Mechanical Vibration/Background & Models

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Background & Models

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Some of the theories and mathematical structures you will be learning shortly are entirely classical; in fact, the root of vibrational study is a derivation of Newton's laws of motion. During approximately the last four hundred years there has been an increased interest in this subject as engineers have been reevaluating earlier simpler formulation and design in an effort to maximize stability within their particular creation or area of focus. It is typically the vibration of all parts in a machine that can lead to its eventual failure and we are concerned with gaining the ability to shrink vibrational freedom. There are many applications where vibrations are permitted or even encouraged regarding the system; our efforts are typically reduction, though. The history of the evolution of vibrational study was modernized by the work of Lord Rayleigh in his book The Theory of Sound; a reminder that in essence both vibration and acoustics are intrinsically identical.

We can begin by firstly separating some key terms in order to clarify before there is much room to get lost. The following terms will be used often:


Vibration is a reciprocating motion of an elastic body or virtually any medium forced from a state of equilibrium. A classic example is a bell that has been struck with a hammer. It will periodically change its shape between two extremes. This can be heard as a sound and felt on the surface.
Another example is given in the animation.

Oscillation is a reciprocating movement of a complete structure. An example could be the pad of an orbital sander which swings back and forth without actually changing its shape.

A vibrating structure undergoes deformation, while an oscillating structure does not.

Structures range from simple objects like beams or rods to more complex things like rotor blades or the fuselage of an aircraft. Structures may be composed from several parts, but they can be considered one part in their analysis.

Systems are more general and abstract. They can loosely be defined as a grouping of parts, such as an assembly with different, distinguishable parts that do however interact closely with each other.

All structures are systems, not vice-versa.
A damped spring-mass system.

Environments are external to the system but have the ability to interact with them and influence the system's behavior.

Modeling is the act of representing certain properties of physical structures or systems by means of mathematical formulation, simulation or the like.

Inertia is the property of a mass which makes it resist any change in motion.

Stiffness is the resistance of an elastic body to deformation by an applied force. The stiffer a structure the more resistant it is to deformation.

Damping is the effect that reduces the amplitude of oscillations or vibrations over time by dissipating the kinetic energy that caused the movement into heat (see animation).

Prerequisites · Idealization & Formulation