeneous boundary conditions. The accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature. It is shown that good accuracy can be obtained even with a relatively small number of nodes.

Keywords: Numerical method, Integral equations, Green's function, Vibration, Timoshenko beam.

1 INTRODUCTION

The vibration analysis for structures is a very important field in engineering and computational mechanics. These dynamic problems are classically described by a system of ordinary differential equation associated with a set of boundary conditions. Mostly, these problems cannot be solved or are difficult to solve analytically. Alternatively, the numerical methods can provide approximate solutions. Numerical methods such as finite element (Thomas and Abbas (1975); Jang and Bert (1989); Klasztorny (1992); Cleghorn and Tabarrok (1992); Friedman and Kosmatka (1993); Boukhalfa and Hadjoui (2010); Hamedani et al. (2012)), finite difference (Popplewell and Chang (1996); Laura and Gutierrez (1993); Fu-le and Zhi-zhong (2007)), differential quadrature (Liu and Wu (2001); Bert et al. (1988); Bert and Malik (1996); Janghorban (2011); Mirtalaie et al. (2012); Rajasekaren (2013)), the dynamic stiffness matrix (Banerjee (1998, 2001); Li et al. (2004); Hashemi and Richard (1999)) and some other methods have been used in solving free vibration problems of structures.

The present paper follows the approach that was first proposed by Hajdin (1958), and later also used in further works by Krajcinovic and Herrmann (1970) and Hajdin and Krajcinovic (1972). The main point of proposed method for numerical solutions of the ordinary differential equations contains in choosing of the highest derivative in these equations for the basic unknown. The unknowns are determined through the corresponding integral equations which kernels are influence function for bending moment of a beam.

In the present paper, the above approach is applied to a set of second-order ordinary differential equations of variable coefficients, with arbitrary boundary conditions, and their application to the free vibration of beams. In the formation of the system of linear algebraic equations from which can be obtained a characteristic equation of natural frequencies, we use in this paper statically-kinematic analogy that exists between the generalized displacements and cross section forces in the appropriate fictitious beam. In that case, the generalized displacements can be determined by known methods of structural analysis used to determine the cross section forces. In contrast to FEM where the displacement field is prescribed, in the present method, as already mentioned, the field of highest derivative of function is assumed, so the solution is obtained through numerical integration. With a relatively small number of points it can be achieved satisfactory accuracy of the results.

2 THEORETICAL CONSIDERATIONS

Consider the problem governed by a system of homogeneous ordinary differential equations of second order, with suitable boundary conditions at both ends, z = 0 and z = L. This system of equations can be written in the matrix form as follow

where A, B, C and D are known N x N matrix function of z, q is a vector whose elements are N unknown functions (displacement parameters)

and is circular frequency, which corresponds to the solution for the non-trivial case. Primes in equation (1) denote differentiation with respect to z.

The second derivatives of any component of vector q, shall be denoted

In addition to differential equation (3) the following boundary conditions are given

The solution of Eq. (3) may be written as (Byron and Fuller 1992)

where is the integration variable and is Green's function corresponding to the differential equation (3), which obeys homogeneous boundary conditions (4). Function , known as kernel of integral equation (5), is defined by

If is understood as external transverse load distributed along the axis of the beam, and as a bending moment and a transversal force at the left end of the beam, respectively, and as bending moment along the beam, than integral equation (5) defines the dependence between the cross section bending moment and external load of the fictitious cantilever beam of length l, fixed at the right end, Fig. 1. From this it follows that Green's function is the influence function for the bending moment of a cantilever beam, well known to civil engineering.

Differentiating equation (5) we get

where

is influence function for transversal force of the cantilever beam.

3 NUMERICAL SOLUTION

The values of definite integral in equation (5) can be expressed approximately, using method for numerical integration. Unknown function , which by static analogy with beam in bending represents the load, will be shown in the form of polygonal line with characteristic values at selected points k (k = 0, 1, 2,.., M) of the equidistant spacing

where is linear function which exists only along parts between the point k, where it takes the value 1, and adjacent nodes, where it takes the value 0, Fig. 2. The ordinate of function at point k = 0, is determined by linear extrapolation, so that .

Substituting Eq. (9) into Eq. (5) we obtain

where

represents influence line for bending moment in section z of cantilever beam, caused by unit triangular load, Fig. 3.

The functional equation (10) may be transformed into a system of algebraic equations if the argument z is assigned at discrete points j , in which the integration is performed

where

and

Taking into account that we have the additional members

Differentiating equation (10) with respect to z, and applying the previous transformations, in a similar way we obtain

where

Also, from equation (3) it follows

Equations (12), can be represented in matrix form as

or explicitly

where and (j = 1, 2,..., M) are the vectors of nodal displacements and "load" of the element, all of which, in general, has N components.

The matrix forms of Eqs. (16) and (19) are given by

Conditions that differential equations (1) are satisfied at all discrete points j (j = 1, 2,..., M), can be expressed as

where

Substituting eqs. (20), (22) and (23) into Eq. (24) yields

from which we obtain

where

Taking into account equations (21) and (22) we can establish a connection between the values of the vector q and q' on the right and left end of the element

or

Substituting (27) into (30) we get

At the beam ends, depending on the support conditions, boundary conditions are given by the forces and/or displacements. Consider the most general case of mixed boundary conditions at both ends of the beam. Geometric quantities (displacement parameters), as before, will be marked with q and static (cross-section forces) with r. For the left and right ends of the beam, we introduce generalized vectors

Between vectors and , and and can be established a connection

Multiplying the left side of the matrix equation (31) with matrix and taking into account the equations (33) and (34), we obtain

or

where

This is a system of 2×N linear homogeneous algebraic equations for the 4×N unknowns. If homogeneous boundary conditions at the ends of the beam, which number is 2×N, enter in the equation (36) we get the homogeneous system of 2×N equations with 2×N unknowns. This system, representing an algebraic eigenvalue problem, can have a nonzero solution only when the determinant of the equation system vanishes.

4 NUMERICAL EXAMPLES

In this section, the introduced method will be employed in analyzing the free vibration of beams with different boundary conditions. Two numerical examples are presented to demonstrate the applicability of the proposed method, and numerical results by the present study are compared to those reported by other researchers.

4.1 Example 1

As the first illustrative problem, consider the transversely vibrating uniform Timoshenko beam. The free harmonic vibrations of beam are defined by homogeneous differential equations of motion

and appropriate end conditions

• The geometric boundary conditions:

• The natural boundary conditions:

Bending moment (M):

Shear force (Q):

In above equations and denote the amplitudes of the sinusoidally varying transverse displacement and flexural rotation, the circular frequency, the material density, F the cross sectional area, I the second area moment of inertia about the neutral axis of the beam cross section, EI and kGF the flexural and shear rigidity of beam, respectively and the mass per unit length. Primes denotes differentiation with respect to coordinate .

Displacement state vector , Eq. (2), consisting of 2 displacement parameters

and matrix A, B, C, D and E are as follows

For the cantiliver glass-epoxy composite beam of Han et al (1999), clamped at the left end

and for the structural and material properties as follows:

in Table 1 are listed the first six natural frequencies. In the same table, the results of Han et al (1999) are also given for comparison.

Close agreements in natural frequencies are obtained between the proposed method and method of Han et al. (1999). For k = 12, the differences between the two methods are less than 3%. As can be seen from Table 1, a small number of interior points are required for a good prediction of frequencies.

4.2 Example 2

The governing partial differential equations of harmonic motion, with circular frequency , for the laminated composite Timoshenko beam exhibiting coupled flexure - torsion free natural vibration are given by

where , and are transverse deflection, bending rotation and twist angle of the beam, respectively. Beside the above differential equations, geometric and natural boundary conditions must be taken into account

• The geometric boundary conditions:

• The natural boundary conditions:

Bending moment (M):

Shear force (Q):

Torque (T):

Displacement state vector q, Eq. (2), is given by

Matrix A, B, C, D and E are as follows

The beam model studied by Mirtalaie et al (2012) is solved in this study as an illustrative example, by using proposed method. It is a cantilever glass-epoxy composite beam with a rectangular cross section with width 12.7 mm and thickness 3.18 mm. Unidirectional plies each having fiber angles of +15 are used in the analysis. The data used for the analysis are as follows:

bending rigidity (EIy) = 0.2865 Nm²;

torsional rigidity (GJ) = 0.1891 Nm²;

bending±torsion coupling rigidity (K) = 0.1143 Nm²;

shear rigidity (kFG) = 6343.3 N;

mass per unit length (m) = 0.0544 kg/m;

mass moment of inertia per unit length (Is) = 0.7770 x10-6 kgm;

length of the beam (L) = 0.1905 m.

Because of symmetry, only half of the beam is considered.

For k = 60, the lowest four natural frequencies of the beam are presented in Table II, where they are compared with the exact results obtained by the method of Prokić (2005 and 2006), for the pinned-pinned boundary conditions. Also, in Table 2, for clamped - clamed and free - free boundary conditions, the results of the present method are compared with the results by Mirtalaie et al (202). The numerical results of the present study are in good agreement (the differences are less than 0.2%) with exact and results by Mirtalaie et al (2012).

4 CONCLUSIONS

In this paper a numerical method is proposed which can be applied to a wide range of problems defined by a set of second-order ordinary differential equations with different boundary conditions. The solution is obtained through numerical integration. The basic mathematical operation is simple and can be readily solved by the application of matrix calculus. Numerical verification demonstrates that the proposed method is reasonably accurate, e.i. the numerical approximations, in most cases, is accurate for quite low values of k. Finally, it can be said that proposed method can serve as a convenient alternative to the similar numerical techniques in the analysis of problems defined by a system of second-order ordinary differential equations, with arbitrary boundary conditions.

The problems which are reduced to differential equations with variable coefficients and some other integration schemes will be investigated in next work.

Acknowledgements

The present work has been supported by The Ministry of Education and Science of the Republic of Serbia (Project No. ON174027).

Corresponding author: [e-mail]

• Banerjee, J.R., (1998). Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Computers & Structures 69: 197 - 208.
• Banerjee, J.R., (2001). Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams. Journal of Sound and Vibration 247: 97 - 115.
• Bert, C.W., Jang, S.K., Striz, A.G., (1988). Two new approximate methods for analyzing free vibration of structural components. AIAAJ 26: 612-618.
• Bert, C.W., Malik, M., (1996). Differential quadrature method in computational mechanics: A review. Applied Mechanics Review 49: 1-27.
• Boukhalfa, A., Hadjoui, A., (2010). Free vibration analysis of an embarked rotating composite shaft using the hp-version of the FEM. Latin American Journal of Solids and Structures 7: 105-141.
• Byron, F.W., and Fuller R.W., (1992). Mathematics of Classical and Quantum Physics, Dover Publications, Inc., New York.
• Cleghorn, W.A., Tabarrok, B., (1992). Finite element formulation of a tapered Timoshenko beam for free lateral vibration analysis. Journal of Sound and Vibration 152: 461 - 470.
• Friedman, Z., Kosmatka, J.B., (1993). An improved two-node Timoshenko beam finite element. Computers & Structures 47: 473 - 481.
• Fu-le, L., Zhi-zhong, S., (2007). A finite difference scheme for solving the Timoshenko beam equations with boundary feedback. Journal of Computational and Applied Mathematics 200: 606 - 627.
• Hajdin, N., (1958). A method for numerical solution of boundary value problems. Trans. Civ. Engng. Dept 4: 1-58.
• Hajdin, N., Krajcinovic, D., (1972). Integral equations method for solution of boundary value problems of structural mechanics, Part I. Ordinary differential equations. Int. J. Numer. Meth. Engng. 4: 509-522.
• Hajdin, N., Krajcinovic, D., (1972). Integral equations method for solution of boundary value problems of structural mechanics, Part II. Elliptic partial differential equations. Int. J. Numer. Meth. Engng. 4: 523-539.
• Hamedani, S.J., Khedmati, M.R., Azkat, S., (2012). Vibration analysis of stiffened plates using Finite Element Method. Latin American Journal of Solids and Structures 9: 1-20.
• Han, S.M., Benaroya, H., Wei, T., (1999). Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration 225: 935-988.
• Hashemi, S.M., Richard, M.J., (1999). A dynamic finite element (DFE) for coupled bending - torsional vibrations of axially loaded beams. Proceedings of the 40th AIAA/ASME/ ASCE/AHS/ ASC structures, structural dynamics and materials conference and exhibit, AIAA/ASME/AHS adaptive structures forum and AIAA forum on non-deterministic approaches, AIAA-99-1402, 1707 - 1715.
• Jang, S.K., Bert, C.W., (1989). Free vibration of stepped beams: Exact and numerical solutions. Journal of Sound and Vibration 130: 342 - 346.
• Janghorban, M., (2011). Static and free vibration analysis of carbon nano wires based on Timoshenko beam theory using differential quadrature method. Latin American Journal of Solids and Structures 8: 463-472.
• Klasztorny, M., (1992). Finite beam elements of Euler and Timoshenko type. Comput. Meth. Civil Eng. 2: 7 - 32.
• Krajcinovic, D., Herrmann, G.,(1970). Numerical solution of the dynamic stability problems. Int. J. Numer. Meth. Engng. 2: 551-561.
• Laura, P.A.A., Gutierrez, R.H., (1993). Analysis of vibrating Timoshenko beams using the method of differential quadrature. Shock and Vibration 1: 89 - 93.
• Li, J., Shen, R., Hua, H., Jin, X., (2004). Bending - torsional coupled dynamic response of axially loaded composite Timoshenko thin-walled beam with closed crosssection. Composite Structures 64: 23 - 35.
• Liu, G.R., Wu, T.Y., (2001). Vibration analysis of beams using the generalized differential quadrature rule and domain decomposition. Journal of Sound andVibration 246: 461 - 481.
• Mirtalaie, S.H., Mohammadi, M., Hajabasi, M.A., Hejripour, F., (2012). Coupled lateral-torsional free vibrations analysis of laminated composite beam using differential quadrature method. Word Academy of Science, Engineering and Technology 67: 117-122.
• Popplewell, N., Chang, D., (1996). Free vibrations of a complex Euler - Bernoulli beam. Journal of Sound and Vibration 190: 852 - 856.
• Prokić, A., (2005). On triply coupled vibrations of thin-walled beams with arbitrary cross section. Journal of Sound and Vibrations 279: 723-737.
• Prokić, A., (2006). On fivefold coupled vibrations of Timoshenko thin-walled beams. Engineering Structures 28: 54-62.
• Rajasekaran, S., (2013). Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods. Applied Mathematical Modelling 37: 4440-4463.
• Thomas, J., Abbas, B.A.H., (1975). Finite element model for dynamic analysis of Timoshenko beam. Journal of Sound and Vibration 41: 291 - 299.

A numerical method for free vibration analysis of beams

A. Proki ć ^I; M. Be š evi ć ^II; D. Luki ć ^III

Publication Dates

• Publication in this collection
  06 May 2014
• Date of issue
  Dec 2014