20.14 Mathematical Tricks — Linear Damping Approximations

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20.14.1 Viscous Damping

In the Hooke’s Law expression, F ¼ 2kx; it is common practice to approximate hysteresis of oscillatory

motion by letting k become a complex coefficient. This is also standard practice in a variety of fields, such

as the description of lossy electromagnetic media. No doubt the practice has been further popularized by

the standard approach of solving electrical engineering ac circuit problems by means of phasors, the

technique developed by Steinmetz (1893).

We recognize in the expression xðtÞ ¼ x0 ejvt ¼ x0 cos vt þ jx0 sin vt that harmonic variation is

contained in the real part (or alternatively the imaginary part) of the complex exponential form. Using

Newton’s Second Law, and representing the spring constant by k ejd with d ,, 1 (small damping), we

obtain the damped harmonic oscillator equation

mx€ þ kx þ ðjkdÞx ¼ 0 ð20:38Þ

where the approximations cos d ! 1 and sin d ! d have been employed.

However, since x_ ¼ jvx; and

k

m ¼ v2; Equation 20.38 can be rewritten as

x€ þvdx_ þv2x ¼ 0 ð20:39Þ

We thus see that the damping constant vd ¼ v=Q ¼ 2b permits us to express the logarithmic decrement

D in terms of the angle d with which x lags F; i.e., D ¼ bT ¼ pd: (Note that we are making no distinction

here between the periods with and without damping, since the difference is small and hard to measure.) If

b were independent of frequency, then d would be inversely proportional to the frequency, which is rarely

realized in practice.

20.14.2 Hysteretic Damping

Equation 20.39 does not properly represent some of the most important engineering systems. For those

labeled “hysteretic,” we must use a different form for the complex spring constant. We assume that

F ¼ 2ðkcomplexÞx ¼ ðk þ jhÞx where h is a real constant. Since dx=dt ¼ jvx; this yields the equation of

motion

x€ þ

h

mv

x_ þv2x ¼ 0 ð20:40Þ

Since h is assumed to be a true constant (independent of frequency), the lag angle between displacement

and force is given by

d ¼

h

k ¼

1

Q ¼

h

mv2 ð20:41Þ

Damping Theory 20-43

© 2005 by Taylor & Francis Group, LLC

which is seen to be inversely proportional to the square of the frequency. (Note that d here is the same as

a in Figure 20.6.) It should be noted that the complex form for the spring constant is not simply obtained

using the common theory of viscoelasticity. Such theory requires a multitude of relaxation times

(stretched exponentials) (Speake et al., 1999).

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