Natural Frequencies of Multi-Irregular Span Beams under Elastic Supports by Modal Analysis Method

Abstract

Evaluating the natural frequencies of multi-span beams with elastic supports play a major role in vibration designing and optimizing of many structures such as bridges, railways ,pipes and so on The continuity of the boundary conditions ,state space and numerical methods are normally used to investigate the vibration characteristics of such structures .Unfortunately ,such methods lead to high size matrix in dealing with the boundary value problem as the number of spans increase. In the present work, the problem is solved analytically by using Modal Analysis techniques in which the continuous system is discreteized to finite degree of freedoms in terms of the generalized coordinates A proper shape function are employed for describing the system dynamical behavior and satisfying the boundary conditions .In the present method the size of the resulting Eigen matrix depends on the number of mode chosen regardless of the number of spans. With this method wide variety of support configurations can be treated. The validly and convergence of the present method for calculating the natural frequencies is carefully checked by comparing with the exact values for two-span beams with different boundary conditions . It is found that using only (5) modes for the assumed solution gives only 2% error for two span simply supported and free ends beam , however for clamped ends the error is 8% .The present method is further checked by comparing with the Finite Element method the results show good agreements where the error is not increases 1% .The results of the natural frequencies of up to (10) equal and unequal spans beams under different boundary conditions and support stiffness are presented .The results showed that the natural frequencies can be highly controlled by proper choosing of the structure parameters and support stiffness.