Coupled Oscillator: Double Pendulum

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} = \widehat{A} \, \underline{x}(t) + \widehat{B} \, \underline{u}(t) = \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} + \widehat{0} }

Let's plug some numbers. Knowing 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=32} , and assuming 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 m_1=3} , 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=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=l_2=16} , the constants defined previously become,

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=1024 \quad B=256 \quad C=2048 \quad D=256 \quad E=512}

Hence, the state space matrix is,

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{8}{3} & 0 & \dfrac{2}{3} & 0 \\ 0 & 0 & 0 & 1 \\ \dfrac{8}{3} & 0 & -\dfrac{8}{3} & 0 \\ \end{bmatrix} \begin{Bmatrix} \theta_1 \\ \theta_1^{'} \\ \theta_2 \\ \theta_2^{'} \end{Bmatrix} }

Eigenvalues & Eigenvectors

The eigenvalues and eigenvectors are easily obtained with the help of a TI-89 calculator. First, we consider the 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 \widehat{A}} 's identity matrix,

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 [\lambda I-A] = \begin{bmatrix} \lambda & 1 & 0 & 0 \\ -\dfrac{8}{3} & \lambda & \dfrac{2}{3} & 0 \\ 0 & 0 & \lambda & 1 \\ \dfrac{8}{3} & 0 & -\dfrac{8}{3} & \lambda \\ \end{bmatrix} }

Once we define the 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 \widehat{A}} matrix, the eigenvalues are determined by using the eigVl() function,

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 \lambda_1= 2 \mathbf{i}}

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 \lambda_2= -2 \mathbf{i}}

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 \lambda_3= 1.1547 \mathbf{i}}

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 \lambda_4= -1.1547 \mathbf{i}}

On the other hand, we use the eigVc() function to find the eigenvectors,

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 k_1 = \begin{bmatrix} -0.2 \mathbf{i} \\ 0.4 \\ 0.4 \mathbf{i} \\ -0.8 \\ \end{bmatrix} \quad k_2 = \begin{bmatrix} 0.2 \mathbf{i} \\ 0.4 \\ -0.4 \mathbf{i} \\ -0.8 \\ \end{bmatrix} \quad k_3 = \begin{bmatrix} -0.29277 \mathbf{i} \\ 0.33806 \\ 0.58554 \mathbf{i} \\ -0.67621 \\ \end{bmatrix} \quad k_4 = \begin{bmatrix} 0.29277 \mathbf{i} \\ 0.33806 \\ 0.58554 \mathbf{i} \\ 0.67621 \\ \end{bmatrix} }

Standard Equation

Now, we plug the eigenvalues and eigenvectors to produce the standar equation,

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 x = c_1 k_1 e^{\lambda_1 t} + c_2 k_2 e^{\lambda_2 t} + c_3 k_3 e^{\lambda_3 t} + c_4 k_4 e^{\lambda_4 t}}

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 x = c_1 \begin{bmatrix} -0.2 \mathbf{i} \\ 0.4 \\ 0.4 \mathbf{i} \\ -0.8 \\ \end{bmatrix} e^{2 \mathbf{i}} + c_2 \begin{bmatrix} 0.2 \mathbf{i} \\ 0.4 \\ -0.4 \mathbf{i} \\ -0.8 \\ \end{bmatrix} e^{-2 \mathbf{i}} + c_3 \begin{bmatrix} -0.29277 \mathbf{i} \\ 0.33806 \\ 0.58554 \mathbf{i} \\ -0.67621 \\ \end{bmatrix} e^{1.1547 \mathbf{i}} + c_4 \begin{bmatrix} 0.29277 \mathbf{i} \\ 0.33806 \\ 0.58554 \mathbf{i} \\ 0.67621 \\ \end{bmatrix} e^{-1.1547 \mathbf{i}} }

Matrix Exponential

The matrix exponential is,

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 \dot{\bar{z}}=\hat{A}\bar{z}=TAT^{-1}\bar{z}}

where

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 \hat{A} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -\dfrac{8}{3} & 0 & \dfrac{2}{3} & 0 \\ 0 & 0 & 0 & 1 \\ \dfrac{8}{3} & 0 & -\dfrac{8}{3} & 0 \\ \end{bmatrix} } ,

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 T^{-1} = [k_1|k_2|k_3|k_4] = \begin{bmatrix} -0.2 \mathbf{i} & 0.2 \mathbf{i} & -0.29277 \mathbf{i} & 0.29277 \mathbf{i} \\ 0.4 & 0.4 & 0.33806 & 0.33806 \\ 0.4 \mathbf{i} & -0.4 \mathbf{i} & 0.58554 \mathbf{i} & 0.58554 \mathbf{i} \\ -0.8 & -0.8 & -0.67621 & 0.67621 \\ \end{bmatrix} } ,

so

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 T = (T^{-1})^{-1} = [k_1|k_2|k_3|k_4]^{-1} = \begin{bmatrix} 1.25 \mathbf{i} & 0.625 & -0.625 \mathbf{i} & -0.3125 \\ -1.25 \mathbf{i} & 0.625 & 0.625 \mathbf{i} & -0.3125 \\ 0.853913 \mathbf{i} & 0.73951 & 0.426956 \mathbf{i} & 0.369755 \\ -0.853913 \mathbf{i} & 0.73951 & -0.426956 \mathbf{i} & 0.369755 \\ \end{bmatrix} }

Again, we can resort to the TI-89 calculator. As it is mentioned above, the matrix exponential is obtained by typing eigVc(a)^-1*a*eigVc(a), where a is the 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 \widehat{A}} matrix. 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 \dot{\bar{z}} = \hat{A}\bar{z} = TAT^{-1}\bar{z} = \begin{bmatrix} 2 \mathbf{i} & 0 & 0 & 0 \\ 0 & -2 \mathbf{i} & 0 & 0 \\ 0 & 0 & 1.1547 \mathbf{i} & 0 \\ 0 & 0 & 0 & -1.1547 \mathbf{i} \\ \end{bmatrix} \bar{z} }