20.12 Hysteretic Damping

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20.12.1 Physical Basis

The model of simple harmonic oscillation with viscous damping assumes dissipation from an externally

acting force. It is not suited to a conceptual understanding of hysteretic damping. To accommodate

internal friction requires more than a single mass connected to the elastic component responsible for

restoration. Two systems are pedagogically useful in this regard, one being a long-period physical

pendulum (mechanical), and the other being the oscillator used by Ruchhardt to measure the ratio of

heat capacities of a gas (mainly thermodynamic). Because of widespread confusion concerning the

difference between viscous and hysteretic damping, both cases are presented here. The treatments are

provided as evidence for the premise that hysteretic damping is the more important case for applied

physics and engineering.

It is common knowledge that the damping of a mechanical oscillator results from the conversion of

mechanical energy into thermal energy. One might expect, then, that a direct consideration of

thermodynamics could yield conceptual understanding of the underlying physics. Although an ideal gas

is rarely considered in this context, there is a classic experiment which speaks to its relevance. It is the

ingenious technique used first in 1929 by Ruchhardt to measure g; the ratio of heat capacity at constant

pressure to that at constant volume (Zemansky, 1957).

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20.12.2 Ruchhardt’s Experiment

Consider a piston of mass m moving in a cylinder of cross-sectional area A; alternately compressing and

expanding a volume of ideal gas V0 about the residual pressure P0: Assume that there is no sliding friction

between the piston and the cylinder. A small displacement x of the mass results in volume change

DV ¼ V 2 V0 ¼ Ax: There is a restoring force F ¼ ADP; where the pressure difference DP relates to DV

through an assumed adiabatic process; i.e., the period of the motion is assumed too short for appreciable

heat transfer into and out of the gas. Using PVg ¼ constant; one obtains

gP0Vg21

0 DV þ Vg

0 DP ¼ 0 ð20:23Þ

from which one obtains

mx€ þ

gP0A2

V0

x ¼ 0 ð20:24Þ

This is the equation of motion of a simple harmonic oscillator. There is no damping because of the

assumed adiabatic process. By measuring the period T ¼ 2p=v ¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V0m=gP0A2

p

, one can estimate g:

Historically, it appears that such measurements slightly underestimate g; which can be understood as

follows.

The ideal gas equation of state PV ¼ NkT yields, through differentiation

P0xA þ V0

F

A ¼ NkDT

mx€ þ

P0A2

V0

x ¼

NkA

V0

DTðtÞ ¼ FdðtÞ ð20:25Þ

Notice the difference between Equation 20.24 and Equation 20.25. In Equation 20.25, damping is

possible (a type of “negative drive” term) from temperature variations associated with heat transfer

during traversal of the cycle. If it were possible for the oscillation to be isothermal (DT ¼ 0 at very low

frequency, essentially quasistatic), then the frequency would be lower than that of the adiabatic case, since

g . 1 is missing from Equation 20.25. In the isothermal case, there would also be no damping, since the

heat into the gas during compression would be balanced by that which leaves during expansion. The only

way to get damping is for the paths of compression and expansion in a plot of pressure vs. volume to

separate, i.e., for there to be hysteresis. Reality must correspond to something between the two extremes

of adiabatic and isothermal, with experiment obviously favoring adiabatic. The process must depart

somewhat from adiabatic, however, since there is damping, which Equation 20.25 shows to derive from

temperature variations yielding hysteresis. It is interesting to look at the temperature variations relative

to a “driving force,” F0d

ðtÞ: In the Ruchhardt experiment, there must be small variations DT 0ðtÞ that lag

behind xðtÞ: (These are not the reversible temperature variations of the adiabat, onto which the DT 0ðtÞ are

superposed.) By comparing with Equation 20.25, the right-hand zero of Equation 20.24 may be replaced

with a damping force that can be written in terms of the velocity as

F 0dðtÞ / DT0ðtÞ ! 2

c

v

x_ ð20:26Þ

where c ¼ constant. Notice that the multiplier on the velocity is not simply a constant, but rather a

constant divided by the angular frequency. The use of velocity is mathematically convenient, but the

magnitude of the velocity (speed) is not expected to be a first order influence on the temperature changes

of hysteresis type. The derivative of x with respect to time not only shifts the phase by 908, which

accommodates the lag with which heat is transferred, but it also introduces a frequency multiplier

through the chain rule. Thus, to make damping proportional to the velocity would cause increased

dissipative heat flow and thus increased damping as the frequency is increased. Since this does not

happen, and lest we introduce a nonphysical term into the equation, it is necessary to divide by the

frequency. Replacing the right-hand-side zero of Equation 20.24 with Equation 20.26, we obtain the

Damping Theory 20-37

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modified equation of motion, with damping

mx€ þ

c

v

x_ þ

gP0A2

V0

x ¼ 0 ð20:27Þ

Additional justification for the form of the damping term in Equation 20.27 can be realized by looking at

cases where there is negative damping, i.e., c , 0: Such is true when the gas is caused to cycle as an

engine. An illustrative case study was that of a low temperature Stirling engine (Peters, 2002a, 2002b,

2002c), in which reasonable agreement between theory and experiment was realized through the use of

an equation based on the same arguments used to derive Equation 20.27.

It is seen that a straightforward modeling of Ruchhardt’s experiment to include damping yields an

equation of motion that is in the form of hysteretic damping. It appears that, for many systems in which

the dissipation is dominated by internal friction, hysteretic damping is a near universal form.

20.12.3 Physical Pendulum

In the paper by Speake et al. (1999), one finds the

following statement:

the logarithmic decrement ðQ21Þ varied as

the inverse of the square of the frequency.

We interpreted this as evidence that, in Cu –

Be over this frequency range, the imaginary

component of Young’s modulus was independent

of frequency, contrary to that which

was predicted by the Maxwell model.

To fit their theory with experiment, they used a

“modified” Maxwell model with a distribution of

time constants that ranged from 30 sec to more

than 4000 sec. Motivation for their continued

modeling efforts derived partly from the observation

by Kuroda (1995) that anelasticity was

cause for some of the huge errors that have been

present in estimates of the Newtonian gravitational

constant, G, by the time-of-swing method.

Although it gives agreement with their particular

experiment, the model of Speake et al. (1999) does

not have the blessing of Occam’s razor. Moreover,

their claim that damping derived primarily from

their flex pivot of Cu – Be may not be true. Other studies suggest that the material defining the axis of a

long-period pendulum is for many cases no more important (and sometimes much less important) to the

damping than the material from which the pendulum proper is constructed. A model which also agrees

with experiment of the type they conducted, but which is simpler, is now presented.

Illustrated in Figure 20.13 is an idealized long-period mechanical oscillator which could be labeled a

“physical pendulum.” The top and bottom masses are the same, M; assumed to be much greater than the

mass of the connecting structure, which is represented by the curved line.

A primary mechanism for internal friction damping can be understood by looking at the external

forces acting on the pendulum, which are pictured in the “negative displacement” (b) case. The upward

“normal” force that acts through the pivot (usually a knife-edge) is opposed by the pair of bob-weights

situated left and right, respectively, of the axis of rotation. As the pendulum swings alternately between

positive and negative displacements, the structure undergoes periodic flexure. It should be pointed out

that internal friction could still be operative throughout the structure even without net bending; i.e., there

M

h ΘI

ΘI

M

(a) positive displacement (b) negative displacement

FIGURE 20.13 Idealized physical pendulum used to

develop the modified Coulomb damping model.

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could be complementary pieces of the structure undergoing compression and tension. Even if the

oscillator were in the weightlessness of space, a drive torque would result in dynamic reactionary forces

that give rise to damping by this means.

Assume that the masses are separated a distance 2L and the axis of rotation is DL above the geometric

center. Applying Newton’s Second Law, with the lower mass causing a restoring torque and the upper

mass a “destoring” torque, yields

u€1 þ

g

2L

1 þ

DL

L

􀀏 􀀐

u1 2

g

2L

1 2

DL

L

􀀏 􀀐

u2 ¼ 0; u2 ¼ u1 þ h ð20:28Þ

(Note: Equation 20.28 can be rewritten to accommodate larger displacements, where elastic nonlinearity

gives rise to unusual behavior. The amplitude trend of the period is opposite to that of the gravitational

nonlinearity, thus providing for improved isochronism. For details refer to Peters, 2003a, 2003b, 2003c).

The difference in displacement of the masses involves an elastic term proportional to u1 and a

dissipative term that depends on its time rate of change, i.e.

h ¼ c u1 cos d 2

u_1

v

sin d

􀁻 !

; v ¼

ffiffiffiffiffiffiffi

g

DL

L2

s

ð20:29Þ

where c is a dimensionless constant. This result can be obtained by the complex exponential Steinmetz

(phasor) method. The equation is consistent with the common assumption that stress and strain are

related through a complex constant. The angle d is the phase angle with which h strain) lags behind u1

(stress). To describe the motion of the lower mass, we can ignore the elastic part of h; since it does not

contribute to the damping (or if the rod does not bend, assuming there still is damping as noted

previously). We thus remove the subscript, and after some algebra obtain the result

u€þ

a

v

u_ þv2u ¼ 0; a ¼

gc

2L

sin d; for DL ,, L ð20:30Þ

which can also, in terms of Q ¼ 2pE=ð2DEÞ; be expressed as

u€þ

v

Q

u_ þv2u ¼ 0; Q ¼

2L

gcd

v2; d ,, 1 ð20:31Þ

If, as a material property, d is independent of frequency, then Q is quadratic in the frequency; i.e., the

damping of the pendulum due to internal friction is inversely proportional to the square of the

frequency — even though the internal friction (determined by d) is itself frequency-independent. It is

important to note that the frequency dependence of internal friction is not to be equated with the

frequency dependence of the Q of the oscillator, even though internal friction is frequently stated as

simply 1=Q: This will be discussed in greater detail in Section 20.13.2.

20.12.3.1 Test of Q Dependencies

The dependence of Q on frequency and length in Equation 20.31 was tested experimentally with a

physical pendulum. Two Pb spheres, each of mass approximately 1 kg, were each drilled through a

diameter to allow the insertion of the shaft of an aluminum alloy arrow (length approx. 70 cm) of the

type commonly used with compound hunting bows. A second hole was drilled perpendicular to the first

and tapped for a set screw. The shaft was sawed into two pieces, which were rigidly rejoined around a

carbon – steel knife-edge using force fit and epoxy to machined protuberances above and below the knifeedges.

The knife-edges extend perpendicularly outward on opposite sides of the arrow at its center.

20.12.3.2 Simple Method to Measure Damping

Although an SDC sensor could have been employed instead, the experiments to be described were

performed with a measurement technique that warrants description because of its novel simplicity — yet

it is reasonably accurate. To measure both period and damping, a small “flag” was super-glued to the top

of the upper shaft. This flag was a small, thin, U-shaped piece of plastic in which the upper legs of the U

were about 1 mm wide, with a spacing between centers of about 0.5 cm. An infra-red photogate of the

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type used in general education laboratories was mounted so the flag would trip the photogate during

pendulum oscillation. Two different timing measurements were then performed, using a Pasco Smart

Timer. In every run, the pendulum was displaced initially about 108 by hand and then released. There was

no need for precision initialization.

In the pendulum mode of the timer, the period was directly measured. For this case, the photogate was

positioned, relative to the U-shaped flag (for which one vertical arm is slightly longer than the other), so

as to be interrupted only once by the pendulum per pass. In the time-interval mode, the flag was

positioned so that both arms interrupted the photogate beam. The reciprocal of this time of interruption

proved to be a reasonable measure of the instantaneous speed of the pendulum at the position of the

photogate, which was that of maximum kinetic energy. The time intervals were recorded manually for

traversals separated by one period, through five cycles of oscillation. These numbers were then typed into

Excel and their reciprocals graphed. A trendline (using the option to print the slope) was applied to the

near linearly declining graph. The decrement of this line (fractional decrease per cycle) proved to be a

good approximation to the logarithmic decrement of the motion, which could have been estimated with

exceptional precision by means of the other techniques mentioned in this chapter.

In the first set of experiments, the sphere on the lower shaft was maintained at a constant distance from

the knife-edge, while the mass on the upper shaft was positioned at increasingly greater distances from

the knife-edge to lengthen the period. Over the full range of periods considered, the distance between the

two masses changed by a small amount around its nominal value of 67 cm. The results of this first study

are shown in the left graph of Figure 20.14, where the log-decrement has been plotted vs. the square of the

period. The Q of the pendulum ðp=DÞ may be calculated for any value of the period using the indicated

slope of 0.0004. For example, the Q at a period of 10 sec was 76, this being near the shortest period

considered. Near the other extreme of T ¼ 35 sec; Q ¼ 6: At the shortest possible periods, damping due

to air drag would begin to become important.

The reasonable fit of the linear regression vs. period squared is consistent with the prediction by

Equation 20.31 that Q should be quadratic in the frequency.

The Equation 20.31 also indicates that the log-decrement should be proportional to the reciprocal of

the distance, L; between the masses. To test this prediction, the period of the pendulum was measured as a

function of mass separation, also using the smart timer. In generating the data for the right graph of

Figure 20.14, the period was maintained constant at 20 sec. For every datum, the top sphere was always

only slightly closer to the knife-edge than the lower sphere. At 0.049, the intercept of the trendline differs

enough from zero, relative to the size of the error bars, to imply a systematic error. Possible sources of the

error include: (i) the masses are of finite size, rather than being points as assumed by the model, and (ii) a

nonnegligible mass from parts other than the spheres. Nevertheless, the data show a clear size dependence

of the Q:

Log. Decr. vs square of period

(L ~ 67 cm)

Log. Decr. vs 1/(mass separation)

(T =20s)

y = 0.00040 x + 0.015

R2 = 0.96

0

Δ

Δ

500 1000 1500

T2 (s2)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.45

0.4

0.35

0.25

0.2

0.15

0.1

0

0.3

0.05

y = 5.5 x + 0.049

R2 = 0.96

1 / L (cm−1)

0 0.01 0.02 0.03 0.04 0.05 0.06

FIGURE 20.14 Results of experiments to test the dependencies of Q on (i) frequency and (ii) length of pendulum.

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The experiments just described do not permit one readily to isolate the source of the damping, which,

for the cases in Figure 20.14, had the knife-edge resting on silicon wafers (integrated circuit stock

material). It is not known to what extent the dissipation was dominated by strain in the knife-edge –

silicon interface or by flexure of the aluminum arrow. Although the model that generated equation 20.31

assumed only the latter, there is nevertheless theoretical and experimental basis for model acceptance,

regardless of the details of the damping.

20.12.3.3 Highly Dissipative, yet Hard Materials

The same pendulum was used to demonstrate some counterintuitive features of internal friction

damping by replacing the Si wafers with various materials. When very soft material, such as lead, was the

support for the knife-edges, there was a significant increase of the damping, as expected. It was also

found, however, that cast iron increased the log-decrement (10-sec period) by more than 40%. The same

was also true of ceramic PZT wafers of the type used to ignite gas grills by striking the wafer impulsively.

Both the cast iron and the lead – zirconate – titanate samples are very hard, so the internal friction must

derive from large defect densities in which atomic disorder is a sensitive function of stress. Some other

hard interfaces, such as steel on glass, or steel on sapphire, did not show a difference from steel on Si,

which suggests that the dominant source of damping for the pendulum in all these cases was flexure of

the aluminum shaft.

The observation involving cast iron is consistent with its known excellent damping properties at higher

frequencies — important, for example, to engine blocks. Some magnesium alloys have also been

developed that have excellent damping characteristics without seriously sacrificing strength.

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