Double pendulum

Schematic drawing of a double pendulum.

Trajectory of an idealized double pendulum

The double pendulum is a popular model for demonstrating chaotic processes. It is also one of the simplest non-linear dynamic systems , which shows chaotic behavior. Another pendulum of length with mass is attached to the mass of a pendulum with the length . The derivation of the equation of motion for calculating the movement of the double pendulum can be simplified if one assumes rigid, massless pendulum rods and freedom from friction. ${\displaystyle m_{1}}$ ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$ ${\displaystyle m_{2}}$

A characteristic of a chaotic system is that there are initial conditions , so that another experiment with almost identical initial conditions , which differ only by an infinitesimal perturbation , shows a different behavior after a short time. This sensitive dependency can be characterized by calculating Lyapunov's exponents of the trajectories . ${\displaystyle x_{i}}$ ${\displaystyle x_{i}+\Delta x}$ ${\displaystyle \Delta x}$

Derivation of the equations of motion

If and denote the lengths of the (massless) connecting rods, and the pendulum masses, the deflection from the perpendicular and the acceleration due to gravity, then one finds for the positions of and : ${\displaystyle l_{1}}$ ${\displaystyle l_{2}}$ ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$ ${\displaystyle \theta _{1},\theta _{2}}$ ${\displaystyle g}$ ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$

${\displaystyle {\begin{aligned}x_{1}&=l_{1}\sin(\theta _{1})\\y_{1}&=-l_{1}\cos(\theta _{1})\end{aligned}}}$

and

${\displaystyle {\begin{aligned}x_{2}&=l_{1}\sin(\theta _{1})+l_{2}\sin(\theta _{2})\\y_{2}&=-l_{1}\cos(\theta _{1})-l_{2}\cos(\theta _{2}).\end{aligned}}}$

This allows the speeds of the masses, which are necessary for the next step, to be determined, where the time derivative of is: ${\displaystyle {\dot {\theta _{1}}}}$ ${\displaystyle \theta _{1}}$

${\displaystyle {\begin{aligned}u_{1}&={\frac {\partial x_{1}}{\partial t}}={\dot {\theta }}_{1}l_{1}\cos(\theta _{1})\\v_{1}&={\frac {\partial y_{1}}{\partial t}}={\dot {\theta }}_{1}l_{1}\sin(\theta _{1})\end{aligned}}}$

and

${\displaystyle {\begin{aligned}u_{2}&={\frac {\partial x_{2}}{\partial t}}={\dot {\theta }}_{2}l_{2}\cos(\theta _{2})+u_{1}\\v_{2}&={\frac {\partial y_{2}}{\partial t}}={\dot {\theta }}_{2}l_{2}\sin(\theta _{2})+v_{1}.\end{aligned}}}$

Using the Lagrange formalism where the kinetic energy of the two pendulum masses and their potential energy in the constant gravitational field is, with ${\textstyle L=T-V}$ ${\displaystyle T}$ ${\displaystyle V}$

${\displaystyle {\begin{aligned}T_{1}&={\frac {1}{2}}m_{1}(u_{1}^{2}+v_{1}^{2})\\T_{2}&={\frac {1}{2}}m_{2}(u_{2}^{2}+v_{2}^{2})\\V_{1}&=m_{1}gy_{1}\\V_{2}&=m_{2}gy_{2}\end{aligned}}}$

you get

${\displaystyle {\begin{aligned}T&=T_{1}+T_{2}={\frac {1}{2}}m_{1}{\dot {\theta }}_{1}^{2}l_{1}^{2}+{\frac {1}{2}}\left(m_{2}{\dot {\theta }}_{2}^{2}l_{2}^{2}+m_{2}{\dot {\theta }}_{1}^{2}l_{1}^{2}+2m_{2}{\dot {\theta }}_{1}l_{1}{\dot {\theta }}_{2}l_{2}\cos(\theta _{1}-\theta _{2})\right)\\V&=V_{1}+V_{2}=-m_{1}gl_{1}\cos(\theta _{1})-m_{2}gl_{1}\cos(\theta _{1})-m_{2}gl_{2}\cos(\theta _{2}).\end{aligned}}}$

This then results for the Lagrange function

${\displaystyle {L={\frac {1}{2}}(m_{1}+m_{2}){\dot {\theta }}_{1}^{2}l_{1}^{2}+{\frac {1}{2}}m_{2}{\dot {\theta }}_{2}^{2}l_{2}^{2}+m_{2}{\dot {\theta }}_{1}l_{1}{\dot {\theta }}_{2}l_{2}\cos(\theta _{1}-\theta _{2})+(m_{1}+m_{2})gl_{1}\cos(\theta _{1})+m_{2}gl_{2}\cos(\theta _{2})}.}$

Using the Euler-Lagrange equation

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\theta _{i}}}}}-{\frac {\partial L}{\partial \theta _{i}}}=0}$

is obtained after a few transformations

${\displaystyle {\begin{aligned}{\ddot {\theta }}_{1}&=-{\frac {m_{2}}{m_{1}+m_{2}}}{\frac {l_{2}}{l_{1}}}({\ddot {\theta }}_{2}\cos(\theta _{1}-\theta _{2})+{\dot {\theta }}_{2}^{2}\sin(\theta _{1}-\theta _{2}))-{\frac {g}{l_{1}}}\sin(\theta _{1})\\{\ddot {\theta }}_{2}&=-{\frac {l_{1}}{l_{2}}}({\ddot {\theta }}_{1}\cos(\theta _{1}-\theta _{2})-{\dot {\theta }}_{1}^{2}\sin(\theta _{1}-\theta _{2}))-{\frac {g}{l_{2}}}\sin(\theta _{2})\end{aligned}}}$

the angular accelerations for and , which describe the evolution of the pendulum. ${\displaystyle \theta _{1}}$ ${\displaystyle \theta _{2}}$

In the equations of motion, angle functions of the state variables and also derivatives appear. So it is a non-linear system. In the special case of small deflections as initial conditions , however, the equations of motion can be simplified using the small-angle approximation . Then, for example, other special cases such as or with analytical approaches can be considered that have an approximately harmonic solution; this can also be determined analytically. ${\displaystyle (\sin ,\cos )}$ ${\displaystyle \theta _{1}(t=0),\theta _{2}(t=0)\approx 0}$ ${\displaystyle m_{1}\ll m_{2}}$ ${\displaystyle m_{1}\gg m_{2}}$

Solution of the equations of motion

The equations of motion for the generalized coordinates and represent a non-linear system of two coupled differential equations , which cannot be solved analytically. With four known initial values ( ) it can be solved with numerical methods . The initial deflections (e.g. 30 ° and 30 ° ) and the initial speeds (e.g. and ) are entered and the evolution of the pendulum is then calculated. ${\textstyle {\theta _{1}}}$ ${\textstyle {\theta _{2}}}$ ${\displaystyle \theta _{1},\theta _{2},{\dot {\theta }}_{1},{\dot {\theta }}_{2}}$ ${\textstyle 0\mathrm {\frac {rad}{s}} }$ ${\textstyle 0\mathrm {\frac {rad}{s}} }$

Using trigonometry , the angles and can be converted into the Cartesian coordinates of the mass points. ${\displaystyle \theta _{1}}$ ${\displaystyle \theta _{2}}$ ${\displaystyle (x_{1},y_{1},x_{2},y_{2})}$

Applications

• The Blide uses the energetic exchange between the masses of the pendulums to accelerate a small weight on the outer pendulum from the potential energy of a large weight on the inner pendulum.
• A church bell with a clapper forms a double pendulum, but with an additional restriction for the angle . ${\displaystyle \theta _{2}}$

Evaluation of the chaotic behavior

There are a number of ways of looking at the chaotic behavior of the double pendulum. Often a statement about chaotic behavior can be made by means of the simplest calculations. Examples are the maximum Lyapunov exponent (MLE) or bifurcation diagrams .

Maximum Lyapunov exponent

The MLE is the so-called maximum Lyapunov exponent ( maximum Lyapunov exponent ) and describes the “strength” of the chaotic behavior. It is part of the Lyapunov spectrum which contains all Lyapunov exponents (one per degree of freedom). It is assumed that the system has a disturbance in the direction of the MLE and since it shows the greatest growth, it is to be expected that the MLE will dominate the evolution of the system after a certain time. A positive MLE usually indicates a chaotic system. It is calculated with:

${\displaystyle \lambda _{\mathrm {max} }(t)={\frac {1}{t}}\ln \left({\frac {\|\delta (t)\|}{\|\delta (t=0)\|}}\right)}$

In two experiments with an initial separation of in the initial conditions or even less, this difference increases exponentially and causes the trajectories to diverge. The separation (the natural logarithm of the above equation) can then be plotted against time in a semi-logarithmic diagram. Then the slope is determined by means of linear regression and this gives the approximated MLE. ${\textstyle \delta (t=0)\backsim {\mathcal {O}}(10^{-8})}$

Bifurcation diagram

Representation of the bifurcation of a double pendulum with the same lengths, the same masses, for at . The red line shows the bifurcation of the double pendulum from harmonic oscillations to chaotic oscillations. ${\textstyle g=9{,}81\mathrm {\frac {m}{s^{2}}} }$ ${\displaystyle \theta _{1}=\theta _{2}}$ ${\displaystyle {\dot {\theta _{2}}}=0}$

Bifurcation diagrams are a way of compressing complex information about the phase space of a dynamic system into a two-dimensional, visualizable plot. Usually, the qualitative change in the behavior of a system is examined by means of the variation of a suitable parameter. For example, the ratio of the masses, the ratio of the lengths, the acceleration due to gravity or the initial conditions can be used for the double pendulum. By continuously changing the selected bifurcation parameter, the system is checked for stability (periodic, quasi-periodic solutions) or for chaos .

If one chooses the initial angles as the bifurcation parameter, the qualitatively variable behavior of the double pendulum can be illustrated very well. To do this, the two angles are simultaneously increased piece by piece and the double pendulum is integrated (calculated) again for each increment. This data can then be used to illustrate how the system vibrates. So you have a four-dimensional phase space made up of. Conveniently, the angular velocities oscillate around zero, although with indefinite amplitude. Therefore, it is to be expected that both cross zero again and again. For a harmonically oscillating system (periodic solution) the zero crossings are at fixed points, since the system always ends its upward and downward movement and swings back at certain points ( ). This is comparable to a normal rigid pendulum . Conversely, it is therefore to be expected that the chaotically oscillating system shows an angular velocity of zero at all possible points ( ). If you then consider a "slice" from phase space separately, for example angular velocity , you can represent the bifurcation of the behavior two-dimensionally by plotting the angular velocity against the variable initial condition (see right). ${\textstyle \theta _{1}=\theta _{2}}$ ${\textstyle \theta _{1},\theta _{2},{\dot {\theta _{1}}},{\dot {\theta _{2}}}}$ ${\textstyle {\dot {\theta _{1}}},{\dot {\theta _{2}}}}$ ${\displaystyle {\dot {\theta _{1}}},{\dot {\theta _{2}}}}$ ${\textstyle \theta _{1},\theta _{2}}$ ${\textstyle \theta _{1},\theta _{2}}$ ${\textstyle {\dot {\theta _{2}}}=0}$ ${\textstyle {\dot {\theta _{1}}}}$

See also

• Magnetic pendulum
• Multi-pendulum
• KAM theorem
• Church bell

Web links

Commons : Double pendulums  - collection of images, videos and audio files

• Java double pendulum (English)
• Double pendulum simulation in Java and Python (German)
• Double pendulum - limits of the simulation shows that the movement can only be simulated for a short period of time
• Derivation of the differential equations to describe the double pendulum

Individual evidence

1. ↑ Wolf, A., Swift, JB, Swinney, HL, & Vastano, JA (1985). Determining lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, Vol. 16 No. 3: 285-317