Analysis Of Transverse Vibration Characteristics Of Functionally Graded Beams With Variable Cross-Sections By The Differential Quadrature Method

With the rapid development of society,the safety and stability standards of building structures have been improved,especially under special circumstances,higher requirements have been placed on the performance of building structural materials.So materials with special properties are bound to become a new research field about building materials.Meanwhile,the structural safety and stability of highperformance materials are also the focus of research on this type of problem.As a numerical calculation method,the differential quadrature method has simple formula,convenient use,less calculation and high precision,coupled with the rapid development of computer technology,DQM has gradually become a common technical method for solving mathematical problems in engineering.Therefore,this paper mainly uses the differential quadrature method to study the transverse vibration characteristics of a functionally graded beam with variable cross-sections whose material properties change continuously along the axial direction.The relevant numerical results are obtained in order to provide theoretical guidance and help for engineering practice.Firstly,based on Euler-Bernoulli and Timoshenko beam theories,the mechanical models of free vibration and stability of the axially functionally graded beam with variable cross-sections and steady-state harmonic vibration were established respectively and the corresponding mathematics are derived by combining the constitutive relationship of the beam systems of control differential equations.Secondly,according to the basic principle of the differential quadrature method,the eigenvalue problem of the free vibration and stability differential equation(set)of the axially functionally graded beam with variable cross-sections is transformed into the eigenvalue problem of the linear algebraic equations satisfying the corresponding boundary conditions;Similarly,the two-point boundary value problem of the steady-state harmonic vibration differential equation(set)of the axially functionally graded beam with variable cross-sections is transformed into the two-point boundary value problem of the linear algebraic equation system that satisfies the corresponding boundary conditions.Finally,the numerical results of the transverse vibration characteristics of the axially functionally graded beam with variable crosssections are obtained by programming calculations and corresponding parametric analysis studies are carried out for the obtained numerical results.Since the discrete way of nodes in DQM has a close relationship about the determination of the weight coefficient matrix and the calculation accuracy of the differential equation,this paper proposes a non-uniform way of nodes:the dispersion of nodes in a geometric sequence.The geometric way of points can suitably arrange more dense and non-uniform nodes at both ends of the beam,avoiding drastic changes in the displacement and internal forces near the beam ends due to changes in the rigidity of the areas near the ends of the beam.Meanwhile,the geometric sequence of nodes is beneficial to reduce the truncation error of the weight coefficient matrix and improve the calculation accuracy of the numerical results of the DQM.Numerical results show that the geometric sequence of nodes proposed in this paper is either to calculate the natural frequency and critical load during free vibration and stability or to calculate the displacement and internal force about the steady-state harmonic vibration of the axially functionally graded beam with variable cross-sections,satisfactory and stable numerical results can be obtained.Compared with the uniform node,the geometric sequence node improves the calculation accuracy,stability and convergence of the numerical results of the DQM,while retaining the geometric characteristics and calculation accuracy of the Chebyshev polynomial root node.The number of node elements n is the only parameter for the calculation accuracy of the uniform node and Chebyshev polynomial root node to control algorithm.During the number of node elements n is small,the error between the actual value and the calculated value about the uniform and Chebyshev polynomial root node is large.The geometric sequence of nodes can also reduce the numerical dissipation of the algorithm and improve the calculation accuracy of the numerical results by adjusting the size of the common ratio q1,which is a significant advantage of the geometric sequence node compared to others.Based on the superiority of the geometric sequence of nodes form algorithm and the good calculation accuracy of the numerical results,the calculation of the steady-state harmonic dynamic characteristics of the axially functionally graded beam with variable cross-sections using the DQM under the geometric sequence of nodes is further studied.Parametric analysis of resonance load excitation frequency,non-uniform parameters and taper coefficients can provide a certain theoretical reference for engineering practice.