Geometry
\[ L = \sqrt{a^2+b^2}, \qquad \tan\alpha = \frac{b}{a} \]Dynamics in \(\theta\)
\[ J(\theta)\,\ddot{\theta} + \frac{1}{2}\,\frac{\partial J(\theta)}{\partial\theta}\,\dot{\theta}^2 + c\,L^2\cos^2(\alpha+\theta)\,\dot{\theta} + k\,L\sin(\alpha+\theta)\bigl(a - L\cos(\alpha+\theta)\bigr) = 0 \] \[ \text{where}\quad J(\theta) = \bigl( m_A \sin^2(\alpha+\theta) + m_B \cos^2(\alpha+\theta) \bigr)\,L^2 \]Linearized dynamics in \(\theta\)
\[ (m_A b^2 + m_B a^2)\,\ddot{\theta} + c\,a^2\,\dot{\theta} + k\,b^2\,\theta = 0 \]Dynamics in \(x\)
\[ m(x)\,\ddot{x} + \frac{1}{2}\,\frac{\partial m(x)}{\partial x}\,\dot{x}^2 + c\,\frac{(a-x)^2}{\textcolor[rgb]{0.494,0.722,0.851}{b^2+2ax-x^2}}\,\dot{x} + k\,x = 0 \] \[ \text{where}\quad m(x) = m_A + m_B\,\frac{(a-x)^2}{\textcolor[rgb]{0.494,0.722,0.851}{b^2+2ax-x^2}} \]Linearized dynamics in \(x\)
\[ \left(m_A + \frac{a^2}{b^2}m_B\right)\ddot{x} + c\,\frac{a^2}{b^2}\,\dot{x} + k\,x = 0 \]Coordinate change
\[ \text{nonlinear:}\quad a-x = L\cos(\alpha+\theta) \] \[ \text{linear:}\quad x = b\,\theta \]