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Application of Lagrange Equations to 2D Double Spring- Pendulum in Generalized Coordinates

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Application of Lagrange Equations to 2D Double Spring- Pendulum in Generalized Coordinates RUHUNA JOURNAL OF SCIENCE Vol 10 (2): 120-134, Dec 2019 eISSN: 2536-8400 Faculty of Science http://doi.org/10.4038/rjs.v10i2.78 University of Ruhuna Application of Lagrange equations to 2D double spring- pendulum in generalized coordinates N.O. Nenuwe Department of Physics, Federal University of Petroleum Resources, P. M. B. 1221, Effurun, Delta State, Nigeria, Nigeria Correspondence: [email protected] https://orcid.org/0000-0002-3112-3869 Received: 31st December 2018; Revised: 30th July 2019; Accepted: 31st December 2019 Abstract In this study, the Lagrange’s equations of motion for a 2D double spring-pendulum with a time dependent spring extension have been derived and solved approximately. The resulting equations are also solved numerically using Maple, and plots of motion for the pendulum bobs m1 and m2 are presented and compared. It was observed that motion along the x-axis is characterized by sine wave function while motion along y-axis is characterized by cosine wave function with slightly changing amplitudes. Change in stiffness constant, angle of deflection, mass of pendulum bob and spring length were found to have significant effect on the dynamics of the double spring-pendulum. The periodic and chaotic behaviour noticed in this study is consistent with current literature on spring-pendulum systems. Keywords: Lagrange equations, double spring-pendulum. 1 Introduction The two dimensional (2D) double pendulum is a typical example of chaotic motion in classical mechanics. The pattern of its motion is well known to change drastically as the energy is increased from zero to infinity (Biglari and Jami 2016). However, at low and very high energies the system represents coupled harmonic oscillators, and can be considered as an integrable system. But, at intermediate energies, the system is known to exhibit chaotic features. Double spring-pendulum is a classical mechanical system consisting of two bobs of mass m1 and m2 fixed to the ends of two weightless elastic springs with stiffness constants k1 and k2, and the angle of deflections q1 and q2 , respectively. The second spring is connected to the first mass as shown in Fig. 1. The dynamics of double spring-pendulum appears to be scanty in literature, though there are reports on single and double spring pendulum systems. Marcus et al. (2016) studied the order-chaos-order transition of Faculty of Science, University of Ruhuna 120 Sri Lanka N.O. Nenuwe Application of Lagrange equations to double spring-pendulum spring pendulum using the Hamiltonian formulation. Numerical analysis of the equations of motion for double pendulum was reported by Smith (2002) using Maple soft. de Sousa et al. (2017) reported on the energy distribution in spring pendulum. Lewin et al. (2015) numerically analyzed the dynamics of single and double pendulum using MATLAB. Also, double pendulum numerical analysis with Lagrangian and Hamiltonian equations of motions using MATLAB was reported by Biglari and Jami (2016). Despite these studies, no reports to the best of my knowledge have been made on double spring-pendulum with time dependent extension in spring length using Lagrangian formulations. Also, it is well known that physical systems can be described by their Lagrangian, and from the Lagrangian function one can obtain second order differential equations of motion describing such dynamic systems. In most cases the exact solution cannot be obtained for these Lagrange equations of motion, and this leads to employing alternative numerical approach to solve such equations (Baleanu et al. 2015). The interest in this study is to analytically obtain the equations of motion for a double spring-pendulum with time dependent spring-extension. Hence, Lagrangian formulation of mechanics is used to derive the equations of motion for the system. The resulting Lagrange’s equations are solved approximately and numerically using MAPLE software. Fig. 1. Schematic diagram of a double spring-pendulum 2 Derivation of the Lagrange’s equations of motion Suppose the positions of mass m1 and m2 at any time in space is expressed in Cartesian coordinates as (x1, y1) and (x2, y2), unstretched lengths of the Ruhuna Journal of Science 121 Vol 10(2): 120-134, December 2019 N.O. Nenuwe Application of Lagrange equations to double spring-pendulum springs are l1 and l2, and the springs extend by 1 (t) and 2 (t) when the respective masses are attached as shown in Figure 1. At the point of suspension, the positions of the bobs are given by the following equations: x1( t ) ( l 1 1 ( t ))sin q 1 ( t ) (1) y1() t ( l 1 1 ())cos t q 1 () t (2) x2() t ( l 1 1 ())sin t q 1 () t ( l 2 2 ())sin t q 2 () t (3) y2() t ( l 1 1 ())cos t q 1 () t ( l 2 2 ())cos t q 2 () t (4) The total kinetic energy (T) of the system is given by: 112 2 2 2 T mxtyt1 1()()()() 1 mxtyt 2 2 2 (5) 22 1 2 2 2 T(,,,) q q m1 1 ()( t l 1 1 ()) t q 1 () t 2 2 2 2 2 2 2 (6) 1()t 2 ()( t l 1 1 ()) t q 1 ()( t l 2 2 ()) t q 2 () t 1 m22 12 ( tt ) ( ) 2( l 112212 ( tl ))( ( tqtqt ) ( ) ( ) cos( qq 12 ) 2 2(l2 2 ()()()2( t 1 t q 2 t l 1 1 () t 2(t ) q 1 ( t ) sin( q 1 q 2 ) By taking a plane at distance (l1+l2) below the point of suspension of Figure 1 as a reference level, the potential (V) energy of the system is then given by: V(,) q m1 g l 1 l 2 ( l 1 1 ())cos() t q 1 t (7) mglll2 1 2 ( 1 1 ( t ))cos qtl 1 ( ) ( 2 2 ( t ))cos qt 2 ( ) 11 k22( t ) k ( t ) 221 1 2 2 From this, the Lagrangian function for the system is given by: LTV 112 2 2 2 2 2 2 Lq(,,,) qmtl11111 ()( ())() tqt m 21 () t 2111 ()( tl ())() tqt 22 (8) 22 (l222 ( tqt )) ( )+ 2 12 ( tt ) ( ) 2( l 112212 ( tl ))( ( tqtqt ) ( ) ( ) cos( qq 12 ) 2(l2 2 ( ttqt ) 1 ( ) 2 ( ) 2( l 1 1 ( ttqt ) 2 ( ) 1 ( ) sin( qq 1 2 ) mglll11211 ( ( t ))cos qt 1 ( ) mglll 21211 ( ( t ))cos qt 1() 1 1 (l ( t ))cos q ( t ) k2 ( t)() kt 2 2 2 22 1 1 2 22 Ruhuna Journal of Science 122 Vol 10(2): 120-134, December 2019 N.O. Nenuwe Application of Lagrange equations to double spring-pendulum The Lagrange’s equations (Murray 1967, Goldstein et al. 2000, Martin and Salomonson 2009) associated with the generalized coordinates q12( t ), q( t ), and are given by: 1(t ), 2 ()t d L L d L L 0, 0, dt q1 q 1 dt q 2 q 2 (9) d L L d L L 0, 0 . dt1 1 dt 2 2 Now differentiating equation (8) accordingly, and substituting into equation (9) gives four Lagrange’s equations of motion for the system; one equation for each degree of freedom (i.e., qq1, 2, 1 and 2 ). They are as follows: (mml12111 )( ( tqtml )) ( ) 222 ( ( t ))cos( qqqtm 122 ) ( ) 2 sin( qq 122 ) ( t ) 2 (10) 2(mml121111 )( ( t )) ( tqtml ) ( ) 222 ( ( t ))sin( qqqt 121 ) ( ) 2m2 cos( q 1 q 2 ) 2 ( t ) q 2 ( t ) m 1 m21 gqsin m2( l 2 2 ( t )) q 2 ( t ) m 2 sin( q 1 q 2 ) 1 ( t ) 2 2m2 cos( qq 1211 ) ( tqtml ) ( ) 211 ( ( t )sin( qqqt 121 ) ( ) (11) 2m2 ( l 2 2 ( t )) 2 ( t ) q 2 ( t ) m 2 g sin q 2 ml222 ( t ) sin( qqqtmm 122 ) ( ) 121 ( tm ) 2 cos( qq 122 ) ( t ) 2 (12) 2m2 sin( qq 1 2 ) 2 ( tqt ) 2 ( ) mml 1 2 1 1 ( tqt ) 1 ( ) 2 ml222 ( t ) 1 cos( qqqtmgq 122 ) ( ) 1 cos 1 cos qkt 211 ( ) ml211 ( t ) sin( qqqtm 121 ) ( ) 2 cos( qq 121 ) ( tmt ) 22 ( ) 2 (13) 2m21211 sin( qq ) ( tqtml ) ( ) 222 ( t ) l 11 ( t ) cos( qqqt 121 ) ( ) m2 gcos q 2 k 2 2 ( t ) Equations (10) – (13) represent a pair of coupled second order differential equations describing the unconstrained motion of a double spring-pendulum. Generally, equations of motion can be represented in matrix form as: Mpt()()()() cpt12 cpt ft (14) Where, c1 and c2 are the damping coefficient and stiffness matrices. Rearranging equation (14), one obtains the mass matrix M and rest matrix R in the representation given by equation (15): Mp()() t R t (15) Ruhuna Journal of Science 123 Vol 10(2): 120-134, December 2019 N.O. Nenuwe Application of Lagrange equations to double spring-pendulum where: p()() t M1 R t (16) and q1 q pt() 2 (17) 1 2 Substituting equations (10), (11), (12) and (13) into (15), we obtain (mmlt1211 )( ( )) mltqq 222 ( ( ))cos( 12 )0 mqq 212 sin( ) 00m( l ( t )) m sin( q q ) (18) 2 2 2 2 1 2 M 0 m l ( t ) sin( q q ) m m mcos( q q ) 2 2 2 1 2 1 2 2 1 2 m2 l 1 1( t ) sin( q 1 q 2 ) 0 m 2 cos( q 1 q 2 ) m 2 q1 q pt() 2 (19) 1 2 2(mml )( ( t )) ( tqtml ) ( ) ( ( t ))sin( qqqt )2 ( ) 2 m cos( qq ) ( tqt ) ( ) 121111 222 121 2 1222 m1 m 2 gsin q 1 (20) 2m cos( q q ) ( t ) q ( t ) m ( l ( t )sin(q q ) q2 ( t ) 2 m ( l ( t )) ( t ) q ( t ) 2 1 2 1 1 2 1 1 1 2 1 2 2 2 2 2 m22 gsin q R 2m sin( qq ) ( tqtmml ) ( ) ( tqtml ) 22 ( ) ( t ) 1 cos( qqqt ) ( ) 21222 12111 222 122 m1 gcos q 1 cos q 2 k 1 1 ( t ) 2m sin( qq ) ( tqtml ) ( ) ( t ) l ( t ) cos( qqqtmgqkt )2 ( ) cos ( ) 21211 222 11 121 2 222 The matrix equations in (18), (19) and (20) represent a four dimensional system of equations of motion for the double spring-pendulum in generalized coordinates: qq1, 2, 1 and 2 .
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