Potsdam, August 006 Report E-Project Henriette Laabsch 7685 Toni Luhdo 7589 Steffen Mitzscherling 7540 Jens Paasche 7575 Thomas Pache 754
Introduction From 7 th February till 3 rd March, we had our laboratory experiment time with the main topic 'Oscillations', overseen by members of the Nonlinear Dynamics group. Our experiments were divided into two parts. In the first part we dealt with electrical oscillation between resistances and capacitors in different circuits. In the second part we used computer simulations of double pendulum, the Lorentz system, oscillator dynamics for the Van der Pol oscillator and synchronization of a selfsustained oscillator by external force. Table of Contents I Basic Theory 03-05 I. Oscillators: from linear to nonlinear 03-04 I. Phase Space 04 I.3 Chaotic Oscillations 05 I.4 Resonance 05 I.5 Synchronisation of Oscillations 05 II Van der Pol Oscillator and syncronization 06-6 II. Oscillation Dynamics for Small Amplitude Oscillations 06- II. Relaxation Oscillations -3 II.3 Hard Excitation 4 II.4 Synchronization 5-6 III Alternating Oscillator 7-0 III. Ignition- & Extinction-Voltage 7 III. Voltage-Current-Course 7 III.3 Period Duration of Various RC-Combinations 9 III.4 Temporal Voltage Course of a RC-Combination 9-0 IV Chaos -7 IV. Solution of the Double Pendulum with Euler-Lagrange -3 IV. Lorenz System 4-7
I Basic Theory We have oscillatory processes everywhere in our surrounding (mechanical oscillations: violin, harp; electrical oscillations: alternating circuit). There are three types of oscillation: harmonic, non-harmonic and chaotic. Oscillations can occur with other phenomenon such as resonance and synchronization. I. Oscillators: from linear to nonlinear The main feature of the harmonic oscillation is the linear differential equation x x 0 0 with the squared angular frequency as a constant. Examples for harmonic oscillations are the simple pendulum with small deviations and the pendulum clock. In the following we discuss the conservation laws of both examples. At the point with the largest amplitude the mass is resting. Therefore, the energy of the system is only determined by its potential energy V = m g l sinα. When α = 0 the point of mass has its maximum angular velocity α and no m elevation. Thus, it has only the kinetic energy T = α. Consequently, the energy conservation is given by E = V + T = m g l sinα + m α. Due to friction and damping ideal pendulums do not exist in nature. On the other hand self-sustained oscillators, e.g. a pendulum clocks, exist in nature. The loss of energy caused by the friction is equalized by the potential energy of the lifted weight. 3
The second Newton's law for an ideal simple oscillation can be written as ml d dt mg sin d dt g l sin 0 For an analytical solution of this equation the term sin shall be substituted by 3 5 its Taylor expansion sin.... For small amplitudes the terms of 3! 5! higher order in this expansion will disappear and sin. Using the higher order terms in the linear equation will lead to an advanced differential equation like 0 Duffing equation: 6 0. Setting 6 finally leads to the 0 0 I. Phase Space: The plane of spatial position and velocity of a system span the phase space. It can be used to analyse all possible states of a system. Each point in the phase space is unique for a state of the system. Trajectories in phase space (time dependent change in spatial position and velocity of the system) cannot intersect each other. A picture of the phase portrait may give qualitative information about the dynamics of the system (attractors & deflectors). An attractor can be viewed as the final trajectory in the phase space which the system reaches after evolving over a long time. Attractors exist in different types. The fixed point attractor is a point where a system evolves towards. The limit cycle is a periodic orbit of the system which is isolated, often a closed trajectory. There is also the limit torus, which is the three-dimensional limit cycle, for example two alternate frequencies and the strange attractor, which has non-integer dimension or the dynamics on the attractor are chaotic. 4
I.3 Chaotic Oscillations: An autonomous continuous dynamical system needs at least three dimensions in his phase space to exhibit chaos. One example for chaos is the double pendulum with large deviations. It has already four dimensions, the spatial positions of its two masses and its two velocities. A nonchaotic nonlinear oscillator can be forced to a chaotic movement by introducing external influences. In the case of the Duffing oscillator, the oscillation can be influenced by a brake or an external force acting with system independent frequency that way, it is added a additonal function f. I.4 Resonance: The tendency of a system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration (its resonant frequency) than it does at other frequencies is called resonance. Examples are the acoustic resonances of musical instruments, or an opera singer with a very high and loud voice (high frequency) is able to break glass. The high frequency force the glass into oscillation, but the glass is not able to swing as fast as the excitation frequency. Consequently, it breaks. I.5 Synchronization of Oscillations: Synchronization is, when two (or more) oscillators are in phase (both move in a similar manner) or in anti-phase (they swing in opposite directions; the first pendulum is in leftmost position and the second is on rightmost position). In synchronized systems very accurate measurements are able to reveal an infinitesimal phase shift. Under certain set up conditions, two wrist clocks are in synchronisation, when they are connected with a supporting medium, which transmits the oscillations. Damping causes the loss of synchronisation when the distance between the two clocks gets too far. In a mechanical matter, two clocks, one placed in Africa the other in Europe, are not able to be in synchronization. 5
II Van der Pol Oscillator and synchronization II. Oscillation Dynamics for Small Amplitude Oscillations: In dimensionless variables the Van der Pol oscillator can be described as: x a x x x 0...with the non-linear damping term a x. II..) The fixed points of the Van der Pol equation are found by setting its derivatives to zero: x y 0 y a x y x 0 The fixed point is (0,0). The stability conditions of the fixed points can be obtained using the Jacobi matrix: A = 0 axy a( x ) x= 0 y= 0 It leads to: A = 0 a The characteristic polynomial then is given by λ aλ + = 0 The solution of this equation produces the Eigenvalues, which are: a λ = ± a 4, Clearly, when a < 0, the zero fixed point is stable. The system undergoes Hopf bifurcation at a = 0, hence the limit cycle is produced. 6
II..) By setting a 0 (black) the Van der Pol oscillator reduces to a harmonic oscillator and the trajectory becomes a circle in the phase space ( x ; x ) of the radius with the initial conditions x ; x 0. For a, the trajectories move on the limit cycle, no matter which initial condition is chosen as shown by different colors. x& x 7
II..3) Choosing two initial conditions and setting a 0 will lead to different radii of the trajectory in the phase space. Describing a harmonical oscillation as shown below: x& x Initial conditions: ) x ; x 0 (blue) ) x 0 ; x,5 (red). 8
They are producing concentric circles but due to different initial conditions the x t -courses are phase shifted as shown below. x(t) t 9
II..4) Setting a and than much larger than the orbit gets distorted from a circle (now the oscillaton behaves nonlinear). We see different limit cycles (black and red line). x& x 0
II..5) Now the damping term a x is replaced by a x v in the Van der Pol equation and we get a limit-cycle in the phase space with the shape of a circle of radius. x& x
II. Relaxation Oscillations: II..) Setting a much larger than, for example a=0, it reaches a strongly non-linear limit. We observe that the limit-cycle consists of extremely slow charging and sudden discharging sequences. x& x
II..) For this phase portrait the course x t is shown in the following figure: x(t) t It can be observed a slow evolution followed by a rapid jump for all courses x t. 3
II.3 Hard Excitation: II.3.) In the following figure two limit-cycles can be observed. The outer, stable one is situated where the red, green and black curves are coming together. The second, inner one, which is unstable, can be expected to exist somewhere between the beginnings of the green and blue curve. x& x Additionally there is a stable fixed point at the origin of the phase space. The reduced form of the Van der Pol equation applied in the above figure is following: x x x 4 x x 0 With settings for this observation are: 7 ; 0, ; β = 0. The final behaviour of the trajectory depends on the initial conditions. If this value is within the unstable cycle, the trajectory goes to the stable fixed point. However, if the initial conditions are outside the unstable cycle, the trajectory approaches to the stable limit cycle with radius larger than that of the unstable one. 4
II.4 Synchronization of a Self-Sustained Oscillator: II.4.) We can use the Lissajous figure to test whether the phases of two systems are synchronized or not in the following way. Output Input When the Van der Pol oscillator is locked by the external force, one can obtain a simple curve in the Lissajous figure as shown above with =,00. Otherwise, if they are not synchronized, a complex curve is obtained without pronounced simple relationship between each other. Output Input 5
Changing both the argument A and the frequency of the external force simultaneously, one can determine the region when the Van der Pol system is synchronized by the external forcing. The graphic shows the relationship between A and the different -values. Ωlow Ω high A 0,95,07 0,5 0,9,09 0,6 0,88, 0,7 0,86, 0,8 0,84,3 0,9 0,8,4 0,78,8, 0,7,,5 0,6,5,7 0,55,8 Arnold-Tongue,5,5 A 0,5 0 0 0, 0,4 0,6 0,8,,4 Omega As you can see here in the picture that you have a high and low frequency-limit for each amplitude of the external force, where in between two systems are synchronized. When the parameters are not in the area limited by the frequency borders, the Van der Pol oscillator is not synchronized by the external force. 6
III. Ignition & Extinction Voltage: III Alternating Oscillator A fluorescent lamp together with a voltmeter was connected parallely to a voltage source. Then vollage was varied using the stat and then ignition (U I ) and extincting voltages (U E ) were measured. U I in V U E in V 44.7 8. 44. 7. 44.3 6.4 44. 5.7 44.4 5.8 So the average value of these voltages are U I = 44.34V and U E= 6.64V III. Voltage-Current-Course: An ammeter was added to the connection of Practice A in order to measure the current for increasing and decreasing voltages. increasing voltage: U in V I in ma 44..06 46..4 48..3 50..3 55..57 60.3.8 65..04 70..9 75..53 77.7.66 decreasing voltage: U in V I in ma 44..06 4. 0.96 40. 0.86 38. 0.76 36. 0.66 34. 0.57 3. 0.48 8. 0.8 6. 0.6 7
U-I characteristic were: U-I-Curve 85 75 65 U in V 55 45 35 5 0 3 I in ma The equation of the linear regression-curve of the decreasing voltage is: V U ( I) = 0.4 I +. 58V ma In comparison with the equation U I R V I U 0 it can be determined: V R V = 0.4 = ˆ 0. 4kΩ and U 0. 58V ma = Looking at the linear regression-curve of increasing voltage it results: V U ( I) = 0.839 I +. 39V and therefore ma R V = 0. 839kΩ and U. 39V 0 = Compared to the real value of R V 7 k there's a deviation of R V 3 k which comes to 7.65% of the real resistance. 8
III.3 Time Duration of Various RC-Combinations: For different time RC combinations the voltage was fixed at U 60V * it has been measured three times the duration of 0 periods of each RC-combination. Then time period was calculated by measuring of 0 oscillations. R in M C in F 0 T in s T I II III II III,0,0 8.50 8,50 8,85 8.85 8.00 8,00,85.85 0.5 0,5,0 4,59 4.59 4,75 4.75 4,85 4.85 0,47 0.47 0.5 0,5,0 0. 0, 0.00 0,00 9,3 9.3 0,98 0.98,0,0 9.03 9,03 8,87 8.87 9,09 9.09 0,90 0.90 0.5 0,5 4,0 4 35.8 35,8 35.03 35,03 35.00 35,00 3,5 3.5 time of RC combinations 4 3 T in s 0 0 3 RC in s The period duration of one oscillation consists of the charging time t C and the discharging time t D. T t C t D t C and t D can be determined in the following way: U I U 0 U * U E t C C and t U E U D D 0 U * U I 9
...where C R V C and D R C Now the values can be calculated to: t C ins t D ins T ins 0.06.5.57 0.03 0.38 0.4 0.06 0.76 0.8 0.03 0.76 0.78 0..5.6 III.4 Temporal Voltage Course of a RC-Combination: An oscillograph has been added to the RC-combination ( R = 0.5 MΩ ; C = 0. µf ). This way we could observe the term U t - voltage against time. The charging and discharging time now have been determined to the following values reading them off the oscillograph: T ins t C ins t D ins 0.06 0.053 0.007 According to the former used equations in Practice C the values have been calculated as well: T ins t C ins t D ins 0.0407 0.0378 0.009 That means the measured values show a 47,53% deviation compared to the calculated ones. The current can be determined with the following equation and has been calculated to: t τ D I = ( U I U 0 ) e = 0.568mA R V 0
IV Chaos The double pendulum is one of the most simple systems which create a chaotic movement. In case of large amplitudes the system will move on a non-predictable trace. If the amplitude is small, the pendulum will behave like a harmonic oscillator. Of course, both pendulums are disturbing each other. At one time, they oscillate inphase and some time later in opposite phase. If only one pendulum is moving, it will transmit the energy to the other one until it stops, then the process turns back. This fact is used in high buildings. To prevent them against oscillation for example caused by strong wind or earthquakes they have a damped pendulum in their top. This pendulum absorbs the energy that might lead to oscillation and secures the building against destruction in this way. The Fernsehturm of Berlin uses a huge bar, fixed with tree steel cables. The mass of that pendulum is.5 tons.
IV. Solution of the Double Pendulum with Euler-Lagrange: The coordinates of the two masses m and m are x = l ϑ y x y sin( ) = l ϑ cos( ) = l sin( ϑ ) + l sin( ) ϑ = l cos( ϑ ) l cos( ) ϑ...where ϑ is the angel of the upper, ϑ the angel of the lower part of the double pendulum. With these terms, the kinetic and the potential energy ( T and V ) are given by: T T V V m = l ϑ cos( ϑ) + ϑ sin( ϑ) m = l ϑ cos( ϑ ) + l ϑ sin( ϑ ) + l ϑ cos( ϑ ) + l ϑ sin( ϑ ) ( l cos( )) = m g ϑ ( l ϑ ) l cos( )) = m g cos( ϑ Taking the following interrelations into account sin ( ϑ ) + cos ( ϑ) = cos( ϑ + ϑ ) = cos( ϑ )cos( ϑ ) sin( ϑ )sin( ϑ ) a simplification of the energy-terms is possible and we got T m m l m l m l l cos V m m g l cos m g l cos Now we can determine the Lagrangian of the system
L T V and set up the Euler-Lagrange differential equation L q i d dt L q i 0 The double pendulum has two free variables. So q i stands for and. The results are 0 m l l sin m m gl sin l m l l cos m l l sin 0 m l l sin sin cos m gl sin m l We can simplify the problem by using only small angles and only regarding the first term of the Taylor development of sin. So we set all constant values to and approximate sin. This way, we get two solutions t 0 4i e i t e i t 0 4i e i t e i t t 0 4i e i t e i t 0 4i It can be seen clearly, that angel # depends on angel velocity of # and around of course. This is the mathematically way to show, that both pendulums are interacting. In case of small amplitudes, we can calculate the coordinates of the both masses depending on the time. For high amplitudes we can not approximate sin and it is impossible to solve the equations. This state is called chaos. e i t e i t 3
IV. Lorenz System: In 963 the meteorologist Edward N. Lorenz noticed chaotic behaviour in simulated weather phenomenons. Doing small deviations to the initial conditions he got heavy variance of the results. It was Lorenz himself, who found the equations that describe the system. x s y x y rx y xz z xy bz...where s, b and r are given parameters. For the whole duration of the experiment we set b 8 3 and s 0. To find the fixed points, we set the derivations to zero. For r, b const. and s const. we get x = ± b r y = ± b r ( ) ( ) z r The point (0,0,0) is also the steady state of the system. The nonlinear terms are xz and xy. When we set r the phase diagram nearly looks like a simple line. Only one stable equilibrium is obtained at the origin. 4
With increasing r ( r ) we get more complex spirals. black: r 5 blue: r 6 yellow: r 0 green: r 4 When <r<4.74, the origin is unstable and the system has two stable stationary states with the expressions of the fixed points on the previous page. When 4.74<r, for example r=8, the two stationary states become unstable and one obtains the chaotic dynamics. 5
The Lorenz attractor in three dimensions z x y The x-z-projection z x 6
Relations to the fluid dynamics One can relate the movement of the double pendulum to motion of fluids. There is a fixed point depending on the speed, when the motion turns from laminar to turbulent flow. For r we get the simple line and for r we will get more complex spirals. So r is the critical Speed of the flowing water. You can compare the line with laminar flowing water and the spirals with it s fixed points as turbulent flow. Roughly speaking, the variable x measures the rate of convective overturning, the variable y measures the horizental temperature variation, and the variable z measures the vertical temperature variation. The three parameters, s, r and b are respectively proportional to the Prandtl number, the Rayleigh number, and some physical properties of the region under consideration. r 8, s 0, b 8 3 x orange (x/y/z)-(//) blue (x/y/z)-(//0) black (x/y/z)-(/0/0) t This diagram shows, that three trajectories start very closely at the beginning. After some oscillations, their ways split off and each trajectory follows it s own direction and one cannot determine the distance between these three trajectories any more. In terms of nonlinear science, this is the so-called the sensivitity on the initial conditions. 7