Vibration Dynamics

In this chapter, we review the dynamics of vibrations and the methods of deriving the equations of motion of vibrating systems. The Newton–Euler and Lagrange methods are the most applied methods of deriving the equations of motion. We will show how to use these methods to derive and simplify the equations of motion of vibrating systems in detail. The translational and rotational equations of motion for a rigid body, expressed in the global coordinate frame G, indicate how the resultant of the external forces and moments applied on the rigid body, measured at the mass center C, makes the rigid body move. The equations of motion of an n-DOF linear vibrating system can always be arranged in matrix form of a set of second-order differential equations, in terms of a set of generalized coordinates of the system. The set of equations will be expressed in terms of square matrices of [m], [c], [k] for the mass, damping, and stiffness matrices, respectively. The sum of kinetic and potential energies, E = K + P, is called the mechanical energy. Mechanical energy of an energy conserved system is a constant of motion, and hence, its time derivative is zero. We will show how this principle can be used to derive the equations of motion.