Symplectic Eigenvalue Analysis Method For Mechanical Problems Of Beams

Most mechanical problems of beams,including static bending,vibration,and stability,from a mathematical point of view,can be reduced to the problem of solving differential equations under various boundary conditions.To find a uniform and effective method of such problems has long been the goal that mechanical and structural engineers worked on.Based on the symplectic dual solution system in applied mechanics,this dissertation proposes a new symplectic eigenvalue analysis method of dealing with the study of mechanical problems of beams.The key idea is to transform the typical mechanical problems into the symplectic eigenvalue problem,and to obtain the solution to those problems by computing the eigenvalues and eigenvectors.The main issues addressed in this dissertation are:(1)For the static case,this dissertation incorporates the singularity functions to propose a new hybrid method—symplectic-singularity method—for obtaining particular solutions due to external loads applied on the beam.This method is general and reliable,and it is applicable for obtaining the particular solutions of complex loads such as continuous distributed loads,non-continuous distributed loads and singular loads.(2)For the natural vibration,the graphs that illustrate the relationships between the symplectic eigenvalues and the circular frequencies are called eigenvalue spectra.This dissertation provides a series of continuous or discrete eigenvalue spectra.The frequency equations and modes are obtained for various boundary conditions.The mode vectors are composed of transverse deflection,bending rotation,shear force,and bending moment,and are called the full mode shape vectors.In addition,these vectors are verified to be orthogonal.(3)For the static buckling,this dissertation describes the property of the singularity analysis in the symplectic system by the curves of the relationship between axial force and natural vibration frequency.Furthermore,there are two special points on these curves:one is the mode for free vibration of beam without any axial force,and the other is the mode of static bucking,and other points represent natural vibration modes of beams with particular axial forces.(4)For bending of complex beam structures such as beams with varying parameters,multi-span beams,continuous beam,this dissertation proposes a symplectic transfer matrix method by incorporating the symplectic solutions and the transfer matrix method.In addition,this dissertation interprets the mathematical structure of the transfer matrix.This dissertation unifies the solution procedure by the eigenvalue analysis and tries to interpret the physical meanings of the symplectic eigenvalues and eigenvectors—the symplectic eigenvalue represents the attenuation rate of the functions of space,and the symplectic eigenvector fixes the differentiation relations among variables.This dissertation places an emphasis on the eigenvalue analysis.This expands the scope of application of eigenvalue analysis which highlights the nature of system in traditional structural mechanics.In addition,this provides a powerful tool for gaining new insight into analytical solutions and qualitative analysis of beam problems.The new symplectic eigenvalue analysis method puts the application of the symplectic dual solution system in applied mechanics to beam problems,and makes such application systematic,comprehensive,and thorough.This method is a new methodology which is parallel to the single-variable method.It brings new blood into the field of structural mechanics which is an old subject and has a broad prospect of application.