EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2019 Simulated inverted pendulum using analytical mechanics CARL CLAUSON MALTE SLOMA KTH SKOLAN FÖR TEKNIKVETENSKAP EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2019 Simulering av inverterad pendel med analytisk mekanik CARL CLAUSON MALTE SLOMA KTH SKOLAN FÖR TEKNIKVETENSKAP Contents 1 Abstract 2 2 Sammanfattning 3 3 Introduction 4 3.1 Background . 4 3.2 Purpose . 4 3.3 Problems . 5 3.3.1 Double Pendulum . 5 3.3.2 Kapitza Pendulum . 6 3.3.3 Kapitza Pendulum with elliptic oscillation. 7 4 Theory 8 4.1 Newtonian Mechanics . 8 4.2 Functionals and Variational Calculus . 8 4.3 Lagrange's equations . 10 5 Method 11 5.1 Software . 11 5.1.1 Maple with Sophia package . 11 5.1.2 Matlab . 11 5.2 General method . 11 5.3 Double pendulum . 12 5.3.1 Lagrangian mechanics . 12 5.3.2 Solving with Euler-Lagrange equations . 13 5.4 Kapitza pendulum . 14 5.4.1 Lagrangian mechanics . 14 5.4.2 Solving with Euler-Lagrange equations . 14 5.5 Kapitza pendulum with elliptical oscillation . 15 5.5.1 Lagrangian mechanics . 15 5.5.2 Solving with Euler-Lagrange equations . 15 6 Results 16 6.1 Double pendulum . 16 6.2 Kapitza pendulum . 17 6.2.1 Minimum angular velocity to maintain stability . 17 6.2.2 Maximum initial angle to maintain stability . 20 6.3 Kapitza pendulum with elliptical oscillation . 22 6.3.1 Maximum width of the orbit to maintain stability . 22 6.3.2 Influence of width in macro oscillations . 24 A Appendix code 29 A.1 Double pendulum . 29 A.2 Kapitza's pendulum . 30 A.3 Kapitza's pendulum with elliptical oscillation . 31 1 1 Abstract Analytical mechanics is an alternative to Classical (Newtonian) mechanics for calculating the movement of particles and systems of particles. In this paper we analyze three different mechanical systems using analytical mechanics. We then implement this into the maple based software "Sophia" to further analyze how changing some of the parameters alter the behavior of the system. One of our main focuses in the report is studying the stability of an inverted pendulum. In 1951 the Russian physicist Pyotr Kapitza, figured out that the pendulum could be stabilized in the upside-down position by forcing it into a rapid vertical oscil- lation. We examine the boundaries of this stability and explore if the pendulum can instead be stabilized using an elliptical oscillation. 2 2 Sammanfattning Analytisk mekanik ¨arett alternativ till klassisk (Newtonsk) mekanik f¨oratt ber¨aknar¨orelsenav partiklar och system av partiklar. I den h¨arrapporten anal- yserar vi tre olika mekaniska system med analytisk mekanik. Vi implementerar sedan detta i den maple-baserade programvaran "Sophia" f¨oratt ytterligare analysera hur ¨andringarav vissa parametrar f¨or¨andrarsystemets beteende. Ett av v˚arahuvudsyften med den h¨arrapporten ¨aratt studera stabiliteten hos en inverterad pendel. Ar˚ 1951 fann den ryska fysikern Pyotr Kapitza att pendeln kunde stabiliseras i dess upp-och-ner-l¨agegenom att tvinga den till en snabb vertikal sv¨angning. Vi studerar gr¨anserna f¨ordenna stabilitet och unders¨oker om pendeln ist¨alletkan stabiliseras genom en elliptisk sv¨angning. 3 3 Introduction 3.1 Background Newtonian mechanics has throughout its existence been used to determine vec- tor quantities of forces, momenta and motion in a certain system. When scien- tists discovered that the Newtonian mechanics was not enough to solve certain types of mechanical problems, analytical mechanics was invented which can be defined as a collection of closely related alternative formulations of Newtonian mechanics. Analytical mechanics utilizes scalar properties of motion, i.e usually the systems total potential energy and kinetic energy and not vector quantities of accelerations and forces of individual particles. Lagrangian mechanics is an analytic method which is ideal for calculating the motion of different systems with various types forces, for example pendulums. In the present work we will restrict the forces acting on the system to conservative fields. A regular pendulum is in it's ground state stable and will all the time strive to keep the center of mass as low down as possible. In 1951 the Russian physicist Pyotr Kapitza, figured out that the pendulum could be stabilized in the upside- down position by forcing the pendulum into a quick vertical oscillation. This pendulum is one of threee systems that will be studied in this thesis. 3.2 Purpose The methods of analytical mechanics will be applied to model the three different physical systems using the dynamic tool for Maple; Sophia [2]. To compute the motion of these systems using regular Newtonian mechanics is possible but even with only two degrees of freedom it becomes very complicated. Therefore will the selected systems firstly be modeled quite easily using analytical mechanics and then use these methods to solve and analyze three different problems. The systems are a regular double pendulum, and then two different versions of the Kapitza pendulum [3]. 4 3.3 Problems 3.3.1 Double Pendulum Problem 3.29 from Apazidis [3]. Figure 1: Double pendulum. 'Consider a double pendulum consisting of two bars OA and AB of masses m1 and m2 and lengths l1 and l2 respectively that are free to rotate in the vertical plane according to the figure 1. Choose the following numerical values of the parameters m1 = 3 kg, m2 = 1 kg, l1 = l2 = 1 m and calculate and plot the trajectory of end B by means of the Sophia and Graphics packages. Choose the following initial conditions q1(0) = 1 rad,q _1(0) = 0 rad, q2(0) = 1:5 rad, q_2(0) = 0 rad. Calculate then the trajectory of the point B for slightly different initial conditions and show the sensitive dependence of the motion of the system on initial conditions by comparing the two trajectories.' 5 3.3.2 Kapitza Pendulum Figure 2: Kapitza pendulum. Consider an inverted pendulum consisting of one bar OA of mass m1 and length l1 that is free to rotate in the vertical plane according to the figure. The bar will be forced into a harmonic vertical oscillation which will stabilize the pendulum. In order to change the behaviour of the pendulum, parameters will be changed and examined. The stability of the pendulum will be studied by deciding the minimum angular velocity ! and the maximum initial angle to maintain stability. The parameters of the pendulum is described in the method section. 6 3.3.3 Kapitza Pendulum with elliptic oscillation. Figure 3: Kapitza pendulum with elliptic oscillation. Consider an inverted pendulum consisting of one bar OA of mass m1 and length l1 that is free to rotate in the vertical plane according to the figure. The bar will be forced into a harmonic oscillation around an ellipse which will stabilize it vertically. In order to change the behaviour of the pendulum, the parameters will be changed and examined. The stability of the pendulum will be studied by deciding the maximum re- lationship between the height and the width of the ellipse in order to maintain vertical equilibrium. The effect of an increasing width will also be studied, causing macro oscillations. 7 4 Theory 4.1 Newtonian Mechanics From Newton's laws of motions, the concept of energy can be derived. When describing a conservative mechanical system, energy becomes a key factor to solve the often non-linear differential equations for the system. In classical mechanics the energy of the system can be divided into two parts. The kinetic energy is defined as m(v · v) T = (4.1) 2 for a particle with all of it's mass distributed in one single point. When talking about larger and more complicated structures than particles, e.g a bar, the kinetic energy will consist of two parts, translational and rotational: m(v · v ) !T I! T = G G + (4.2) tot 2 2 Where the inertia I and the angular velocity ! per definition Apazidis [3]. Potential energy, is defined per the Cambridge Dictionary as the "energy stored by something because of position (as when an object is raised), because of its condition (as when something is pushed or pulled out of shape), or in chemical form (as in fuel or an electric battery)" [5]. 4.2 Functionals and Variational Calculus A functional is a mathematical operator that uses functions as its input argu- ments and returns a scalar. One fundamental problem of variational calculus is to find a real function y(x) of a real variable x so that the functional of I[y] becomes an extreme value. Z x2 I[y] = fy(x); y0(x); xdx (4.3) x1 I[y] is the functional of y with fixed endpoints x1 and x2. The purpose of the functional is to determine those functions y(x), which uses the given values y1 = y(x1) and y2 = y(x2) as endpoints and makes the functional I[y] an extremum. In other words, for which functions y(x) the functional I[y] assumes a maximum, minimum or a saddle point. Now assume Z x2 I[β] = fy(x; β); y0(x; β); xdx (4.4) x1 8 where y(x; β) = y(x) + βη(x) with η(x1) = η(x2) = 0. This implies that y(x) is different from y(x; β) but have the same boundary conditions, i.e. multiple paths are possible between the two endpoints. Figure 4: Example of functions with same boundary conditions. The variation of I is given by ( ) dI Z x2 @f dy @f dy0 δI = dβ = dx + 0 dβ: (4.5) dβ x1 @y dβ @y dβ When the second term in the integral is integrated by parts ! ! x2 Z x2 @f d dy Z x2 dy d @f @f dy dx = − dx + (4.6) 0 0 0 x1 @y dx dβ x1 dβ dx @y @y dβ x1 the conclusion is that the boundary terms do not contribute because dy/dβ = η(x).