Investigation On An Analytical Approach Of Stochastic Response Of A Cantilever Beam With A Lumped Mass

Nonlinear vibration of a cantilever beam is one of the most typical nonlinear vibration problems.A lot of practical engineering vibration problems can be modeled by the nonlinear vibration of a beam.Many structures can be simplified as a cantilever beam structure with a lumped mass,such as high-rise buildings with various local masses,offshore drilling platforms,water tower,towers and bridge piers.An analytical approach of stochastic response of a cantilever beam with a lumped mass is investigated in the present dissertation.The main work is as follows:(1)On the basis of the predecessors,this paper derived the vibration equation of a cantilever beam with a lumped mass.In order to remove the nonlinear acceleration term from the equation,Taylor series expansion method was proposed to replace the acceleration nonlinear term by reformulating the vibration equation.The applicability of the proposed method was also studied.(2)The Gauss-Legendre path integration solution was adopted to obtain the probability density function(PDF)of the cantilever beam.The results were compared with the results of the equivalent linearization method and Monte Carlo simulation.The applicability of the Gauss-Legendre path integration method was verified.Both the time step of the path integration method and the selection of integral area were further studied on the accuracy of the adopted method.(3)A parametric study was conducted on the PDF distribution of the response of the cantilever beam using different location of mass blocks,different dimensions of mass blocks,and cantilever beams geometry parameters.Finally,two different section size cantilever beams in practice were further studied using the proposed solution procedure.The stationary and non-stationary PDF distributions were investigated.The reliability of the cantilever beams was also discussed.