Vibration Kinematics

In this chapter, we study (1) mechanical elements of vibrating systems, (2) physical causes of mechanical vibrations, (3) kinematics of vibrations, and (4) simplification methods of complex vibrating systems. Mechanical vibration is a result of periodical transformation of kinetic energy K to potential energy P. When the potential energy is at its maximum, the kinetic energy is zero, and when the kinetic energy is at its maximum, the potential energy is minimum. A fluctuation in kinetic energy appears as a periodic motion of a massive body, and this is the reason we call such energy transformations the mechanical vibrations. The kinetic and potential energies are stored in physical elements. Any element that stores kinetic energy is called the mass or inertia, and any element that stores potential energy is called the spring or restoring element. If the total value of mechanical energy E = K + P decreases during vibrations, there is also a phenomenon or an element that dissipates energy. The element that causes energy to dissipate is called damper. Any periodic motion Multi DOF system is characterized by a period T, which is the required time for one complete cycle of vibration, starting from and ending at the same conditions. The frequency f is the number of cycles in one T = 1∕f. In theoretical vibrations, we usually work with angular frequency ω[rad∕s], and in applied vibrations, we use the cyclic frequency f[Hz]. The fundamental equation of vibration is the free and undamped equation of motion of a mass m attached to a linear spring with stiffness k. The solution of the fundamental equation of vibration can be expressed by a harmonic function. Any sine or cosine or their combination is called a harmonic function. We will also review Fourier transform. The purpose of Fourier series is to decompose a periodic function into its harmonic components.