插值矩阵法分析FGM矩形截面梁的自由振动OA

The lateral free vibration characteristics of Timoshenko beams made of functionally graded materials(FGM)along the thickness direction are analyzed by the interpolation matrix method(IMM).Firstly,based on Hamilton's principle,the motion differential equation system for functionally graded material Timoshenko beams is derived.Then,relying on the principle of the interpolation matrix method,the eigenvalue problem of the motion differential equation system for Timoshenko beams is transformed into a standard generalized algebraic equation eigenvalue problem.Finally,the QR method is adopted to solve the generalized algebraic equation system,thereby obtaing the natural frequency of the transverse free vibration of Timoshenko beams.In this paper,the first 10 natural frequencies and their corresponding mode functions are presented simultaneously.Numerical examples show that the greater the number of discrete elements along the beam length,the higher the calculation accuracy.When the number of discrete elements is doubled,the convergence between the calculated values and exact solutions in this paper is accelerated.Specifically,when the number of discrete elements n is set to 40,the IMM approximate solutions and exact solutions in this paper achieve full convergence,which verifies the feasibility and accuracy of the IMM and confirms its ability to meet the practical engineering requirements.Meanwhile,the calculation results also indicate that the height-to span ratio of Timoshenko beams(δ=h/L)exerts a relatively minor impact on the low-order natural frequencies of functionally graded material Timoshenko beams,but has a more significant impact on the high-order natural frequencies.