Page 85 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

72 Harmonically Excited Vibration Chap. 3

Figure 3.7-1. Energy dissipated by viseous damping.
Damping properties of materials are listed in many different ways, depending
on the technical areas to which they are applied. Of these, we list two relative
energy units that have wide usage. First of these is specific damping capacity,
defined as the energy loss per cycle divided by the peak potential energy U:

(3.7-6)
u
The second quantity is the loss coefficient, defined as the ratio of damping
energy loss per radian divided by the peak potential or strain energy U:

V = 2ttU (3.7-7)
For the case of linear damping, where the energy loss is proportional to the
square of the strain or amplitude, the hysteresis curve is an ellipse. When the
damping loss is not a quadratic function of the strain or amplitude, the hysteresis
curve is no longer an ellipse.
Example 3.7-1
Determine the expression for the power developed by a force F = FQsin(ö>i 4- </>)
acting on a displacement x = Xqsin o)t.

Solution: Power is the rate of doing work, which is the product of the force and velocity.
sin + (/))cos o)t

f
= (ia3i()FQ)[cos (f) • sin o)t cos cot + sin <) ' cos^wr]
= \o)X^ffi^)[sin 4>+ sin(2(ot + (/>)]

The first term is a constant, representing the steady flow of work per unit time. The
second term is a sine wave of twice the frequency, which represents the fluctuating