Vibrations are closely related to sound, which takes the form of "pressure waves."

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To start the investigation of the mass-spring-damper we will assume the damping is negligible and that there is no external force applied to the mass (that is, free vibration).

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The same waves can induce the vibration of other structures, such as the ear drum.

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Such vibrations can be caused by imbalances in the rotating parts, uneven friction, the meshing of gear teeth, and so forth.

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The natural frequency and damping ratio are not only important in free vibration, but also characterize how a system will behave under forced vibration.

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Written in this form we can see that the vibration at each of the degrees of freedom is just a linear sum of the mode shapes.

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Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker.

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The figure also shows the time domain representation of the resulting vibration.

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The fundamentals of vibration analysis can be understood by studying the simple mass-spring-damper model.

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We can view the solution of a vibration problem as an input/output relation—where the force is the input and the output is the vibration.

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Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely.

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The frequency response of the mass-spring-damper therefore outputs a high 7 Hz vibration even though the input force had a relatively low 7 Hz harmonic.

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Hence, the solution to the problem with a square wave is summing the predicted vibration from each one of the harmonic forces found in the frequency spectrum of the square wave.

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Note: In this article the step by step mathematical derivations will not be included, but will focus on the major equations and concepts in vibration analysis.

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The plot of these functions, called "the frequency response of the system," presents one of the most important features in forced vibration.

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Consequently, one of the major reasons for vibration analysis is to predict when this type of resonance may occur and then to determine what steps to take to prevent it from occurring.

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The amplitude of the vibration “X” is defined by the following formula.

Forced vibration is when an alternating force or motion is applied to a mechanical system.

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The following are some other points in regards to the forced vibration shown in the frequency response plots.

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