Indian Journal of Pure & Applied Physics Vol. 49, October 2011, pp. 657-664 Practical method for thermal compensation of long-period compound pendulum Branislav Popkonstantinovic 1, Ljubomir Miladinovic 2, Miodrag Stoimenov 3, Dragan Petrovic 4, Nebojsa Petrovic 5, Gordana Ostojic 6 & Stevan Stankovski 7* 1-5Mechanical Engineering Faculty, University of Belgrade, Serbia 6,7 Faculty of Technical Sciences, University of Novi Sad, Serbia E-mail: 1 [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; 7*[email protected] Received 3 September 2010; revised 7 April 2011; accepted 25 August 2011 The methods for compensation of the clock pendulum thermal dilatation presented in literature are based on combined analytical and experimental procedures. As a rule, the mass center position is always compensated analytically with sufficiently large factor of safety, while the refinements are obtained experimentally. An analytical method has been studied for the pendulum thermal compensation which considers not only the mass center but also the pendulum mass moments of inertia of the first and the second order. Fast and efficient mathematical method as well as the practical constructive solutions are presented which allow the technically acceptable thermal compensation of the long period compound pendulum. The proposed calculation procedures are iterative and fast converging, which make them suitable not only for the design of vintage “wood rod - lead bob” pendulums, but also for the modern pendulums which feature rods thermally compensated by invar alloy. Moreover, the proposed principles and methods are useful in all scientific measurements where the effect of thermal noise must be minimized or even eliminated. Keywords: Clock, Thermal compensation, Dilatation, Compound pendulum, Temperature 1 Introduction The problem of the pendulum thermal Among various disturbances, thermal dilatations compensation can be solved either experimentally or are the phenomena which most severely deteriorate theoretically. Experiments are founded on the “trial the uniformity of the clock rate. This paper considers and error” approach, are time-consuming and may be the effect of the pendulum temperature dilatation to expensive. The first experimental consideration and the clock rate error and explains the principles by results, reported by Derek Roberts 13 in his remarkable which this detrimental effect can be minimized. historical book “Precision Pendulum Clocks-The All the materials of which pendulums can be made Quest for Accurate Timekeeping” were contributed expand by heat. Consequently, every pendulum by the English celebrated clockmakers of the 17 th naturally goes slower in hot weather than in cold. century. Although the lengthening of the rod is far too small to Despite the fact that the problem of thermal measure, except by most delicate experiments, it compensation of oscillators, especially of the clock contributes to measurable differences. For example, a pendulums, is more than 200 years old, it still common iron wire pendulum ( T = 2 s) yields a represents the respectable and venerable subject of difference of a minute per week between moderate research in modern science and technique. Thus, the spring weather (15 °C) and summer heat (30 °C). For effect of temperature noise on the linear dimension the same conditions, a brass pendulum yields the variations in new broadband seismometers, clarifies which elements of the device are most sensitive to same difference in five days, while for a wooden rod ambient temperature variations and determines the the difference equals one minute in three weeks. It is level of noise generated by all the elements has been important to note that the basic principle for the studied 2. Moreover, the general problem of oscillation temperature compensation is basically simple: the frequency and condition control is also very important pendulum should be made of different materials in modern science and technology and were presented whose temperature dilatations cancel each other out and analyzed by Tangsrirat at al 1. Furthermore, so that the swing period remains unchanged. Linear Tangsrirat and Worapong 4 proposed the novel thermal expansion coefficients for some materials are sinusoidal oscillator and universal biquadrate filter presented Table 1. However despite of simplicity, the with independent tuning of the natural angular realization of this idea may be quite difficult. frequency and the bandwidth. 658 INDIAN J PURE & APPL PHYS, VOL 49, OCTOBER 2011 Table 1 — Linear thermal expansion coefficients for some determination of gravitational constant using the time- materials of-swing method. Woodward 9,12 and Matthys 11 had Lead AluminiumBrass Iron Wood Invar analyzed various methods and technical solutions for the thermal compensation of pendulum and proposed a[K −1]10 −6 28-29.2 23.8 18.4 10.5-12.3 4.0- 4.5 1.2-1.5 important practical advice for the design of high precision pendulum clocks. Kaushal 10 reviewed the Agatsuma et al 3. were the first to directly measure role played by the pendulum and its variants in the the thermal fluctuation of a pendulum in an off- development of mathematical sciences. In particular, resonant region using a laser interferometric the case of a pendulum harmonic oscillator in physics gravitational wave detector, concluding that the is highlighted in the context of mathematical, measured thermal noise level corresponds to a high conceptual, conventional and engineering disciplines quality factor of the order of 10 5 of the pendulum. Jie 5 besides the one in mathematical physics. Gretarsson et al . introduced the weighting function by 14 and coauthors considered the problem of pendulum considering the nonlinear least-squares fitting method thermal noise in large, advanced interferometers for and the correlation method and developed a method to 15 detection of gravity waves. Peter R Saulson have calculate the thermal noise influence on the period analyzed the thermal noise in mechanical experiments measurement in a torsion pendulum. Consequently, and constructed models for the thermal noise spectra they obtained a rigorous formula of thermal noise of systems with more than one mode of vibration and limit and uncertainty estimation for the pendulum evaluated a model of a specific design of pendulum period measurement. The significance of this suspension for the test masses in a gravitational-wave contribution lies in the precise determination of the interferometer. Finally, it is necessary to consider the Newtonian gravitational constant using the time-of- complexity of product development and swing method. Cumming et al 6. studied the finite production 16,17. element modeling and associated analysis of the loss A novel, fast and efficient, simplified calculation as in quasi-monolithic silica fiber suspensions for future well as the practical design solution have been advanced gravitational wave detectors. They proposed in the present paper which yields the emphasized that the thermal noise of the detector technically acceptable pendulum temperature suspension is an important noise source at operating compensation. Moreover, the proposed principles and frequencies between 10 and 30 Hz approximately, methods should be of use in all scientific which results from a combination of thermo elastic measurements where the effect of thermal noise must damping, surface and bulk losses associated with the be minimized. suspension fibers. They concluded that its effects can be reduced by minimizing the thermo-elastic loss and optimization of pendulum dilution factor via 2 Long-period Compound Pendulum appropriate choice of suspension fiber and attachment A physical pendulum which is almost perfectly geometry. Dzhashitov and Pankratov 7 considered the balanced, is a simple way to make a long-period control possibility of interconnected mechanical and pendulum. As shown in Fig. 1, this can be thermal processes in non-linearly perturbed dynamic accomplished by fastening two bobs (weights) on systems for irregular motions and parametric both ends of the pendulum rod in such a way that its temperature perturbations. It was shown that the pivot is slightly dislocated from the pendulum center choice of thermal parameters and mechanical of mass. Since this configuration has large inertia but subsystems as well as the introduction of mechanical a small restoring force, the pendulum swing period control subsystem is proportional to the temperature can be adjusted to very high values. Compound gradient between its elements into the feedback which pendulums with long-swinging period find use in can provide both regularization of oscillations and measurement of gravity, designs of seismometers and control of oscillations at parametric temperature vibration isolation. In addition, they are suitable perturbations. Jie and Dian-Hong 8 have considered oscillators for metronomes, large mechanical clocks the environment temperature variation and (tower and public time keepers) as well as for some inhomogeneity of background gravitational field and types of stationary timepieces. proposed an improved correlation method to The design of the temperature-compensated long- determine the variation period of a torsion pendulum period pendulum is presented in this paper through with high precision. This analysis is significant for the three key phases: POPKONSTANTINOVIC et al .: THERMAL COMPENSATION OF LONG-PERIOD COMPOUND