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Vibration usually becomes a concern when it's amplitudes grow large enough to cause either excessive stress, or if it disturbs the people in, on or near the vibrating object(s). As far as most structures are concerned, vibration will disturb the people around the structure long before stress becomes an issue. There are many items of equipment (balances, microscopes, cameras, transmission equipment etc.) that are very sensitive to vibration.

Modal analyses are important in machines where there is likely to be cyclic out of balance forces, such as in rotating machinery (engines, electric & pneumatic motors, generators, industrial equipment, etc.) and fluid flow applications (due to alternating vortex shedding). The chief aim of any vibration analysis is to ensure that the system is not subject to a dangerous resonant condition during the range of operation. A point to note is that although the response of the system is time dependant, any excitation will be harmonic, and the solution may be obtained using the eigenvalue approach. It is important to note that many applications fall in a category beyond this range, and full dynamic analyses are required.

If a system is given some initial disturbance, then it will vibrate at some frequency known as it's natural frequency. The natural frequency of a system is defined as the frequency at which the system oscillates if the forcing function is identically zero. If harmonic loading is applied, the solution becomes transient in nature, but modal analyses can still be carried out for systems.

You may recall from elementary vibrations lectures that the square of the natural frequency is referred to as an eigenvalue. For a single mass-spring system, there is one eigenvalue, for distributed mass systems (all practical applications), an infinite number of eigenvalues exist. The lowest natural frequency, usually referred to as the fundamental frequency, has the lowest potential or strain energy, and hence the reason why it is often regarded as the 'lazy mode'.

The fundamental frequency is usually the one of most interest to design engineers, as most systems are designed to operate below it. Oftentimes, an operating frequency is higher than the fundamental, hence as the equipment speeds up or slows down, it experiences a momentary shudder period as it passes through the resonance zone. There is a corresponding mode shape which describes the displacement of the system due to the vibration.