Dynamic Analysis Of Complex Planar Framed Structures With Damping Using Mode Superposition Method

The dynamic analysis of framed structures has always been a concern in practical engineering and also attracts attentions from many researches.The role of the damping in structure vibration is critical in structural dynamic studies.It’s known that the increasing of the damping in a structure will reduce its response to a given excitation.Thus if the damping in a structure is increased there will be a reduction of vibration,noise,fatigue of structure.So it’s necessary to take the effects of damping in consideration in the dynamic research of structures.To reduce the cost of computations and get the more accurate results,the lump mass-spring-damping system is used to model the joint the dynamic characters of which can’t be ignored in the planar framed structures in this paper,and simultaneously the continuous parameter model will be used to efficiently model the practical structure.It’s noted that the method of reverberation-ray matrix(MRRM)and the mode superposition method(MSM)will respectively be applied to solve the natural vibration characteristics and the dynamic response of damped structures.Firstly,we obtain the real natural frequencies and real natural mode of complex planar framed structures with damping by MRRM.Compared to the dynamic stiffness method,MRRM could avoid the null mode problem.And then the further works will be descripting the complex planar framed,deriving the mode orthogonality conditions with respect to the mass,stiffness and damping properties of the structure,constructing the mode superstition method to obtain the transient responses of the damped planar frames subjected to joint or/and member external force excitations.We compile the corresponding program to calculate numerical examples to validate our proposed method by comparing the results with those from the finite element method(FEM)and analyze the effects of proportional damping coefficient on the transient responses.Secondly,the MRRM is extended to solve the natural vibration characteristics of complex planar frame structures with non-proportional damping.Due to the introduction of damping in the governing equations,the natural frequency and the natural modes need to be expressed in the form of complex numbers.Therefore,we introduce a rooting method suitable for complex solutions--Müller method.The numerical examples are provided to verify the correctness and feasibility of the method compared to the computational results of ANSYS and the effects of damping on the natural frequencies and modes will be stated.Finally,we reduce the forced governing equations from second-order differential equation to first-order differential equation.On the basis of the above section,the orthogonality of complex modes about the combinations with respect to the mass,stiffness and damping of structures will be derived.Subsequently,the analytical formula for dynamic response of the complex planar frame structures with damping could be deduced by constructing the mode superstition method.The reliable of the method is demonstrated by the numerical examples calculated and the effects of the damping coefficient on the transient responses are also studied.