Practical Method for Thermal Compensation of Long-Period Compound Pendulum

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 PENDULUM 659

• Assumption of the pendulum swinging period, If the swing period T = 4 s has to be chosen for any suitable materials and concept design. reason, the length of the equivalent mathematical • Aalculation of temperature compensation. pendulum is l ≈ 4 m approximately which would be • Check of preliminary solution for residual errors, prohibitive for indoor use. That is the main reason for necessary refinements and finalization of the adopting the preliminary pendulum configuration as technically acceptable design of the thermally shown in Fig. 1. The length of the exposed pendulum compensated pendulum. can be chosen to be small enough, its mass disposition can still possess a large moment of inertia and 3 Assumptions and Preliminaries sufficiently small restoring force. Consequently, the The first assumption pertains to the pendulum swing period of that compound pendulum can be materials which are suitable for thermal adjusted to be almost arbitrarily long. compensation: the pendulum rod will be made of As shown in Fig. 2, let us assume that pendulum wood (white deal) and the bob of lead. Since the bobs are of prismatic shape with rectangular cross- linear thermal expansion coefficient of the white deal section. If the cross-section dimensions ( m × n) are −6 −1 is very small in the grain direction ( αW = 4 10 K ), known, the bob’s linear mass density µ0 can be −6 3 and the metal lead is extremely large ( αPb = 29 10 defined as: µ0 = ρPb ⋅m⋅n, where ρPb = 11.34 g/cm is K−1), temperature dilatations of these materials cancel the volumetric mass density of lead. If the linear mass each other out efficiently. The wooden pendulum rod density µ0 of the pendulum bob is known, its mass M should be thoroughly dried and saturated with oil or can be calculated as: M = µ0 ⋅ lB, where lB is the length some antiseptic fluid and varnished to prevent damp of the bob. Let us also assume that the upper and absorption. lower pendulum rods consist of two pairs of wooden

Fig. 1 — Long period compound pendulum Fig. 2 — Compound Pendulum Thermal compensation 660 INDIAN J PURE & APPL PHYS, VOL 49, OCTOBER 2011

cylindrical sticks. Stick diameters dW1 and dW2 of the Now, we are going to calculate the approximate upper and lower pendulum rods are different but value of λ and suppose that λ is the same for both known and their lengths lW1 and lW2 have to be parts of the compound pendulum. If we assume the determined. cross-section dimensions (m × n) of the pendulum The realistic mathematical models of the thermal bob: m = 20 mm , n = 30 mm, bob mass can be 3 pendulum compensation require complex and time- estimated at M = 2 kg. ( ρPb = 11.34 g/cm ). Let us also consuming calculations. Wooden pendulum rod and assume that the masses of each part of pendulum rod its bob of lead should expand longitudinally in segments are the same: m = 0.4 kg. ( ρW = opposite directions in such a way that the pendulum 0.65 g/cm 3). The coefficients for linear thermal swing period remains unchanged. However, the expansion for the white deal and lead are known resulting accuracy of such a difficult and intricate (T.1). Finally, the pendulum bob and rod length ratio analysis could be paralleled by a much simpler can be determined as: λ = 0.305. procedure. Instead of demanding the thermal invariability of the pendulum mass moment of inertia 4 Simplified Calculations (of the second order), we can calculate and attain only If the compound pendulum as shown in Fig. 2 is the fixed position of the pendulum mass center and perfectly balanced, its swing period T would be obtain a technically acceptable, if not perfect, infinite. Should one dislocate the pivot point away solution. from the pendulum center of mass for the distance x, The position of the pendulum mass center is the swing period would be limited to T. Our task is to determined by its (static) mass moment of inertia of determine one particular value of “ x” for a predefined the first order S: value of T = 4 s. Considering the formula given in Eq. (4) for the oscillations period T of the compound l a W(1+ W θ ) pendulum: Sm= + Ml(W (1 + a W θ ) 2 …(1) l a J Pb(1+ Pb θ ) T = 2π ; …(4) −),θ =t − t 0 ; 2 gξC M

In formula given in Eq. (1), lW and lPb are the the following parameters must be determined as the lengths of the pendulum rod and bob, respectively at functions of the distance x: temperature t0; aW and aPb are linear thermal S1, S 2 ~ mass moments of inertia of the first order of expansion coefficients for wood and lead; m and M the upper and lower pendulum bobs, respectively: are the masses of wooden rod(s) and bob of lead, respectively. Formula given in Eq. (1) describes S as a 1 2 S1=−µ 0 a(2 − alx )( − ) function of temperature θ = t −t0. Derivative (d S/ dθ) is 2 defined by the expression given in Eq. (2) and 1 2 S=µ a(2 − alx )( + ) …(5) represents the change rate of the pendulum mass 22 0 moment of inertia of the first order S per unit change of temperature θ. S = S 1 + S 2 ~ total mass moment of inertia of the first order of both pendulum bobs: dS mla⋅WW ⋅ Mla ⋅ PbPb = +⋅⋅−M lW a W ; …(2) SSS a ax dθ 2 2 =+=1 22µ 0 (2 − ) …(6) Derivative given in Eq. (2) can be nullified (d S/ dθ) All mass moments of inertia of the first order are = 0 for one particular ratio a = (lPb /l W) given by determined for an axis passing through the pivot Eq. (3): point; M = M 1 + M 2 ~ total mass of both pendulum bobs : l a 2M+ m a =Pb = W ⋅ …(3) MM= + M = 2 alµ ; …(7) lw a Pb M 1 2 0

This means that the position of the pendulum mass ξC ~ position of the pendulum mass center : center will be invariant to the changes of temperature S if the ratio λ of the pendulum bob and rod lengths ξ = =(2 − a ); x …(8) satisfies the condition given in Eq. (3). C M POPKONSTANTINOVIC et al .: THERMAL COMPENSATION OF LONG-PERIOD COMPOUND PENDULUM 661

Positive direction of coordinate ξC is as shown in The length of the upper pendulum rod is l1 = l – x 2 = Fig. 2. 100 cm – 11.28 cm = 88.72 cm. The length of the J01 ~ mass moments of inertia of the upper lower pendulum rod is l2 = l + x 2 = 100 cm + 11.28 pendulum bob about an axis passing through its centre cm = 111.28 cm. The length of the upper bob is of mass; P1 ~ Steiner’s component of the upper L1 = a ⋅ l 1 = 0.305 ⋅88.72 cm = 27.06 cm and its mass pendulum bob mass moment of inertia about an axis is M 1 = ρPb ⋅ m ⋅n⋅l1 = 11.34 ⋅2⋅3⋅27.06 = 1.841 kg. passing through the pivot point : The length of the lower bob is L2 = a ⋅ l2 = 0.305 ⋅ 1 1 111.28 cm = 33.94 cm and its mass is M2 = ρPb ⋅ m ⋅ n J=µ alxP33( − ); = µ aalx (2)( −− 23 ) …(9) 0112 0 1 4 0 ⋅ l2 = 11.34 ⋅ 2 ⋅ 3 ⋅ 33.94 = 2.309 kg.

J1 ~ mass moments of inertia of the second order of 5 Refinements the upper pendulum bob about an axis passing It should be noted that the compensation coefficient through the pivot point: “a” was calculated in Sec 3 for the previously assumed pendulum rods and bob masses ( M = 2 kg, µ0 2 2 3 JJP1=+= 01 1 aa( +− 3(2 alx ) )( − ) …(10) m 12 = 0.4 kg). Since the values assumed differ from the values obtained in Sec 4, it would be reasonable to re- J02 ~ mass moments of inertia of the lower calculate and correct the coefficient “a” and check pendulum bob about an axis passing through its centre and iterate the calculations presented in Sec 4. of mass; P2 ~ Steiner’s component of the lower However, instead of repeating the whole procedure pendulum bob mass moment of inertia about an axis until the necessary level of accuracy is achieved, we passing through the pivot point: can compute new masses of the upper and lower pendulum rods, for which the value of the coefficient 133 1 23 J02=µ 0 alxP( + ); 2 = µ 0 aalx (2)( −+ ) …(11) “a” as well as the values of the pendulum bob masses 12 4 remain unchanged. This computation can be J2 ~ total mass moments of inertia of the lower accomplished by Eq. (15) which is derived directly pendulum bob about an axis passing through the pivot from Eq. (3): point:   aPb µ0 2 2 3 mi=λ −2  M i ; i = 1,2 …(15) JJP2=+= 02 2 aa( +− 3(2 alx ) )( + ) …(12) a 12 Wb 

J = J 1 + J 2 ~ total mass moment of inertia of both And thus, the new value for the mass of the upper pendulum bobs about an axis passing through the pendulum rod is: pivot point: m1 =(0.305 ⋅( 29 4) −⋅ 2) 1.841 = 0.389kg µ0 2 222 JJJ=+=1 2 ala( +− 3(2 al ) )( + 3 x ) …(13) 6 and for the lower one: For the sake of computational simplicity, the m =0.305 ⋅ 29 4 −⋅ 2 2.309 = 0.488 kg masses of all wooden pendulum parts will be 2 ( ( ) ) neglected at this instance. Since Eq. (14) is derived neglecting the masses of After the substituting Eqs (7), (8) and (11) in the wooden pendulum rods, the swing period of the Eq. (4), the quadratic Eq. (14) is obtained: real pendulum, whose rods masses are taken into 3Tg2 (2−= ax )π 22 ( a +− 3(2 al ))( 222 + 3) x …(14) account which differs from the idealistic one. Thus, it is necessary to determine the swing period T of the Since parameters T, g, a, l are known (T = T 0 = 4 s, real pendulum, the difference from the designed value 2 g = 9.8060 m/s , a = 0.305, l = 1 m), Eq. (14) can be T0 = 4 s and to correct the error. After taking into readily solved. Since the first solution of Eq. (14) is account the masses of the pendulum rods, oscillation too large x1 = 2.981 m, it is rejected as impractical. period of the pendulum is calculated as T = 3.9384 s. The second solution x2 = 0.1118 m = 11.28 cm is Then, the difference T0 – T = 0.0616 s is added to the accepted as practical and suitable. designed T0 = 4 s and Eq. (14) is solved ( x = 10.82 Now, it is possible to compute the preliminary cm) for the corrected value T0 = 4.0616 s. The new dimensions of the long-period compound pendulum. value of x determines new values of l1, l 2, m 1, m 2, L 1, 662 INDIAN J PURE & APPL PHYS, VOL 49, OCTOBER 2011

Table 2 — Iiteration results

x m1 m2 M1 M2 J ξC T [m] [kg] [kg] [kg] [kg] [kg ⋅m2] [m] [s]

1 0.1118 0.389 0.482 1.841 2.309 3.437 0.1775 3.9384 2 0.1082 0.391 0.486 1.851 2.300 3.420 0.1703 4.0101

L2, M 1 and M2, and the computation proceeds as previously explained. The results of both iterations are shown in Table 2. It is necessary to clarify the dimensioning and construction of the pendulum rods. As shown in Fig. 2, the upper and lower pendulum rods consist of two wooden sticks with circular cross-sections. Diameters of these cross-sections must be determined so that the total mass of each pendulum rod satisfies the calculated value. It is easy to arrive at the value of d = 21 mm, which is a perfectly suitable diameter for Fig. 3 — Thermally uncompensated pendulum all sticks of which the pendulum rods are made. Moreover, this value is large enough to allow the way that their swing periods are identical and equal finest mass adjustment of the pendulum rods. exactly 4 s ( T = 4 s) at an exact temperature ( t = It is clear that this simplified iteration works very 15 °C). Since both clocks have the same pendulum well. The temperature-compensated long-period rate and identical gear trains, they will measure, compound pendulum with the swing period of indicate and keep time in an identical way at the T = 4.0101 s was adopted as the final solution. The specified reference temperature. Let us change the iteration stopped here because the error ( ε = 0.0101 s) temperature for ∆t = +10 °C and ∆t = −10 °C and was so minor that it could be easily corrected by the determine the variation of the swing period for both empirical adjustment of the clock rate. pendulums and the corresponding clock rate errors. The quality of the pendulum (oscillator) thermal 6 Critical Evaluation of the Results compensation can be evaluated and determined by Once the solution for the compound pendulum comparing the results from Table 3. In both cases, the thermal compensation is obtained, it is necessary to suspension spring dilatation is neglected due to small critically evaluate the results as well as the significance. corresponding design and construction. This will be To make this comparison more comprehensible and realized by comparing rate errors of two clocks with evident, the variation of the swing period for the identical gear trains but different pendulums. The first uncompensated and an equivalent, but thermally of these clocks is equipped with the long period compensated, pendulum is expressed by the compound pendulum whose thermal compensation is corresponding daily and weekly clock error for the performed using the proposed procedures, while the same temperature alteration. In essence, the daily and second one is supplied with an equivalent, but weekly clock errors represent the accumulation of the thermally uncompensated pendulum (Fig. 3). The clock rate error during each day and week, uncompensated pendulum has a lead bob the mass of respectively, and are given in Tables 4 and 5. Since M = 9.5 kg, and a wooden rod the mass of m = 1.8 kg. the period of the pendulum swing is 4 s, the daily and The effective length of the pendulum rod is adjusted weekly clock errors are obtained by multiplying the to l = 4.09393 m to obtain the swinging period of T = clock rate error (pendulum frequency error) with the 4.00 s. Since these measures, dimensions and masses factors of 21600 and 151200, respectively. are common for pendulums used in many large tower The following conclusions can be drawn from the clocks, the assumed uncompensated ordinary results presented in Tables 3-5: pendulum is suitable for comparison with other pendulum designs. 1 Errors generated by uncompensated pendulum are Let us assume the conditions under which this more than 10 times higher than errors generated critical evaluation will be performed. Firstly, let us by the equivalent, but thermally compensated suppose that both pendulums are adjusted in such a compound pendulum. POPKONSTANTINOVIC et al .: THERMAL COMPENSATION OF LONG-PERIOD COMPOUND PENDULUM 663

Table 3 — Variation of the swing period for the thermally assumption of pendulum dimensions and materials; uncompensated and compensated pendulums (2) Calculation of the thermal compensation Rate error T[s], T[s], T[s], coefficient “ a”; (3) Simplified calculation of the t = 15 °C t = 5 °C t = 25 °C distance “ x” between the pivot point and pendulum Uncompensated 4.00 4.000080211 3.999919782 center of mass; masses of the bobs are taken into Compensated 4.00 4.000007468 3.999992533 account; masses of all wooden parts are neglected in that instance; (4) Preliminary determination of Table 4 — Daily accumulated error for thermally uncompensated and compensated pendulums pendulum rods and bobs dimensions and masses; (5) Calculation of the new value of the upper and lower E E E Daily Error [s/day], [s/day], [s/day], pendulum rods masses for which the value of the t = 15 °C t = 5 °C t = 25 °C coefficient “a” as well as the values of the pendulum uncompensated 0.00 +1.73255 −1.73271 bobs masses remain unchanged; (6) Calculation of the compensated 0.00 +0.161309 −0.161287 swing period T of the real pendulum, without Table 5 — Weekly accumulated error for thermally neglecting the rod masses; calculation of the error ε = uncompensated and compensated pendulums T – T 0; (7) Repetition of steps 3, 4, 5 and 6, until the Weekly Error E[s/week], E[s/week], E[s/week], error ε becomes sufficiently small. t = 15 °C t = 5 °C t = 25 °C Obviously, this computational procedure is based Uncompensated 0.00 +12.12785 −12.12897 on an assumption of a simplified pendulum model. In Compensated 0.00 +1.12916 −1.12901 case of necessity to take into account various other small parts, which have been omitted from this 2 Thermal compensation of the long-period analysis, the resulting pendulum model would be pendulum is not perfect since the effects of slightly more intricate. However, the computational thermal dilatations are not annihilated completely. procedure would be quite similar to the proposed one. Daily and weekly errors are shown in Tables 4 Finally, it must be emphasized that this procedure is and 5 are still observable and measurable. applicable not only to the designs of vintage “wood 3 The sign of errors of the thermally compensated rod-lead bob” pendulums, but also to the modern compound pendulum indicates that the pendulums which have rods thermally compensated compensation insufficient. Thus, the value of the by invar alloy. Moreover, since the linear thermal compensation coefficient “ a” should be slightly expansion coefficients for invar alloy is higher. approximately 3 times smaller than that of the wood, 4 Despite the thermal compensation imperfections, the thermal compensation of the invar pendulum rod clock rate errors are low enough to be technically would be even easier to accomplish. acceptable. 5 It is important to emphasize that the small References residual rate error of a slightly undercompensated 1 Tangsrirat, Worapong, Tanjaroen, Wason, Indian J Pure & pendulum is practically useful since it is always Appl Phys, 48(05) (2010) 0975. 2 Kislov K V, Seis Inst, 46 (2010) 21. diminished by the alteration in buoyancy caused 3 Agatsuma K, Uchiyama T, Yamamoto K, Ohashi M, by the variations in air density. Kawamura S, Miyoki S, Miyakawa O, Telada S & Kuroda K, 6 If the level of accuracy is not virtually and Phys Rev Lett , 104 (2010) 015701. practically sufficient, the complete procedure can 4 Tangsrirat, Worapong, Indian J Pure & Appl Phys, 47(11) (2009) 0975. be repeated with a better estimation of the 5 Jie L, Cheng-Gang Sh & Dian-Hong W, Class Quantum compensation coefficient “ a”. Reiteration can be Grav, 26 (2009) 195005. performed several times until the clock rate error 6 Cumming A, Heptonstall A, Kumar R, Cunningham W, becomes satisfactorily small or even totally Torrie C, Barton M, Strain K A, Hough J & Rowan S, Class eliminated. Quantum Grav. 26 (2009) 215012. 7 Dzhashitov V E & Pankratov V M, J Compt & Syst Sci Int , 48 (2009) 481. 7 Conclusions 8 Jie L & Dian-Hong W, Rev Sci Inst, 79 (2008) 094705. The computation procedure proposed in this paper 9 Woodward P, Woodward on Tiem (Group 5: Pendulum and comprises the following key steps: their suspensions; 1. Compensation of Pendulums; 3. A bimetallic compensator), Bill Taylor and British Horological (1) Assumption of the pendulum constructive Institute, UK, 2006. geometry and swing period T0; preliminary 10 Kaushal R S, Indian J Pure & Appl Phys, 43(07) (2005) 0975. 664 INDIAN J PURE & APPL PHYS, VOL 49, OCTOBER 2011

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