The Study On Energy Finite Element Method For Variable Thickness Beam And Plate Structures Varing As A Power Function

Variable thickness beam and plate structures can effectively save materials,improve the utilization rate of materials and reduce the weight of the structure,and is widely used in ship structures,aerospace structures and building structures.When the variable thickness structure is used in ships or aerospace structures,due to the lightweight design requirements,the structure becomes thinner,and the running speed is getting higher and higher,it is easy to induce high-frequency vibration inside the structure,which may damage its own structure and the electronic devices it carries and affect the stable running of the whole system.Energy finite element method is an ideal method for high frequency dynamic response analysis.At present,in the energy finite element method,the analysis of high-frequency dynamic response of structures with variable thickness is based on the energy density governing equation of uniform structures,which will inevitably lead to errors.Therefore,it is very important for the safe running of the structure to study the EFEA for predicting the high frequency dynamic response of the structure with variable thickness.Based on the theory of EFEA,this thesis derives the energy density governing equation of power-law variable thickness beam and plate structure,and studies the energy transfer relationship at the coupling boundary when the variable thickness structure is coupled with the constant thickness structure,thus the EFEA model of power-law variable thickness beam,plate and their coupling structures is established.The main research contents and conclusions of this thesis are as follows:(1)The EFEA model of the power-law variable thickness beam is established.According to the geometric acoustic approximation method,the displacement fluctuation solution of the bending deformation of the beam with variable thickness is obtained,and then the relationship between the energy density averaged by time and energy flow averaged by time of the beam is deduced from the displacement solution.Combining the energy balance relationship and energy dissipation relationship in the micro-element,the energy density governing equation of the beam structure with power-law variable thickness is obtained.Then the method of discretization and numerical solution of energy density governing equation by Galerkin weighted residual method is analyzed.Taking a beam with variable thickness under harmonic excitation as an example,the energy density distribution of each node on the beam is obtained by using the EFEA modal developed in this thesis,and the correctness of the model is proved by comparing with the results of finite element method and RFF4-5 numerical algorithm.(2)The EFEA model of the coupling structure with uniform thickness beam and powerlaw variable thickness beam is established.For the vibration problem of the coupling beam,when solving the coupling relationship,it is assumed that the uniform beam is a semi-infinite structure,and the energy transfer relationship at the coupling node is solved by setting coupling nodes at the coupling point.Combined with the energy density governing equation of the power-law variable thickness beam,the energy finite element model of the coupling power-law variable thickness beam structure is established,and the energy density results of each node on the beam can be obtained by solving.The effects of excitation frequency and the ratio of cross-section length to coupling position on energy transfer coefficient are also discussed.(3)The EFEA model of the power-law variable thickness plate is established.By exploring the relationship between the average energy density and energy flow in time and space and the energy balance relationship in the micro-element,the energy density governing equation of the variable thickness plate is derived.The matrix expression of the energy density governing equation is studied by using Galerkin weighted residual method and Green’s formula.The vibration characteristics of a power-law variable thickness plate structure are analyzed by using the power-law variable thickness plate EFEA model in this thesis.By comparing with the energy density distribution of a uniform thickness plate structure,it is found that the energy density of a power-law variable thickness plate structure presents asymmetric distribution along the thickness change direction.The energy density near the small end of the variable thickness plate is greater than that at the big end,and the energy density at the small end of the power-law variable thickness plate structure has a significant energy gathering effect,which is consistent with the conclusion of acoustic black holes.On this basis,the influence of different excitation frequencies on the structural vibration characteristics is analyzed.The results show that with the increase of frequency,the acoustic black hole effect gradually weakens at the small end of the power-law variable thickness plate.(4)The EFEA model of the coupling structure with uniform plate and power-law variable thickness plate is established.By solving the energy reflection and transmission coefficients at the coupling node and combining the energy density governing equation of the power-law variable thickness plate,the EFEA model of the coupled power-law variable thickness plate is established.The EFEA method is used to analyze the vibration problems of coupled power-law variable thickness plate structures and uniform plate structures with the same size.The results show that in the coupled power-law variable thickness plate,the energy density will suddenly change at the coupling joint,and then it will continue to decrease in the power-law variable thickness plate.There is no coupling joint in the uniform plate,so there is no sudden change in energy density.Compared with the coupled power-law variable thickness plate,the increase of transmission distance makes the energy loss effect of damping more obvious.By observing the energy density of nodes at the coupling joint,it is found that with the increase of frequency,the abrupt change amplitude of bending wave energy density at the structural coupling becomes smaller.