Vibration is the motion of a particle or a body or system of connected bodies displaced from a position of equilibrium. Most vibrations are undesirable in machines and structures because they produce increased stresses, energy losses, because added wear, increase bearing loads, induce fatigue, create passenger discomfort in vehicles, and absorb energy from the system. Rotating machine parts need careful balancing in order to prevent damage from vibrations.
Vibration occurs when a system is displaced from a position of stable equilibrium. The system tends to return to this equilibrium position under the action of restoring forces (such as the elastic forces, as for a mass attached to a spring, or gravitational forces, as for a simple pendulum). The system keeps moving back and forth across its position of equilibrium. A system is a combination of elements intended to act together to accomplish an objective. For example, an automobile is a system whose elements are the wheels, suspension, car body, and so forth.
A static element is one whose output at any given time depends only on the input at that time while a dynamic element is one whose present output depends on past inputs. In the same way we also speak of static and dynamic systems. A static system contains all elements while a dynamic system contains at least one dynamic element.
A physical system undergoing a time-varying interchange or dissipation of energy among or within its elementary storage or dissipative devices is said to be in a dynamic state. All of the elements in general are called passive, i.e., they are incapable of generating net energy. A dynamic system composed of a finite number of storage elements is said to be lumped or discrete, while a system containing elements, which are dense in physical space, is called continuous.
The analytical description of the dynamics of the discrete case is a set of ordinary differential equations, while for the continuous case it is a set of partial differential equations. The analytical formation of a dynamic system depends upon the kinematic or geometric constraints and the physical laws governing the behaviour of the system.
Classification of Vibrations
Vibrations can be classified into three categories: free, forced, and self-excited. Free vibration of a system is vibration that occurs in the absence of external force. An external force that acts on the system causes forced vibrations. In this case, the exciting force continuously supplies energy to the system. Forced vibrations may be either deterministic or random (see Fig.). Self excited vibrations are periodic and deterministic oscillations. Under certain conditions, the equilibrium state in such a vibration system becomes unstable, and any disturbance causes the perturbations to grow until some effect limits any further growth. In contrast to forced vibrations, the exciting force is independent of the vibrations and can still persist even when the system is prevented from vibrating.
The outputs of a vibrating system, in general, depend upon the initial conditions, and external excitations. The vibration analysis of a physical system may be summarised by the four steps:
1. Mathematical Modelling of a Physical System
2. Formulation of Governing Equations
3. Mathematical Solution of the Governing Equations
4. Physical interpretation of results
Components of Vibration
1. Stiffness element: Sometimes it requires finding out the equivalent spring stiffness values when a continuous system is attached to a discrete system or when there are a number of spring elements in the system. Stiffness of continuous elastic elements such as rods, beams, and shafts, which produce restoring elastic forces, is obtained from deflection considerations.
The stiffness coefficient of the rod is given by k =EA/l
The cantilever beam stiffness is k =3EI/l3
The torsional stiffness of the shaft is K =GJ/l
2. Mass or inertia elements: The mass or inertia element is assumed to be a rigid body. Once the mathematical model of the physical vibrating system is developed, the mass or inertia elements of the system can be easily identified.
3. Damping elements: In real mechanical systems, there is always energy dissipation in one form or another. The process of energy dissipation is referred to in the study of vibration as damping. A damper is considered to have neither mass nor elasticity. The three main forms of damping are viscous damping, Coulomb or dry-friction damping, and hysteresis damping. The most common type of energy-dissipating element used in vibrations study is the viscous damper, which is also referred to as a dashpot. In viscous damping, the damping force is proportional to the velocity of the body. Coulomb or dry-friction damping occurs when sliding contact that exists between surfaces in contact is dry or have insufficient lubrication. In this case, the damping force is constant in magnitude but opposite in direction to that of the motion. In dry-friction damping energy is dissipated as heat.
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