Study Of The Natural Vibration Characteristics Of Axial-loaded Beams With Variable Section

Elastic beam is a common component widely used in a lot of structures.Accurate calculation of its natural vibration characteristics has impo rtant engineering application value for reducing its dynamic effect under dynamic loads.For the non-uniform elastic beam with the effect of axial force,its natural vibration characteristics are still not completely solved.For the rotating beam with variable section,it can also be regarded as the non-uniform beam with the action of axial tension,but the influence of?I(?)~2?/(?)t~2 should be considered for the study based on Timoshenko beam theory.Therefore,the transverse free vibration of the rotating beam is further analyzed in this paper.The specific research contents and results are as follows:(1)Based on the Euler–Bernoulli beam theory,the natural frequencies and corresponding mode shapes of the non-uniform beam with arbitrary concentrated elements and under axial load are studied.After the power series solution of the free vibration differential equation of the non-uniform beam element is obtained,the transfer matrix method is applied to establish the state vector equation for both ends of the beam.Then the frequency equation is obtained by combining the boundary conditions.The effect of the number of terms in the power series solution on frequency accuracy is investigated in this paper.The influence of factors such as spring stiffness,cross-section variation coefficient and axial force on the dimensionless natural frequency is also studied.Results indicate that the required number of terms in the power series depends on the cross-section variation coefficient and axial force when the frequency accuracy is invariable.(2)Based on Timoshenko beam theory and the concept of physics neutral surface of beam,the governing differential equations of free vibration of axially-loaded composite beam which material properties vary along the beam height are derived by using Hamilton principle.Then the power series solutions of the differential equations are obtained.Then frequency equation is obtained based on the boundary conditions.The effects of the parameters such as span-depth ratio,gradient index,axial force and cross-section variation coefficient on the natural vibration characteristics of functionally graded beam are studied.The results show that the dimensionless natural frequency and the critical load of functionally graded beam decreases with the increase of the gradient index k,and the cross-section variation coefficient.(3)Based on Timoshenko beam theory,a power series method is used to solve the differential equation of bending vibration of rotating beam with eccentric concentrated mass.The corresponding frequency equation is obtained by combining with the boundary conditions.The effects of span-depth ratio,rotating speed,cross-section variation coefficient,concentrated mass and its eccentricity on natural frequency are studied.The comparison with the results obtained based on Rayleigh beam theory shows that the influence of shear deformatio n on the natural frequency of the beam gradually increases with the decrease of span-depth ratio,while the influence of shear deformation on the natural frequency of the beam decreases with the increase of rotating speed.