The fourier spectral element method for vibration analysis of general dynamic structures

THE FOURIER SPECTRAL ELEMENT METHOD FOR VIBRATION ANALYSIS OFGENERAL DYNAMIC STRUCTURESbyXUEFENG ZHANGMay 2012Advisor: Dr. Wen LiMajor: Mechanical EngineeringDegree: Doctor of PhilosophyThe Fourier Spectral Element Method (FSEM) was proposed by Wen Li on the vibrationof simple beams (Li, 2000), and was extended to the vibration of rectangular plates (Li, 2004).This dissertation proposes a revised formulation on the vibration of rectangular plates withgeneral boundary conditions, and extends the FSEM on the vibration of general triangular plateswith elastic boundary supports. 3-D coupling formulation among the plates and beams is furtherdeveloped. A general dynamic structure is then analyzed by dividing the structure into coupledtriangular plates, rectangular plates, and beams. The accuracy and fast convergence of FSEMmethod is repeatedly benchmarked by analytical, experimental, and numerical results from theliterature, laboratory tests, and commercial software.The enabling feature of FSEM method is that the approximation solution satisfies boththe governing equation and the boundary conditions of the beam (plates) vibration in an exactsense. The displacement function composes a standard Fourier cosine series plus severalsupplementary functions to ensure the convergence to the exact solution including displacement,bending moment, and shear forces, etc. All the formulation is transformed into standard formsand a set of stored matrices ensure fast assembly of the studied structure matrix. Since the matrix