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PART I: ESSENTIALS Introduction Definitions, aims and general concepts Basic features of a vibrating system, and further concepts Tutorial questions Single degree-of-freedom systems Basic equation of motion Free vibration response Equivalent spring stiffnesses for various structural and mechanical systems Response to harmonic excitation Tutorial questions Systems with more than one degree of freedom Introductory remark Equations of motion Techniques for assembling the stiffness matrix The flexibility formulation of the equations of motion and assembly of the flexibility matrix Determination of natural frequencies and mode shapes The flexibility formulation of the eigenvalue problem Worked examples The modal matrix Orthogonality of eigenvectors Generalized mass and stiffness matrices Worked examples Modal analysis Worked example Tutorial questions Continuous systems Introduction Transverse vibration of strings Axial vibration of rods Flexural vibration of beams Orthogonality of natural modes of vibration Dynamic response by the method of modal analysis Finite-element vibration analysis The finite-element formulation Stiffness and consistent mass matrices for some common finite elements Assembly of the system equations of motion References
PART II: GROUP-THEORETIC FORMULATIONS Basic concepts of symmetry groups and representation theory Symmetry groups Group tables and classes Representations of symmetry groups Character tables Group algebra Idempotents Applications References Rectilinear models Introduction A Shaft-disc torsional system A Spring-mass extensional system Conclusions Plane structural grids Introduction Rectangular configurations Square configurations Conclusion High-tension cable nets Basic assumptions and geometric formulation Outline of computational scheme Illustrative examples Symmetry-adapted functions Symmetry-adapted flexibility matrices Subspace mass matrices Eigenvalues, eigenvectors and mode shapes Summary and concluding remarks References Finite-difference formulation for plates General finite-difference formulation for plate vibration Group-theoretic implementation Application to rectangular and square plates Finite-difference equations for generator nodes of the basis vectors Symmetry-adapted finite-difference equations and system eigenvalues Concluding remarks References Finite-element formulations for symmetric elements Group-theoretic formulation for finite elements Coordinate system, node numbering and positive directions Symmetry-adapted nodal freedoms Displacement field decomposition Subspace shape functions Subspace element matrices Final element matrices Concluding remarks References