20.13 Internal Friction

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20.13.1 Measurement and Specification of Internal Friction

Mechanical spectroscopy is a popular means for measuring internal friction of materials (Fantozzi,

1982). Typically, a torsion pendulum is used to stress harmonically a sample and the lag of the response

(strain), relative to the stress, provides the loss tangent and thus the internal friction. In such

experiments, it is widespread practice to report internal friction as Q21: There can be confusion because

of this practice, depending on the nature of the measurement technique, i.e., whether one actually

measures Q as opposed to measuring something proportional to the stress – strain lag angle. If Q is

obtained from an oscillatory free decay, using the logarithmic decrement defined as follows, then there is

no problem.

D ¼ ln

xn

xnþ1 ¼ bT ¼

p

Q ð20:32Þ

Here, xn and xnþ1 are adjacent turning point amplitudes separated by one period of the motion, T: In

practice, it is very difficult to adjust a mechanical system to oscillate over a wide frequency range. The

widest range known to the author, for a mass – spring system, involved the work of Gunar Streckeisen

(1974), in which a vertical seismometer using the LaCoste spring was adjusted to have periods in the

range between 7 and 140 sec. Because of the difficulties in attaining a wide range of eigenmodes, internal

friction is typically determined with a specimen that does not oscillate. We now consider that case.

20.13.2 Nonoscillatory Sample

In the typical torsional pendulum used to measure internal friction, the sample is of very small mass.

Such a pendulum was built, for example, around the original version of the fully differential capacitive

sensor, to study magnetoelastic wires (Atalay and Squire, 1992). As with many delicate instruments, the

Atalay and Squire instrument was of the type labeled “inverted.” A silk fiber at the top of the specimen

was used to provide minimal tension in the sample. They used one linear rotary differential capacitance

Damping Theory 20-41

© 2005 by Taylor & Francis Group, LLC

transducer (LRDCT) (Peters, 1989) in the drive mode to provide a known stress to the delicate

magnetoelastic sample and a second LRDCT to measure the strain magnitude and the angle by which it

lags behind the stress because of an elasticity. As such, they were measuring the lag angle and not Q; as

will now be shown.

Without an inertial term, the sample response x to a periodic external force F is governed by

F ¼ Kx ¼ ðk þ jzÞx ¼ F0ejvt ð20:33Þ

so that the transfer function is given by

x

F ¼ k21 2 j

z

k2 ð20:34Þ

from which it is seen that the measurement does not yield Q21 but rather the lag angle z=k; where k is

constant. Perhaps the measured angle, which is an indicator of the internal friction, has been called Q21

because k ¼ mv20

for an oscillator of frequency v0; and Q ¼ mv20

=z for the freely decaying oscillator. Bear

in mind, however, that this expression for k does not apply to the nonoscillatory measurement just

described. There is a frequency square difference between such a measurement and what would be

measured if an adjustable oscillator were being considered.

An example of the importance of this issue is found in the article by Lakes and Quakenbush (1996), in

which one reads from the abstract the following statement:

The damping, tan d; followed a n2n dependence, with n < 0:2; over many decades of frequency n:

This dependence corresponds to a stretched exponential relaxation function, and is attributed to a

dislocation-point defect mechanism. It is not consistent with a self-organized criticality dislocation

model which predicts tan d / A22: Dislocation damping in metals is relevant to development of

high damping metals, the behavior of solders and of support wires in Cavendish balances.

The present arguments suggest that the experiment by Lakes and Quackenbush is (1996) not in strong

disagreement with the SOC model; that the magnitude of the exponent difference between theory and

experiment is really 0.2 and not 1.8 as they have indicated.

20.13.3 Isochronism of Internal Friction Damping

It is well known that, in the viscous damping free-decay case, the frequency of oscillation is lowered by

damping according to

v1 ¼

ffiffiffiffiffiffiffiffiffiffiffi

v20

2 b2

q

¼ v0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 ð2Qv Þ22

q

ð20:35Þ

and the resonance frequency of the driven oscillator is lowered even further (Marion and Thornton, 1998).

It is not well known how difficult it is to measure this damping “red-shift,” which brings in features of the

Heisenberg uncertainty principle. Additionally, it is not well known that extensive damping experiments

suggest that the frequency may not, for some systems, depend on the damping at all; i.e., the oscillator

is isochronous. Isochronism cannot be realized with a linear homogeneous differential equation, but it

can be realized with a nonlinear form that is obtained by modifying the damping term as follows:

v

Q

x_ !

p

4

v

Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2½xðtÞ􀀉2 þ ½x_ðtÞ􀀉2

q

sgnðx_Þ ð20:36Þ

where sgnðdx=dtÞ is the algebraic sign of the velocity — it causes the equation of motion to be nonlinear

even if the square root term were not present. For small damping, the square root term can be shown to be

equal to the time-dependent amplitude of the motion multiplied by the angular frequency.

Other damping types are possible and are indicated in Peters (2002a, 2002b, 2002c) (…universal…)

where evidence is also provided for harmonic distortion in the waveform because of the nonlinearity. It is

shown in Peters and Pritchett (1997) that the oscillation is isochronous.

20-42 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

For large values of Q; the lag angle (radian measure) is given by d ¼ 1=Q: Researchers usually measure

d and specify the magnitude of the internal friction as Q21: As noted previously, Q is proportional to

frequency for the viscous damped oscillator. Thus, for viscous damping, the internal friction is inversely

proportional to the frequency.

For hysteretic damping we obtain the result

tan d ¼ a ¼

h

k ð20:37Þ

where the variables are defined in Equation 20.19. For small damping in which tan d ¼ d ¼ Q21; we find

that the internal friction for hysteretic damping is inversely proportional to the square of the frequency,

since h is constant and k ¼ mv2:

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