About

Double pendulum with external drive

The simulation demonstrates a frictionless, mathematical double pendulum that either:

1. starts passively from given initial conditions,
2. or is driven periodically by an external source.

The massless, rigid pendulum rods are shown as straight lines. Masses are concentrated in the center of the pendulum bobs.

When the simulation file is opened, the pendulum is in horizontal position. The length of the secondary pendulum 2 (yellow) is one half that of the primary one 1 (blue): L₂/L₁ = 0.5. The mass of both is equal: m₂/m₁ = 1. Their initial speed is zero. There is no external drive: A=0.

Start initiates the calculation of movement under the influence of gravity. The path of the secondary yellow pendulum bob (2) is shown in red for a limited time period. Stop freezes the movement until a new start. Clear deletes traces, Reset reestablishes the default initial conditions.

The primary blue bob can be drawn with the mouse to create different initial angular positions, with stretched pendulum and zero initial velocity.

In the right window a phase space diagram y₁´(y₁) is shown for the blue bob of the primary pendulum (for a simple pendulum this would be a closed periodical curve, and would be a circle for small arcs of oscillation). The length of the traces is limited to 2500 points of calculation.

The speed of the animation can be varied with slider Speed.

The length L₁ of the primary pendulum is kept constant. Slider L₂/L₁ changes the length of the secondary pendulum. The window size is adjusted to twice the maximum size of the double pendulum. L₂/L₁ = 0 results in a single pendulum, with both bob masses coinciding.

Slider m₂/m₁ determines the mass relation of the pendulum bobs. m₂/m₁= 0 results in a single primary pendulum. In the simulation the orientation of the fictive massless secondary pendulum stays constant, while it will briskly oscillate for any finite mass.

The movement of the double pendulum is chaotic and shows no periodicity. Yet it is strictly deterministic: after reset the same path will be resumed (start without clearing the trace).

In a real pendulum friction would diminish the amplitude of oscillations. Friction is neglected in this model.

With A > 0 a periodic external torque is acting upon the primary pendulum, with amplitude and direction as shown by the black arrow. Slider delta changes the drive frequency. With delta =1 the drive frequency is equal to that of the single primary pendulum (m2 = 0) at small arcs. For all initial conditions the driven double pendulum shows a rich variety of deterministic chaotic movements.

The 3D-Phase space button opens another window with two rotating 3D frames for the three dimensional phase spaces of both pendulums.

y₁´´(y₁ , y´₁) und y₂´´(y₂ , y₂´)

The scalings of the 3D frames are adjusting automatically to the amplitudes.

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH- Foundation; lookang

http://iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejss_model_e_Double_pendulum_drivenwee/e_Double_pendulum_drivenwee_Simulation.xhtml

Briefing Document: Double Pendulum HTML5 JavaScript Simulation Model

1. Overview

This document provides a detailed review of the "Double Pendulum HTML5 JavaScript Simulation Model" as described on the Open Educational Resources / Open Source Physics @ Singapore website. The simulation is an interactive educational tool designed to demonstrate the complex dynamics of a double pendulum, including both passive motion from initial conditions and motion driven by an external periodic force. It's built using Easy JavaScript Simulations (EjsS) and is accessible via web browsers and as a mobile app.

2. Main Themes and Key Concepts

• Deterministic Chaos: A central theme is the exploration of deterministic chaos. The double pendulum exhibits chaotic behavior – its motion is unpredictable in the long term, despite being governed by deterministic equations. The simulation highlights this with the statement, "The movement of the double pendulum is chaotic and shows no periodicity. Yet it is strictly deterministic: after reset the same path will be resumed..."
• Nonlinear Dynamics: The simulation emphasizes that the chaotic behavior arises from the nonlinear nature of the differential equations that govern the pendulum's motion. The document explicitly states, "The decisive origin of chaotic behavior is the existence of more than one nonlinear differential equation."
• Initial Conditions and Sensitivity: The simulation illustrates how extremely small changes in the initial conditions can lead to drastically different trajectories over time. The document notes, "Yet when trying to create the same initial condition by moving the pendulum with the mouse, the path will develop differently, as it depends critically on the exact value of the initial conditions."
• Simple vs. Double Pendulum: The simulation enables a comparison between a simple pendulum (which has periodic motion, especially for small oscillations) and a double pendulum (which exhibits complex, chaotic motion). Setting L2/L1 = 0 or m2/m1 = 0 transforms the simulation into a simple pendulum.
• External Drive: The inclusion of an external drive allows for the study of forced oscillations and further investigation into chaotic behavior. The document explains that with A > 0, "a periodic external torque is acting upon the primary pendulum."
• Phase Space Diagrams: The simulation also includes phase space diagrams, which visually depict the relationship between a pendulum's position and momentum, helping to illustrate the periodic nature of simple pendulum motion and the complex nature of double pendulum motion.
• Frictionless Model: The model is a mathematical one, and therefore frictionless, the document states "In a real pendulum friction would diminish the amplitude of oscillations. Friction is neglected in this model."

3. Simulation Features and Functionality

• Interactive Controls: The simulation features a number of interactive controls, such as sliders for adjusting parameters like the length ratio (L2/L1), mass ratio (m2/m1), and drive frequency (delta), as well as the amplitude of the driving force (A).
• Start/Stop/Reset: Buttons to start, stop, and reset the simulation are available, allowing users to observe trajectories, freeze the movement, and return to default settings. A "Clear" option is also provided to delete previous traces.
• Mouse Interaction: The user can drag the primary pendulum bob (blue) to create different initial angular positions, demonstrating the impact of varied initial conditions on the trajectory.
• Visualizations: The simulation displays the movement of the pendulum bobs, with the path of the secondary bob (yellow) traced in red for a limited time period. A phase space diagram and 3D phase space visualisations are available in separate windows.
• Adjustable Speed: The speed of the animation can be controlled with a slider.
• 3D Phase Space: A button opens another window with rotating 3D frames for the three-dimensional phase spaces of both pendulums, y1''(y1, y'1) and y2''(y2, y'2).

4. Mathematical Model

• Differential Equations: The document outlines the underlying mathematical model, stating that a simple pendulum is described by "one nonlinear ordinary second order differential equation," while the double pendulum is described by "two nonlinear, coupled, ordinary differential equations of second order." The text provides equations of both systems and shows they are interconnected.
• Nonlinearity: The key is that the equations for the double pendulum contain nonlinear and coupling terms, leading to the chaotic behavior.
• External Driving Force: The document shows how the equation of the primary pendulum is modified to include the external driving force. It's introduced as f1_drive = f1 + A cos(delta*t).
• No Friction: As stated previously, there is no friction in the simulation.

5. Activities and Learning Goals

The simulation includes suggestions for activities designed to enhance learning:

• E1: Demonstrating deterministic chaos by observing that traces coincide after a reset.
• E2: Investigating the effect of increasing animation speed.
• E3: Observing the effect of changing the length ratio (L2/L1).
• E4: Studying the behavior of the system as a simple pendulum (setting L2/L1 ~ 0) and exploring the relationship between the motion and phase space diagram.
• E5 & E6: Attempting to stabilize the pendulum in specific initial positions.
• E7: Examining the effect of changing mass ratios (m2/m1).
• E8 & E9: Exploring the impact of periodic driving force and its frequency.
• E10: Changing parameters while the simulation is running.
• E11: Observing the long-term dependence of the trace from slight variations of the starting condition.
• E12: Suggestion on how to create a physical double pendulum with periodic drive, using a DC motor and sine wave generator.

6. Target Audience

The simulation is suitable for a wide audience, including students learning about:

• Classical mechanics
• Nonlinear dynamics
• Chaos theory
• Differential equations
• Mathematical modeling
• Physics education at Secondary and Higher education levels

7. Credits and Licensing

• Creators: The simulation was created by Dieter Roess and later recreated in JavaScript by lookang.
• Licensing: The content is licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License. There is a specific note regarding commercial use of EasyJavaScriptSimulations Library and they can be contacted at This email address is being protected from spambots. You need JavaScript enabled to view it..

8. Key Quotes

• "The movement of the double pendulum is chaotic and shows no periodicity. Yet it is strictly deterministic: after reset the same path will be resumed..."
• "The decisive origin of chaotic behavior is the existence of more than one nonlinear differential equation."
• "Yet when trying to create the same initial condition by moving the pendulum with the mouse, the path will develop differently, as it depends critically on the exact value of the initial conditions."

9. Conclusion

The Double Pendulum HTML5 JavaScript Simulation Model is a powerful and versatile educational tool. It provides an interactive and visually engaging way to explore complex concepts in physics and mathematics, particularly related to chaotic behavior, nonlinear dynamics, and the sensitivity of initial conditions. The simulation's detailed mathematical model, interactive features, and suggested learning activities make it a valuable resource for educators and students alike.

 

Double Pendulum Simulation Study Guide

Quiz

1. What are the two modes of operation for the double pendulum simulation?

• The simulation allows the double pendulum to start passively from given initial conditions or be driven periodically by an external source. The passive mode allows for observation of motion resulting from gravity and initial conditions, whereas the external drive mode allows for exploration of forced, chaotic behavior.

1. How are the pendulum rods and masses represented in the simulation?

• The pendulum rods are represented as massless, rigid straight lines, while the masses are concentrated at the center of the pendulum bobs. This simplification helps to isolate the core dynamics of the double pendulum system, and ignores real-world effects of rod mass.

1. What does the phase space diagram in the simulation represent?

• The phase space diagram shows the relationship between the angular position (y1) and angular velocity (y1') of the primary pendulum's bob. For a simple pendulum, it would show a closed periodic curve, while for the double pendulum, it can demonstrate chaotic behavior.

1. What happens when the ratio L2/L1 is set to 0 in the simulation?

• When L2/L1 is set to 0, the secondary pendulum's length becomes zero, effectively resulting in a single pendulum system. Both bobs coincide at the primary pendulum, and the system will behave as a standard pendulum.

1. What does the slider m2/m1 control, and what happens when it is set to 0?

• The slider m2/m1 controls the mass ratio between the secondary and primary pendulum bobs. When it is set to 0, the secondary bob effectively has no mass, and this will leave a single primary pendulum bob.

1. Describe the behavior of the double pendulum system regarding determinism and periodicity.

• The double pendulum exhibits chaotic behavior, lacking periodicity, yet its motion is strictly deterministic. This means that given the same initial conditions, the same path will always be reproduced, but slight changes will dramatically alter the long-term path of the pendulum.

1. What effect does friction have on the movement of a real pendulum, and how is it treated in the simulation?

• Friction in a real pendulum would cause the amplitude of its oscillations to diminish over time. This simulation neglects friction to focus on the core dynamics of the double pendulum, allowing users to explore the complex motion more clearly.

1. How does the addition of external torque affect the double pendulum, and what parameter controls its frequency?

• The addition of a periodic external torque, controlled by the amplitude parameter A, can cause the double pendulum to exhibit a variety of chaotic movements. The frequency of the external drive is controlled by the delta slider.

1. According to the document, what is the fundamental reason for the chaotic behavior of the double pendulum?

• The chaotic behavior of the double pendulum stems from the presence of two coupled, nonlinear, ordinary differential equations. It is the coupling and nonlinearity that gives rise to the unpredictable and sensitive nature of its motion.

1. What two parameters does the document mention can be changed while the simulation is running?

• According to the document, the parameters that can be changed while the simulation is running include the drive frequency using the delta slider and the mass ratio of the bobs with the slider m2/m1.

Quiz Answer Key

1. The simulation allows the double pendulum to start passively from given initial conditions or be driven periodically by an external source. The passive mode allows for observation of motion resulting from gravity and initial conditions, whereas the external drive mode allows for exploration of forced, chaotic behavior.
2. The pendulum rods are represented as massless, rigid straight lines, while the masses are concentrated at the center of the pendulum bobs. This simplification helps to isolate the core dynamics of the double pendulum system, and ignores real-world effects of rod mass.
3. The phase space diagram shows the relationship between the angular position (y1) and angular velocity (y1') of the primary pendulum's bob. For a simple pendulum, it would show a closed periodic curve, while for the double pendulum, it can demonstrate chaotic behavior.
4. When L2/L1 is set to 0, the secondary pendulum's length becomes zero, effectively resulting in a single pendulum system. Both bobs coincide at the primary pendulum, and the system will behave as a standard pendulum.
5. The slider m2/m1 controls the mass ratio between the secondary and primary pendulum bobs. When it is set to 0, the secondary bob effectively has no mass, and this will leave a single primary pendulum bob.
6. The double pendulum exhibits chaotic behavior, lacking periodicity, yet its motion is strictly deterministic. This means that given the same initial conditions, the same path will always be reproduced, but slight changes will dramatically alter the long-term path of the pendulum.
7. Friction in a real pendulum would cause the amplitude of its oscillations to diminish over time. This simulation neglects friction to focus on the core dynamics of the double pendulum, allowing users to explore the complex motion more clearly.
8. The addition of a periodic external torque, controlled by the amplitude parameter A, can cause the double pendulum to exhibit a variety of chaotic movements. The frequency of the external drive is controlled by the delta slider.
9. The chaotic behavior of the double pendulum stems from the presence of two coupled, nonlinear, ordinary differential equations. It is the coupling and nonlinearity that gives rise to the unpredictable and sensitive nature of its motion.
10. According to the document, the parameters that can be changed while the simulation is running include the drive frequency using the delta slider and the mass ratio of the bobs with the slider m2/m1.

Essay Questions

1. Explain the concept of deterministic chaos using the double pendulum as an example. Discuss how small changes in initial conditions can lead to significantly different outcomes, and relate this to the simulation's behavior when manually repositioning the primary bob.
2. Compare and contrast the mathematical models for a simple pendulum and a double pendulum, focusing on the number of differential equations, the presence of nonlinearity, and how these factors contribute to the different behaviors observed in the simulation.
3. Describe the effect of the external drive on the behavior of the double pendulum. Include a discussion of the parameters involved in the external drive, including amplitude and frequency, and how they influence the observed motion.
4. Using specific examples from the simulation, describe and analyze the concept of phase space. Include a discussion of how phase space diagrams can reveal the qualitative differences between the behavior of simple and double pendulums.
5. The document mentions that the double pendulum system is simplified by neglecting friction. Discuss how the inclusion of friction might alter the behavior of the double pendulum and consider the challenges of modeling such effects in a simulation.

Glossary of Key Terms

• Deterministic Chaos: A system that behaves unpredictably despite having no random elements; its behavior is determined by its initial conditions but is highly sensitive to them.
• Double Pendulum: A pendulum with another pendulum attached to its end, creating a more complex system of motion.
• Phase Space: A space in which all possible states of a system are represented, each point corresponding to a unique state (e.g., position and velocity).
• Nonlinear Differential Equations: Equations that do not follow the principle of superposition; they can be difficult to solve and can lead to complex behaviors such as chaos.
• Coupled Differential Equations: A set of differential equations that depend on each other; the variables in one equation also appear in another.
• External Torque: A twisting force applied to an object from an external source.
• Periodicity: Describes motion that repeats itself at a regular time interval.
• Initial Conditions: The set of parameter values at the very start of a simulation, which significantly impact the system's behavior over time.
• Angular Velocity: The rate at which an object rotates or revolves about an axis, measured in radians per second.
• Amplitude: The maximum extent of a vibration or oscillation, measured from the position of equilibrium.
• Trigonometric Function: Functions such as sine and cosine which relate the angles of a triangle to the lengths of its sides.
• Friction: A force that resists motion between two surfaces that are in contact. In this simulation it is neglected.

App

https://play.google.com/store/apps/details?id=com.ionicframework.dpendulumapp262907&rdid=com.ionicframework.dpendulumapp262907 

 

Sample Learning Goals

[text]

Description

       Double pendulum with external drive

 

The simulation demonstrates a frictionless, mathematical double pendulum that either:

 

1. starts passively from given initial conditions,
2. or is driven periodically by an external source.

The massless, rigid pendulum rods are shown as straight lines. Masses are concentrated in the center of the pendulum bobs.

When the simulation file is opened, the pendulum is in horizontal position. The length of the secondary pendulum 2 (yellow) is one half that of the primary one 1 (blue): \( \frac{L_{2}}{L_{1}} = 0.5 \)  The mass of both is equal:  \( \frac{m_{2}}{m_{1}} = 1 \). Their initial speed is zero. There is no external drive: A=0.

Start initiates the calculation of movement under the influence of gravity. The path of the secondary yellow pendulum bob (2) is shown in red for a limited time period. Stop freezes the movement until a new start. Clear deletes traces, Reset reestablishes the default initial conditions.

The primary blue bob can be drawn with the mouse to create different initial angular positions, with stretched pendulum and zero initial velocity.

In the right window, a phase space diagram \(y_{1}^{'} versus y_{1} \) is shown for the blue bob of the primary pendulum (for a simple pendulum this would be a closed periodical curve, and would be a circle for small arcs of oscillation). The length of the traces is limited to 2500 points of calculation.

The speed of the animation can be varied with slider Speed.

The length L₁ of the primary pendulum is kept constant. Slider  \( \frac{L_{2}}{L_{1}}  \) changes the length of the secondary pendulum. The window size is adjusted to twice the maximum size of the double pendulum. \( \frac{L_{2}}{L_{1}} =0 \) results in a single pendulum, with both bob masses coinciding.

Slider  \( \frac{m_{2}}{m_{1}}  \) determines the mass relation of the pendulum bobs.  \( \frac{m_{2}}{m_{1}} = 0 \) results in a single primary pendulum. In the simulation, the orientation of the fictive massless secondary pendulum stays constant, while it will briskly oscillate for any finite mass.

The movement of the double pendulum is chaotic and shows no periodicity. Yet it is strictly deterministic: after resetting the same path will be resumed (start without clearing the trace).

In a real pendulum, friction would diminish the amplitude of oscillations. Friction is neglected in this model.

With A > 0 a periodic external torque is acting upon the primary pendulum, with amplitude and direction as shown by the black arrow. Slider delta changes the drive frequency. With delta =1 the drive frequency is equal to that of the single primary pendulum (m2 = 0) at small arcs. For all initial conditions, the driven double pendulum shows a rich variety of deterministic chaotic movements.

The 3D-Phase space button opens another window with two rotating 3D frames for the three-dimensional phase spaces of both pendulums.

y₁´´(y₁ , y´₁) and y₂´´(y₂ , y₂´)

The scalings of the 3D frames are adjusting automatically to the amplitudes.     

Model

       Formulas

Without external drive a pendulum has constant energy, as determined by the initial conditions. At rest (ω_A = 0) and at height y_A above the lowest point y₀ = 0 the initial energy is 100% potential energy: 

 

E_{pot_max} = mg ( y_A - y₀ )

 

In movement potential and kinetic energy are exchanged continuously with constant sum. For a single pendulum, all energy is kinetic at the lowest point y = 0.

Ekin_max = m v²/2 = m (rω)²/2 = mg (y_A - y₀) = E_{pot_max}

For the double pendulum, the initial potential energy is distributed between the kinetic and potential energy of both bobs, while the secondary one oscillates around the primary one.

 

The simple pendulum is described by one nonlinear ordinary second order differential equation with two initial conditions (angle and angular momentum):

It is nonlinear because of the nonlinear trigonometric function that connects oscillation angle with angular velocity.

The second order differential equation is equivalent to the description by two first-order differential equations, the first of which (1) is linear, the second one (2) nonlinear.

The nonlinearity lies in the simple relation between angle and angular velocity, described by f.

With α angle of oscillation, ω angular velocity, g gravity acceleration, L pendulum length:

\(\frac{d\alpha^{2}}{dt^{2}} = -\frac{g}{L} sin \alpha \)

which implies

(1)  \( \frac{d\alpha}{dt} = \omega \)

(2) \( \frac{d\omega}{dt} = f(\alpha) \)

 \( f (\alpha) = -\frac{g}{L} sin \alpha \)

The double pendulum is described by two nonlinear, coupled, ordinary differential equations of second order with four initial conditions (angle and angular momentum of both pendulums). This is equivalent to four equations of first order, all of which are nonlinear because of trigonometric functions and the quadratic coupling terms shown in red:

 

(1)  \( \frac{d\alpha_{1}}{dt} = -\omega_{1} \)

(2)  \( \frac{d\omega_{1}}{dt} = f_{1}(\alpha_{1}, \alpha_{2}, \omega_{1}, \omega_{2}) \)

 where \( f_{1}(\alpha_{1}, \alpha_{2}, \omega_{1}, \omega_{2}) = -\frac{-\frac{g}{L_{1}}((m_{1}+m_{2}) sin(\alpha_{1})- m_{2} sin(\alpha_{2}) cos(\alpha_{1}- \alpha_{2}))
-m_{2} sin(\alpha_{1}-\alpha_{2})(\frac{L{2}}{L_{1}}\omega_{2}^{2}+\omega_{1}^{2} cos(\alpha_{1}-\alpha_{2})))}{m_{1}+m_{2} sin^{2}(\alpha_{1}-\alpha_{2}) )}\)

 

(3)  \( \frac{d\alpha_{2}}{dt} = \omega_{2}\)

(4) \( \frac{d\omega_{2}}{dt} = f_{2}(\alpha_{1}, \alpha_{2}, \omega_{1}, \omega_{2})\)

where \( f_{2}(\alpha_{1}, \alpha_{2}, \omega_{1}, \omega_{2}) = -\frac{g}{L_{2}}sin (\alpha_{2})+\frac{L_{1}}{L_{2}} \omega^{2} sin(\alpha_{1}-\alpha_{2}) - \frac{ \frac{L_{1}}{L_{2}} cos(\alpha_{1}-\alpha_{2} ) (-\frac{g}{L}(m_{1}+m_{2}) sin(\alpha_{1})- m_{2}sin\alpha_{2} cos (\alpha_{1}-\alpha_{2}) -m_{2} sin(\alpha_{1}-\alpha_{2})( \frac{L_{2}}{L_{1}}\omega_{2}^{2}+\omega_{1}^{2} cos(\alpha_{1}-\alpha_{2})   ) }{(m_{1}+m_{2} sin(\alpha_{1}-\alpha_{2}) sin(\alpha_{1}-\alpha_{2})) }   \)

 

The equations for the second derivatives \(f_{1}\) and \(f_{2}\) are far more complicated than for the single pendulum. They depend on the relations of the pendulum lengths and masses, and on the angles and angular velocities of both pendulums. The fact is that there are two coupled nonlinear differential equations is the deeper cause of the deterministic chaotic behavior. When restarting the movement with exactly the same initial conditions (as the computer does with Reset), the same trace is reproduced. Yet when trying to create the same initial condition by moving the pendulum with the mouse, the path will develop differently, as it depends critically on the exact value of the initial conditions. This is best observed when superimposing traces of two oscillations (choose pause but not clear, draw bob and start).

The chaotic behavior remains when some of the nonlinear terms are removed or changed, as long as the nonlinear character of the equations is preserved (the equations then no longer model the double pendulum). The decisive origin of chaotic behavior is the existence of more than one nonlinear differential equation.

With m₂ = 0 (0r L₂ = 0) the first equation is reduced to that of the single pendulum, the second equation becomes identical to zero. The simulation then shows the periodic, generally nonlinear, oscillation of a single pendulum.

With drive the differential equation for the primary pendulum is

f_{1_drive} = f₁ + A cos(delta*t)

where A is the amplitude of the drive, delta its frequency in relation to that of the free pendulum at small amplitudes.     

Activities

E1: When you open the simulation, the double pendulum will start its oscillation. Stop it after some time with Reset, which leaves the old trace and resets the original initial condition. Start again, and observe that traces coincide (deterministic chaos).

Reset after a few oscillations, draw the blue bob very slightly and try to put it into the vertical position again. Can you do that exactly enough to reproduce the old trace for more than a few oscillations?

 

E2: Increase Speed, which increases the time between calculated points. This will increase the maximum trace length, as the number of calculated points shown is constant. Use Clear when traces become too complex.

 

E3: Choose Reset and change the length of the secondary pendulum with slider L₂/L₁. What happens?

 

E4: Set L₂/L₁ ~ 0 after Reset. Now you simulate a simple pendulum. Investigate when its oscillation is approximately linear. Is it always periodical? What does the phase space diagram tell you?

 

E5: As E4; try to pull the bob as exactly as possible into its upper position. Can you stabilize it there? Study repeated oscillations from the highest starting position. Can you reproduce them?

 

E6: As E5; position the bob near to its stable point (y ~0). Compare the phase space diagram to that of E5.

 

E7: Change the mass relation of the bobs via Slider m₂/m₁:  Do that for different length relations L₂/L₂.

 

E8:  Add periodic drive (A > 0).

 

E9:  Change the drive frequency with slider delta

 

E10:  You can change parameters while the simulation is running.

 

E11:  After Reset and Clear let the pendulum perform a few oscillations;  then choose Reset without Clear, which leaves the old trace. Pull the blue bob very slightly from its position and restart. Observe the long-term dependence of the trace from slight variations of the starting condition (a₁).

 

E12: Consider how you can realize a double pendulum with periodic drive! One suggestion: 

clamp the end of the primary pendulum to the axis of a small DC motor that is driven by the amplified signal of a sine wave generator.     

Video

[text]

 Version:

1. this JavaScript version is recreated by lookangon 15 January 2017 as a response to the request by Prof padyala radhakrishnamurty. http://weelookang.blogspot.sg/2017/01/double-pendulum-html5-javascript.html
2. created by Dieter Roess in April 2009, This simulation is part of “Learning and Teaching Mathematics using Simulations – Plus 2000 Examples from Physics” ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG     

Other Resources

https://www.geogebra.org/m/DpmwJPzg Angular Momentum Collision by ukukuku

Frequently Asked Questions about the Double Pendulum Simulation

1. What is a double pendulum, and how does this simulation model it? A double pendulum consists of two pendulums connected end-to-end. The simulation models this as two rigid, massless rods with concentrated masses (bobs) at the ends, connected in a way that allows for free rotation in a 2-dimensional plane. The simulation allows for both passive (starting from initial conditions) and driven (with external periodic force) motion. It neglects friction for simplicity. The lengths and masses of each pendulum as well as the initial conditions can be modified. The simulation aims to demonstrate the complex, often chaotic, motion of such a system.
2. What are the key parameters that can be adjusted in the simulation, and what effects do they have? The simulation allows users to modify several parameters:

• Length ratio (L2/L1): Adjusts the length of the second pendulum relative to the first. Changing this affects the complexity and range of motion. Setting L2/L1 to 0 effectively creates a single pendulum.
• Mass ratio (m2/m1): Alters the mass of the second bob relative to the first. Setting m2/m1 to 0 also creates a single pendulum.
• Initial Conditions: The primary bob can be dragged with the mouse to create different initial angular positions, with stretched pendulum and zero initial velocity.
• External Drive (A): When A > 0, the primary pendulum is subjected to a periodic external torque.
• Drive Frequency (delta): Changes the frequency of the external torque, affecting how the driven double pendulum behaves, with delta = 1 being equal to the free pendulum's frequency.
• Speed: Adjusts the speed of the animation.

1. What is meant by 'deterministic chaos' in the context of the double pendulum? Deterministic chaos in the double pendulum refers to the fact that while the system's behavior is governed by precise mathematical equations (deterministic), the motion is often unpredictable in the long term. Small changes in initial conditions lead to dramatically different paths over time. The simulation demonstrates this; while identical initial conditions always produce identical results, slight changes result in very different paths. This means the system isn't random but highly sensitive to starting conditions.
2. What is a 'phase space diagram,' and what does it show in this simulation? The simulation displays a phase space diagram (y1' vs y1), which plots the angular velocity of the primary bob against its angular position. For a simple pendulum, this creates a closed, periodical curve (a circle for small arcs). The phase space for a double pendulum is far more complex reflecting the more complicated motion. The 3D-phase space plots for both pendulums are also available in another window.
3. What do the provided equations represent and what do the simplified equations for the single pendulum show us? The equations show the mathematical relationships that govern the double pendulum's motion, expressed as coupled, non-linear second-order differential equations. These equations are far more complex than the equations for a simple pendulum (which is a single non-linear second order differential equation), highlighting the root cause of the double pendulum's complex and chaotic movement. The single pendulum's simplified equations illustrate how its motion is described by angular velocity and position, with the non-linearity arising from the trigonometric function linking them.
4. How does the external drive affect the double pendulum's movement? The external drive introduces a periodic torque on the primary pendulum, governed by the parameters A (amplitude) and delta (frequency). This can lead to even more complex and chaotic behavior than with the passive motion. Depending on the chosen drive frequency and amplitude, the system can show a variety of complex, non-periodic movements.
5. How can I use this simulation for learning purposes?
6. The simulation offers several learning opportunities, including:

• Observing Chaos: By manipulating the initial conditions, users can see firsthand the unpredictability of chaotic systems.
• Parameter Effects: Observing how changes in mass, length, and drive parameters affect the system's movement.
• Deterministic nature of systems: Although the motion looks chaotic, the simulation makes clear that given the same initial conditions, the same trajectory is followed every time.
• Phase Space: Learning to interpret phase space diagrams to understand system dynamics.
• Real-World Applications: Understanding how models like this relate to actual physical systems, despite simplifications. The provided activities (E1-E12) provide guidance for specific experiments to test how the double pendulum works.

1. What are some limitations of this simulation? The model is simplified in that it excludes friction or air resistance, which would eventually dampen oscillations in real-world conditions. The massless rods and concentrated bob masses are also simplifications, as actual pendulums would have mass distributed. Additionally, while the model is deterministic, the reliance on computational calculations means it is an approximation of the real-world behavior of a double pendulum rather than a literal representation of the equations that are being modeled.