Analytical and numerical methods for vibration analyses (Q2863163)

This book gives a course in structural dynamics for students of graduated schools. Chapter 1 presents the terminology and types of vibrations. In Chapter 2, the author derives the equations of motion for different vibrations such as vibrations of stretched strings, longitudinal vibrations of rods, torsional vibrations of continuous shafts, and flexural vibrations of continuous Euler-Bernoulli and continuous Timoshenko beams, which may be axially loaded or be on an elastic foundation. Here, the author also considers the vibrations of rectangular and circular membranes and plates. Chapter 3 gives exact solutions for natural frequencies and mode shapes in Bessel functions for longitudinal vibrations of conical rods, torsional vibrations of conical shafts, and bending vibrations of single-taped beams and double-taped beams. Some solutions are also obtained by the finite element method. Chapter 4 explains the transfer matrix method for discrete and continuous systems. It presents the torsional vibration analysis of shafting systems and the analysis of flexural vibrations of Euler-Bernoulli and Timoshenko beams with various boundary conditions and with an arbitrary number of concentrated masses.NEWLINENEWLINEIn Chapter 5, the author considers the eigenproblem of free vibrations of structural systems by the Jacobi method, which is the simplest tool for solving this problem. Considered are longitudinal vibrations of rods, flexural vibrations of Euler-Bernoulli and Timoshenko beams, and transformations of element stiffness matrices and mass matrices. In Chapter 6, the author examines free vibrations of longitudinal rods, torsional shafts and flexural beams by using the finite element method. For example, the 12 DOFs Timoshenko beam element is analyzed, but without the stiffness matrix. Chapter 7 is devoted to analytical methods and the finite element method for free vibration analysis of circularly curved Euler and Timoshenko beams. It is shown that the finite element method for curved beams and for straight beams gives the same results.NEWLINENEWLINEThe final Chapter 8 demonstrates that the coupled equations of motion for multi-degree-of-freedom structural systems may be transformed into an equation of motion using the orthogonality conditions among the normal mode shapes. Since each equation of motion is similar to an equation of motion for a single-degree-of-freedom system, first of all it is natural to consider the solution of this equation under an arbitrary load. This can be realized if the damping forces are similar to stiffness forces. (Unfortunately, this is not underlined in this book). Solutions for the equations of motion of multi-degree-of-freedom systems can be obtained by two methods: direct integration and mode superposition. The direct integration is realized by using a numerical step-by-step procedure without transformation of the equations of motion. The mode superposition method is based on a transformation of the equations using the orthogonality condition.NEWLINENEWLINE The appendices contain a list of integrals, the theory of the modified bisection method, the determination of influence coefficients, an exact solution of cubic equations, and an exact solution for simply supported Euler arch.NEWLINENEWLINEUnfortunately, the author omits the case when the matrix stiffness coefficients can be obtained by means of exact integrations such as given in the paper [\textit{A. A. Belous}, ``Kolebaniya i staticheskaya ustojchivost' ploskikh i prostranstvennykh ram'', Raschet Prostranstvennykh Konstruktsij, 211--264 (1955)]; see also the reviewer's book [Dynamical characteristics of viscous-elastic systems with distributed parameters. Saratov: Izdatel'stvo Saratovskogo Universiteta (1977; Zbl 0998.74500)]. The problems of vibration systems with distributed parameters can be solved in space by Laplace transform when automatically accounting for initial conditions. The solutions use the amplitude-phase-frequency characteristics. The reader can find the details of such an approach in the paper [\textit{Yu. N. Sankin}, Tr. Sredn. Mat. Obshch. 8, No. 2, 22--33 (2006; Zbl 1150.74385)]. The meaning of Fig. 1.3, p. 129, is not clear. The signs of \(\alpha\) and \(\beta\) may be negative, so we have nine variants of solutions (see the reviewer's book [loc. cit.]).