Analytical solutions and FEM results for the vibration response of a cantilever beam with two end springs subjected to the support motion

In this paper, the cantilever beam with two end springs subjected to the support motion is considered. This beam problem is analytically studied by using the method of mode superposition in conjunction with the quasi-static decomposition. The Cesàro sum technique can deal with the divergent series due to the illegal treatment of termwise differentiation of displacement response, which is solved by using the mode superposition method without decomposing the quasi-static part. In addition, the numerical method, FEM, is also used to solve this problem. The San Francisco Golden Gate Bridge Tower is a demonstrative case. Finally, a simple case with an analytical solution subjected to a sinusoidal support motion is also derived and FEM results match very well. According to the results of the support motion problem, the total solution is necessary to be decomposed into the quasi-static solution and dynamic-inertia solution in advance. Besides, the real earthquake is introduced by using the 1940 El Centro earthquake for the support motion.