c4dynamics.eqm.derivs.eqm6

c4dynamics.eqm.derivs.eqm6(rb: rigidbody, F: ndarray | list, M: ndarray | list) → ndarray

Translational and angular motion derivatives.

A set of first-order ordinary differential equations (ODEs) that describe the motion of a rigid body in three-dimensional space under the influence of external forces and moments.

Parameters:

• rb (rigidbody) – C4dynamics’ rigidbody object for which the equations of motion are calculated on.

• F (array_like) – Force vector \([F_x, F_y, F_z]\)

• M (array_like) – Moments vector \([M_x, M_y, M_z]\)

Returns:

out (numpy.array) – \([dx, dy, dz, dv_x, dv_y, dv_z, d\varphi, d\theta, d\psi, dp, dq, dr]\)

12 total derivatives; 6 of translational motion, 6 of rotational motion.

Examples

Euler integration on the equations of motion of a stick fixed at one edge:

(mass: 0.5 kg, moment of inertia about y: 0.4 kg*m^2, Length: 1m, initial Euler pitch angle: 80° (converted to radians))

Import required packages:

>>> import c4dynamics as c4d
>>> import numpy as np

Settings and initial conditions

>>> dt = 0.5e-3
>>> t = np.arange(0, 10, dt)
>>> length =  1  # metter
>>> rb = c4d.rigidbody(theta = 80 * c4d.d2r)
>>> rb.mass = 0.5 # kg
>>> rb.I = [0, 0.4, 0]

Main loop:

>>> for ti in t:
...    rb.store(ti)
...    tau_g = -rb.mass * c4d.g_ms2 * length / 2 * c4d.cos(rb.theta)
...    dx = c4d.eqm.eqm6(rb, np.zeros(3), [0, tau_g, 0])
...    rb.X += dx * dt
>>> rb.plot('theta')