Chaotic Motion of the Double Pendulum

Transcription

1 MEGL Mathematical Art and 3D Printing George Mason University: College of Science December 16, 2016

2 Table of Contents 1 The Mathematics 2 Inspiration for the Model Planning the Construction of the Model Implementing the plan 3 A Simulation in Mathematica

3 Equation for the motion of a simple pendulum (without damping and external driving): d 2 θ dt 2 + g L sin θ = 0 where θ = the angle from the downward vertical, g = acceleration due to gravity and L = the length of the pendulum.

4 (Continued) Nondimensionalization: let ω = g L d 2 θ dt 2 + sin θ = 0 and τ = ωt. Then Coverting to a first order system of differential equations gives: dθ dt = v dv dt = d 2 θ dt 2 = sin θ

5 Composed of a simple pendulum with another pendulum hanging from its bob Positions of the bobs: x 1 = l 1 sin θ y 1 = l 1 cos θ x 2 = l 1 sin θ 1 + l 2 sin θ y 2 = l 1 cos θ 1 l 2 cos θ 2...4

6 (Continued) Get the velocities by differentiating with respect to time: dx 1 dt = l dθ 1 1 dt cos θ dy 1 dt = l dθ 1 1 dt sin θ dx 2 dt = l dθ 1 1 dt cos θ dθ l 2 dt cos θ dy 2 dt = l dθ 1 1 dt sin θ dθ l 2 dt sin θ 2...8

7 (Continued) Obtaining the Lagragian, L = T V : V = m 1 gy 1 + m 2 gy 2 (the potential energy of m 1 and m 2 ) T = 0.5m 1 v m 2 v 2 2,

8 (Continued) d 2 θ 1 dt 2 = 3g sin θ 1 g sin(θ 1 2θ 2 ) 2 sin(θ 1 θ 2 )(v 2 2 v 2 1 cos(θ 1 θ 3 cos(2θ 1 2θ 2 ) d 2 θ 2 dt 2 = 2 sin(θ 1 θ 2 )[2v g cos θ 1 + v 2 2 cos(θ 1 θ 2 )] 3 cos(2θ 1 2θ 2 ) where g = 9.81 and the length of both rods has been set to 1

9 (Continued) Letting θ 1 and θ 2 be the angles of the two bobs with respect to the vertical and using the similar method as the simple pendulum, then the system of differential equations will be dθ 1 dt = v 1 dv 1 dt = 3g sin θ 1 g sin(θ 1 2θ 2 ) 2 sin(θ 1 θ 2 )(v 2 2 v 2 1 cos(θ 1 θ 2 3 cos(2θ 1 2θ 2 ) dθ 2 dt = v 2 dv 2 dt = 2 sin(θ 1 θ 2 )[2v g cos θ 1 + v 2 2 cos(θ 1 θ 2 )] 3 cos(2θ 1 2θ 2 )

10 The Mathematics Inspiration for the Model Planning the Construction of the Model Implementing the plan Inspiration for the Model Figure: Double Pendulum Model by gmelenka from Thingiverse Figure: Double Pendulum Model by stevenbtroy

11 Inspiration for the Model Planning the Construction of the Model Implementing the plan Planning the Construction of the Model Parts of the Double Pendulum Model: 1 Vertical Stand with a Base 2 Top Arm of the Double Pendulum 3 Lower Arm of the Double Pendulum 4 Spacers to allow for free motion of the arms without any friction

12 Inspiration for the Model Planning the Construction of the Model Implementing the plan Implementing the plan Software and Hardware used for the Double Pendulum Model First Model Redesign, hence some newer parts The Final Drafts on OpenScad and Makerware Examples

13 Inspiration for the Model Planning the Construction of the Model Implementing the plan Software and Hardware used for the Double Pendulum Model Software: Openscad - for making the parts of the model in 3 dimensions Makerware - for printing the parts Mathematica - for making a simulation of the double pendulum s motion iphone 7 Camera application s Slow Motion feature - to capture the motion of the Double Pendulum Model in slow motion Hardware: bearings various screws nuts and bolts

14 Inspiration for the Model Planning the Construction of the Model Implementing the plan Redesign, hence, some new parts added Newer parts added for the model: A Second part for making one top arm for the pendulum A block spacer A short lower arm

15 A Simulation in Mathematica A Simulation in Mathematica