Effect of non-ideal boundary conditions on the free vibrations of beam with a simply support

Usually, in dynamic analysis of mechanical systems, joints are supposed to be ideal and complete and effects like clearance, local deformations, wear and lubrication are ignored. These types of boundary conditions have small deviations from the ideal conditions and named as non-ideal boundary conditions. In this research the effect of non-ideal boundary conditions on the free vibrations of a Euler-Bernoulli simply supported beam, as a weakly nonlinear system, has been analyzed and investigated. For this purpose, the boundary conditions in clearance supports are assumed to be non-ideal and the equations of vibration of the beam are analyzed by both Lindstedt-Poincaree and multiple scales perturbation techniques and results are compared. It was observed that the results of the two methods are compatible with together. In order to validate the methods, by assuming the perturbation parameter to be zero, the frequency obtained from the two methods is compared with the frequency of the previous references, which have equal results. To investigate the issue, one of the boundary conditions supposed to be non-ideal, In this case the existence of clearance increases the amplitude of the mode shape but depending on the number of mode, can lead to increasing or decreasing natural frequencies. Also, the existence of clearance in a support causes the asymmetry of the shape of modes.