from IPython.display import SVG
SVG(filename='mass_spring_damper.svg')

import sympy as sym
import sympy.physics.mechanics as me

%load_ext sympy.interactive.ipythonprinting

x, v = me.dynamicsymbols('x v')

m, c, k, g, t = sym.symbols('m c k g t')

ceiling = me.ReferenceFrame('C')

o = me.Point('o')
p = me.Point('p')

o.set_vel(ceiling, 0)

p.set_pos(o, x * ceiling.x)
p.set_vel(ceiling, v * ceiling.x)

damping = -c * p.vel(ceiling)
stiffness = -k * p.pos_from(o)
gravity = m * g * ceiling.x
forces = damping + stiffness + gravity
forces

zero = me.dot(forces - m * p.acc(ceiling), ceiling.x)
zero

dvdt = sym.solve(zero, v.diff(t))[0]
dxdt = v
dvdt, dxdt

mass = me.Particle('mass', p, m)

kane = me.KanesMethod(ceiling, q_ind=[x], u_ind=[v], kd_eqs=[v - x.diff(t)])

kane.kanes_equations([(p, forces)], [mass])

M = kane.mass_matrix_full
f = kane.forcing_full
M, f

M.inv() * f