20.16 Zener Model

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20.16.1 Assumptions

The SLM of viscoelasticity provides a sound basis for some damping phenomena, yet it fails badly as an

approximation for hysteretic damping. Its prominence in both the worlds of physics and engineering

warrants the following detailed discussion so that the failure case may be properly documented.

Following the example of Zener, the following linear differential equation relates the stress, s; the

strain, 1; and their first time derivatives:

sðtÞ þt1s_ ¼ E1ð1 þts 1_Þ ð20:42Þ

The t s are relaxation times (subscript 1 meaning at constant strain and subscript s at constant stress),

and E1 is the relaxed elastic modulus (ratio of stress to strain in a very slow process). Nominally, ts . t1;

consistent with strain lagging stress. For periodic variations

sðtÞ ¼ s0 e jvt ; 1ðtÞ ¼ 10 e jvt ð20:43Þ

which, when substituted into Equation 20.42, yields

ð1 þ jvt1Þs0 ¼ E1ð1 þ jvts Þ10 ð20:44Þ

The complex modulus of elasticity is defined by

EC ¼

1 þ jvts

1 þ jvt1

E1 ð20:45Þ

and is seen to relate stress and strain according to

sðtÞ ¼ EC1ðtÞ ð20:46Þ

Damping Theory 20-45

© 2005 by Taylor & Francis Group, LLC

From Equation 20.45, the real and imaginary parts of the modulus are found to be

Real ðECÞ ¼

1 þ v2t1ts

1 þ v2t 2

1

E1 ð20:47Þ

Imag ðECÞ ¼

vðts 2 t1Þ

1 þ v2t 2

1

E1 ð20:48Þ

The independent variable, or “frequency,” for all cases is the convenient dimensionless parameter, vt ¼

v

ffiffiffiffiffiffi

t1ts p :

It is convenient to use polar form, so that

EC ¼ lECl e jd ð20:49Þ

where lECl is obtained by computing the square root of the sum of the squares of the real and imaginary

parts. In this form, it is apparent that d is a lag angle which determines the damping loss for the system.

Moreover, from Equation 20.47 and Equation 20.48, it is seen to obey

tan d ¼

vðts 2 t1Þ

1 þ v2tst1 ð20:50Þ

20.16.2 Frequency Dependence of Modulus and Loss

The essential features of the Zener model are

illustrated in Figure 20.15, where the “unrelaxed”

high-frequency modulus obeys the relation

ðE1E2Þ=ðE1 þ E2Þ ¼ E1ðts =t1 Þ:

In viscous damping models, the damping is

quantified by the product bT; which is equal to

the logarithmic decrement. The logarithmic

decrement is directly proportional to the period

when the damping “constant” b is truly constant.

The graph in Figure 20.16 compares the

logarithmic decrement computed by the standard

model against a case where b ¼ constant. Also

shown in the figure is a set of hysteresis curves

for vt ¼ 10; 1; and 0.1, respectively. Notice that

the damping is large only for vt near 1, in

accord with the bottom plot of Figure 20.15. For

that case, points (a) to (f) and back to (a) are

shown, labels to illustrate work done by the stress

in traversing the hysteresis loop. The algebraic

sign of the work changes around the loop and

the net work done in one cycle is just the area

enclosed by the loop.

For damping based on the Zener (standard linear) model to agree with the simple viscous

approximation, it is necessary that vt .. 1; i.e., the period of the oscillator must be significantly shorter

than the smaller of the relaxation times, as illustrated in the bottom graph of Figure 20.16.

20.16.3 Successes — Models of Viscoelasticity

Viscoelasticity, as an approximation for damping, is evidently quite adequate for some materials.

The assumption of fluid character as a basis for hysteresis is expected to be closest to correct when

E1

E2

dashpot

spring

spring

modulus

s

E1 E2

E1 + E2

E21

E2 (E1 + E2)

E1

0.01 1.0 100

tan d

w p

FIGURE 20.15 Zener Model of anelasticity. Bottom

curves are “frequency” variation of modulus and loss

respectively.

20-46 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

applied to those cases in which variations in strain are almost continuous. The materials of rheological

type for which this appears to be most true are solids built from long chain polymers, i.e., various

plastics. Such materials can yield surprising results, however. Shown in Figure 20.17 are results from a

study that used a nylon monofilament sample (8-lb fishing line). The pair of torsional free-decay

records corresponds to two different temperatures — 290 K (room temperature) and 390 K (above the

glass transition temperature of the nylon). Although a significant increase in the period was observed

as the temperature was increased above the glass transition temperature (changing from 18.2 to

27.8 sec), the logarithmic decrement was found to be almost unchanged. This was not in keeping with

the expectation that softening of the material at the higher temperature would result in significantly

greater damping. The effect is just the opposite of what was mentioned concerning cast iron, which,

though very hard, does not have small damping. Here, a softening does not result in significantly

increased damping.

Although there was some creep observed for both the decays of Figure 20.17, the creep was more

pronounced in the higher temperature case. This is illustrated by the lower curve of the bottom graph,

which is a computer fit in which the secular term necessary for best fit was removed to illustrate the creep.

In both decay cases, the log decrement was calculated by importing the A/D data (Dataq DI-154RS) to

Excel and then using trial and error adjustment of parameters to achieve the best fit.

Although the damping of glasses is normally treated using the theory of viscoelasticity, Granato

(2002) has recently modeled these materials via defects. In his paper, Granato states the following:

“As dislocations carry the deformation in crystals, interstitials are the basic microscopic elements

carrying the deformation in glasses near and above the glass temperature.”

20.16.4 Failure of Viscoelasticity

Unfortunately for the elegant theory of the Zener model that has been presented, there are many

mechanical systems for which the Q is not proportional to frequency, but rather proportional to the

square of the frequency. The logarithmic decrement ðD ¼ p=QÞ has been measured for a host of

Hysteresis

stress

strain

b

f

a

Damping

0.01 0.1

Harmonic Oscillator

with viscous

damping

1. 10. Period

Logarithmic Decrement

c

e

10

1

0.1

d

−

+

+

− +

+

FIGURE 20.16 Characteristics of the Zener model.

Damping Theory 20-47

© 2005 by Taylor & Francis Group, LLC

long-period mechanical oscillators, configured as some form of a pendulum. In all cases, these systems

were described approximately by D ¼ bTaT2 rather than by bTaT: Similar behavior has been noted in

mechanical oscillators other than the pendulum — for example, in the geophysics research of Gunar

Streckeisen and Erhard Wielandt, who are well known for the development of the widely employed STS-1

seismometer. During his pursuit of the Ph.D., Streckeisen (1974) measured the numerical damping

(fraction of critical damping) of a vertical Sprengnether long-period seismometer 5100-V. After

removing the magnet of the velocity transducer (to eliminate eddy currents and reduce viscous air

damping, he found that the numerical damping was proportional to the square of the period between

periods of 7 and 140 sec. He took about 30 measurements over this interval of periods, and showed that

the damping increased from about 0.0008 to about 0.3 — a factor of roughly 400, not 20 as one would

expect for viscous damping. To quote Wielandt (private communication), “the data are very clear.”

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