21.10 Air Influence

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As seen from Figure 21.37, low-frequency motions are likely to be influenced more by internal friction

than by any fluids that interact with the oscillator. The most important fluid is of course air, and a true

delineation between external and internal effects requires that the oscillator be studied in a high vacuum.

It is not enough to just remove most of the air, since the viscosity of gases is surprisingly constant until

the mean free path between collisions becomes a significant fraction of chamber dimensions.

Theoretically, it is possible to roughly estimate air influence, although only in the simplest of

geometries, such as a sphere. In such cases, Equation 21.11 could be used (with accounts for the history

term, using appropriate values for the viscosity and density). It is also possible in some cases to estimate

air influence experimentally, as in the example that follows.

21.10.1 Brass and Solder Rod Pendula

Because of its malleability, the internal friction of solder (lead – tin alloy) is large, compared to that of

much harder brass. A pendulum of each material was studied, both having a length of about 50 cm and a

no correction for history term

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

Log (Q)

3

3.5

4

4.5

FIGURE 21.36 Illustration of how huge errors can occur in damping estimates if one ignores the history term.

Comparison of Theory and Experiment

(Water Damped Pendulum)

Frequency (Hz)

Q

0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

300

350

FIGURE 21.35 Comparison of theory and experiment for a pendulum damped by water.

Experimental Techniques in Damping 21-31

© 2005 by Taylor & Francis Group, LLC

diameter of about 3 mm. The technique used was the photogate method described in Section 21.4.4

(Case 4 above). Unlike the previous study, no lead masses were clamped on the rod — but it used the

same adjustable knife-edge.

Figure 21.39 clearly shows that the internal friction for the solder pendulum is much greater than that

of the brass pendulum.

A nonlinear fit was generated for each decay curve, from which the history of the quality factor was

graphed as a function of velocity amplitude, as shown in Figure 21.40.

Consider the pair of brass curves in Figure 21.40. The large difference in Q at 10 cm/sec (387 compared

to 266) is in stark contrast with their near equality at 50 cm/sec. This is primarily a consequence of air

drag that is quadratic in the velocity at the larger amplitude. It is more important to brass than to solder

because of the small internal friction of the brass.

From the large difference in internal friction of the two materials, a first order correction for air

influence on the solder pendulum is to simply subtract 1=Q of the brass from 1=Q (raw data) of the solder,

to yield the reciprocal Q (corrected) due to internal friction of the solder. This has been done in

Figure 21.41.

0.8

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.85 0.9

f (Hz)

Δf (Hz)

0.95 1

both

added mass only

Theoretical Estimates–Effect on frequency

of Added Mass and Buoyancy

FIGURE 21.38 Example of how fluid properties influence the frequency as well as damping of an oscillator.

ignoring internal friction

ω (rad/s)

0 1 2 3 4 5 6 7

0

Q

350

300

250

200

150

100

50

FIGURE 21.37 Illustration of significant low-frequency errors that result from a failure to recognize the hysteretic

damping component of the pendulum.

21-32 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

From Figure 21.41 it can be seen that the internal friction damping is not simply hysteretic

(constant Q); rather it is a function of amplitude. It can also be seen, from the close proximity of the

solid and dashed curves, that the air influence on the solder pendulum is much less than that of

the internal friction. By contrast, air influence is of comparable magnitude to the internal friction in the

case of the brass pendulum (or even larger, at large amplitude).

A minimum of two frequencies was considered for the study, since the frequency variation of the

damping is different for external and internal frictions. (Note: although the period is a function of

amplitude, the amount of nonisochronism is small compared to the damping changes and is ignored

here.) The periods were matched for the two pendula at each of 2.03 and 2.51 sec. For hysteretic-only

(internal friction) damping, the Q at the shorter period should in theory be 1.53 times that of the longer

period, for both brass and solder. If the damping were viscous only, the factor should be 1.24. In the case

of solder at 10 cm/sec (corrected), the ratio is 1.66 ¼ 131/71, and for brass it is 1.46 ¼ 387/266.

Although the ratio for solder is greater than the expected 1.53, the difference is within experimental

60

50

40

30

20

10

0

0 50 100 150

Time (s)

Velocity (cm/s)

200 250 300

Brass

Solder

Rod Pendula, 2.03 s

Comparison of Brass and Solder Pendula at two different frequencies

40

35

30

25

20

15

10

5

0

0 50 100

Time (s)

Velocity (cm/s)

150 200

Brass

Solder

Rod Pendula, 2.51 s

FIGURE 21.39 Free-decay curves for brass and solder pendula at two different frequencies, showing the larger

internal friction of solder. The velocity is that of the peak value (amplitude) at the top of the pendulum, approx.

22 cm above the axis.

Quality Factors, Rod Pendula

2.03 s

2.51 s

2.03 s

2.51 s

Solder

Brass

400

350

300

250

200

150

100

50

0

0 10 20 30 40 50

Q

Velocity Amplitude (cm/s)

FIGURE 21.40 Illustration of amplitude-dependent damping in a rod pendulum made of (i) brass and (ii) solder.

The two different matched periods of oscillation are indicated in seconds.

Experimental Techniques in Damping 21-33

© 2005 by Taylor & Francis Group, LLC

uncertainty for individual Q values, which from other, more detailed experiments were in the

neighborhood of 5 to 10%.

The ratio for brass (1.46) is between 1.24 and 1.53, as expected, because of the comparable influence of

air and internal friction.

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