Lagging Pendulum. Martin Ga

Transcription

1 02 Lagging Pendulum Martin Ga 1

2 Task A pendulum consists of a strong thread and a bob. When the pivot of the pendulum starts moving along a horizontal circumference, the bob starts tracing a circle which can have a smaller radius, under certain conditions. Investigate the motion and stable trajectories of the bob. 2

3 What does it look like? Stationary point 3

4 Lagging pendulum lag = phase lag regarding Lagging attachment pendulum point by almost π Stationary point 4

5 Apparatus Power source (adjustable ω) Motor Pendulum 5

6 Apparatus R Thread L Bob 6

7 ሷ 16 Analytical theory: circular trajectories ω Assumption: bob moves only in vertical plane R Frame of reference: rotating with the arm α L Force Angular of acceleration: Centrifugal the thread α α = ω 2 0 sin force α + ω 2 R L ω 0 = + sin α cos α g/l Gravitational force Gravity Centrifugal 7

8 Analytical theory: circular trajectories α ሷ [arb. u.] dependence of αሷ on α? Equilibrium points α [rad] ω 0 = g/l αሷ α = ω 2 0 sin α + ω 2 R L + sin α cos α = 0 Equilibrium condition 8

9 Analytical theory: circular trajectories ω 0 = g/l αሷ α = ω 2 0 sin α + ω 2 R L + sin α cos α = 0 Equilibrium condition 9

10 Analytical theory: circular trajectories Equlibrium condition: sin α + ω2 R Centrifugal 2 + sin α cos α = 0 ω 0 L force ω 0 = g/l Two free parameters: ω/ω 0 and R/L R L Natural frequency (Depends on L) dependence of ω/ω 0 10

11 Equilibrium angles 16 1,6 1,2 α [rad] 1 st 0,8 0,4 0-0, nd ω ω 0-0,8-1,2-1,6 R L = 0,2 3 rd trajectory ω 0 = g/l 11

12 Spotted circular trajectories Observed Observed Not observed 12

13 Spotted circular trajectories Observed Observed Not observed Are the trajectories stable? 13

14 Are the trajectories stable? Pendulum lower potential energy We should also add 2 nd dimension We will plot total effective potential energy α ሷ [arb. u.] 2 nd : Lagging α [rad] rd rd 2 nd : Lagging Traditional α [rad] 2 nd trajectory: unstable energy [arb. u.] Traditional Angular acceleration Integration Potential energy 14

15 3D Energy plot in corotating frame 16 3 rd position Lagging position y Energy Unstable (2 nd dimension) Unstable (all dimensions) Traditional position Stable falls down View from side x 15

16 Results Stability analysis does not correspond with the experiment, where the lagging motion (2 nd position) is observed 16

17 Stabilizing effect of Coriolis force 16 Coriolis acceleration: a C = 2ω v Angular velocity of the rotating frame Vector product perpendicular to both ω and v Velocity of the bob in rotating frame Potential energy map a cor Large Coriolis force could act as a centripetal force (that has the same direction) Bob 17 Small

18 Stabilizing effect of Coriolis force 16 Coriolis acceleration: a C = 2ω v Angular velocity of the rotating frame Vector product perpendicular to both ω and v Well-known problem: stability of L4, L5 Lagrange points also stabilized by the Coriolis force Coriolis Singh, Jagadish, force and Jessica could Mrumun Begha. act "Stability as of a equilibrium points in the generalized perturbed restricted centripetal three-body problem." force Astrophysics and Space Science (2011): (that has the same direction) Photo credits: By Lagrange_points.jpg: created by NASAderivative work: Xander89 - Lagrange_points.jpg, CC BY 3.0, Velocity of the bob in rotating frame w/index.php?curid= Potential energy map a cor Bob 18 Large Small

19 How does Coriolis stabilize? Energy [arb. u.] φ Lagging position θ 19

20 How does Coriolis stabilize? Energy [arb. u.] φ Lagging position Stable θ Fits experiment 20

21 Could Coriolis force stabilize 3 rd position? Energy [arb. u.] 3 rd position Unstable θ φ Fits experiment 21

22 Overview: predictions and experiment 16 1 st 2 nd 3 rd trajectory Position: Theoretical predictions: Stable Stable Unstable Experiment: Observed Observed Not observed Qualitative predictions Done Quantitative predictions 22

23 ω [rad.s-1] 16 Lagging angle: theory experiment Theory Experiment θ [deg] 23

24 Task A pendulum consists of a strong thread and a bob. When the pivot of the pendulum starts moving along a horizontal circumference, the bob starts tracing a circle which can have a smaller radius, under certain conditions. Investigate the motion and stable trajectories of the bob. tracing a circle smaller radius 24

25 CONDITIONS OF LAGGING MOTION 1. Parameters of pendulum: R/L, ω/ω 0 2. Initial conditions 25

26 Parameters: ω/ω 0 by R/L α [rad] 1,6 1,2 Equilibrium angles We can calculate critical angular velocity from theory 0,8 potential energy 0,4 ω E[arb. u.] 0 ω 0-0,4-0,8 0 5 inflex point equilibrium θ[rad] -1,2-1,6 Only 1 position Lagging position exists Critical angular velocity (ω c ) 26

27 Parameters: ω/ω 0 by R/L α [rad] 1,6 1,2 Equilibrium angles We can calculate critical angular velocity from theory 0,8 potential energy 0,4 ω E[arb. u.] 0 ω 0-0,4-0,8 0 5 θ[rad] -1,2-1,6 Only 1 position Critical angular velocity (ω c ) Lagging position exists equilibrium inflex point Can be solved only numerically conditions: E α ቚ ω c = 0 2 E α 2 ቚ ω c = 0 27

28 No solution 16 Critical omega theoretical prediction ω c /ω Critical omega tends to diverge when R L 1 0 0,2 0,4 0,6 0,8 1 1,2 R/L 28

29 No solution 16 Critical omega theoretical prediction ω c /ω Critical omega tends to diverge when R L 1 Always-unstable region R L coriolis R L < 1 0 0,2 0,4 0,6 0,8 1 1,2 sum gravity unstable region R/L 29

30 ω [rad.s-1] Critical angular velocity: theory-experiment Theory Experiment (one position) Experiment (two positions) ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 L [m] 31

31 Conditions for parameters 1. Ratio of lengths: R L < 1 R L 2. Angular velocity: ω > ω c Where ω c is a function of R L and ω 0 = g L Conditions: parameters necessary for lagging motion 32

32 Initial conditions: time evolution In the beginning the motion is jerky (trembling) Later, bob stabilizes to lagging motion Stationary point Time-lapse video: trembling lagging Lagging motion: Long-exposure photograph Motion is being stabilized by air drag 33

33 Effect of air drag Bob parameters: mass = 1.29 ± 0.02 g diameter = (1.3 ± 0.1)cm a drag = c = 1 2 CSρ m 1 2 CSρ m v2 Object Effective drag coefficient Bob 0,03 m 1 Thread 0,10 m 2 L (L 0. 5m 1 m) Thread has also a great affect on damping 34

34 Numerical simulation We are unable to solve system analytically Numerical simulation from first principles Inertial frame of reference used Constraint modelled using Lagrange multipliers Air drag of the thread computed (may have different direction than that of the bob) (Spherical coordinates omitted due instabilities near poles) 35

35 Numerical simulation Runge-Kutta-Fehlberg algorithm (4 th order method) Computes both 4 th order and 5 th order Error estimate (difference between 4 th and 5 th order) Adapts stepsize if the error is too big Despite these upgrades: - Numerical instability (violation of constraint) Thread exchanged with very hard spring The lenght of thread stays constant in simulation & spring does not influence character of motion 36

36 Example output from simulation Trembling (initial motion) Lagging motion (relaxed motion) 37

37 θ [rad] Air drag: stabilization Parameters: R/L ω = 4ω 0 θ initial = 0.15 'uhol.txt' using 1: π π φ [rad] Δθ Dissipation ~15s Time Trembling Lagging 'uhol.txt' using 1:3 Stabilization Δφ 0.03 π ~15s Time (Boundary is only qualitative and approximate) 38

38 Simulation: equilibrated angles θ 1,2 1 0,8 0,6 Phase transition (critical ang. velocity) 0,4 0,2 0-0,2-0,4-0,6 0 0,5 1 1,5 2 2,5 Initial angle (in simulation) Changes in θ created by drag are negligible 39

39 Initial conditions: angles 0,3 9 1, , , , , , , Theta/Phi 5, , , , , , , , , , θ [rad] 2, 0, Traditional motion 0, 0,2574 0, 0, , 0,19305 Lagging 0, 0, motion 0, 0,1287 0, 0, , Initial conditions 0, , 0, , 0, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , φ [rad] π 0 π v initial = , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

40 Initial conditions: angles v = 0 [rad] 0,35 0,30 Theta/Phi 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 Theta/Phi 0,5 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,388 2, , , , , , , , , , ,388 2, , , , , , , , , , , , ,376 2, , , , , , , , , , ,376 2, ,364 Traditional 2, , , , , , , , , , , , , , , , , , , , , ,364 motion 2, , , , , , , , , , , , ,352 2, , , , , , , , , , ,352 2, , , , , , , , , , , , ,34 2, , , , , , , , , , ,34 2, , , , , , , , , , , , ,328 0, , , , , , , , , , ,328 2, , , , , , , , , , , , ,316 0, , , , , , , , , , ,316 2, , , , , , , , , , , , ,304 0, , , , , , , , , , , , , , , , , , , , , , ,292 0, , , , , , , , , , ,292 Lagging motion , , , , , , , , , , , ,28 0, , , , , , , , , , , , , , , , , , , , , , ,5 0 φ [rad] 0,5 41

41 Conclusion 1. Lagging motion at a potential maximum Stabilized by Coriolis force 2. Conditions for lagging motion 3. 3D equations of motion Phase diagrams predicted 42

42 Thank you for your attention! Lagging motion at a potential maximum Stabilized by Coriolis force 2. Conditions for lagging motion 3. 3D equations of motion Phase diagrams predicted 43

43 02 Lagging Pendulum Martin Gazo 44

44 APPENDICES 45

45 What is Coriolis force? Bob moves away from the center Coriolis acceleration: a C = 2ω v Δr 46

46 What is Coriolis force? Bob moves away from the center F c Velocity change Δv = ωδr (in tangential direction) Coriolis acceleration: a C = 2ω v Δr Fictitious force acting sideways 47

47 Coriolis on parabolic potential Circular trajectories around equilibrium Ω ω 2 1,5 1 0,5 Centripetal Ω 2 r = C 2 r + 2 ω (Ωr) Centrifugal + Gravity (2 nd order Taylor expansion of potential) Ω Solution: Coriolis angular frequency of oscillation around equilibrium Ω = ω ± ω 2 C ,2 0,4 0,6 0,8 1 1,2 Slope: C 2 /ω 2 No solution for steep potential 48

48 Simulation: Equations of motion 16 Lagrange approach Spherical coordinates θ, φ φ ሷ ~ 1 sin θ Numerically unstable near south pole Cartesian coordinates x, y, z Non-conservative forces (not included in potential energy; e.g. air drag) Lagrange multiplicator λ = m 2L 2 (gz x ሶ + ωrsinωt 2 yሶ ωrcosωt 2 zሶ 2 ω 2 R cosωt x Rcosωt + sinωt(y Rsinωt) +F Vx x R cos ωt + F Vy y R sin ωt + F Vz z 49

49 Fundamental equations 16 Definition of Lagrangian Kinetic and (total) potential energy For Lagrangian applies: If there are other forces (not yet included) then: Arbitrary coordinate Sum of all (generalized) forces which have not been included yet in potential energies 51 51

50 Fundamental equations 16 Definition of Lagrangian Kinetic and (total) potential energy Lagrange multiplier Constraint equation: Determines area where there the thread could be located (because of configuration in which it is) 52 52

51 Fundamental equations 16 Kinetic energy Constraint equation (total) potential energy Lagrange multiplier continuing Other two coordinates will be derived analogically 53 53

52 Fundamental equations 16 Values of forces in particular coordinates Lagrange multiplier - needed to be evaluated External forces - for example drag making derivatives 54 54

53 Let us derive λ first 16 modifying making a derivative once more modifying, marking part of the expression by 55 55

54 ..now,,a 16 Defining Accelerations in particular coordinates 5656

55 ..continuing with λ 16 addition modifying 57 57

56 Unstable motion 16 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 θ avg Highly-unstable - Unstable in φ angle ω/ω 0 Zig-zag motion View from outside 58

57 Variation on textbook problem Special case: R = 0 Unstable R/L = 0 R/L = 0,001 R/L = 0,01 59

58 3D Energy plot 3 rd position Lagging position φ θ = 0 Energy Unstable (2 nd dimension) Unstable (all dimensions) Traditional position Stable 0 θ 60

59 Stability analysis of found positions Energy Total potential energy in the reference frame of arm [Arbitrary units] L = 0,3m R = 0,1m ω = 10rad. s 1 16 Color scale 61