Usage

This is an example of a simple one degree of freedom system: a mass under the influence of a spring, damper, gravity and an external force:

/ / / / / / / / /
-----------------
  |    |     |   | g
  \   | |    |   V
k /   --- c  |
  |    |     | x, v
 --------    V
 |  m   | -----
 --------
    | F
    V

Derive the system:

from sympy import symbols
import sympy.physics.mechanics as me

mass, stiffness, damping, gravity = symbols('m, k, c, g')

position, speed = me.dynamicsymbols('x v')
positiond = me.dynamicsymbols('x', 1)
force = me.dynamicsymbols('F')

ceiling = me.ReferenceFrame('N')

origin = me.Point('origin')
origin.set_vel(ceiling, 0)

center = origin.locatenew('center', position * ceiling.x)
center.set_vel(ceiling, speed * ceiling.x)

block = me.Particle('block', center, mass)

kinematic_equations = [speed - positiond]

force_magnitude = mass * gravity - stiffness * position - damping * speed + force
forces = [(center, force_magnitude * ceiling.x)]

particles = [block]

kane = me.KanesMethod(ceiling, q_ind=[position], u_ind=[speed],
                      kd_eqs=kinematic_equations)
kane.kanes_equations(particles, loads=forces)

Create a system to manage integration and specify numerical values for the constants and specified quantities. Here, we specify sinusoidal forcing:

from numpy import array, linspace, sin
from pydy.system import System

sys = System(kane,
             constants={mass: 1.0, stiffness: 10.0,
                        damping: 0.4, gravity: 9.8},
             specifieds={force: lambda x, t: sin(t)},
             initial_conditions={position: 0.1, speed: -1.0},
             times=linspace(0.0, 10.0, 1000))

Integrate the equations of motion to get the state trajectories:

y = sys.integrate()

Plot the results:

import matplotlib.pyplot as plt

plt.plot(sys.times, y)
plt.legend((str(position), str(speed)))
plt.xlabel('Time [s]')
plt.show()