20.7 Hysteretic Damping

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20.7.1 Equivalent Viscous (Linear) Model

The few mechanical oscillators governed by Equation 20.1 tend to be those in which there is an external

control, such as eddy current damping. For oscillators in which the damping derives from internal friction

of its members, the following linear approximate form of the hysteretic damping model has been used:

mx€ þ

h

v

x_ þ kx ¼ F ð20:19Þ

It should be noted that hysteresis is the cause for all damping; however, engineers have come to use the

term “hysteretic damping” for systems described by Equation 20.19. This equation differs in two important

ways from Equation 20.1. For the viscous damped oscillator, Q is proportional to the frequency, but for the

hysteretic damped oscillator, Q is proportional to the square of the frequency. Also, viscous damping

changes the frequency of the oscillator, since v1 , v0 and, for resonance, the frequency is even lower.

However, the hysteretic oscillator is isochronous, requiring only a single frequency v ¼

ffiffiffiffiffi

k=m p ! vr to

describe all features of the motion. For example, it is easy to show that the oscillator resonates at this

frequency. Off resonance, the response is not the standard Lorentzian. To show this, assume steady state and

use the phasor method given to us by Steinmetz, 1893 (complex exponential form for the variables); i.e.,

F ¼ F0 ejvt and x ¼ x0 ejvt to get the frequency transfer function

kx

F ¼

1

1 2 v2 m

k þ j

h

k

¼

1

1 2 r2 þ ja ¼ Z; with r ¼

v

vr

and a ¼

h

k ¼

1

Q ð20:20Þ

for which the real and imaginary parts are given by

Re Z ¼

1 2 r2

ð1 2 r2Þ2 þ a2

; Im Z ¼

2a

ð1 2 r2Þ2 þ a2 ð20:21Þ

which is expressible in polar form as

Z ¼ lZl e jd ; where lZl ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 2 r2Þ2 þ a2

p and d ¼ 2tan21 a

1 2 r2 ð20:22Þ

It is interesting to compare the steady-state response of the driven, hysteretic damped oscillator with that

of the driven, viscous damped oscillator; i.e., Equation 20.22 compared with normalized Equation 20.7.

A Bode plot comparison (log – log, for the amplitude case) is provided in Figure 20.6. At small values of

the damping parameter a (large Q), there is insignificant difference between the two cases. At large values,

however, the difference is significant.

20.7.2 Examples from Experiment of Hysteretic Damping

The vertical seismometer that was used for several of the present studies is known to decay according

to hysteretic damping. In Section 20.16.4 titled “Failure of Viscoelasticity”, details are provided of the

work by Gunar Streckeisen (1974) that showed this to be true. Decay curves of the instrument are

Damping Theory 20-27

© 2005 by Taylor & Francis Group, LLC

frequently a near-perfect exponential, once corrected for secular drift of the record. Sometimes, this

drift is the result of creep in the spring of the instrument, but it may also be the result of other

factors, such as (i) temperature change, or (ii) barometric pressure variation, or even (iii) tidal

influence. The temperature sensitivity is due to the difference of thermal coefficients of the materials

from which the instrument is constructed, and the pressure variation is a buoyancy effect. Tidal

influence is the smallest of the three, which causes minute accelerations of the crust of the Earth with

a period of about 12 h.

In the discussions which follow, two different decay records are provided. In both cases, the initial

amplitude of oscillation is quite large, being a significant fraction of 1 mm, and the period for the two

cases is different — the first case being 17 sec and the second one 21 sec. The first case time record, shown

in Figure 20.7, contains 9800 points. Once a 12 mV/s (upward) drift was removed, the decay (left curve)

is seen to be “nearly textbook” exponential.

The adjective “nearly” is appropriate because there is a 12% difference in the decay constants defining

the upper and lower turning points (0.0022 top, 0.0025 bottom), which were determined by trial and

error “eyeball” exponential fits using Excel. In this author’s experience, such is the norm for virtually all

mechanical oscillations; perfectly symmetric exponential decays have rarely been seen in the hundreds of

cases studied.

FIGURE 20.7 Free-decay of a vertical seismometer due to hysteretic damping. The period of oscillation is 17 sec.

Z

10

1.0

0.1

0.1

0

1.0 10.

0.1 = a = 1/Q

normalized freq., r

d

−p

−p/2

1

10

0.1

1

10

FIGURE 20.6 Bode plot comparison of steady-state driven system with (i) hysteretic damping (dark curves) and

(ii) viscous damping (light curves).

20-28 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Because there are roughly 150 oscillations in the time record of Figure 20.7, it is not possible to

resolve individual turning points of the motion, but the oscillations are very nearly that of a pure

damped sinusoid, as noted from the right-side graph of the figure. This spectrum was generated with a

4096-point FFT, comprising the first 1090 sec of the time record. The second harmonic is the only

distortion observed, and it is about 65 dB below the fundamental. For the case presented in Figure 20.7,

Q ¼ 80:

Another example of free-decay hysteretic damping is provided in Figure 20.8. As usual, the record was

afflicted by drift, possibly from creep in the spring, in this case a constant rate of 60 mV/s, as observed in

the graph on the right. All of these graphs were produced with Excel, as noted earlier in the discussion of

induced current damping. As with the decay curve of Figure 20.7, the creep-corrected graph on the left

was generated by adding a secular term to the raw data. Once corrected, the decay is a near-perfect

exponential of hysteretic type. We will see other examples (from pendulum studies) in which two

damping mechanisms are simultaneously active in a decay.

The Q values corresponding to Figure 20.7 and Figure 20.8 are consistent with hysteretic damping; i.e.,

80 for the 17-sec oscillation and 52 for the 21-sec oscillation. As noted elsewhere in this document, Qav2

for hysteretic damping as opposed to an exponent of 1 for viscous damping. Of course, one must collect

data over a very much larger range of frequencies to verify this, as was done by Streckeisen (1974).

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