Nonlinear coupling in an asymmetric pendulum
J. Qiuhan, L. Yao, Z. Huijun, W. Yinlong, W. Jianguo, W. Sihui
NNonlinear coupling in an asymmetric pendulum
Qiuhan Jia, Yao Luo, ∗ Huijun Zhou, Yinlong Wang, Jianguo Wan, and Sihui Wang † School of Physics, Nanjing University, Nanjing, P.R. China, 210093 (Dated: July 1, 2020)
Abstract
We investigate the nonlinear effect of a pendulum with the upper end fixed to an elastic rodwhich is only allowed to vibrate horizontally. The pendulum will start rotating and trace a del-icate stationary pattern when released without initial angular momentum. We explain it as am-plitude modulation due to nonlinear coupling between the two degrees of freedom. Though thephenomenon of conversion between radial and azimuthal oscillations is common for asymmetricpendulums, nonlinear coupling between the two oscillations is usually overlooked. In this paper,we build a theoretical model and obtain the pendulum’s equations of motion. The pendulum’smotion patterns are solved numerically and analytically using the method of multiple scales. Inthe analytical solution, the modulation period not only depends on the dynamical parameters,but also on the pendulum’s initial releasing positions, which is a typical nonlinear behavior. Theanalytical approximate solutions are supported by numerical results. This work provides a gooddemonstration as well as a research project of nonlinear dynamics on different levels from highschool to undergraduate students. a r X i v : . [ n li n . C D ] J un . INTRODUCTION The ideal trajectory of a two-dimensional asymmetric pendulum is a Lissajous figure, thesuperposition of two independent simple harmonic motion (SHM). A common example ofasymmetric pendulum is a “Y-suspended” pendulum which was invented twice for scientificand recreational purposes.
When the frequency ratio of the two oscillations ω /ω is arational number, the trajectory will be stationary. Otherwise, when ω /ω is not exactlya rational number, the motion is quasi-periodic and the trajectory varies with time dueto a growing phase drift. Singh et al. described that in an asymmetric two-dimensionalpendulum, the quasi-periodic motion shifts from planar to elliptical and back to planaragain. The period that the pendulum returns to planar motion was related to the strengthof symmetry breaking introduced with an additional spring.Once linear coupling is introduced to an oscillation system with two degrees of freedom,for instance, by connecting two identical pendulums with a weak spring, two normal modesare formed, whose frequencies are usually different from the pendulums’ natural frequencies. The resultant motion described by linear superposition of the two normal modes with slightlydifferent frequencies may give rise to a “beat” motion, in which the amplitudes of thependulums varies slowly and energy is transferred cyclically between the two pendulums.If we look at a real asymmetric pendulum, perfect independent motions are quite unlikelyto happen. The phenomenon of conversion between radial and azimuthal oscillations is com-mon for asymmetric pendulums. However, nonlinear coupling between the two oscillationsis usually overlooked. In this paper, we study the dynamics of a quasi-two-dimensionalpendulum with weak nonlinear coupling.The pendulum is shown in Fig. 1(a). Suspend a bob on a string from the end of anelastic rod. The other end of the rod is supported with another taut string to avoid verticaldeflection. If the pendulum is released in a plane parallel to the rod, the radial oscillation willspontaneously convert into a motion that shifts from planar to elliptical motion and back toplanar again. An experimental video clip and an animation are included in supplementarymaterial 1. A typical pattern of this quasi-2D pendulum’s motion is shown in Fig. 1(b).At first sight, the pattern resembles a proceeding Lissajous figure with varying phase. ALissajous figure is plotted in Fig. 1(c) for comparison. We see that in a Lissajous figure,the amplitudes in x and y directions are fixed, while the quasi-2D pendulum amplitudes2 X 𝑥𝑥 𝑐𝑐 𝑦𝑦 𝑐𝑐 m 𝑙𝑙𝑜𝑜 𝑧𝑧 𝑐𝑐 𝑥𝑥 (a) xy (b) xy (c) FIG. 1. (a)The asymmetric pendulum. The upper end of the string is fixed to an elastic rod onlyallowed to vibrate horizontally. The generalized coordinates are also defined. (b) A typical patternderived from numerical results. (c) A Lissajous figure for comparison. in both directions change alternately. Amplitude change and energy transfer between theazimuthal ( x ) and radial ( y ) directions are apparently consequences of coupling between thetwo degrees of freedom.In this paper, we build a theoretical model and simplify the pendulums equations ofmotion into two-dimensional. The pendulums motion is solved numerically and analyticallyusing the method of multiple scales. The difficulty in this problem is that the coupling isnonlinear, and the motion cannot be simply decomposed into two normal modes. Fortu-nately, nonlinearity in this problem is small and can be treated as perturbation to the twoindependent oscillations on radial and azimuthal directions. In the analytical solution, eachmotion is comprised of two oscillations with slightly different frequencies. We introduce thenonlinear modulation period and modulation depth to describe the feature of the motion.The modulation period and modulation depth we derived not only depends on dynamicalparameters, but also on the pendulums initial releasing positions. The analytical solutionsare consistent with numerical solutions with good accuracy when nonlinear effect is weak.This problem has aroused extensive interest among students as a popular competitionproblem in the 2018 International Young Physicists Tournament and China UndergraduatePhysics Tournament. The phenomenon described in this problem is common in many two3imensional asymmetric pendulums. The advantage of this experimental apparatus is thatit has appealing visual effects and the strength of coupling and other oscillation parametersare controllable thus can easily be compared to theoretical results. The solution in thispaper provides a theoretical explanation for the phenomenon. This work provides a gooddemonstration experiment as well as a research project from high school to undergraduatestudents.
II. THEORETICAL MODELA. Equation of Motion
We write the Lagrangian of the pendulum in terms of Cartesian coordinates. As shownin Fig. 1(a), the origin O is taken as the bobs equilibrium position when the rod has nodeflection. The bobs coordinates x c , y c , z c are taken with respect to O . x and y are thebobs relative coordinates with respect to the rods oscillatory end. For small deflection,the horizontal deflection of the end is approximately one dimensional and denoted as X .Therefore, x c = X + x, (1) y c = y, (2) z c = l − (cid:112) l − ( x + y ) (cid:39) l ( x + y ) , (3)where l is the string length. In this problem, we consider a hard rod whose natural frequencyis much higher than that of the pendulum. Moreover, as the rod is slender and the deflectionis small, we can model the rod using Euler-Bernoulli beam theory. More specifically, therod is treated as a cantilever beam bent by a force at the free end. Actually, the deflectionat any point is approximately proportional to the force, and the strain energy and kineticenergy are both quadratic. Hence, The total kinetic energy of the bob and the rod is T = 12 m ( ˙ x c + ˙ y c + ˙ z c ) + 12 M ∗ ˙ X , (4)where m is the mass of the bob, M ∗ ˙ X is the rod’s kinetic energy expressed in term of aneffective mass M ∗ . 4he potential energy is V = mgz c + 12 kX , (5)where the rod’s potential energy is given in term of an effective elastic coefficient k .The Lagrangian is therefore L = 12 m (( ˙ x + ˙ X ) + ˙ y ) + 12 M ∗ ˙ X − kX − mg l (cid:0) x + y (cid:1) + m l ( x ˙ x + y ˙ y ) . (6)The equations of motion obtained using Euler-Lagrange (E-L) equations with x , y and X are ¨ X + ¨ x + ω y x + 1 l ( x ˙ x + x ˙ y + xy ¨ y + x ¨ x ) = 0 , (7a)¨ y + ω y y + 1 l ( y ˙ x + y ˙ y + xy ¨ x + y ¨ y ) = 0 , (7b)(1 + γ ) ¨ X + ¨ x + ω X X = 0 , (7c)where ω y = gl , ω X = km , γ = M ∗ m .Rearrange the highlighted terms, Eqs. (7a) and (7b) can be rewritten as¨ X + ¨ x + ω y x + dd t ∂ v z ∂ ˙ x − ∂ v z ∂x = 0 , (8a)¨ y + ω y y + dd t ∂ v z ∂ ˙ y − ∂ v z ∂y = 0 . (8b)We see that all the nonlinear terms come from the vertical motion which is usuallyneglected concerning the pendulum’s short-term behavior. When long-term behavior isconsidered, the effects of these small terms accumulate over time, and they modulate theamplitudes on x and y directions periodically. We will treat the effects of these smallterms as perturbation. Before doing so, we firstly simplify the pendulum’s motion into twodimensional by considering the “undisturbed” solution. B. Simplification: Two Dimensional Model
When the nonlinear terms are ignored in Eq. (7), the pendulum’s equations of motionfor the undisturbed system (the generating system) is reduced to( ¨ X + ¨ x ) + ω y x = 0 , (9a)¨ y + ω y y = 0 , (9b)(1 + γ ) ¨ X + ¨ x + ω X X = 0 . (9c)5n Eq. (9b), we see that y is independent of x and the its solution is simple harmonicmotion. x and X are still coupled. Write the trial solution as x = x e λt , X = X e λt , (10)and substitute them into Eqs. (9), we find that λ X + ( λ + ω y ) x = 0 , (11a) (cid:0) (1 + γ ) λ + ω X (cid:1) X + λ x = 0 . (11b)Nontrivial solution exists only if the determinant equals zero. Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ λ + ω y (1 + γ ) λ + ω X λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (12a)that is, γλ + ( ω X + (1 + γ ) ω y ) λ + ω X ω y = 0 . (12b)The solutions of this equation are λ − (cid:39) − (1 − κ ) ω y ≡ − ω x , (13a) λ (cid:39) − γ (1 + κ ) ω X ≡ − ω (cid:48) , (13b)where κ ≡ mgkl (cid:28) ω x is an effective naturalfrequency of the bob’s azimuthal oscillation which is slightly lower than the pendulum’snatural frequency and ω (cid:48) is a much higher frequency of the magnitude order of the rod’snatural frequency.Substitute each λ in Eqs. (13) into Eq. (9a), we have X x = ω y + λ − λ , = κ − κ , when taking λ − . κ κ γ − λ + . (14)For the solution with lower frequency ω x , the bob’s oscillation amplitude is much greaterthan that of the rod’s, since κ (cid:28)
1. For the solution with high frequency ω (cid:48) , they are of the6ame magnitude. Normally the bob’s displacement is much greater than the displacement ofthe rod’s end, therefore we will neglect the high frequency solution. Thus, Eq. (14) becomes X = κ − κ x . (15)So, X = κ − κ x. (16)Substitute Eq. (16) into Eqs. (9), the equations are reduced to two dimensional as¨ x + ω x x = 0 , (17a)¨ y + ω y y = 0 . (17b)Apparently, for the ”undisturbed system”, the asymmetry is simply described by the differ-ence in frequencies of azimuthal and radial oscillations introduced by the elastic rod.Retaining the nonlinear coupling terms, we obtain the following equations of motion¨ x + (1 − κ ) x = − (1 − κ )( x ˙ x + x ˙ y + xy ¨ y + x ¨ x ) , (18a)¨ y + y = − ( y ˙ y + y ˙ x + xy ¨ x + y ¨ y ) . (18b)In this equation and below, for simplicity, the variables x and y represent x/l and y/l , and t represents ω y t .Notice that in Eqs. (18), there is only one parameter, the coupling coefficient κ , whichdetermines the frequency difference between the two directions. Also note that κ is inde-pendent of the effective mass of the rod M ∗ . The mass of the rod has little effect on thissystem because the kinetic energy of the rod is negligible compared to its elastic potentialenergy, when we only consider the low frequency motion. III. NUMERICAL AND ANALYTICAL RESULTSA. Numerical Result
Applying NDSolve method in Wolfram Mathematica to solve Eqs. (18), we obtain theresults as shown in the Fig. 2. In the calculation, κ = 0 .
01, and the pendulum is releasedfrom x (0) = 0 . , y (0) = 0 . , x (cid:48) (0) = y (cid:48) (0) = 0. Figs. 2(a) and 2(b) show the oscillations on7
00 400 600 800 t - - x (a)
200 400 600 800 t - - y (b) - - x - - y (c) 𝜎𝜎 𝑦𝑦 𝜎𝜎 𝑥𝑥 (d) FIG. 2. (a) and (b) show the oscillations on x (the azimuthal) and y (the radial) directions. (c)shows the trajectory projected on the horizontal plane. (d) is the frequency spectrum of motionsin x and y directions. x (the azimuthal) and y (the radial) directions respectively. In the figures, the oscillationon x and y directions resembles that of a “beat” motion, in which the amplitudes on x and y directions are modulated periodically. The time span in the figures is twice the amplitudemodulation periods T . Within 2 T , the pendulum returns to initial planar motion and thetrajectory forms a complete stationary pattern as shown in Fig. 2(c). We will see that thependulum’s phase evolution period is 2 T . 8 a) (b) (c) (d)(e) (f) (g) (h) (i) FIG. 3. These figures show how the motion evolves in a period. The yellow ellipses depict shortterm motion and the arrows on the ellipses depict the rotation direction of the bob. The red pointsare the points of tangency and the red arrows beside the points show how these points move. In (a),the bob is released from the upper-right point. Then the bob rotates ”elliptically”. The directionof rotation is clockwise from (a) to (e) and counter-clockwise from (e) to (i), which is the same asthe rotation of the ellipse axes.
To illustrate the phase and amplitude evolution more clearly, we plot the evolution dia-grams within 2 T in Fig. 3. The background blue pattern is the same as in Fig. 2(c) whichclearly shows the stationary envelope of the trajectory. And the yellow curves are the shortterm trajectory as the bob moves back and forth within one oscillation. Within each oscil-lation, the trajectory is nearly closed since the rate of procession is very small. So we canapproximate them as ellipses gradually rotating and transforming to depict the short termmotion. The red points are the points of tangency of the ellipse and the envelope curve,and the arrows show how the points of tangency move. After the bob is released at theupper-right point in Fig. 3(a), the planar motion gradually becomes elliptical and both themajor axis and the bob rotate clockwise, see the animation. The proceeding rate increasesand reaches maximum when the major axis of the ellipse passes the x-axis in Fig. 3(c).Then the proceeding rate begins to decrease and finally the oscillation returns to planarin the position symmetric to the initial position at time T , as shown in Fig. 3(e). At thetime, the amplitude for y oscillation returns to maximum, and the amplitude for x oscilla-9ion returns to minimum. Afterwards, the pendulum reverses its rotation counter-clockwisefrom Fig. 3(e) to Fig. 3(i). As we havent considered the effect of damping, the motion isquasi-periodic: the pendulum repeats this motion pattern cyclically similar to the movingLissajous figure with slightly different frequencies. Hence, the duration from Fig. 3(a) toFig. 3(i) is the period of phase modulation 2 T .Then we perform Fourier transformation to x and y oscillations and obtain the frequencyspectra in Fig. 2(d). In the figure, the spectra of x and y oscillations are indicated in solidblue line and dashed yellow line respectively. We see that both oscillation spectra compriseof several discrete peaks which are evenly spaced. We will show analytically that the spacingis twice the frequency difference of the main peaks σ y − σ x . Both the main peak frequenciesof x and y oscillations are slightly lower than the natural frequency of undisturbed motion,which equals 1 in dimensionless Eq. (18b). The discrete spectra of each oscillation withspacing 2( σ y − σ x ) give rise to amplitude modulation featured as the beat phenomenon withperiod T = π σ y − σ x ) . The main peak frequency difference σ y − σ x gives rise to phase shiftingwith period 2 T = πσ y − σ x .Moreover, the results obtained from the simplified Eqs. (18) show no observable differencewhen we substitute the same parameters into the original equations (7). This proves thatour simplification is reasonable. To see how nonlinear coupling affects the pendulums motionpatterns, we utilize an analytical method for further study. B. The Method of Multiple Scales
The method of multiple scales was first introduced by Peter A. Sturrock in 1957 anddeveloped by Nayfeh and others. The underlying idea of the method of multiplescales is to regard the motion as a superposition of motions in multiple time scales whichare independent variables. Firstly, we introduce independent time scale variables accordingto T n = ε n t ( n = 0 , , , · · · ) , (19)10here ε is a dimensionless small quantity. It follows that the derivatives with respect to t become expansions in terms of the partial derivatives with respect to T n according todd t = ∂∂T d T d t + ∂∂T d T d t + ∂∂T d T d t + · · · , = ∂∂T + ε ∂∂T + ε ∂∂T + · · · , = D + ε D + ε D + · · · , (20)where D n ≡ ∂∂T n . One assumes that the solution can be represented by an expansion in theform x ( t, ε ) = m +1 (cid:88) n =1 ε n x n ( T , T , T , · · · , T m ) , (21)where m is the the order to which we need to carry out the expansion. Here we take m = 2.Substituting Eq. (20) and Eq. (21) into Eq. (18) and equating the coefficients of ε to thepower of 1, 2 and 3 separately, we obtainD x + (1 − κ ) x = 0 , (22a)D y + y = 0 , (22b)D x + (1 − κ ) x = − D x , (23a)D y + y = − D y , (23b)and D x + (1 − κ ) x = − D x − D x − D x − (1 − κ ) (cid:2) x (D x ) + x (D y ) + x y D y + x D x (cid:3) , (24a)D y + y = − D y − D y − D y − (cid:2) y (D y ) + y (D x ) + y x D x + y D y (cid:3) . (24b)The solution of Eqs. (22) is x = A ( T , T ) e i (1 − κ ) T + cc, (25a) y = B ( T , T ) e iT + cc, (25b)11here A ( T , T ) and B ( T , T ) are complex amplitudes to be solved and cc denotes thecomplex conjugate of the preceding terms. Substituting Eqs. (25) into Eq. (23a), we obtainD x + (1 − κ ) x = − i (1 − κ )D Ae i (1 − κ ) T , (26a)D y + y = − i D Be iT . (26b)To eliminate secular terms of x and y , we haveD A = 0 & D B = 0 . (27)Because we have included all the information of general solution in the first order solutionEqs. (25), we should not consider general solution in higher order solutions, or else thecoefficients will be underdetermined. Nayfeh in his book has a further discussion. Hence,the general solution of the second order equations is 0. In addition, from Eqs. (27) weknow that particular solution of Eqs. (26) equals 0. Taking general and particular solutionstogether, we have x = y = 0 . (28)Substituting Eqs. (25), Eqs. (27) and Eq. (28) into Eq. (24a), we obtainD x + (1 − κ ) x = (cid:0) − i (1 − κ )D A + 2(1 − κ ) A ¯ A (cid:1) e i (1 − κ ) T + (1 − κ ) (cid:104) (1 − κ ) A e i (3 − κ ) T + AB e i (3 − κ ) T + ¯ AB e i (1+ κ ) T (cid:105) + cc. (29)To eliminate secular terms of x , we have − i (1 − κ )D A + (1 − κ ) A ¯ A = 0 . (30)Notice that A is independent of T . For convenience, we write A in the polar form A ( T ) = 12 a ( T ) e iθ ( T ) , (31)where a and θ are real functions of T . Substituting Eq. (31) into Eq. (30) and separatingthe result into real and imaginary parts, we obtainD a = 0 , (32a)D θ = − a (1 − κ ) . (32b)12t follows that a is a constant and hence a = a , (33a) θ = − a (1 − κ ) T + θ , (33b)where a and θ are real constants. Returning to Eq. (31), we find A ( t ) = 12 a e i ( − − κ ε a t + θ ) , (34)where we have used T = ε t . Similarly, we can deduce B ( t ) = 12 b e i ( − ε b t + φ ) , (35)where b and φ are real constants. Substituting for A and B from Eqs. (34) and (35) intoEq. (29) and setting the general solution as 0, we obtain x = 1 − κ (cid:104) (1 − κ ) a (1 − κ ) − σ x e i (3 σ x t +3 θ ) + a b − κ − ( σ x + 2 σ y ) e i (( σ x +2 σ y ) t + θ +2 φ ) + a b − κ − (2 σ y − σ x ) e i ((2 σ y − σ x ) t − θ +2 φ ) (cid:105) + cc, (36)where σ x ≡ (1 − κ ) − − κ ε a (cid:39) − ε a − κ,σ y ≡ − ε b . The parameters σ x and σ y , as we will see in Eqs. (39), are just the primary peak frequenciesin the spectra. Normally, σ x and σ y approximately equal 1, and κ (cid:28)
1; thus the third termin Eq. (36) is much greater than the others. We may neglect the small terms and simplifythe solution as x = 14 (1 − κ ) a b − κ − (2 σ y − σ x ) e i ((2 σ y − σ x ) t − θ +2 φ ) + cc. (37)Then x = εx + ε x + ε x = 12 ae i ( σ x t + θ ) + 14 (1 − κ ) ab − κ − (2 σ y − σ x ) e i ((2 σ y − σ x ) t − θ +2 φ ) + cc, (38)13here a ≡ εa and b ≡ εb . Similarly, we can find solution for y. According to the initialcondition x (cid:48) (0) = 0 , y (cid:48) (0) = 0, we have θ = φ = 0. Hence the solution is x = a cos σ x t + (1 − κ ) ab b − a − κ cos(2 σ y − σ x ) t, (39a) y = b cos σ y t + (1 − κ ) a b a − b + 4 κ cos(2 σ x − σ y ) t. (39b)We find that x (or y ) oscillation is the superposition of two harmonic components with asmall angular frequency difference 2( σ y − σ x ). The superposition of the two componentsresults in an amplitude modulation whose period is given by T = 2 π σ (cid:39) πa − b + 2 κ , (40)where ∆ σ ≡ σ y − σ x . When only consider the leading terms in Eqs. (39a) and (39b), wemay approximate the angular frequency difference between x and y oscillations as σ y − σ x .This means that the period of phase shift is π ∆ σ = 2 T , twice the period of the amplitudemodulation as discussed above. And ∆ σ can be seen as the average angular speed of pro-cession.Notice that a and b are not the initial coordinates x and y . To acquire the initialposition, we set t = 0 in Eqs. (39) and obtain x = a + (1 − κ ) ab b − a − κ (41a) y = b + (1 − κ ) a b a − b + 4 κ (41b)In analytical calculations, the parameters ( a, b ) are solved from ( x , y ) according toEqs. (41). The solution is multi-valued and unrealistic solutions are eliminated. Thus,numerical and analytical results can be compared. We plot the simulative and analyticaltrajectories in Figs. 4(a) and 4(b) for κ = 0 . , x = 0 . , y = 0 .
08 in both numericaland analytical solutions. As we can see, there are only subtle difference between the twotrajectories, see Figs. 4(a) and 4(b). However, when we take κ = 0 . , and larger initialdisplacements x = 0 . , y = 0 .
12, the difference becomes obvious, see Figs. 4(d) and 4(e).Then we plot frequency spectra of the numerical results for these parameters, see Figs. 4(c)and 4(f). For each oscillation, there are several discrete peaks. The main and secondarypeaks for each oscillation correspond to the two terms in analytical result Eqs. (39a) and(39b). When the amplitude is small, as shown in Fig. 4(c), only the secondary peak is14 - - - (a) - - - - (b) xy ω Intensity (c) - - - - (d) - - - - (e) 𝜎𝜎 𝑥𝑥 𝜎𝜎 𝑦𝑦 Δ𝜎𝜎Δ𝜎𝜎Δ𝜎𝜎 Δ𝜎𝜎 Δ𝜎𝜎 (f)
FIG. 4. (a) and (b) show the numerical and analytical trajectory diagrams for κ = 0 . , x =0 . , y = 0 .
08, respectively. (c) is the frequency spectrum of (a). (d) and (e) show the numericaland analytical trajectory diagrams for κ = 0 . , x = 0 . , y = 0 .
12, respectively. (f) is thefrequency spectrum of (d). σ x and σ y denote the frequencies of the highest peaks of x and y motion. competitive to the main peak in each oscillation, and the effect of other peaks are too weakto be observed. Under this circumstance, analytical results fit well with numerical results.However, when the amplitude is increased, as shown in Fig. 4(f), the effect of other minorpeaks are not negligible and the first order analytical solution will lose accuracy.Now we study how coupling coefficient κ and initial amplitudes affect the nonlinearcoupling. To describe it more clearly, we label the intensities of primary and secondary peakas h x and h (cid:48) x for x oscillation. Also, we label the dimensionless frequencies of the primaryand secondary peaks as σ x and σ (cid:48) x ( σ y and σ (cid:48) y ) for x ( y ) oscillation. Then, we introduce ζ x ≡ h (cid:48) x /h x . (42)If the motion only comprises two harmonic components, ζ x is the modulation depth, i.e.15 nalyticalNumerical κ ζ x (a) AnalyticalNumerical κ σ x (b) AnalyticalNumerical κ σ x (c) AnalyticalNumerical κ Δσ (d) FIG. 5. Modulation depth ζ x , peak frequencies σ x , σ (cid:48) x , and frequency difference ∆ σ with respectto κ for x = 0 . , y = 0 . the ratio of modulation amplitude to carrier amplitude in terms of amplitude modulation.In this case, the modulation depth ζ x can be found from the ratio of the coefficients of thesecond term and first term in Eqs. (39): ζ x = (1 − κ ) b b − a − κ . (43)Thus, we can verify the accuracy of the analytical results. Fig. 5(a) shows how ζ x dependson κ for x = 0 . , y = 0 .
06, where κ varies from 0.01 to 0.1. We find that ζ x becomessmaller as κ increases, indicating that nonlinear effect is weaker for larger κ .Moreover, we have studied how the main peak frequency σ x and secondary peak frequency σ (cid:48) x change with κ . Observing Eqs. (39), we see that σ (cid:48) x = 2 σ y − σ x = σ x + 2∆ σ, (44a) σ (cid:48) y = 2 σ x − σ y = σ y − σ, (44b)which is also verified by the numerical result, see Fig. 4(f). To compare the analytical and16 nalyticalNumerical α ζ x (a) AnalyticalNumerical α σ x (b) AnalyticalNumerical α σ x (c) AnalyticalNumerical α Δσ (d) FIG. 6. Modulation depth ζ x , peak frequencies σ x , σ (cid:48) x , and frequency difference ∆ σ with respectto α for κ = 0 . , x = α, y = 2 α . numerical results, we have Figs. 5(b)-5(d). Although σ x and σ (cid:48) x change very little with κ ,the modulation frequency ∆ σ increase drastically. This means that the average processionspeed is significantly increases for greater κ .To see how amplitudes affect the pendulum motion, we plot modulation depth ζ x , peakfrequencies σ x and σ (cid:48) x as well as frequency difference ∆ σ in Figs. 6(a)-6(d) as functions ofinitial positions. We take κ = 0 .
01, and initial positions x = α, y = 2 α , and α variesfrom 0.01 to 0.06. In Figs. 6(b) and 6(c), both σ x and σ (cid:48) x decrease as α increases, butthe modulation frequency ∆ σ is nonmonotonic: it reaches a minimum near α = 0 .
04 inFig. 6(d). Fig. 6(a) shows that the modulation depth is greater for larger α , as nonlinearityis stronger for larger amplitudes. The analytical results also gradually lose accuracy for largeramplitudes due to stronger nonlinearity. The regular motion we have solved is valid only inthe range of weak nonlinearity. In addition, when the rod’s natural frequency is comparableto pendulum’s natural frequency, the pendulums motion will become far more complicatedthan we have presented. Finally, we have carried an experiment to verify the dependence17f modulation frequency on the coupling coefficient and amplitudes. The experiment dataincluded in supplementary material 2 match our theoretical trends well. IV. CONCLUSION
The phenomenon of conversion between radial and azimuthal oscillations described in thispaper is common for asymmetric pendulums. However, nonlinear coupling between the twooscillations is usually overlooked. We explain it as amplitude modulation due to nonlinearcoupling. The pendulums motion patterns are solved numerically and analytically. Theamplitude modulation period T is explicitly expressed in terms of coupling coefficient andamplitudes. The amplitude dependence of T is a typical nonlinear behavior. The advantageof this experimental apparatus is that it has appealing visual effects and the strength ofcoupling and other oscillation parameters are controllable thus can easily be compared totheoretical results. The method of multiple scales we introduce can easily be followed byundergraduate students. This work provides a good demonstration as well as a researchproject of nonlinear dynamics on different levels. V. ACKNOWLEDGMENT
The authors are grateful to Mr. Lintao Xiao for instructive discussion and proofreading. ∗ Permanent address: Division of Engineering and Applied Science, California Institute of Tech-nology, California, US, 91125 † [email protected] Whitaker and J. Robert, “Types of Two-Dimensional Pendulums and Their Uses in Education,”Science & Education , 401-415 (2004). T. B. Greenslade, “Devices to Illustrate Lissajous Figures,” Phys. Teach. , 351-354(2003). R. J. Whitaker, “A note on the Blackburn pendulum,” Am. J. Phys. , 330-333 (1991). Jorge Quereda et al. , “Calibrating the frequency of tuning forks by means of Lissajous figures,”Am. J. Phys. , 517 (2011). P. Singh et al. , “Study of normal modes and symmetry breaking in a two-dimensional pendu-lum,” eprint: arXiv:1806.06222 (2018). Richard P. Feynman,
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