Here we will derive the equations of motion for the classic mass, spring, damper system under the influence of gravity. The following figure gives a pictorial description of the problem.

In [1]:

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

Out[1]:

Start by loading in the core functionality of both SymPy and Mechanics.

In [2]:

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

We can make use of the pretty printing of our results by loading the SymPy printing extension.

In [3]:

%load_ext sympy.interactive.ipythonprinting

We'll start by defining the variables will need for this problem. $x$ is the distance of the particle from the ceiling, $v$ is the speed of the particle, $m$ is the mass of the particle, $c$ is the damping coefficient of the damper, $k$ is the stiffness of the spring, $g$ is the acceleration due to gravity, and $t$ represents time.

In [4]:

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

In [5]:

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

Now, we define a Newtonian reference frame that represents the ceiling which the particle is attached to, $C$.

In [6]:

ceiling = me.ReferenceFrame('C')

We will need two points, one to represent the original position of the particle which stays fixed in the ceiling frame, $o$, and the second one, $p$ which is aligned with the particle as it moves.

In [7]:

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

The velocity of point $o$ in the ceiling is zero.

In [8]:

o.set_vel(ceiling, 0)

Point $p$ can move downward in the $x$ direction and it's velocity is $v$ in the downward direction.

In [9]:

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

There are three forces acting on the particle. Those due to the acceleration of gravity, the damper, and the spring.

In [10]:

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

Out[10]:

(-c⋅v + g⋅m - k⋅x)*c_x

Now we can use Newton's second law, $0=F-ma$, to form the equation of motion of the system.

In [11]:

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

Out[11]:

$$- c \operatorname{v}{\left (t \right )} + g m - k \operatorname{x}{\left (t \right )} - m \frac{\partial}{\partial t} \operatorname{v}{\left (t \right )}$$

We can then form the first order equations of motion by solving for $\frac{dv}{dt}$ and introducing the kinematical differential equation, $v=\frac{dx}{dt}$.

In [12]:

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

Out[12]:

$$\begin{pmatrix}\frac{- c \operatorname{v}{\left (t \right )} + g m - k \operatorname{x}{\left (t \right )}}{m}, & \operatorname{v}{\left (t \right )}\end{pmatrix}$$

Forming the equations of motion can also be done with the automated methods available in the Mechanics package: LagrangesMethod and KanesMethod. Here we will make use of Kane's method to find the same equations of motion which we found manually above. First define a particle which represents the mass attached to the damper and spring.

In [13]:

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

Now we can construct a KanesMethod object by passing in the generalized coordinate, $x$, the generalized speed, $v$, and the kinematical differential equation which relates the two, $0=v-\frac{dx}{dt}$.

In [14]:

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

Now Kane's equations can be computed which returns $F_r$ and $F_r^*$.

In [15]:

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

Out[15]:

$$\begin{pmatrix}\left[\begin{smallmatrix}c \operatorname{v}{\left (t \right )} - g m + k \operatorname{x}{\left (t \right )}\end{smallmatrix}\right], & \left[\begin{smallmatrix}m \frac{\partial}{\partial t} \operatorname{v}{\left (t \right )}\end{smallmatrix}\right]\end{pmatrix}$$

The equations are also available in the form $M\frac{dq,u}{dt}=f(\dot{u}, u, q)$ and we can extract the mass matrix, $M$, and the forcing equations, $f$.

In [16]:

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

Out[16]:

$$\begin{pmatrix}\left[\begin{smallmatrix}1 & 0\\0 & m\end{smallmatrix}\right], & \left[\begin{smallmatrix}\operatorname{v}{\left (t \right )}\\- c \operatorname{v}{\left (t \right )} + g m - k \operatorname{x}{\left (t \right )}\end{smallmatrix}\right]\end{pmatrix}$$

Finally, we can form the first order differential equations of motion $\frac{dq,u}{dt}=M^{-1}f(\dot{u}, u, q)$, which is the same as previously found.

In [17]:

M.inv() * f

Out[17]:

$$\left[\begin{smallmatrix}\operatorname{v}{\left (t \right )}\\\frac{- c \operatorname{v}{\left (t \right )} + g m - k \operatorname{x}{\left (t \right )}}{m}\end{smallmatrix}\right]$$