An analog simulation experiment to study free oscillations of a damped simple pendulum
AAn analog simulation experiment to study free oscillations of adamped simple pendulum
Ivan Skhem Sawkmie , and Mangal C. Mahato , ∗ Department of Physics, North-Eastern Hill University, Shillong-793022, India (Dated: March 18, 2019)
Abstract
The characteristics of drive-free oscillations of a damped simple pendulum under sinusoidalpotential force field differ from those of the damped harmonic oscillations. The frequency of oscil-lation of a large amplitude simple pendulum decreases with increasing amplitude. Many prototypemechanical simple pendulum have been fabricated with precision and studied earlier in view ofintroducing them in undergraduate physics laboratories. However, fabrication and maintenance ofsuch mechanical pendulum require special skill. In this work, we set up an analog electronic sim-ulation experiment to serve the purpose of studying the force-free oscillations of a damped simplependulum. We present the details of the setup and some typical results of our experiment. Theexperiment is simple enough to implement in undergraduate physics laboratories.
PACS numbers: ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] M a r . INTRODUCTION The determination of the period of a pendulum is a common physics experiment per-formed in schools and junior colleges. Using the small amplitude approximation, theamplitude-independent period of oscillation of the pendulum is related to the accelerationdue to gravity in the laboratory. Although the air resistance ultimately brings the pendu-lum to a stop, its effect on the period of oscillation is small and hence usually ignored. Anequivalent experiment is also performed using an LCR circuit. The voltage oscillation isinitialized with an ac voltage drive and then the drive is switched off at an appropriate timeto observe decaying voltage oscillations in time using an oscilloscope. However, these areessentially damped harmonic oscillations.An experiment in which the amplitude of oscillation of the pendulum is large is anadvanced step. The sinusoidal force experienced by this simple pendulum (derived from thegravitational potential) cannot be approximated to be harmonic. The period of oscillation ofthe simple pendulum depends on its amplitude. In the undamped case the period is given interms of elliptic integrals[1]. The effect of air resistance and other kinds of dampings have alsobeen investigated theoretically[2] and experimentally in real mechanical simple pendulum[3–5]. The damping force could depend linearly (Stokesian), and/or quadratically on velocity.The mechanical fulcrum could also contribute to velocity independent damping[5]. Thereis a large number of similar investigations on the subject as reported in this journal (Am.J. Phys.) as well as elsewhere [6–17]. However, introducing mechanical simple pendulumin college and university teaching laboratories may not be as simple for it requires specialmechanical skills and infrastructure. In this work we propose an electronic circuit equivalentof the underdamped simple pendulum experiment. The experimental setup is simple enoughto fabricate, implement and maintain with relatively small expenditure in any undergraduateteaching laboratory.The study of simple pendulum has pedagogic relevance because it has exact analogiesin, and can be considered as the prototype of, many other phenomena of physical interest[18, 19]. Apart from a particle moving on the surface of a sinusoidal potential [20, 21],some of the examples being the experimental study of the chaotic motion of a mechanicalpendulum [3, 4], ionic motion in a superionic conductor, and a phase motion in a Josephsonjunction [22, 23]. The structure of the equations of motion for these phenomena is identical2nly the physical significance of the parameters differ.The equation of motion of a simple pendulum is exactly equivalent to a particle movingin a medium of friction coefficient γ along a potential V ( x ) = − V cos( kx ) and driven bythe external periodic force F ( t ) = F sin( ωt ). The equation of motion of the pendulum(identifying x with the angular displacement θ , etc) is thus given by m d xdt = − γ dxdt − V k sin( kx ) + F sin( ωt ) . (1.1)Here the system is considered to be underdamped ( γ << ω ) where ω = (cid:113) kV m is thenatural frequency of free oscillation (at small amplitude). The driven pendulum, when thedamping is small, has so far not found exact analytical description [24]. Therefore, it isquite educative to experimentally study the free ( F = 0) oscillation of a simple pendulumeven if only in an equivalent electronic circuit.The dimensionless form of the equation of motion is: d xdt = − γ dxdt − sin( x ) + F sin( ωt ) . (1.2)Here, all parameters are dimensionless and written by taking m, k , and V as independentparameters and setting m = k = V = 1. Note that, in these dimensionless units, the natural(angular) frequency ω turns out to be 1. The free oscillation of a damped pendulum isdescribed by setting F = 0. We expect the amplitude of the free but damped oscillation todecay as Ae − γ t and also the frequency of oscillation to decrease with amplitude[2] unlike inan LCR circuit where the frequency ω = (cid:113) ω − γ is independent of amplitude.In the following, we give the details of the experimental setup. A brief explanation of asimilar experimental setup can also be found in Ref. [21]. We then present the results fromour experiment in graphical form. We also provide a brief discussion on the results. II. EXPERIMENTAL SETUP
Our experimental setup is shown in Fig. 1 in a block diagram form. The sinusoidal inputvoltage used in our experiment is taken from the Agilent 33500B series waveform generator.The waveforms are recorded using the InfiniiVision MSO-X 3014A oscilloscope from AgilentTechnologies. However, any reasonable waveform generator and oscilloscope (for example,Keysight InfiniiVision 1000 X-Series DSO (50MHz, 2 Ch)) can be used to carry out theexperiment. 3he electronic circuit shown in Fig. 1 is designed and set up to simulate an equation,given below, similar to the equation of motion given by Eqn. (1.1) or its dimensionless formEqn. (1.2). Here, the system is initially driven periodically by an external periodic inputcurrent I inp ( t ) = I sin( ωt ), derived from an input voltage V in ( t ). However, to obtain thefree oscillation we finally switch off the drive.The Kirchhoff condition at A, as in Eqn. (1.1), is given by the equation: R C C d V out dt = − (cid:26) R C R B + C + R C R A (cid:27) dV out dt − U V sin (cid:18) V out V (cid:19) + V in ( t ) R (2.1) FIG. 1: Block diagram to simulate Eqn. (2.1). Here the parameters R = 5 . K Ω when V in ≈ R = 5 . K Ω, R = 10 . K Ω, R A = 1 M Ω, R B = 470 K Ω, C = 1 . nF and C = 10 . nF are fixed parameters whereas C is a variable parameter (for example, C is set equal to 212 . pF for γ = 0 . Here we have taken V = 1volt, U = V R volt /ohm, m = R C C A volt − sec , k = V volt − and V in ( t ) = V in sin( ωt ) volt, where V in is the amplitude of the input (signal)voltage, ω = 2 πf ( f is the frequency of the periodic input current). Eq. (2.1) is written indimensionless units [25] by setting the parameters m = 1, U = 1 and k = 1. The equationwith reduced variables denoted again by the same symbols, corresponding to Eq. (2.1) iswritten as d V out dt = − (cid:26) R C R B + C + R C R A (cid:27) dV out dt − sin( V out ) + I sin( ωt ) (2.2)Comparing Eqs. (2.2) and (1.2), we see that these two equations are similar with dampingcoefficient γ = { R C R B + C + R C R A } ( R /m ) . and I = V in R R V . Changing the value of C γ since other terms contributing to the value of γ are kept fixed. Inour experiment, the output voltage V out ( t ) is analogous to the trajectory x ( t ) of Eqn. (1.2).For convenience, we give in, Table I, the relationship between the dimensionless unitsand dimensioned units for a few physical quantities. TABLE I: This table shows the relationship between the value in dimensioned and dimensionlessunits. The parameters m , U and k are as defined in the text with the parameter values given inthe caption of Fig. 1.Dimensioned units Dimensionless units Dimensionless valuet(1 secs) t = tm . U − . k − C = Cm . kU . R = RU − k − V = Vk − I = I U k The electronic circuit (Fig. 1) is a weakly damped periodically forced nonlinear (feedback)oscillator that simulates a second order ordinary differential equation giving the solutionin the form of an output voltage, V out ( t ), in response to an input current, I inp ( t ). Notethat our drive frequencies are not very different from the characteristic small-amplitudenatural frequency of the oscillator. For a periodic I inp ( t ) of frequency f , one is expectedto obtain a periodic V out ( t ) of the same frequency given the parameter γ suitably fixed. Inthe experiment, we choose a suitable frequency f = f f for which the amplitude of V out ( t ) ismaximum. We set this frequency f = f f and the amplitude of the input periodic current I = 0 . V in ≈ V out ( t ) to oscillate. We then switch off the inputperiodic current I inp ( t ). Since the oscillator is underdamped, V out ( t ) oscillates freely with anexponentially diminishing amplitude [2], as a solution of the equation: d V out dt = − (cid:26) R C R B + C + R C R A (cid:27) dV out dt − sin( V out ) (2.3)5herefore, for our final free oscillation experiment we use the same electronic circuitshown by the block diagram in Fig. 1 with the drive I inp ( t ) switched off giving the solutionof equation (2.3). The circuit essentially consists of two integrator segments, the outputof one is fed as input to the second at point B through a resistor R and two negativefeedbacks as input at point A. One (upper) feedback simulates the damping term γ dV out dt through the capacitor C . The other feedback gives the sinusoidal force term sin( V out )derived from the periodic potential. Of course, the equation (2.1) satisfied at point Aassumes the components to be ideal. However, the characteristic parameter values of thereal components may differ slightly from the ideal values and hence, for example, the value of γ may need adjustment. Also, the other feedback term representing sinusoidal force sin( V out )to be evaluated instantaneously of the continuously changing argument V out needs carefulconsideration.The IC AD534 gives the sine of the input signal and from the datasheet, it works accordingto the equation: V sine = 10 sin( π × V z
10 ) (2.4)where V sine is the output signal from the sine converter and V z is the input signal to the sineconverter, where V z can go from −
10V to +10V. However, when we use the circuit designas specified in the datasheet, we find that the IC works best only when the input signalis in the range from 0V to 1.165V out of the maximum 10V as specified in the datasheet.Therefore, we modify the parameters related to AD534 so that we can go beyond 1.165V.We use trial and error method and arrive at a conclusion that, for different combinations ofthe parameters related to AD534, the output from the sine converter should be of the form V sine = 4 . π × V z V z can go from -5V to +5V. We found that even if V z goes to ± V sine . Therefore, we have a sine converterwhere the argument θ = ( π/ × ( V z /
5) is in the range [ − π , π ].The sine converter block AD534 of Fig. 1 is tasked to obtain sin( θ ), θ = V out which maytake values larger than ± π . However, as mentioned above, it was found that using onlythe block AD534, without the block [Arg], the sine converter converts only roughly upto arange of argument, − π/ < θ < π/ IG. 2: Block diagram for the argument of sine to get the full potential of Fig. 2, in order to get the full potential, − cos( θ ), with − π ≤ θ < π . Noting that therange of argument − π/ ≤ θ ≤ π/ − ≤ sin( θ ) ≤ θ < − π/ θ > π/ − π/ ≤ θ ≤ π/ θ = V out ( t )) can have anyvalue in the range: − π ≤ θ ≤ π . The schematic shown in Fig. 3 allows the input signalto go through unchanged only when the input signal is in the range − π/ < θ < π/ θ (cid:48) , which is theoutput signal at point M in Fig. 3, therefore remains unchanged, θ (cid:48) = θ . The schematicshown in Fig. 4 accepts the input signal to be converted only when the input signal is inthe range π > θ > π/ θ is not in this range the output signal is set zero. ThisChannel 2 converts the allowed value of θ to θ (cid:48) = ( π − θ ) so that the output signal (= θ (cid:48) )from this schematic (at point M) is either in the range 0 < θ (cid:48) < π/ θ (cid:48) = 0. Similarly,the schematic shown in Fig. 5 accepts the input signal to be converted only when the inputsignal is in the range − π < θ < − π/ θ to θ (cid:48) = − π − θ so that the output signalfrom this schematic (at point M) is either in the range 0 > θ (cid:48) > − π/ θ (cid:48) = 0. The output7 IG. 3: Schematic diagram representing Channel 1 from Fig. 2. from these three channels are then added using an adder shown in Fig. 2. The output fromthe adder becomes the input argument to the sine converter at point K. Thus, using thesethree complementary channels simultaneously, we have an input signal (argument) suitablefor the sine converter in the whole range − π ≤ θ < π . Therefore, in Fig. 1, the transformedvoltage V K at point K appears in the limited range of − π/ ≤ V K < π/ − π ≤ V out ( t ) < π .We have used an inverter with gain= π after point K so that from Eqn. 2.5, V z is givenby V z = V K × π . This is done in order for the voltage V K to be the right argument in thesine converter. In the circuit of Fig. 1, we have also used an inverter with gain= . in orderto compensate for the scale factor=4.6 given in Eqn. 2.5. Hence, the voltage at point P is − sin( V out ), as envisaged.In order to verify the efficacy of the module [Arg] described above we plot, in Fig. 6,the input voltage to the module [Arg] (represented by V out ( t ) in Fig. 1) and the final signalobtained from the sine converter at point P. The output signal is fitted with a function8 IG. 4: Schematic diagram representing Channel 2 from Fig. 2. IG. 5: Schematic diagram representing Channel 3 from Fig. 2. A B C D V ou t ( t ) , s i n ( V ou t ( t )) t -sin(3.03sin(2 π out (t))V out (t) FIG. 6: A plot between the input signal to the sine converter ( V out ( t )) and the output signal fromthe sine converter (sin( V out ( t ))) along with the fitted curve − sin(3 .
03 sin(2 π . t − . − sin(3 .
03 sin(2 π . t − . θ = ± π/ V out ( t )), are because of the switchings fromone channel to the other. For example, when the signal V out ( t ) goes from < π/ > π/ ≈ µsec , it takes ≈ µsec for the channel 2 to become fullyfunctional. At ≈ π/
2, channel 1 tries to go to zero where the response time is ≈ µsec butchannel 2 wants to change the input signal to π − θ where its response time is again ≈ µsec and hence we see an abrupt dip and rise (with a total response time of ≈ µsec ) in theoutput signal before channel 2 activates as shown by the point A in Fig. 6. Similarly, forthe spikes at points B, C and D. The frequencies used in our experiment are around 5300Hz(time period ≈ . µsec ) and since the total response time of a spike is ≈ µsec , thesespikes do not affect our overall experimental results.As a small remark, when the amplitude of the output signal V out ( t ) is very close to π/ V out ( t )) has a flattened profile the system cannot unambiguously decidewhether to go to the channel 1, 2, or 3 of the circuit shown in Fig. 2. Because of this, theoutput at point M, for this particular amplitude, fluctuates. However, this is not a common11ccurrence when the amplitude of V out ( t ) is slightly > + π/ < + π/
2) or slightly < − π/ > − π/
2) by even a small finite value, this problem of fluctuation disappears. Therefore,this special rare occurrence does not concern us in our experiment.
III. EXPERIMENTAL RESULTS
As mentioned earlier, for a given damping coefficient γ , initially the system is periodicallydriven with a frequency so that we can have a maximum amplitude of response V out ( t ). Forall values of γ we take the drive current amplitude I = ( V in R /V R ) same and ≈ . γ = 0 . ω = 0 .
784 in dimensionless units. The external drive isthen switched off at an instant when the response V out was close to maximum and the freeoscillation V out ( t ) was measured. Note that all the data points, for example, V out , t , etc. areobtained from the oscilloscope readings which are subsequently converted into dimensionlessunits and presented as measured values in our results. Figs. 7, 9-11, show our experimentalresults for various measured values of γ , all in dimensionless units. A. Variation of amplitude with time
Figure 7 shows the free oscillation of the output signal voltage ( V out ( t ))(blue line) asa function of time for γ = 0 . V out ( t (cid:48) ) can start at any instant of time t (cid:48) not necessarily equal to 0. However,in Fig. 7, we have shifted the origin of time so that the free oscillation of V out ( t ) startsat the shifted time t = 0, either from a maximum or a minimum of V out as desired. Themeasured maxima and minima of the oscillation are shown by the brown points. Themaxima of the plot are fitted with the equation 1 . e − γ (cid:48) t and the minima are fitted with − . e − γ (cid:48) t where t = t (cid:48) − t + or t = t (cid:48) − t − , according as whether we consider, respectively,the maxima or the minima of V out , for the effective damping coefficient γ (cid:48) ≈ . γ . Here, t (cid:48) = t + = 4092 . V out starts at a maximum (=1.57448) and t (cid:48) = t − = 4095 . V out . For both the fitting functions, the plots areshown by the green lines with the general equation V mout e − γ (cid:48) t , where V mout is the extremum12 V ou t ( t ) t γ =0.0795 FIG. 7: Plot of the free oscillation of the output signal voltage ( V out ( t )) (blue line) as a functionof time when γ = 0 . value of V out at t = 0. We see that the fits are quite good. Note that V mout , t + and t − canbe different for different sets of experiment. The exponentially decreasing amplitude of theunderdamped system reaches the minimum measurable value after a certain time (say t d ).As we increase the γ value, we see that the number of measurable cycles of free oscillationalso decreases and the corresponding time duration t d upto which the free oscillations canbe measured unambiguously shortens, as expected.Fig. 8 shows the free oscillation of the output signal voltage ( V out ( t )) along with the inputsignal voltage ( V in ( t )) as a snapshot from the oscilloscope, illustrating the free oscillationsshown in Fig. 7. The physical quantities in Fig. 8 are in units of volt and sec whereas inFig. 7, we have used dimensionless units.In Fig. 9, we show how the amplitudes of oscillation decay with time t for several values of γ = 0 . , . , . , . . γ = 0 . t + , t − and t (cid:48) from the oscilloscope have different values foreach γ and for each set of experiment. However, when the figures are plotted as a functionof t = t (cid:48) − t + or t = t (cid:48) − t − (instead of as a function of t (cid:48) ), all the curves begin with t = 0(instead of beginning with t (cid:48) = t + or t (cid:48) = t − ), for all γ . The curves are thus automaticallytime shifted appropriately. Of course, the curves begin with different amplitudes V mout as13 IG. 8: Plot of the free oscillation of the output signal voltage ( V out ( t )) (yellow line) and the inputsignal voltage ( V in ( t )) (green line) as a function of time when γ = 0 . shown in the figure. All the fitted curves have the same form V mout e − γ (cid:48) t . The inset of Fig.9 shows the magnified picture of the curves at large t values when the amplitude is small.The curve fittings appear to be good even for these large times. However, a semilogarithmicplot should give a better check.The time varying amplitude shown in Fig 9 for different γ values are replotted in thesemilogarithmic graph of the magnitudes of both the maxima and minima of V out as afunction of time t in Fig. 10. The curves are then fitted with straight lines having slopesequal to − . γ (= − γ (cid:48) ). The fittings are quite good in the entire range of t except forthe large t values or for small amplitude values. In the small amplitude range also one canfit roughly by straight lines but with smaller slopes than γ (cid:48) . Theoretically, to conform tothe damped harmonic oscillator case, the slope of the fitted curve even for the small t rangeshould have been γ in place of γ (cid:48) . The discrepancy could be because of approximate values ofused parameters of the components in the expression for γ = { R C R B + C + R C R A } ( R /m ) . . Asnoted earlier, the IC’s are not ideal and wires do have a parasitic capacitance and resistance,and hence the calculated values of a damping coefficient γ may not be the same as thatactually appearing in the circuit. Also, the differing slope of the fitted curves in the tworegions of small and large amplitude could be because of variation of frequency of oscillationas a function of its amplitude unlike the amplitude independent constant frequency in the14 V ou t ( t ) t γ =0.0523 γ =0.0795 γ =0.1051 γ =0.1204 γ =0.1381 -0.25 0 0.25 55 75 95 FIG. 9: Plot of the measured peak of the amplitudes of the output signal voltage ( V out ( t )) as afunction of time for γ = 0 . , . , . , . . ≈ V mout e − γ t ) is shown by the thick lines. -5-4-3-2-1 0 1 0 40 80 120 160 l og ( V ou t ( t )) t γ =0.1381 γ =0.1204 γ =0.1051 γ =0.0795 γ =0.0523 FIG. 10: Semilogarithmic plot of the amplitude of Fig. (9) for γ = 0 . , . , . , . . -2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 0 40 80 120 V ou t ( t ) t Simple PendulumHarmonic Oscillator -0.08 0.08 100 120 140 (b) -5-2 1 0 50 100 150 (a)
FIG. 11: Plot of the free oscillation of the output signal voltage ( V out ( t )) (blue line) as a function oftime when γ = 0 . In Fig. 11 we compare a sample trajectory V out ( t ) for γ = 0 . V out ( t ) at smallamplitudes as in the inset Fig. 11b and the damping coefficient γ = 0 . × . γ value changes from 0 . × . . × . B. Change in frequency of free oscillation with amplitude
Figure 12 shows the measured angular frequency of free oscillation of the analog circuitmodel simple pendulum as a function of the amplitude for γ = 0 . , . , . , . . ω Amplitude (V ) γ =0.1381 γ =0.0523 γ =0.0795 γ =0.1051 γ =0.1204Exact solutionKittels Approximation FIG. 12: Plot of the angular frequency of free oscillation ( ω ) as a function of the measured peak ofthe amplitudes of the output signal voltage ( V out ( t )) for γ = 0 . , . , . , . . frequency of oscillation which increases as the amplitude of the free oscillation decreases.This is to be compared with the amplitude independent frequency ( ω = (cid:113) − γ ) of oscil-lation of a damped harmonic oscillator. In Fig. 12, the angular frequency of oscillation fora particular γ value does not differ too much from other γ values. Here we have comparedthe experimental results with the oscillation of a simple pendulum using the approximateexpression used by Kittel et al [24] ω ω ≈ − x
16 (3.1)and also with the exact solution [1, 17] ω ω ≈ π K ( k ) (3.2)where K ( k ) = (cid:90) π dφ (cid:112) − k sin φ (3.3)and 17 = sin x IV. DISCUSSION AND CONCLUSION
We have presented solutions to the differential equation of motion describing the force-free oscillations of a damped simple pendulum using an analog electronic circuit. Naturally,the pendulum is highly nonlinear and oscillates under the full sinusoidal potential force field.We have presented the details of how to obtain the sinusoidal force field.Since exact analytical solutions to the motion of damped simple pendulum are not avail-able, it is worth examining experimentally the expected approximate solutions. Our ex-perimental results show correct qualitative trends when compared with the exact analyticalresults for the undamped simple pendulum. The results also illustrate how oscillations of asimple pendulum differ from those of a harmonic oscillator. The experiment could be usefuland educative at the undergaduate level to make a clear distinction between a pendulumwith a large amplitude of oscillation and the one usually learnt with a small amplitudeapproximation.In our model, the damping is taken to be proportional to the instantaneous velocity.The coefficient of damping is considered constant and, in our circuit model, determined bythe parameters of various components used in the circuit. When this calculated dampingcoefficient is compared with the damping coefficient calculated from the usual amplitudedecay exponent we find a discrepancy of about 8%. There could be many factors causing thisdiscrepancy and the nonideal nature of components used in the circuit could be one. Also,18he measured amplitude dependent frequency of oscillation differs from the theoreticallycalculated one by a similar percentage. At present, we do not have any plausible explanationfor this discrepancy, however.As explained earlier, Squire pointed out that in case of a real pendulums [5], at leasttwo damping terms are generally needed. For a rigid pendulum, apart from the dampingterm which is linear in velocity, air drag is always present contributing to a damping termquadratic in velocity. However, in our model we consider only the linear damping term tokeep the circuit as simple as possible so that the experiment can be replicated easily evenin undergraduate teaching laboratories. [1] A. Sommerfeld, Lectures on Theoretical Physics: Mechanics, Levant Books (Indian reprint),2003.[2] K. Johannessen, ”An analytical solution to the equation of motion for the damped nonlinearpendulum,” Eur. J. Phys. , 3 (1989).[4] J. A. Blackburn, and G. L. Baker, ”A comparison of commercial chaotic pendulums,” Am. J.Phys. , 9 (1998).[5] P. T. Squire, ”Pendulum damping,” Am. J. Phys. , 984 (1986).[6] L.P. Fulcher and B.F. Davis, ”Theoretical and experimental study of the motion of the simplependulum,” Am. J. Phys. , 51-55 (1976).[7] W.P. Ganley, ”Simple pendulum approximation,” Am. J. Phys. , 73 (1985).[8] L.H. Cadwell and E.R. Boyco, ”Linearization of the simple pendulum,” Am. J. Phys. , 979(1991).[9] M.I. Molina, ”Simple linearizations of the simple pendulum for any amplitude,” Phys. Teach. , 489 (1997).[10] R.B. Kidd and S.L. Fogg, ”A Simple Formula for the Large-Angle Pendulum Period,” Phys.Teach. , 81 (2002).[11] L.E. Millet, ”The Large-Angle Pendulum Period,” Phys. Teach. , 162 (2003).[12] R.R. Parwani, ”An approximate expression for the large angle period of a simple pendulum,” ur. J. Phys. , 37 (2004).[13] G.E. Hite, ”Approximations for the period of a simple pendulum, Phys. Teach. , 290 (2005).[14] A. Bel´endez, J. J. Rodes, T. Bel´endez and A Hern´andez, ”Approximation for a large-anglesimple pendulum period,” Eur. J. Phys. L25L28 (2009).[15] M. Turkyilmazoglu, ”Improvements in the approximate formulae for the period of the simplependulum,” Eur. J. Phys. , 1 (2011).[21] Ivan Skhem Sawkmie and M.C. Mahato, ”Stochastic resonance in a sinusoidal potential sys-tem: An analog simulation experiment,” arXiv: 1608.06138 [nlin.CD].[22] C. M. Falco ”Phase-space of a driven, damped pendulum (Josephson weak link),” Am. J.Phys. , 733 (1976).[23] A. Barone, and G. Paterno, Physics and Applications of the Josephson Effect, A Wiley-Interscience publication, 1939.[24] C. Kittel, W.D. Knight, and M.A. Ruderman, Mechanics, Berkley Physics Course - Volume1, McGraw-Hill Book Company, Inc., New York, Ch. 7, 1965.[25] E. A. Desloge, ”Relation between equations in the international, electrostatic, electromagnetic,Gaussian, and Heaviside-Lorentz systems,” Am. J. Phys. , 601 (1994).[26] M. Abramowitz, and I. A. Stegun (ed) Handbook of Mathematical functions, Dover Publica-tions, Inc., New York, 1965., 601 (1994).[26] M. Abramowitz, and I. A. Stegun (ed) Handbook of Mathematical functions, Dover Publica-tions, Inc., New York, 1965.