N-Mass Pendulum
The pendulum system can be represented by a second order system of differential equations. To find the equations that guides this motion, we first need the Lagrangian.
\[\mathcal{L}=T-U\]The second order component (angular acceleration) of the nth mass is guaranteed to be linear. It is given by:
\[\frac{\mathrm{d}}{\mathrm{d}t}(\frac{\partial \mathcal{L}}{\partial \dot{\theta_n}}) - \frac{\partial \mathcal{L}}{\partial \theta_n} = 0\]The x and y coordinates of the first mass (closest to the base) is given by:
\[x = L_{1} \sin{\left(\theta_{1}{\left(t \right)} \right)}\] \[y = - L_{1} \cos{\left(\theta_{1}{\left(t \right)} \right)}\]In general the coordinates of the ith mass are...
\[x_i = x_{i-1} + L_{1} \sin{\left(\theta_{i}{\left(t \right)} \right)}\] \[y_i = y_{i-1} - L_{1} \cos{\left(\theta_{i}{\left(t \right)} \right)}\]The kinetic and potential energy can be obtained by:
\[T = \sum_{i=1}^n \frac{1}{2} m_i (x_i^2 + y_i^2)\] \[U = \sum_{i=1}^n m_i g y_i\]For example, the 3 mass system:
\[\begin{aligned} \mathcal{L} ={}& 0.5 L_{1}^{2} m_{1} \left(\frac{d}{d t} \theta_{1}{\left(t \right)}\right)^{2} + L_{1} g m_{1} \cos{\left(\theta_{1}{\left(t \right)} \right)} \\ &+ g m_{2} \left(L_{1} \cos{\left(\theta_{1}{\left(t \right)} \right)} + L_{2} \cos{\left(\theta_{2}{\left(t \right)} \right)}\right) \\ &+ g m_{3} \Big(L_{1} \cos{\left(\theta_{1}{\left(t \right)} \right)} + L_{2} \cos{\left(\theta_{2}{\left(t \right)} \right)} \\ &\qquad + L_{3} \cos{\left(\theta_{3}{\left(t \right)} \right)}\Big) \\ &+ 0.5 m_{2} \Big(L_{1}^{2} \left(\frac{d}{d t} \theta_{1}{\left(t \right)}\right)^{2} \\ &\qquad + 2 L_{1} L_{2} \cos{\left(\theta_{1}{\left(t \right)} - \theta_{2}{\left(t \right)} \right)} \\ &\qquad\qquad \frac{d}{d t} \theta_{1}{\left(t \right)} \frac{d}{d t} \theta_{2}{\left(t \right)} \\ &\qquad + L_{2}^{2} \left(\frac{d}{d t} \theta_{2}{\left(t \right)}\right)^{2}\Big) \\ &+ 0.5 m_{3} \Big(L_{1}^{2} \left(\frac{d}{d t} \theta_{1}{\left(t \right)}\right)^{2} \\ &\qquad + 2 L_{1} L_{2} \cos{\left(\theta_{1}{\left(t \right)} - \theta_{2}{\left(t \right)} \right)} \\ &\qquad\qquad \frac{d}{d t} \theta_{1}{\left(t \right)} \frac{d}{d t} \theta_{2}{\left(t \right)} \\ &\qquad + 2 L_{1} L_{3} \cos{\left(\theta_{1}{\left(t \right)} - \theta_{3}{\left(t \right)} \right)} \\ &\qquad\qquad \frac{d}{d t} \theta_{1}{\left(t \right)} \frac{d}{d t} \theta_{3}{\left(t \right)} \\ &\qquad + L_{2}^{2} \left(\frac{d}{d t} \theta_{2}{\left(t \right)}\right)^{2} \\ &\qquad + 2 L_{2} L_{3} \cos{\left(\theta_{2}{\left(t \right)} - \theta_{3}{\left(t \right)} \right)} \\ &\qquad\qquad \frac{d}{d t} \theta_{2}{\left(t \right)} \frac{d}{d t} \theta_{3}{\left(t \right)} \\ &\qquad + L_{3}^{2} \left(\frac{d}{d t} \theta_{3}{\left(t \right)}\right)^{2}\Big) \end{aligned}\]The equations can be quite massive, but they will be linear in terms of angular acceleration. We can treat the Lagrangian equations of motion as a set of linear equations to solve for each \(\ddot{\theta_n}\).
\[\begin{bmatrix}\ \ddot{\theta_1} \\ \ddot{\theta_2} \\ \ddot{\theta_3} \\ ... \\ \ddot{\theta_n} \end{bmatrix} = \begin{bmatrix}\ f(\theta_1, \dot{\theta_1}, \theta_2, \dot{\theta_2}, \theta_3, \dot{\theta_3} ... , \theta_n, \dot{\theta_n}) \\ f(\theta_1, \dot{\theta_1}, \theta_2, \dot{\theta_2}, \theta_3, \dot{\theta_3} ... , \theta_n, \dot{\theta_n}) \\ f(\theta_1, \dot{\theta_1}, \theta_2, \dot{\theta_2}, \theta_3, \dot{\theta_3} ... , \theta_n, \dot{\theta_n}) \\ ... \\ f(\theta_1, \dot{\theta_1}, \theta_2, \dot{\theta_2}, \theta_3, \dot{\theta_3} ... , \theta_n, \dot{\theta_n}) \end{bmatrix}\]These equations can be re-paramaterized by setting \(\dot{\theta_n} = z_n\). The following dynamical system is obtained where \(\vec{s_0}\) is the initial conditions.
\[ \dot{\vec{s}} = f(\vec{s}) \] \[ \vec{s} = \begin{bmatrix} \theta_1 \\ z_1 \\\theta_2 \\ z_2 \\ ... \\ \theta_n \\ z_n \end{bmatrix}, f(\vec{s}) = \begin{bmatrix} z_1 \\ f_1(\theta_1, z_1, \theta_2, z_2, ... , \theta_n, x_n) \\ z_2 \\ f_2(\theta_1, z_1, \theta_2, z_2, ... , \theta_n, x_n) \\ ... \\ z_n \\ f_n(\theta_1, z_1, \theta_2, z_2, ... , \theta_n, x_n) \end{bmatrix} \]The system is then numerically integrated using the 4th order Runge-Kutta method to achieve adequate accuracy. The plot of energy over time shows minimal change in the conservative system. A perfect integration scheme would hold the energy constant.
