The Analysis Of Forced Vibration Response Of Modified Timoshenko Beam Theory And Its Application

The research on transverse vibration of the beam can date back to the 18th century.The Euler-Bernoulli beam theory was presented by Leonhard Euler and Jacob Bernoulli based on the famous classical assumptions.Later,this theory was improved gradually and more advanced beam models were established such as Rayleigh,shear and Timoshenko model.However,based on the classical Timoshenko beam theory,Chen Rong and Wan Chunfeng take into account the influence of the moment of inertia caused by the shear deformation and proposed a modified Timoshenko beam theory.The purpose of this paper is to calculate the natural frequency of modified Timoshenko beam theory and its dynamic response when subjected to forced vibration,then to verify its rationality.This paper systematically introduces and summarizes several beam theories which respectively are Bernoulli-Euler beam theory,Rayleigh beam theory,shear beam theory,Timoshenko beam theory and modified Timoshenko beam theory.Besides,the natural frequency equation and the mode function of the above-mentioned various beam theories are obtained under three kinds of boundary conditions(hinged-hinged,clamped-clamped and clamped-free).Then,the natural frequency of the above-mentioned various beam theories are calculated under three length of the beam and three kinds of boundary conditions and compared with the simulated values calculated by ABAQUS,a finite element software.From the comparison,we derive the regularity of the changes of the natural frequencies calculated by the various beam theories with the factors changes of order,the boundary condition and the length of the beam.It can be concluded that the natural frequency calculated by modified Timoshenko beam theory is more practical and more accurate,and that when the frequency is low,the effect of the modification is very small and can be neglected,but,when the frequency is much high,the effect is very obvious.The orthogonality of the modes is the key point to solve the dynamic response problems while using the mode superposition method.This paper deduces the vibration orthogonality of the classical Timoshenko beam theory and modified Timoshenko beam theory.After that,the respective generalized mass and generalized force are obtained,and further the respective time functions are obtained.Finally,the displacement response of midspan when subjected to a simple harmonic load calculated by two kinds of beam theory is obtained and compared with the ABAQUS simulation value.By comparison,the moment of inertia caused by shear deformation has a greater impact when the beam tends to be deep beam,resulting in the calculation error of Timoshenko beam is increasing,while the error of modified Timoshenko beam has been stable at a small level.This is consistent with the conclusion obtained in the previous section that the moment of inertia caused by the shear deformation has a greater effect in the high frequency band.Then,this paper uses the modified Timoshenko beam theory to calculate the midspan displacement response of the bridge structure when subjected to a moving point load.Comparing the solution with the ABAQUS simulation results,we found that the error between the two is very small.This shows that the modified Timoshenko beam theory can be applied to the calculation of bridge structures.Finally,this paper concludes that the modified Timoshenko beam theory has an excellent applicability and higher accuracy for calculating the natural frequency and the displacement response when subjected to forced vibration.