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Basic technical terms concerning the elastic properties, internal friction and their measurements are cited and explained. Any questions, corrections and opinions by E-mail or facsimile are wellcome.

Half width method

Internal friction makes the number of charasteristic vibrations increase, and, therefore, makes the resonance peak broadend. This means that the broadening shows the existence of internal friction of certain mechanism, and its amount can be given by the ratio between the frequency of central resonance peak and the breadth of the broadening.
Since the breadth df is measured at th position of 1/2 of the peak height at the frequency, f, this mathematical procedure is called the method of (half) width at half maximum and the internal friction is calculated by

Since the data of the half width method are taken by scannning the frequency, it takes a relatively long time, but is often used for precision measurement of, for instance, a subtle change because of its high exactness. The measurement can be done manually.

Internal friction

An ideal elastic body without any resistance or interference from outside is expected to keep vibration eternally, but vibration of actual solid materials will be damped unavoidably.
This is because the vibrational energy is gradually turned to heat and the energy is dispatated. What causes the transformation to heat is called the internal friction, which is very sensitive to the structure of the material. Therefore, the structures and its changes in the interieor of materials can be investigated by measuring the internal friction.
Origins of internal friction are various ; for instance, dislocation, lattice transformation, grain boundary, lattice softening, solute interstitial atom, etc. can cause it.
The ratio of the dissipated energy, E, per cycle to the vibrational energy, E, is called the relative damping capacity, and internal friction is defined by the relative damping devided by 2 (one cycle), which is in analogy with the Q value in the electric resonance circuit, that is,

Since the resonance of vibration is generally measured in internal friction, the reciprocal of Q value which express the sharpness of the electric resonance peak, i.e., Q-1 is employed as a measure of internal friction.
Typical methods to measure internal friction are the damping method and the half width method . In the former, the ratio of damping amplitudes at every wave length is measured, and in the latter the ratio of the broardening in frequency to the resonance peak frequency is measured, both of which have no numerical unit.

Dempending upon how to measure and how to calculate mathematically, internal friction has different names as follows.

internal friction=damping constant = tangent delta() = logarithmic damping = internal loss coefficient = damping capacity

For the models for generation of internl friction, see "elastic aftereffect".

Logarithmic damping constant

Internal frictiontypically appears as the damping of free vibrtion of solid body. Therefore, the ratio of successive vibrational amplitudes, Vj/Vj-1 could be used to express the amount of internal friction. In general, vibration dacays exponentially, so that the damping constant is defined as

or in a more general way,


The internal friction Q-1 defined by the energy dissipation during one cycle of vibration, (E/E)/2, is related with the damping constant DC by


Sometimes, damping constant is simply meant by the logarthmic damping constant.

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