Category: 2. Vibration of Simple and Continuous Systems

The present section is concerned with systems possessing two dimensions which are large compared with the third, e.g. plates whose lengths and widths are much greater than their thicknesses. There are numerous applications of vibrating plates in electroacoustical equipment such as loudspeakers, microphones, earphones, ultrasonic transducers, etc. In addition, plates can be found as constituting…

All structural systems such as beams, columns, and plates are continuous systems with an infinite number of degrees of freedom. Consequently, a continuous system has an infinite number of natural frequencies and corresponding mode shapes. Although easier, modeling a structure using a finite number of degrees of freedom provides just an approximation of the behavior…

If there is damping present (as there always is in real systems) the homogenous solution of a harmonically forced vibration system decays away with time. It has to be noted that when damping is included in the mathematical model, the eigenvalues and eigenvectors can be complex numbers, unlike in the undamped case. Although in practice…

By forced vibration, we mean that the system is vibrating under the influence of continuous (external) forces that do not cease. The total response of a multi‐degree of freedom system due to a force excitation is the sum of a homogeneous solution and a particular solution. The homogenous solution depends upon the system properties while…

By free vibration, we mean that the system is set into motion by some forces which then cease (at t = 0) and the system is then allowed to vibrate freely for t > 0 with no external forces applied. First we will consider a free undamped multi‐degree of freedom system, i.e. [R] = [0] and f(t) = 0. Therefore, Eq. (2.22) now becomes…

The simple mass‐spring‐damper system excited by a harmonic force was discussed in the preceding sections assuming a single mass which could move in one axis only. This single‐degree‐of‐freedom system idealization is reasonable when the mass is fairly rigid, the springs are lightweight and its motion can be described by means of one variable. For simple…

3.1 Mass–Spring System a) Free Vibration – Undamped Suppose a mass of M kilogram is placed on a spring of stiffness K newton‐metre (see Figure 2.5a), and the mass is allowed to sink down a distance d metres to its equilibrium position under its own weight Mg newtons, where g is the acceleration of gravity 9.81 m/s2. Taking forces and deflections to be positive upward gives (2.8)…

The motion of vibrating systems such as parts of machines, and the variation of sound pressure with time is often said to be simple harmonic. Let us examine what is meant by simple harmonic motion. Suppose a point P is revolving around an origin O with a constant angular velocity ω, as shown in Figure 2.1. If the vector OP is aligned in…

1 Introduction The vibrations in machines and structures result in oscillatory motion that propagates in air and/or water and that is known as sound. The simplest type of oscillation in vibration and sound phenomena is known as simple harmonic motion, which can be shown to be sinusoidal in time. Simple harmonic motion is of academic…