Stationary Random Vibration Characteristics Of Viscoelastic Timoshenko Beam Under Complicated Boundary Conditions

In recent years,with the development of viscoelastic materials and stochastic process theory,more and more theoretical achievements are applied to engineering structures,making internal damping and external excitations of the systems match actual circumstances.In the present paper,the stationary random vibration characteristics of viscoelastic Timoshenko beam are investigated.(1)According to Alembert principle,it is easy to derive the partial differential equation sets governing the vibration of viscoelastic Timoshenko beam subjected to random excitations.Two types of damping mechanism,i.e.the external viscous damping and the internal viscoelastic damping(Kelvin-Voigt model),are taken into account simultaneously.The excitation forms involve the concentrated force,the distributed force and external moment,and they are ideal white noise,limited white noise and exponential noise in the time domain.(2)The boundary conditions involve in four different systems,i.e.clamped-free,clamped-lumped mass,clamped-lumped mass and spring,and supported-lumped mass and spring.In the light of the method of separation of variables and boundary conditions,the frequency equations,the displacement mode shape function,the angle mode shape function and the orthogonal relation of vibration mode shapes corresponding to each system,can be deduced respectively.With the aid of the orthogonal relation of vibration mode shapes in line with each system to decouple the partial differential equation sets,the stable forced vibration response of the viscoelastic Timoshenko beams can be derived by employing Duhamel integral and the mode superposition method.(3)In terms of the theory of stochastic process,the stationary random transverse vibration of the viscoelastic Timoshenko beam is researched to gain the analytical solutions of statistical properties with respect to three types of stationary stochastic model,and the statistical properties incorporate the autocorrelation functions,the power spectral density functions and the mean square response of displacement,velocity and acceleration of the steady-state displacement response.Owing to two types of damping mechanism are considered concurrently,the products of the modal damping ratio and the natural frequency are not both constants anymore,directly resulting in the infinite integral complex calculations of the mean square response.By using residue integral method and integration by partial fraction,the infinite integral and definite integral are transformed into the expression of the modal damping ratio and the natural frequency,that is,the analytical solution of the mean square response are got.(4)Compared with the random characteristics of the viscoelastic Euler-Bernoulli beam,the simulation results are obtained in the MATLAB software.The results are as follows:1)The decay rate of Euler beam is significantly faster than the Timoshenko beam.2)If the statistical property of the excitation is ideal white noise,the left-clamped Timoshenko system is replaced with Euler-Bernoulli system,however,the supported system is not.3)If the statistical property of the excitation is exponential noise,under the simple boundary the Timoshenko system is replaced with Euler-Bernoulli system,however,under the complex boundary it is not.4)In accordance with the numerical solutions of[Iik]i×k,[I'ik]i×k and[I"ik]i×k,the infinite series of the mean square response can be truncated into a series with finite terms approximately.The truncation method can satisfy the precision requirement,but also can improve the efficiency of calculation.5)Whether the statistical property of the excitation is ideal white noise or exponential nose in the time domain,the maximum value of the mean square response occurred at the free end.