Improved Complex Mode Theory And Its Application In Stochastic Vibration

As everyone knows,for a symmetrical multi-degree-of-freedom system with linear viscous damping,the real mode theory has been unable to uncouple differential equations of motion of the system.And the complex mode theory should be applied to work out the dynamic response.However,the complex algebraic operation is inevitable in the traditional complex mode theory,which causes the results distorted.In this paper,by means of the conjugacy of the complex mode parameters,e.g.the complex characteristic value and vector,the formulae to compute the accuracy solution to the time-domain dynamic response for the system under arbitrary excitation.Based on the improved complex mode theory,the truncation of complex mode superposition is investigated.An accelerating method for the complex mode superposition is proposed,and the time domain analysis of stationary random response in stochastic vibration is carried out.The concrete research contents include the following aspects:First,by making use of the artifice to separate the real and imaginary parts of the complex eigenvalues and vibration shapes,the expressions of the complex mode parameters are deduced,and their respective physical meanings are expounded.According to the conjugacy in the complex mode,the formulae seeking the dynamic response under random excitation are derived.By employing the complex mode expansion of stiffness matrix,An accelerating technique computing the response is obtained for the complex mode superposition after truncation.Moreover,on the basis of the improved complex mode theory,the expressions to calculate the response are get under the simple harmonic and slope step excitations.The respective vibration characteristics are represented afterwards.A five degree of mass-spring-damping system is studied,including the influence of damping on the natural frequency,verifying the validity of the improved complex mode theory and discussion about applicable situations of the truncation and acceleration of the complex modal superposition method in detail.Finally,in accordance with the improved complex mode theory,stationary random responses in stochastic vibration for a general viscous linear damping symmetrical system are researched in time domain.Meanwhile correlation functions and mean square value calculations of white noise and a first order filtered white noise excitation are acquired.Under the condition of white noise excitation,comparison between the time domain analysis and frequency domain analysis method is made as solving the mean square response.The response properties are investigated by virtue of numerical examples.