An integral equation formulation of the coupled vibrations of uniform Timoshenko beams Masa. Tanaka & A. N. Bercin Department of Mechanical Systems Engineering, Shinshu University 500 Wakasato, Nagano 380, Japan EMail: dtanaka@gipwc.shinshu-u.ac.jp Abstract For an asymmetric cross section Timoshenko beam infree vibration, the transverse motions in the two perpendicular directions are coupled with both torsional motion around the axial direction and bending slopes about their planes, this phenomenon results in five coupled partial differential equations. In this study a boundary integral equation fonnulation of coupled vibrations of Timoshenko beams of open cross section with non-collinear shear center and centroid is presented. A new set of fundamental solutions of the problem is derived using the Hormander method. The natural frequencies are obtained and compared with the predictions on the Bemoulli-Euler beam theory. It is seen that the errors can be unacceptably large, especially for the higher order modes, if shear deformation and rotatory inertia effects are neglected. 1 Introduction There has been considerable research interest in the free vibration analysis of elastic beams for which the effects of cross sectional dimensions on the frequencies can not be neglected [1]. The equation of motion of abeam based on the classical Bemoulli-Euler beam theory was derived under the assumption that the deflection is due to flexure only, and the effects of rotatory inertia of the mass and the shear distortion are neglected. The Bemoulli-Euler beam theory of flexural vibrations is known to be inadequate for the aforementioned beams and for higher modes. Corrections due to rotatory inertia and shear in the classical theory were added by Timoshenko [2]. Beams analyzed with such a refined theory are referred to as Timoshenko beams.
128 Boundary Elements When the beam cross section has two axes of symmetry; the shear center and centroid are coincident, therefore the vibration problem can be modeled as a set of uncoupled differential equations for extension along the axis of beam, torsion about the axis and bending in two perpendicular planes. However, when the beam has single axis of cross sectional symmetry or no cross sectional symmetry at all, the elastic axis and mass axis are separated by a distance which results in coupling between bending and torsional vibrations. As will be seen in the next section, in the formulations based on the Timoshenko beam theory bending displacements are further coupled with bending slopes, resulting in five coupled differential equations. Although much work has been proposed for the vibration analysis of Timoshenko beams, the majority of the available research is limited to beams of doubly symmetric cross section and the number of investigations dealing with the coupled vibrations of Timoshenko beams is quite few, considering the importance of the subject. This study presents a boundary integral formulation of coupled vibrations Timoshenko beams of open cross section with non-collinear shear center and centroid. To this end, a new set of fundamental solutions of the problem are derived using the Hormander method [3]. The natural frequencies for an illustrative beam is calculated and the effects of shear deformation and rotatory inertia on the coupled frequencies are investigated. A list of notation is given in the appendix. 2 Basic equations A typical thin walled asymmetric beam of length L is shown in Figure 1. The elastic axis, which is taken to coincide with x axis, is separated from mass axis by offsets e\ and e^. The flexural translations in the z and y directions are denoted by u\ and wg, respectively, while 1/5 represents the torsional rotation around the x axis. Using the Hamilton's principle the differential equations for coupled vibrations of a Timoshenko beam can be derived as follows. The potential energy density KQ and the kinetic energy density TQ of a thin walled beam can be expressed as
Boundary Elements 129 Figure 1: The geometric parameters for a beam of asymmetrical cross section dw, dt dt J " dt dt dt dt (2) d;v \df It is convenient to introduce Lagrangian density, Z/o, given by M) ~ -*0 ~~ ^0 (3) Hamilton's principle takes on the form (4) Performing the integration yields the following coupled partial differential equations of motion
130 Boundary Elements v \ ~\2 -\2 pa j--e^pa -^- = 0 (5) (6) ^ (8) Let Substituting equation (10) into differential equations (5) to (9) yields 4^=o (ii) where the non zero entries of differential operators matrix L are given as, with As is well known, an important step in the application of boundary integral equation method is the derivation of fundamental solution. In what follows, the fundamental solution of the present problem will be derived with the aid of Hormander method.
Boundary Elements 131 3 Fundamental solutions The fundamental solutions of the asymmetric cross section beams, including shear deformation and rotatory inertia, are a set of particular solutions of equation (11) under the action of a set of unit point loads, i.e. the solutions which satisfy the following inhomogenous differential equation %(^C) = -4^% (13) where 5(jt, ) is the Dirac delta function, x and denote the coordinates of the field and source points, respectively, t/*;(*, ) represents the generalized displacements in they direction at the field point x of an infinite beam when a unit point load is applied at the k direction of the source point. Using the Hormander's method [3] the solutions of equation (13) can be represented in the following form (14) in which <P(x, ) is an unknown scalar function. Substituting equation (14) into equation (13) leads to here det L*j denotes the determinant of the L* matrix, and can be expressed as where the constants («=1, 2,..., 6) are given by #1 =%%,-%", + K-3 + 2K-4, (17) KT^, - AC^6 + %,% + ^ - 2K-, 4- AC^ - XT, =-[(^ + ^7X^1^4 + ^5) + ^(^7 - ^9 + aw + ^"7(^1^2 - ^) - KjCJCs 4- ^((x-g +Kg- ^)(^2 + ^3 + ^4) + ^2^3(^4-2K"i) - 7C^ (^2 + ^3) -^"2(^5 +^7) -^(^5
132 Boundary Elements fo -fj- ^ I ^6^7 + * 4 (f 1(^7 + in which *'~ET' **~ El, ' *"'" /2 ' ^ " ^GA ' **~ r ' ^ ^ ^ ^ ' El''' El'* EF ' ' Fourier transform methods are suitable for the solution of equation (15). Hence, defining the Fourier transform pair as LtJ\i Je'~«P(a)dbe Then the Fourier transformation is seen to be -^- (20) and the inverse Fourier transformation yields ^) = Lj^l^da (21) Here A is given by 1 2 3 4 5 6 \^"^) Introducing the variable s = c^ a sixth order polynomial in 5 may be obtained, the polynomial has three unequal real negative Oi, ^» ^3) and three real positive (^4, 5-5, 5-5) roots. Thus the twelve roots of A are seen to be in the form.
Boundary Elements 133 n = 1, 2, 3; ±^G^, m = 4, 5, 6 (23) Having obtained the roots the integration can be performed using the Residue theorem, leading to 3 6 ( >( \ V* 4 -V^(*-0 v^ iv^"(-c-c) (24) where Substituting 0 into equation (14) yields the fundamental solution which can be used in the free vibration analysis of asymmetric Timoshenko beams. 4 Boundary integral equation formulation To derive the boundary integral equations of an asymmetric Timoshenko beam, the governing differential equations of motion are multiplied by the fundamental solution and integrated over the length of the beam. Subsequent integrations by part results in the following integral reciprocity
134 Boundary Elements relation in which Equation (27) provides the transverse displacements, bending slopes, torsional rotation and torsional slope at any interior point due to boundary values of shear forces, bending moments, torques, torque moments, displacements, slopes, and rotations. Considering the two boundary points of a beam, the number of boundary unknowns amounts to 24; for a well posed problem 12 of these unknowns can be determined from the boundary conditions. If the field point is taken to the boundaries 0 and L through a limiting process 10 equations can be obtained from (27), for the remaining 12 unknowns. Differentiating the final component of equation (27) i.e. (/& (DU$) with respect to the field point a supplemental equation can be obtained in the form, with d/d = -d/dx, (29) Using the same limit ing process, two additional expression can be obtained from equation (29). Then the boundary integral equations can be written in the following matrix form Hv-Gt-0 (30) where 12x12 influence matrices H(co) and G(co) consist of entries which are the values of fundamental solution and its derivatives at points 0 and L, and v and t displacement and traction vectors, respectively. After application of the boundary conditions, equation (30) may be rewritten in the form
Boundary Elements 135 B(w)q = 0 (31) Natural frequencies can be found by searching the values of CO that satisfies the following condition det[b(fi>)] = 0 (32) 5 Example application In this section the foregoing natural frequency analysis is applied to an asymmetric cross section beam shown in Figure 1. The material and geometric properties of the beam are: /, = 16680 Nnr, El, = 73480 Nm*, GJ = 10.81 Nm', T = 26.34 Nm\ kga = 896640 N, p = 7800 kgrrt\ A = 2.496x10"' m*, /, = 8.34x10'* m\ /2= 3.67x10' m\ /, =7.57x10' m\ g, = 0.02625m, ^ = 0.02316m, L = 1.5m. Table 1 Natural frequencies (Hz) of a clamped asymmetric cross section beam. Mode Timoshenko theory 1 9124 2 140.82 3 236.74 4 239.30 5 328.43 6 433.26 Bemoulli-Euler theory [4] 98.72 169.43 270.89 373.08 466.64 530.02 Error (%) 6 21 14 55 42 22 The first six natural frequencies with clamped end conditions are given in Table 1 alongside Bemoulli-Euler theory 'exact' results of reference [4]: present results are also exact since no discretization is involved in the analysis, and the differential equations together with associated boundary conditions are satisfied exactly. It can be seen from the results presented that the inclusion of shear deformation and rotatory inertia into the analysis leads to reductions in the coupled frequencies, the reduction being greater for higher modes: although the variation of reduction seems to be non-linear with frequency, this trend is due to the fact that modes 1, 3, and 6 are torsion dominated. Since the difference between Bemoulli-Euler and Timoshenko beam theories is only in the modeling
136 Boundary Elements of the flexural behavior, shear deformation and rotatory inertia effects are more pronounced on the bending dominated natural frequencies. 6 Conclusions An integral equation formulation of the natural vibrations of bending-tors ion coupled uniform asymmetric beams has been presented taking into account the effects of shear deformation and rotatory inertia. A comparison of the predictions of Bernoulli-Euler and Timoshenko beam theories has shown that when the shear deformation and rotatory inertia effects are neglected the errors associated with them become increasingly large as the modal index increases. Timoshenko effects are particularly more pronounced on the bending dominated natural frequencies. References 1. Wagner, H., & Ramamurty, V. Beam vibrations- a review. The Shock and Kz6^rzo/%Dzg^r, 1977, 8, 71-84. 2. Timoshenko, S.P. On the correction for shear of the differential equation for transverse vibration of prismatic bar. Phil. Mag. Series 6, 1921, 41, 744-746. 3. Hormander, L. Linear Partial Differential Operators. Springer-Verlag, Berlin, 1969. 4. Tanaka, Masa. & Bercin, A.N., Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Computers & Structures (to appear) Appendix: Notation A cross sectional area A polar moment of inertia EI^ El2 bending rigidities %2, %4 bending slopes T warping rigidity p density GJ torsional rigidity co circular frequency /i ^ 12 second moments of areas L cofactor matrix of L k cross section shape factor kga shear rigidity