20.9 Air Influence

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Even when operating an oscillator in high vacuum, there is a significant remanent damping that derives

from internal friction. This fact is illustrated in Figure 20.9, which provides data for two different

“simple” pendula. They are simple in the sense that the bob mass is concentrated near the bottom of the

pendulum structure. In the figure, decay time (reciprocal of the decay constant, b) has been plotted

against the natural log of the pressure in mtorr. Pressure reduction was done with a high-quality

roughing pump, and the pressure was measured with (i) a mechanical gauge in the range

8 torr , P , 760 torr and (ii) a thermocouple vacuum gauge for 0 , P , 100 mtorr. In the range

from 100 mtorr to 8 torr, the pressure could not be accurately measured with either of these gauges.

Similarly, pressures below 1 mtorr could not be presently measured, but in similar other experiments

with this pump, and using an ion gauge, it was easy to pump below 0.01 mtorr.

The period of each pendulum was very close to 1 sec, and the starting amplitude of the motion for

every case was about 25 mrad. The heavier pendulum used a pair of pointed steel supports resting on

single-crystalline silicon wafers to provide the axis of rotation. At the bottom of the pendulum was

attached a solid lead ball whose mass was approximately 1 kg. The lighter pendulum was supported by a

steel knife-edge resting on hard ceramic flats, and a large (10.3 cm dia.) lightweight (143 g) hollow metal

sphere was attached at the bottom to provide as much air drag as possible. The motion was measured

with an SDC sensor feeding the computer through a Dataq DI-154RS A/D converter.

Although air damping is evident in Figure 20.9, it is not as influential as one might expect, at least for

the heavy pendulum. Moreover, at atmospheric pressure, it was easy to demonstrate the importance of

nonlinear drag. As also noted in Nelson and Olssen (1986), this form of fluid friction caused a significant

amplitude-dependent damping.

The remanent damping, once air influence is eliminated (pressure below 1 mtorr), is substantial

relative to atmospheric damping, for both pendula. Removing the air increased the Q from 7500 to

3500

3000

2500

2000

1500

1000

500

0

0 5 10 15

Time (S)

heavy pendulum

light pendulum

In (P, mTorr)

y = −60.8 x + 3220

y = −86.1 x + 1470

R2 = 0.95

R2 = 0.98

FIGURE 20.9 Pendulum damping as a function of pressure in a vacuum chamber.

20-30 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

10,100 for the heavy pendulum and, for the light pendulum, the increase was from 1000 to 4600. We thus

see that even a pendulum designed to be heavily influenced by air drag also has significant damping that

depends on the material from which the pendulum is fabricated or on the material upon which it rests.

The difference in internal friction damping between the heavy and light instruments was not expected

to be so great. Although this might be due to the difference in axis-type (points for the heavy instrument

and knife-edge for the light), no systematic effort was made to determine the primary source of the

damping difference. In addition to different axis designs, the means for holding the instruments together

was different. The light pendulum used a large-diameter solid brass wire between the axis and the lower

mass, and the heavy pendulum used an aluminum tube.

Both of the pendula used to generate Figure 20.9 were relatively high-frequency instruments (period of

1 sec). The pivot was located, in each case, near the top end of the instrument. As such, they stand in stark

contrast with the instruments that motivated this paper, where long-period pendula were used. A simple

instrument to demonstrate some of the complexities of long-period instruments is a rod-pendulum of

adjustable period (refer to Figure 20.1 above). The closer the axis to the center, the longer the period and

the greater the influence of internal friction. It is easy to show that the sensitivity of a pendulum to

external forces is proportional to the square of the period. Similarly, the ability to detect influence of

internal configurational change is quadratic in the period.

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