Vol. 2, No. 2 | July – December 2018 Stabilization of Vertically Modulated Pendulum with Parametric Periodic Forces Babar Ahmad∗ Sidra Khan∗ Abstract With the application of Kapitza method of averaging for arbitrary periodic force, a vertically modulated pendulum, with periodic linear forces is stabilized by minimizing its potential energy function. These periodic linear forces are selected in range [-1, 1], further the corresponding stability conditions are compared with that in case of harmonic modulation. Later, a parametric control is defined on some periodic piecewise linear forces, and the nontrivial position is stabilized under different conditions by just adjusting the parameter. Keywords: Kapitza pendulum, fast oscillation, parametric control 1. Introduction A simple pendulum that is suspended under the in- fluence uniform gravitational field has versatile appli- cations in Nonlinear Physics. The Mathematical Rela- tionships and the differential equations associated with pendulum plays an important role in the theory of solu- tions, in the problem of super radiation, in quantum op- tics and the theory of Josephson effects in weak super- conductivity [1]. A simple pendulum has only one sta- ble point i.e. vertically downward position, while a ver- tically modulated pendulum with very high frequency, has upward position also stable. This concept was ini- tialized by Stephenson in 1908.[2, 3, 4]. In 1951, Pjotr Kapitza explained experimentally such kind of extraor- dinary behavior of pendulum in detail, and correspond- ing experimental instrument is known as Kapitza Pen- dulum [5]. In 1960 Landau et al. examined the stability of this system driven by harmonic Force [6]. Later on, Ahmad and Borisenok replaced harmonics force with periodic kicking forces and modified Kapitza Method for arbitrary periodic forces [7]. Ahmad also examined the stability of the system excited by the symmetric forces with comparatively low frequency of fast Oscilla- tions [8]. Later on, the behavior and the stability of a parametrically excited pendulum have been examined [9, 10]. In 2013, Ahmad used parametric periodic lin- ear forces for the horizontal modulated pendulum and discussed its stability by minimizing the potential en- ergy function [11]. In this paper, the stability criterion for vertically modulated pendulum, driven by periodic piecewise linear forces will be discussed. 2. Kapitza Method For Periodic Arbitrary Forces with Zero Mean Consider one dimensional motion of a particle of mass m in conservative system. If U is potential energy func- tion, then its equation of motion is F(x) = −dU dx (1) In this case the system has only one stable point. If a periodic fast oscillating force with zero mean is in- troduced, The system may have more than one stable point. This fast oscillation means that if ω0 = 2π T0 is the frequency due to F1 and ω = 2π T is the frequency due F2 then ω >> ω0. This fast oscillatory force has the Fourier expansion as F2(x,t) = ∞∑ k=0 [ak(x)cos(kwt) + bk(x) sin(kwt)] (2) Here ak and bk are the Fourier coefficients. In Calculus, mean value of a function f(t) is denoted by bar, if T is the time period, then mean is defined as − f= 1 T ∫ T 0 f(x,t)dt (3) The Fourier coefficient a0 is defined as a0 = 2 T ∫ T 0 f2(x,t)dt (4) ∗COMSATS Institute of Information Technology Islamabad, Pakistan Corresponding Email: baber.sms@gmail.com SJCMS | P-ISSN: 2520-0755 | E-ISSN: 2522-3003 c© 2018 Sukkur IBA University - All Rights Reserved 8 Babar Ahmad (et al.), Stabilization of Vertically Modulated Pendulum with Parametric Periodic Forces (8-13) From equation 3 and 4, the mean value of a function is equivalent to Fourier coefficient a0 − f∼= a0(x) (5) The other Fourier coefficients are ak = 2 T ∫ T 0 f2(x,t)cos(kwt)dt bk = 2 T ∫ T 0 f2(x,t) sin(kwt)dt (6) Ignoring friction, we can say that only two forces are acting on the system, hence its equation of motion is mẍ = F1(x) + F2(x,t) (7) Due to these forces, two types of motion namely smooth and small oscillations are observed. So we represent the path of oscillations as the sum of smooth path X(t) and small oscillation ξ(t) x(t) = X(t) = ξ(t) By averaging procedure, the effective potential energy function can be expressed as Ueff = U + 1 4mw2 ∞∑ k=1 a2k + b 2 k k2 (8) For stability of the system, we have to minimize effec- tive potential energy function given by 8 3. The Pendulum Driven by Harmonic Force Consider a pendulum whose pivot point is forced to vi- brate vertically (see Figure 1), under the influence of the harmonic force. The harmonic force is given as f(t) = sin(wt) if 0 ≤ t ≤ T (9) Figure 1: Kaptiza Pendulum with Vertical Oscilla- tion and shown in Figure 2 Figure 2: sin type force And the force acting on the pendulum is f2(φ,t) = mw 2 sin φ×f(t) (10) Its Fourier coefficient is a0 = 0 indicates that its mean value is zero. By using 6, the other Fourier coefficients are: ak = 0 bk = mw 2 sin φ (11) so the effective potential energy is obtained by using 8 Ueff = mgl(−cosφ + w2 4gl sin 2 φ) (12) The following results are obtained after minimizing equation 12 • The downward position φ = 0, is always stable. • Vertically upward position φ = π is stable if w2 > 2gl. • The position φ = arccos(−2gl w2 ) is unstable. 4. Vertically Modulated Pendulum Driven by Periodic Linear Forces Now, replacing the harmonic force with some peri- odic piece-wise linear forces within the range of har- monic forces, Our aim is to stabilize the pendulum at φ = π with low frequency as compared to harmonic force. These periodic linear forces are T-periodical: S(t + T) ≡ S(t). These forces are considered as fol- lowing. f2(φ,t) = mw 2 sin φ×S(t) (13) Inclined Type Force: First of all consider an inclined type force: E(t) = E(t + T), given by equation 14 and illustrated in Figure 3 E(t) = − 2 T t + 1 if 0 ≤ t ≤ T (14) Sukkur IBA Journal of Computing and Mathematical Sciences - SJCMS | Volume 2 No. 2 July – December 2018 c© Sukkur IBA University 9 Babar Ahmad (et al.), Stabilization of Vertically Modulated Pendulum with Parametric Periodic Forces (8-13) Figure 3: Inclined Type Force The force acting on the particle is f(t) = mw 2 sin φ×E(t) (15) The Fourier coefficient a0 = 0, indicates that − E= 0, the other Fourier coefficients are ak = 0 bk = mw 2 sin φ( 2 kπ ) (16) So its potential energy function will be Ueff = U + mw 2 sin 2 φ× 1 π2 ∞∑ k=0 ( 1 k4 ) = U + 0.1097mw 2 sin 2 φ (17) Where φ = 0,π and arccos(− 4.5579gl w2 ) are the ex- tremum of 17. After minimizing 17, we have following results. • The downward position φ = 0, is always stable. • Vertically upward position φ = π is stable if w2 > 4.5579gl. • The point φ = arccos(−4.5779gl w2 ) is unstable. Quadratic Type force: Next, consider a quadratic type force: Q(t) = Q(t + T) (shown is Figure 4), given by equation 18 Q(t) =   1, 0 ≤ t < 3T 8 8 T ( T 2 − t), 3T 8 ≤ t < 5T 8 −1, 5T 8 ≤ t < t (18) Figure 4: Quadratic type force The force acting upon the particle is f(t) = mw 2 sin φ×Q(t) (19) The fast oscillating force in Fourier expansion is given as Q(t) = mw 2 sin φ ∞∑ k=1 ( 2 kπ + 8 π2k2 sin k π 4 ) sin k(wt) With the following Fourier coefficients ak = 0 bk = mw 2 sin φ ∞∑ k=1 ( 2 kπ + 8 π2k2 sin k π 4 ) (20) So the effective potential energy function will be Ueff = U + mw 2 sin 2 φ× 1 4 ∞∑ k=1 1 k2 ( 2 kπ + 8 π2k2 sin k π 4 ) 2 = U + 0.3856mw 2 sin 2 φ (21) Where φ = 0,π and arccos(− 1.2967gl w2 ) are the ex- tremum of above system. With 21, the stability of the system is given as • The point φ = 0, is always stable. • The point φ = π is stable if w2 > 1.2967gl. • The nontrivial position φ = arccos(−1.2967gl w2 ) is unstable. So, it is observed that, the position φ = π is stabilized at lower frequency as compared to harmonic force. Rectangular Type Force: Let’s introduce the rect- angular type force R(t) = R(t + T), and the function R(t) is T-periodic, given in 22, illustrated in Figure 5 R(t) =   1, 0 ≤ t < T 2 −1, T 2 ≤ t < T (22) Sukkur IBA Journal of Computing and Mathematical Sciences - SJCMS | Volume 2 No. 2 July – December 2018 c© Sukkur IBA University 10 Babar Ahmad (et al.), Stabilization of Vertically Modulated Pendulum with Parametric Periodic Forces (8-13) Figure 5: Rectangular type force For vertical modulation, the force acting upon the par- ticle is f(t) = mw 2 sin φ×R(t) The Fourier coefficient a0 = 0 , shows that − f= 0, the other coefficients are ak = 0 bk = mw 2 sin φ( 4 2k − 1 ) (23) Using above coefficients, the Fourier expansion is R(t) = mw 2 sin φ 4 π ∞∑ k=1 1 (2k − 1) sin(2k − 1)wt The effective potential energy is Ueff = U + mw 2 sin 2 φ× 1 4 ( 16 π2 ) 2 ∞∑ k=1 1 (2k − 1)2 = U + 0.4112mw 2 sin 2 φ (24) With the extremum at φ = 0,π and arccos(− 1.2159gl w2 ). After minimizing 24, we have following results • The point φ = 0, is always stable. • The point φ = π is stable if w2 > 1.2159gl. • The nontrivial position φ = arccos(−1.2159gl w2 ) is unstable. From the above examples, it is noticed that, at posi- tion φ = π, the system is stabilized at lower frequency as compared to previous cases. The above results are summarized in Table 4.. It is also observed that, at nontrivial position, as area under the curve increases, the frequency of oscillation decreases. Harmonic and in- clined type force has minimum area so they have maxi- mum frequency as compared to rectangular type force. 5. Parametric Control Next, a parametric control is defined on quadratic type force, to control the non-trivial position φ = π. This force is also T-periodic, Q�(t + T) = Q�(t). The con- trol is defined for 0 < � < 1. This �-parametric force is defined as Q�(t) =   1 0 ≤ t < 1 − �T 2 1 � (− T 2 t + 1) 1 − �T 2 ≤ t < 1 + �T 2 −1 1 − �T 2 ≤ t < T (25) and illustrated in Figure 6 Figure 6: parametric quadratic type force For vertical modulation the force acting upon the par- ticle is f2(φ,t) = mw 2 sin φ×Q�(t) (26) From 25, the other Fourier coefficients are ak = 0 bk = mw 2 sin φ( 2 kπ + 8 �π2k2 sin k π 4 ) (27) Fourier expansion of oscillating force is f2(φ,t) = mw 2 sin φ ∞∑ k=1 ( 2 kπ + 8 (�π2k2) sin k π 4 ) sin kwtφ (28) So, the effective potential energy will be Ueff = U + mw 2 sin 2 φ× 1 4π2 ∞∑ k=1 4 k4 (1 + 1 �kπ sin �kπ) 2 = −mgl cos φ + mw2 sin2 φ.A (29) and A = 1 π2 ∞∑ k=1 1 k4 (1 + 1 �kπ sin �kπ) (30) The effective potential energy 29 has extremum at φ = 0,π,arccos(− 0.5gl w2.A ). After minimizing 29, we have the following results • The system is stable at point φ = 0. • If w2 > 0.5gl w2.A , then the system will be stable at φ = π. Sukkur IBA Journal of Computing and Mathematical Sciences - SJCMS | Volume 2 No. 2 July – December 2018 c© Sukkur IBA University 11 Babar Ahmad (et al.), Stabilization of Vertically Modulated Pendulum with Parametric Periodic Forces (8-13) Table 1: Stability Comparison of different linear forces with harmonic force External Force Position Stability Position Stability Condition sin 0 always π w2 > 2gl Inclined 0 always π w2 > 4.5579gl Quadratic 0 always π w2 > 1.2969gl Rectangular 0 always π w2 > 1.2159gl Table 2: Stability condition of �-parametric force at φ = π 0 < � < 1 Sum A Stability Condition 0.9 0.1320 w2 > 3.7879gl 0.8 0.1607 w2 > 3.1114gl 0.7 0.1956 w2 > 2.5562gl 0.6 0.2357 w2 > 2.1213gl 0.5 0.2793 w2 > 1.7902gl 0.4 0.3239 w2 > 1.5437gl 0.3 0.3664 w2 > 1.3647gl 0.2 0.4029 w2 > 1.2400gl 0.1 0.4287 w2 > 1.1663gl • The nontrivial position φ = arccos(− 0.5gl w2.A ) is unstable. The stability of the system for different values of � is summarized in Table 2. For � = 0.9, the infinite sum A = 0.1320, and the effec- tive potential energy function is Ueff = −mglcosφ + 0.132mw2 sin2 φ At the position φ = π, the system is stable if the condi- tion w2 > 3.7879gl is satisfied, and this value is much greater than all previous results. Next for � = 0.8, the infinite sum A = 0.1606, and the point φ = π is stable is w2 > 3.1114gl, which gives much better result. Similarly, For � = 0.1, the system is stabilized at the same position with the condition w2 > 1.663gl, and this result is better than all considered examples. From above discussed cases, it can be observed that, with the decrease in value of, infinite sum A is increased, thus stabilizing the system at relatively lower frequency at φ = π. It is also observed that, as � → 1, the term A ∼= 0.1098 and the system is stabilized at the position π with the condition w2 > 3.4.5537gl, and this frequency is approximately equal to the inclined type force. Thus, the quadratic type force approaches to inclined type force as � → 1, and the system is stabilized with much greater frequency and is not stable. However, as � → 0, the term A ∼= 0.4386, and the position φ = π is stable if the condition w2 > 1.14gl is satisfied, and this frequency of oscillation is lower than rectangular type force. Observe the Table 2, the rectangular force fall between � = 0.2 and � = 0.1, and for the parametric force with � = 0.1, the frequency of oscillation becomes lower than that in case of rect- angular type force. Hence, by defining the parametric control better results are achieved. 6. Conclusion Using Kaptiza method of averaging for arbitrary pe- riodic forces, the vertically modulated pendulum ex- cited by periodic linear forces is stabilized at φ = π with the frequency w, that was found to be suf- ficiently less relative to the case of harmonic mod- ulation. Moreover, the rectangular type force was found to be the best. The stability conditions at non- trivial position φ = π improves by defining a para- metric control on some of the periodic piecewise lin- ear forces. Hence, by adjusting the parameter, the system is stabilized with less oscillating frequency. Figure 7: Quadratic type force for different values of � Sukkur IBA Journal of Computing and Mathematical Sciences - SJCMS | Volume 2 No. 2 July – December 2018 c© Sukkur IBA University 12 Babar Ahmad (et al.), Stabilization of Vertically Modulated Pendulum with Parametric Periodic Forces (8-13) Figure 8: Ueff is minimum at φ = πifw 2 > 1.14gl Figure 9: Ueff is always minimum at φ = 0 References [1] E. Butikov, ”The rigid pendulum-an antique but evergreen physical model,” European Journal of Physics, pp. 424-441, 1999. [2] A. Stephenson, ”On induced stability,” Philosoph- ical Magzine, pp. 233-236, 1908. [3] A. Stephenson, ”On induced stability,” philosoph- ical Magzine, vol. 17, pp. 756-766, 1909. [4] A. Stephenson, ”On new type of dynamic sta- bility,” Memories and Proceeding of the Manch- ester Litrerary and Philosophical Magzine, pp. 1- 10, 1908. [5] P. L. Kapitza, ”Dynamic stability of pendulum with an oscillating point of suspension,” Journal of Experimental and Theorectical Physics, pp. 588- 597, 1951. [6] E. M. Lifshitz, L. D. Landau and mecanics, Ox- ford, UK: Pergamon Press: butterworth, 2005. [7] B. Ahamd and S. Borisenok, ”Control of effective potential minima for Kapitza oscillator by periodi- cal kicking pulses,” Physics Letter A, pp. 701-707, 2009. [8] B. Ahmad, ”Stabilization of Kapitza pendulum by symmetrical periodical forces,” Nonlinear Dynam- ics, pp. 499-506, 2010. [9] E. Butikov, ”An improved criterian for Kapitza pendulum stability.,” Journal of Physics A: Math- ematical and Theoratical, vol. 44, 2011. [10] E. Butikove, ”Subharmonic resonances of the para- metrically driven pendulum,” Journal of Physics A: Mathematical. Gen, vol. 35, 2002. [11] B. Ahmad, ”Stabilization of driven pendulum with periodic linear forces,” Nonlinear Dynamics, vol. 2013, 2013. Sukkur IBA Journal of Computing and Mathematical Sciences - SJCMS | Volume 2 No. 2 July – December 2018 c© Sukkur IBA University 13