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Research Papers

Barometric Compensation of a Pendulum

[+] Author and Article Information
Douglas S. Drumheller

Hyperbolic Consulting, Cedar Crest, NM, 87008dsdrumh@q.com

A detailed description of the project is posted on the Internet at http://www.trin.cam.ac.uk/clock/.

J. Appl. Mech 79(6), 061009 (Sep 17, 2012) (7 pages) doi:10.1115/1.4006766 History: Received July 15, 2011; Revised April 11, 2012; Accepted April 22, 2012; Published September 17, 2012; Online September 17, 2012

For centuries, accurate pendulum clocks were of strategic scientific and military importance. Environmental factors such as temperature and pressure affect the rate of a pendulum. Without temperature compensation, accuracy is limited to about 10 s per day. With it, however, a secondper month is possible. Barometric compensation is required for additional improvement. Indeed, highly accurate pendulum clocks, such as Shortt-Synchronome Pendulums, are placed in controlled partial vacuums to avoid this unsolved problem. We model the pendulum as a Duffing oscillator to account for the effects of circular deviation, buoyancy, virtual mass, and aerodynamic drag. The application of routine perturbation methods reveals a simple and elegant solution to the problem of barometric compensation, an arc of motion where a balance is achieved. The results are confirmed by comparison to experiment.

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References

Figures

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Figure 3

Normalized amplitude R versus frequency ω¯ for η¯=0, 0.25, 0.5, …, 2 and γ¯=0.75

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Figure 4

Escapement angle Θ¯ɛ versus frequency ω¯ for η¯=0, 0.25, 0.5, …, 2 and γ¯=0.75

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Figure 5

Going sensitivity K, s/day/(kg/m3 ), for Matthys’ [11] clock when Q = 12,000 (solid) and Q = 1200 (dashed). Barometric compensation exists on the bold contours.

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Figure 6

Trinity Clock data for air density versus semi arc at the new nominal semi arc of 48 mrad. The solid lines have a slope corresponding to our predicted value of /dρa at this new semi arc. The dot-dash line corresponds to the value reported by the Trinity website for the old semi arc of 53 mrad.

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Figure 7

Trinity Clock data for air density versus going at the new nominal semi arc of 48 mrad. The solid lines have a slope corresponding to our predicted value of K at this new semi arc. The dot-dash line corresponds to the value reported by the Trinity website for the old semi arc of 53 mrad.

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Figure 1

The pendulum geometry

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Figure 2

Comparison of Matthys’ data [11] for the decaying motion of a free pendulum (squares) and Eq. 20 (solid). Based on these data, Matthys [11] reports a Q of 11,900 above ϑ=1 deg and 14,250 below (we divide his values by 2 due to a difference in his definition of Q).

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