Bond Graph model

The schematic diagram of the system NonlinearMSD is displayed in Figure 4.1. The system comprises

• a rigid foundation,
• a rod of length $L$ hinged at the left-hand end and
• a linear spring of stiffness $k$ attached to the rigid foundation a distance $L$ from the hinge and to the free end of the rod.

The spring is unstretched when the rod makes an angle $\theta = \alpa = \frac{pi}{3}$ with the foundation.

Using elementary geometry, the effectice angular spring generates a torque $\tau$ given by (4.1)

$$\tau = -2kl^2 \cos{\frac{\theta}{2}} (\sin{\frac{\theta}{2}}-\sin{\frac{\alpha}{2}})$$ (4.1)

The acausal bond graph of system NonlinearMSD is also displayed in Figure 4.1 (on page ). This shows the three bond graph components representing the friction R, the inertia I and the spring C components. The non-linear spring characteristic is given explicitly^4.1.

The (nonlinear) system ordinary differential equation is given by MTT in Section (on page ). This is a special case of the general non-linear ordinary differential equation:

$$\begin{cases}\dot{x} = f(x,u)\\ y = g(x,u) \end{cases}$$ (4.2)