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Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation

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Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation Mathematical and Computational Applications Article Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation Wojciech Smiechowicz´ 1, Théo Loup 1 and Paweł Olejnik 2,* 1 The International Faculty of Engineering, Lodz University of Technology, 36 Zwirki˙ Str., 90-001 Lodz, Poland; [email protected] (W.S.);´ [email protected] (T.L.) 2 Department of Automation, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 1/15 Stefanowskiego Str., 90-924 Lodz, Poland * Correspondence: [email protected] Received: 23 September 2019; Accepted: 15 October 2019; Published: 16 October 2019 Abstract: This paper considers three dynamic systems composed of a mathematical pendulum suspended on a sliding body subjected to harmonic excitation. A comparative dynamic analysis of the studied parametric mutations of the rigid pendulum with inertial suspension point and damping was performed. The examined system with parametric mutations is solved numerically, where phase planes and Poincaré maps were used to observe the system response. Lyapunov exponents were computed in two ways to classify the dynamic behavior at relatively early stage of forced responses using two proven methods. The results show that with some parameters three systems exhibit a very similar dynamic behavior, i.e., quasi-periodic and even chaotic motions. Keywords: parametric pendulum; Lyapunov exponents; quasi-periodic behavior; Chaos 1. Introduction Mechanical engineering or other fields always require elaboration based on numerical computations. Each mechanical system associated with periodic excitation behaves unpredictably. Excitations can come from the internal imperfections of the system, such as parametric asymmetry, eccentricity or external forces such as wind, road defects, etc. Any chaotic behavior is undesirable, and their consequences can lead to the destruction of the system. One method of qualitative determination of the state of the system is based on computations of Lyapunov exponents. Qualitatively, it informs about the form of the dynamic response of the system. These exponents are certain characteristic numerical values, creating a spectrum that helps in the qualitative assessment of the dynamics of the system. They determine the convergence or divergence of infinitely close trajectories that begin close to each other. To know the nature of system dynamics, all we need to know is the sign of the Lyapunov exponent λ defined as follows: 1 δZ(t) λ = lim lim ln (1) t δZ0 0 t δZ0 !1 ! j j The Lyapunov exponent in Equation (1) is denoted by λ. Any positive value in the limit accounts for chaos in the system, while non-positive proves the regularity. There are many numerical methods of computing Lyapunov exponents, e.g., Wolf method, Rosenstein method, Kantz method, method based on neural network modification, synchronization method, and others, see [1–6]. There are two main approaches to numerical assessment in the literature: with known motion equations and a time series approach. For example, the first method of computing Math. Comput. Appl. 2019, 24, 90; doi:10.3390/mca24040090 www.mdpi.com/journal/mca Math. Comput. Appl. 2019, 24, 90 2 of 15 Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 2 of 15 of computing the full spectrum of Lyapunov exponents is presented in [3], the calculation of the the full spectrum of Lyapunov exponents is presented in [3], the calculation of the largest Lyapunov largest Lyapunov exponent is given in [5,6], and the determination of the spectrum of Lyapunov exponent is given in [5,6], and the determination of the spectrum of Lyapunov exponents can be found exponents can be found in [7]. in [7]. From a practical point of view, Lyapunov exponents are used in various fields of science, such From a practical point of view, Lyapunov exponents are used in various fields of science, such as: as: rotor systems [8], electricity systems [9], aerodynamics [10]. rotor systems [8], electricity systems [9], aerodynamics [10]. 2.2. The Parametric Pendulum and ItsIts MutationsMutations FigureFigure1 1 shows shows the the three three analyzed analyzed dynamical dynamical systems. systems. The The first first system system presented presented in in Figure Figure1a is1a a is a mathematical pendulum of the length attached to the moving slider of the point-focused mass mathematical pendulum of the length l0 attached to the moving slider of the point-focused mass M. The. The inertial inertial slider slider moves moves horizontally horizontally back back and and forth forth along along the thex -axisx-axis and and is is subjected subjected toto harmonicharmonic ( ) forcingforcingF (t)==F0 coscos! t, where, whereF0 denotes denotes amplitude amplitude of the of force,the force, and ! andstates thestates frequency the frequency of forcing. of Inforcing. the second In the system second depicted system depicted in Figure 1inb theFigure pendulum 1b the pendulum of the length of thel0 is length replaced byis replaced a parametric by a pendulum,parametric pendulum, the length ofthe which length is of described which is bydescribed the function by theS function(t) = l0 +(Asin) =(!+()t), where A, whereis the amplitude is the amplitude of the periodic of the periodic change change of the lengthof the length about aboutl0. The generalized. The generalized coordinates coordinates of first of andfirst secondand second systems systems are: the are: angle the angle' between between the pendulum the pendulum and vertical and vertical axis and axis the and displacement the displacementx of the slider. of the In slider. the third In systemthe third shown system in Figureshown1 cin the Figure first rigid1c the pendulum first rigid of pendulum constant lengthof constant is replaced length by is anreplaced elastic by form, an elastic where form,k and wherec describe and linear describe stiffness linear and damping,stiffness and respectively. damping, respectively. The generalized The coordinatesgeneralized forcoordinates the third systemfor the can third be noted:system dynamicalcan be noted: elongation dynamical of the elongation spring, the of angle the spring,' between the theangle pendulum between and the vertical pendulum axis and and the vertical displacement axis andx theof thedisplacement slider. of the slider. (a) (b) (c) FigureFigure 1.1.The The analyzed analyzed systems: systems: (a) rigid(a) rigid pendulum, pendulum, (b) first (b parametric) first parametric mutation, mutation, (c) second parametric(c) second mutationparametric in mutation an elastic in form. an elastic form InIn orderorder to to simplify simplify mathematical mathematical modelling modelling of theof problemthe problem the following the following assumptions assumptions are made: are made: there is no friction in the inertial slider; •• there is no friction in the inertial slider; masses of the pendulum’s swinging body and slider are treated as point-focused masses; •• masses of the pendulum’s swinging body and slider are treated as point-focused masses; pendulum arm of a constant or variable length is massless. •• pendulum arm of a constant or variable length is massless. DynamicalDynamical analysis of the systemsystem mutations is preceded by derivation of equations of motion. ToTo findfind them,them, oneone cancan useuse NewtonNewton laws,laws, however,however, thisthis approachapproach requiresrequires aa betterbetter knowledgeknowledge ofof thethe system.system. A more universal and simplersimpler approachapproach basesbases onon derivingderiving themthem fromfrom thethe Euler–LagrangeEuler–Lagrange equationequation [[11].11]. 2.1. Mathematical Model of Rigid Pendulum 2.1. Mathematical Model of Rigid Pendulum The kinetic energy of the analyzed two-degrees-of-freedom (2 DoF) system is the sum of kinetic The kinetic energy of the analyzed two-degrees-of-freedom (2 DoF) system is the sum of kinetic energies of the slider (Ts) and the pendulum (Tb): energies of the slider () and the pendulum (): 11h . 2 . 2 . 2 2i =T = Ts++ Tb == [Mx ++(m x + +22x'lcos cos+' + ' l .)]. (2)(2) 22 Math. Comput. Appl. 2019, 24, 90 3 of 15 In the assumed configuration of the model the potential energy of the analyzed system is as follows: U = mgl cos '. (3) − Referring to Figure1, the following denotations have been used in Equations (2) and (3): M—mass . of the sliding body, m—mass of the mathematical pendulum, x—linear displacement and x—velocity . of the sliding body, '—angular displacement and '—angular velocity of the pendulum, l—constant length of the rigid pendulum (Figure1a), g—gravitational constant. The Lagrangian L = T U satisfies the Euler–Lagrange equation for any of the assumed generalized − coordinates. Therefore, one writes: ! d @L @L . = Qi, where qi = ['(t), x(t)], Qi = [0, F0 cos !t]. (4) dt @qi − @qi Substituting the Equations (2) and (3) for the kinetic and potential energy into the Lagrangian, respectively, it allows to derive the equation of motion for each of the generalized coordinates. The number of coupled equations of motion depends on the number of generalized coordinates. The equations of motion for the analyzed 2 DoF system are as follows: for the generalized coordinate ' (pendulum angle): • .. .. x g ' + cos ' + sin ' = 0; (5) l l for the generalized coordinate x (displacement of the slider): • .. .. 2 (M + m)x + ml' cos ' ml' sin ' = F cos !t. (6) − 0 The equations of motion (5)–(6) can be algebraically decoupled with respect to the second derivate: . 2 .. mg sin ' cos '+ml' sin '+F cos !t x = 0 , M+m sin2 ' (7) .. .. g ' = x cos ' sin '. − l − l 2.2. Mathematical Model of First Mutation of the Rigid Pendulum The kinetic energy of the analyzed two-degree-of-freedom model with parametric pendulum is equal to the sum of the kinetic energy of the slider and kinetic energy of the pendulum body: 1 . 2 1 . 2 . 2 T = Mx + m x + S(t) sin ' + S(t)' cos ' + S(t) cos ' + S(t)' sin ' . (8) 2 2 − The potential energy of the analyzed system is as follows: U = mgS(t) cos '. (9) − The Lagrangian satisfies the Euler–Lagrange equation for any of the assumed generalized coordinates.
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