Energy Methods & Generalized Forces

Energy Methods Review

• For undamped, autonomous, 1-DOF systems, the total energy (T + V) remains constant.
• Therefore the rate of change of energy is zero, expressed as $\dot{T} + \dot{V} = 0$.
• The equation of motion (EOM) can be derived using: $\frac{d}{dt}(\frac{\partial T}{\partial \dot{q_i}}) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} = Q_i$
• Note that the partial derivative of potential energy with respect to the generalized velocity is zero which can be expressed as: $\frac{\partial V}{\partial \dot{q_i}} = 0$

Generalized Forces

• Virtual work determines the generalized forces $Q_i$
• Virtual work equation: $\delta W = \sum_{i=1}^{n} \vec{F_i} \cdot \delta \vec{r_i} + \sum_{i=1}^{t} \vec{M_i} \cdot \delta \vec{\theta_i}$
• Virtual displacement/rotation must align with constraints.
• This results in $\delta W = \sum_{i=1}^{n} Q_i \delta q_i$
• $Q_i$ can be calculated using $Q_i = \frac{\delta W}{\delta q_i}$

Simple Pendulum with Damping Example

• Kinetic Energy: $T = \frac{1}{2} m (l \dot{\theta})^2$
• Potential Energy: $V = mgl(1 - cos\theta)$
• Virtual Work: $\delta W = \vec{F_d} \cdot \delta \vec{r}$
• Damping Force: $\vec{F_d} = -b \vec{v} = -b(l \dot{\theta} \hat{e_\theta})$
• Displacement: $\delta \vec{r} = l \delta \theta \hat{e_\theta}$
• Virtual Work Equation: $\delta W = -bl^2 \dot{\theta} \delta \theta$
• Generalized Force: $Q_\theta = -bl^2 \dot{\theta}$
• Applying Lagrange's Equation: $\frac{d}{dt}(\frac{\partial T}{\partial \dot{\theta}}) - \frac{\partial T}{\partial \theta} + \frac{\partial V}{\partial \theta} = Q_\theta$
• $\frac{\partial T}{\partial \dot{\theta}} = ml^2 \dot{\theta}$
• $\frac{\partial T}{\partial \theta} = 0$
• $\frac{\partial V}{\partial \theta} = mglsin\theta$
• Results in: $\ddot{\theta} + \frac{b}{m} \dot{\theta} + \frac{g}{l} \theta = 0$
• For small $\theta$, the equation of motion is: $\ddot{\theta} + \frac{b}{m} \dot{\theta} + \frac{g}{l} sin\theta = 0$

Cart with Pendulum Example

• Kinetic Energy: $T = \frac{1}{2}M\dot{x}^2 + \frac{1}{2}m(\dot{x} + l\dot{\theta}cos\theta)^2 + \frac{1}{2}m(l\dot{\theta}sin\theta)^2$
• Potential Energy: $V = mgl(1 - cos\theta)$
• Lagrangian: $L = \frac{1}{2}M\dot{x}^2 + \frac{1}{2}m(\dot{x}^2 + 2\dot{x}l\dot{\theta}cos\theta + l^2\dot{\theta}^2) - mgl(1 - cos\theta)$
• Generalized Forces: $\delta W = F \delta x$
• $Q_x = F$
• Lagrange's Equations:
  • $\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}}) - \frac{\partial L}{\partial x} = Q_x$
  • $\frac{d}{dt}(\frac{\partial L}{\partial \dot{\theta}}) - \frac{\partial L}{\partial \theta} = Q_\theta$
• First Equation:
  • $\frac{\partial L}{\partial \dot{x}} = M\dot{x} + m\dot{x} + ml\dot{\theta}cos\theta$
  • $\frac{\partial L}{\partial x} = 0$
  • $(M + m)\ddot{x} + ml\ddot{\theta}cos\theta - ml\dot{\theta}^2 sin\theta = F$
• Second Equation:
  • $\frac{\partial L}{\partial \dot{\theta}} = ml\dot{x}cos\theta + ml^2\dot{\theta}$
  • $\frac{\partial L}{\partial \theta} = -ml\dot{x}\dot{\theta}sin\theta - mgsin\theta$
  • $ml\ddot{x}cos\theta - ml\dot{x}\dot{\theta}sin\theta + ml^2\ddot{\theta} = -ml\dot{x}\dot{\theta}sin\theta - mgsin\theta$
  • $\ddot{x}cos\theta + l\ddot{\theta} + gsin\theta = 0$

Linearization

• Assuming small $\theta, \dot{\theta}$:
  • $(M + m)\ddot{x} + ml\ddot{\theta} = F$
  • $\ddot{x} + l\ddot{\theta} + g\theta = 0$
• Matrix Form:
  • $\begin{bmatrix} M+m & ml \ m & l \end{bmatrix} \begin{bmatrix} \ddot{x} \ \ddot{\theta} \end{bmatrix} + \begin{bmatrix} 0 & 0 \ 0 & g \end{bmatrix} \begin{bmatrix} x \ \theta \end{bmatrix} = \begin{bmatrix} F \ 0 \end{bmatrix}$
• Which is similar to: $M\ddot{\vec{q}} + K\vec{q} = \vec{f}$
  • Where: $\vec{q} = \begin{bmatrix} x \ \theta \end{bmatrix}$