A Novel Numerical Method For Structural Mechanics Based On Chebyshev Polynomials Theory

The development of aerospace engineering prompts the applications of new materials and new structures,which causes new challenges in mechanics.Although the finite element method has been widely used in engineering fields as a mature numerical method.However,there are still some defects.For example,it is tedious to mesh repeatedly when the connections between sub-structures change.It is also not convenient to specify the material properties when analyzing the structures made of functionally graded materials which vary in one or several directions.Inspired by some practical conundrum,a novel numerical method for structural dynamics,called Chebyshev element method,is proposed in this paper.Its theories are presented,and element equations for some typical structures are derived.Then this method is implemented to investigate the dynamics characteristics of a micro-endmill,axially functionally graded beams,and a double-beam system.Utilizing the properties of Chebyshev polynomials and adopting the Gauss-Lobatto sampling method,the inner-product matrix and weighted inner-product matrix in spectralTchebychev technique are re-derived in order to make them diagonal.The spectralTchebychev is promoted and extended to two-dimension problems.Take an Euler-Bernoulli beam,for instance,the procedures of Chebyshev element method are illustrated.In this method,deformations in a structure are approximated by truncated Chebyshev expansions,and Gauss-Lobatto sampleing is adopted for discretization.The Lagrange of a system can be calculated by the improved spectral-Tchebychev technique.In this way,discrete governing equations can be derived directly by applying Lagrange's equation.Projection matrices are applied to handle both linear boundary conditions and compatibility conditions between adjacent elements.Element equations for Timoshenko beam,spinning Timoshenko beam,Kirchhoff rectangle plate,Mindlin rectangle plate,etc.are derived,and then are verified by comparing with analytical results or numerical solutions in literature.It is found the overall convergence is approximately exponential,and the method possesses a high degree of accuracy and weak dependence on meshing.Besides,the dynamics of a micro-endmill is also studied in order to investigate the influences of its geometry parameters.Then the proposed method is applied to analysis the dynamics of axially functionally graded(AFG)beams.A uniform discrete governing equation for uniform and nonuniform AFG beams are derived,not only for Euler-Bernoulli model but also for Timoshenko model.We validated their competence by comparing the results with those in literature.Then,we studied the free vibration of an AFG beam spinning with constant angular speed about its longitudinal axis.Its whirling frequencies,critical speeds and mode shapes are calculated in order to investigate the influences of axially functionally graded material(FGM).Results show that the axially FGM has significant influences on the whirling frequencies,critical speeds and mode shapes.Especially for cantilever beam,by choosing an appropriate material gradient index,higher whirling frequencies and critical speeds than those of homogeneous beams can be obtained.These results inspire us a new approach of improving the dynamic performance of a spinning beam,i.e.using optimized axially FGM.The dynamic effects of cable attachment on beams with various boundary conditions are also investigated using the Chebyshev element method.The beam-cable system is modeled as a double-Timoshenko-beam system connected by springs at several points.By utilizing high order Chebyshev polynomials as basis functions and meshing the system according to the locations of its connections,precise numerical results of the natural frequencies and mode shapes can be obtained using only a few elements.The effects of cable's parameters and layout of connections are studied.The results show that the modes of the beam-cable coupled system can be classified into two types: beam mode and cable mode,according to their main deformation.To make sure its first mode is the beam mode,cable's parameters should be in a reasonable range.In addition,the locations of connections prove a considerable influence on system's natural frequencies.