Vibration of Discrete and Continuous Systems

The theory of vibration of single degree of freedom systems serves as one of the fundamental building blocks in the theory of vibration of discrete and continuous systems. As will be shown in later chapters, the concepts introduced and the techniques developed for the analysis of single degree of freedom systems can be generalized to study discrete systems with multi-degree of freedom as well as continuous systems. For this book to serve as an independent text, several of the important concepts and techniques used in the analysis of single degree of freedom systems are briefly discussed in this chapter. First, the methods of formulating the kinematic and dynamic equations are reviewed in the first three sections. It is also shown in these sections that the dynamic equation of a single degree of freedom system can be obtained as a special case of the equations of the multi-degree of freedom systems. The free vibrations of the single degree of freedom systems are reviewed in Sections 1.4 and 1.5, and the significant effect of viscous, structural, and Coulomb damping is discussed and demonstrated. Section 1.6 is devoted to the analysis of the forced vibrations of single degree of freedom systems subject to harmonic excitations, while the impulse response and the response to an arbitrary forcing function are discussed, respectively, in Sects. 1.7 and 1.8. Section 1.9 discusses the importance of the linear theory of vibration covered in this book in the analysis of nonlinear systems.

The differential equations of motion of single and multi-degree of freedom systems can be developed using the vector approach of Newtonian mechanics. Another alternative for developing the system differential equations of motion from scalar quantities is the Lagrangian approach where scalars such as the kinetic energy, strain energy, and virtual work are used. In this chapter, the use of Lagrange’s equation to formulate the dynamic differential equations of motion is discussed. The use of Lagrange’s equation is convenient in developing the dynamic relationships of multi-degree of freedom systems. Important concepts and definitions, however, have to be first introduced. In the first section of this chapter, we introduce the concept of the system generalized coordinates, and in Sect. 2.2, the virtual work is used to develop the generalized forces associated with the system generalized coordinates. The concepts and definitions presented in the first two sections are then used in Sects. 2.3, 2.4, 2.5, and 2.6 to develop Lagrange’s equation of motion for multi-degree of freedom systems in terms of scalar quantities such as the kinetic energy, strain energy, and virtual work. An alternate approach for deriving the dynamic equations of motion using scalar quantities is Hamilton’s principle which is discussed in Sect. 2.7. Hamilton’s principle can be used to derive Lagrange’s equation and, consequently, both techniques lead to the same results when the same set of coordinates is used. The use of conservation of energy to obtain the differential equations of motion of conservative systems is discussed in Sect. 2.8, where general conservation theorems are developed and their use is demonstrated using simple examples.

Mechanical systems in general consist of structural components which have distributed mass and elasticity. Examples of these structural components are rods, beams, plates, and shells. Our study of vibration thus far has been limited to discrete systems which have a finite number of degrees of freedom. As has been shown in the preceding chapters, the vibration of mechanical systems with lumped masses and discrete elastic elements is governed by a set of second-order ordinary differential equations. Rods, shafts, beams, and other structural components on the other hand are considered as continuous systems which have an infinite number of degrees of freedom. The vibration of such systems is governed by partial differential equations which involve variables that depend on time as well as the spatial coordinates.

The approximate methods presented at the end of the preceding chapter for the solution of the vibration problems of continuous systems are based on the assumption that the shape of the deformation of the continuous system can be described by a set of assumed functions. By using this approach, the vibration of the continuous system which has an infinite number of degrees of freedom is described by a finite number of ordinary differential equations. This approach, however, can be used in the case of structural elements with simple geometrical shapes such as rods, beams, and plates. In large-scale systems with complex geometrical shapes, difficulties may be encountered in defining the assumed shape functions. In order to overcome these problems, the finite element (FE) method has been widely used in the dynamic analysis of large-scale structural systems. The FE method is a numerical approach that can be used to obtain approximate solutions to a large class of engineering problems. In particular, the FE method is well suited for problems with complex geometries.

In addition to the matrix-iteration method discussed in Chap. 3, there are several other computer methods that are widely used for solving the eigenvalue problem of vibration systems. Among these methods are the Jacobi method and the QR method. In these methods, which are based on the similarity transformation, a series of transformations that convert a given matrix to a diagonal matrix which has the same eigenvalues as the original matrix are used. Not every matrix, however, is similar to a diagonal matrix, and therefore we find it appropriate to devote several sections of this chapter to discuss the similarity transformation before we briefly discuss the computer methods used for solving the eigenvalue problem of vibration systems. Several definitions will be used repeatedly throughout the development presented in this chapter. Some of these definitions are summarized below.