Investigation On Application Of Numerical Manifold Method To Dynamics Of Beams And Mass Matrix

Beams,as the most fundamental and commonly used components in the engineering structure,are widely used in many different types of structures,play a vital role in aviation,machinery,civil construction and many other fields.So far,many individual beams and composite beams,such as track beams and foundation beams,have been emphasized on the effects in the analysis of the dynamic responses of beam structures.Therefore,the dynamic reaction and vibration characteristics of beam structures are critical and meaningful to the engineering safety and reliability.Up to now,using diagonally-lumped mass matrix of finite element method in analysing dynamic reaction and vibration characteristics of beam structure has become more common.However,none of existing mass lumping schemes that are mathematically rigorous(ignored mass of rotational degree of freedom)are available to maintain the lumped mass matrix symmetric and positive definite,which have difficulty in generating symmetric and positive definite matrices,causing great inconvenience to analyses in both the time and frequency domain.This paper bases on the numerical manifold method,within the framework of the partition of unity(PU).The PU functions and local approximations on the patches are retrieved from Hermitian interpolations.Next,applying the compactness of the weight function and the local approximation on the physical patch to construct the diagonal block mass matrix.To avoid repeated computations in a manifold element,the system mass matrix is formed by assembling the element mass matrices.In addition,compared with commonly used Row-sum method and diagonal scaling procedure(HRZ)in mass lumping scheme,the proposed method could achieve higher precision and efficiency,especially for higher order modes.Besides,this method has a solid mathematical foundation,is accordingly more reliable than the above methods and will have a wider prospect.The main works in this thesis are as follows:1)The diagonally-lumped mass matrix is derived based on the finite element method and the numerical manifold method,which is rigorous in mathematics.2)The results from the proposed method,Row-sum,HRZ,and the consistent mass representation are compared by taking the Euler-Bernoulli beam as an example.3)The dynamic response and vibration results under multiple boundary types,simply-supported,cantilever,clamped,sliding-supported,are compared.4)Dynamic calculation in the time domain and frequency domain of the beam structure are carried out respectively.5)Based on the Euler-Bernoulli beam theory,this thesis has presented a way that variable section beam can be modeled as a series of segments which connected by continuity condition.Consistent mass matrix(CMM)and diagonally-lumped mass matrix(DLMM)is used in analysing the dynamic response of the Winkler foundation beam and the Pasternak foundation beam.Then the results between the proposed mass lumping method and traditional mass lumping method(including CMM and DLMM)are compared.