systems.builtin.deterministic.continuous.FifthOrderMechanicalSystem

systems.builtin.deterministic.continuous.FifthOrderMechanicalSystem(
    *args,
    **kwargs,
)

Fifth-order mechanical system - extremely high-order dynamics.

WARNING: This is an artificially complex system designed for testing high-order integration schemes. Physical systems rarely exceed third order.

Physical Interpretation:

Could represent: - Flexible manipulator with multiple vibration modes - Actuator with nested control loops (each adding an order) - Academic test case for high-order integration

Mathematical Formulation:

State: x = [q, q’, q’‘, q’’’, q⁽⁴⁾] where: - q: Position [m] - q’: Velocity [m/s] - q’‘: Acceleration [m/s²] - q’’’: Jerk [m/s³] - q⁽⁴⁾: Snap (fourth derivative) [m/s⁴]

The system evolves according to: q⁽⁵⁾ = f(q, q’, q’‘, q’’’, q⁽⁴⁾, u)

Dynamics:

q⁽⁵⁾ = -(k/m)q - c₁q’ - c₂q’’ - c₃q’’’ - 0.01q⁽⁴⁾ - g + u/m

This includes: - Stiffness term: -kq (like a spring) - Multiple damping terms at each derivative level - Gravity: -g - Control input: u/m

Parameters:

m : float, default=1.0 Mass [kg] k : float, default=1.0 Stiffness coefficient [N/m] c1 : float, default=0.1 First-order damping (velocity damping) [N⋅s/m] c2 : float, default=0.05 Second-order damping (acceleration damping) [N⋅s³/m] c3 : float, default=0.01 Third-order damping (jerk damping) [N⋅s⁵/m] g : float, default=9.81 Gravitational acceleration [m/s²]

State Space:

State: x = [q, q’, q’‘, q’’’, q⁽⁴⁾] (5D) Control: u = [force] (1D) Output: y = [q, q’] (position and velocity)

Equilibrium:

Static equilibrium (balancing gravity): q_eq = -mg/k (compressed by gravity) All derivatives zero u_eq = mg (supporting weight)

See Also:

SymbolicPendulum2ndOrder : More typical second-order system CoupledOscillatorSystem : More realistic multi-DOF system