NATURAL FREQUENCIES OF BEAM-MASS SYSTEMS IN TRANSVERSE MOTION FOR DIFFERENT END CONDITIONS

In this study, an Euler-Bernoulli type beam carrying masses at different locations is considered. Natural frequencies for transverse vibrations are investigated for different end conditions. Frequency equations are obtained for two and three mass cases. Analytical and numerical results are compared with each other.


INTRODUCTION
Transverse vibrations of beam-mass systems have been investigated by many researchers. Approximate and exact analysis were used to calculate the natural frequencies. The effects of mass, rotary inertia and springs were investigated [1][2][3][4][5][6][7]. Gürgöze and Batan [8] considered the numerical solution of the transcendental frequency equation. The characteristic equation was obtained using Rayleigh-Ritz method [9] and free vibrations were analyzed using Laplace transform and Rayleigh-Ritz method [10]. Maurizi and Belles [11] compared two fundamental theories of beam vibrations. Chai and Low [12] investigated the natural frequencies of a beam with a mass near the beam's ends. Low et al. [13] found that the results of experiments and the theory did not match well for beams of large slenderness ratio for centre loaded beams. Hamdan and Abdel Latif [14] compared Rayleigh-Ritz, Galerkin, Finite Elements and Exact solutions and showed finite elements method was preferable due to numerical stability and accuracy and these methods had a reasonably good accuracy and convergence rate for small attached inertia values. Özkaya et al. [15] analyzed nonlinear free and forced vibrations of a beam-mass system by considering five different sets of boundary conditions by considering the effects of the location and the magnitude of the mass on the natural frequencies. Different assumed shape functions to obtain the kinetic and potential energies of the three classical beams carrying a concentrated mass were presented [16,17]. Low et al. [18] presented both experimental and theoretical results using Rayleigh-Ritz procedure and showed that the correlation between theory and experiments was much improved when stretching effects were included. Auciello and Nole [19] determined the free vibration frequencies of a beam composed of two tapered beam sections with different physical characteristics with a mass at its end. Özkaya and Pakdemirli [20] obtained the frequencies for the clamped-clamped beam with mass and searched approximate solutions for free and forced non-linear vibrations using a perturbation method. The solutions were compared with the results of both analytical and artificial neural network method [21]. Naguleswaran [22] presented the frequency equations for all the combinations of the classical boundary conditions and for various magnitudes and positions of a single particle mass. Öz [23] and Özkaya [24] calculated the frequencies of a beam carrying mass using FEM and analytical methods, and compared with other solutions. Turhan [25] considered the problem with a single mass for various classical end conditions using approximate methods and showed that resulting formulae can be put in reasonably simple forms in the special cases where the beam is symmetrically supported.
In this study, an Euler-Bernoulli type beam carrying masses on different locations is considered. Natural frequencies for transverse vibrations are investigated for different end conditions. Analytical and numerical results are compared with each other.

EQUATIONS OF MOTION
In this section, equations of motion for different cases will be derived. The Lagrangian of the system consisting of n masses can be written as follows where n denotes number of concentrated masses, ρA is the mass per unit length of the beam, w m+1 is the displacement of the different portions of the beam which are separated by concentrated masses, M m is the concentrated mass at location m, EI is the flexural rigidity of the beam, ( . ) and ( )′ are derivatives with respect to time and spatial variables. The first two terms in equation (1) are the kinetic energies of the beam and concentrated masses respectively, the last term is the elastic energy due to bending of the beam. Invoking Hamilton's principle and substituting the Lagrangian from equation (1), performing the necessary algebra, we finally obtain the following set of linear differential equations 0 Fixed-fixed ends: Simple-fixed ends: Fixed-free ends: General end conditions for masses are as follows where α p is the ratio of concentrated masses to the mass of the beam. R is the radius of inertia. Also 1 , 0 . After inserting non-dimensional parameters, we obtain the equations of motion and boundary conditions for masses as follows 0 These equations will be solved analytically for different end conditions and masses in the next section.

ANALYTICAL SOLUTION
for the solution of equation (10), where cc stands for complex conjugate and ω is the natural frequency of the vibrations. Inserting equation (12) into equation (10), 0 is obtained. The end conditions are as follows Simple-simple ends: Fixed-fixed ends: Simple-fixed ends: Fixed-free ends: The solution for equation (13) yields the mode shapes where ω = k . If the assumed solution (19) is arranged using the end conditions in equations (13)-(17), then frequency equations are obtained for each end conditions. Frequency equations were given for a single mass (n=1) and for some end conditions in references [15,20,21,23]. For a simple-simple beam with two concentrated masses, the frequency equation is The symbolic calculations for 3 and more masses are very difficult. That's why numerical methods will be better for calculating frequencies of beams having 3 or more concentrated masses. The frequencies calculated with equations (20) and (21) and Finite Element Methods (FEM) [23] will be given in the next section.

NUMERICAL SOLUTIONS
Numerical values for the natural frequencies for the first five modes will be given in this section. In Tables 1-8, the first five frequencies are presented for simplesimple, fixed-fixed, simple-fixed and fixed-free boundary conditions. The frequencies are calculated for beams having two and three masses from equations (20) and (21) for analytical and from equations in reference [23] for FEM solutions. Analytical and FEM results are close to each other as shown in the tables. It is difficult to find the frequency equations (determinants) for three and more concentrated mass systems, that's why the numerical solutions will be simpler and faster for these cases.   Table 4. Natural frequencies of a fixed-fixed beam with three masses Exact

CONCLUDING REMARKS
The transverse vibrations of an Euler-Bernoulli type beam carrying concentrated masses are investigated using analytical and numerical methods. The natural frequencies are calculated for several boundary conditions and the comparison of frequencies is presented. FEM and analytical solution are close to each other. Since it is tedious and difficult to obtain the frequency equations, FEM will be appropriate to calculate the natural frequencies of beams having three and more concentrated masses.