Vibrations are fluctuations of a mechanical or structural system about an equilibrium position. Vibration is a very important phenomenon in mechanical and structural systems. If vibrations are uncontrolled they can cause catastrophic situations. Vibration of machine tools or machine tool chatter can lead to improper machining of parts. Structural failure can occur because of large dynamic stresses developed during earthquakes or even wind-induced vibration.
Most human activities involve vibration in one form or other. For example, we hear because our eardrums vibrate and see because light waves undergo vibration. Breathing is associated with the vibration of the lungs and walking involves (periodic) oscillatory motion of the legs and hands. Human speech requires the oscillatory motion of larynges (and tongues). Early scholars in the field of vibration concentrated their efforts on understanding natural phenomena and developing mathematical theories to describe the vibration of physical systems. In recent times, many investigations have been motivated by the engineering application of vibrations, such as the design of machines, foundations, structures, engines, turbines, and control systems.
For example, the excessive vibrations of pumps, compressors, turbo machinery, and other machines can induce vibrations of the surrounding structure, leading to inefficient operation of the machines while the noise produced can cause human discomfort. They can be introduced, with beneficial effects, into systems in which they would not naturally occur. Vehicle suspension systems are designed to protect passengers from discomfort when traveling over rough terrain. In rotating machinery, we need to use vibration isolators to protect structures from excessive forces developed in their operation.
Vibrations are initiated when an inertia element is displaced from an external source. A restoring force or moment pulls the element back toward equilibrium. When work is done on the block which is shown in Figure 1 to displace it from its equilibrium position, potential energy is developed in the spring. When the block is released the spring force pulls the block toward equilibrium with the potential energy being converted to kinetic energy. In the absence of neoconservative forces, this transfer of energy is continual, causing the block to oscillate about its equilibrium position.
Consider a pendulum system that is released from a position above its equilibrium position then the moment of gravity force pulls the particle, the pendulum bob, back toward equilibrium with the potential energy being converted to kinetic energy. In the absence of non-conservative forces, the pendulum will oscillate about the vertical equilibrium position. See the following figure.
Even a simple mechanical system such as a pendulum is nonlinear. We simplify the model because of its complexity. If we include every single variable that influences the analyzed system than we would get correct results but it would take an enormous amount of time to get that solution. In general, all systems are nonlinear but we simplify them to get linear or approximated results. The results of the linear analysis are still good results and it can help us to solve or prevent vibrations in the system.
