Coupled Oscillator: Double Pendulum

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By Jimmy Apablaza By Jimmy Apablaza

This problem is described in Page 321-322, Section 7.6 of the A first Course in Differential Equations textbook, 8ED (ISBN 0-534-41878-3).

Figure 1. Coupled Pendulum.‎

Problem Statement

Consider the double-pendulum system consisting of a pendulum attached to another pendulum shown in Figure 1.

Assumptions:

  • the system oscillates vertically under the influence of gravity.
  • the mass of both rod are neligible
  • no dumpung forces act on the system
  • positive direction to the right.

The system of differential equations describing the motion is nonlinear

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (m_1+m_2)l_1^2\theta_1^{\prime\prime} + m_2l_1l_2\theta_2^{\prime\prime}cos(\theta_1-\theta_2) + m_2l_1l_2(\theta_2^{\prime})^2sin(\theta_1-\theta_2) + (m_1+m_2)l_1gsin\theta_1 = 0}
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2l_1^2\theta_2^{\prime\prime} + m_2l_1l_2\theta_1^{\prime\prime}cos(\theta_1-\theta_2) - m_2l_1l_2(\theta_1^{\prime})^2sin(\theta_1-\theta_2) + m_2l_2gsin\theta_2 = 0}


In order to linearize these equations, we assume that the displacements Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_2} are small enough so that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cos(\theta_1-\theta_2)\approx1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sin(\theta_1-\theta_2)\approx0} . Thus,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (m_1+m_2)l_1^2\theta_1^{\prime\prime} + m_2l_1l_2\theta_2^{\prime\prime} + (m_1+m_2)l_1g\theta_1 = 0}
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2l_1^2\theta_2^{\prime\prime} + m_2l_1l_2\theta_1^{\prime\prime} + m_2l_2g\theta_2 = 0}

Solution

Since our concern is about the motion functions, we will assign the masses Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} , the rod lenghts Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_1} , and gravitational force Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} constants to different variables as follows,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=(m_1+m_2)l_1^2 \quad B=m_2l_1l_2 \quad C=(m_1+m_2)l_1g \quad D=m_2l_1^2 \quad E=m_2l_2g}

Hence,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\theta_1^{''} + B\theta_2^{''} + C\theta_1 = 0}
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\theta_2^{''} + B\theta_1^{''} + E\theta_2 = 0}

Solving for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle theta_1^{''}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle theta_2^{''}} we obtain,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_1^{''} = - \left ( \dfrac{B}{A} \right ) \theta_2^{''} - \left ( \dfrac{C}{A} \right ) \theta_1}
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_2^{''} = - \left ( \dfrac{B}{D} \right ) \theta_1^{''} - \left ( \dfrac{E}{D} \right ) \theta_2}

Therefore,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_1^{''} = - \left ( \dfrac{CD}{AD+B^2} \right ) \theta_1 - \left ( \dfrac{BE}{AD+B^2} \right ) \theta_2 }
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_2^{''} = \left ( \dfrac{BC}{AD+B^2} \right ) \theta_1 - \left ( \dfrac{AE}{AD+B^2} \right ) \theta_2}

State Space

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \theta_1^{'} \\ \theta_1^{''} \\ \theta_2^{'} \\ \theta_2^{''} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ \dfrac{-CD}{AD+B^2} & 0 & \dfrac{-BE}{AD+B^2} & 0 \\ 0 & 0 & 0 & 1 \\ \dfrac{BC}{AD+B^2} & 0 & \dfrac{-AE}{AD+B^2} & 0 \\ \end{bmatrix} \begin{Bmatrix} \theta_1 \\ \theta_1^{'} \\ \theta_2 \\ \theta_2^{'} \end{Bmatrix} }
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