Simulation

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Simulation#

Plotting#

We’ll examine how to plot a simple mass-spring-damper system. We’ll first define the system

# System parameters
m = 1.0  # mass
c = 0.5  # damping coefficient
k = 2.0  # spring constant

# Define the system of ODEs
def mass_spring_damper(t, y):
    x, v = y
    dxdt = v
    dvdt = (-c * v - k * x) / m
    return [dxdt, dvdt]

We then define our initial conditions, and the range we would like to simulate.

import numpy as np # For storing out solutions in np.arrays

# Initial conditions
y0 = [1.0, 0.0]  # initial displacement and velocity
# Time span
t_end = 15
t_span = (0, t_end)
t_eval = np.linspace(*t_span, 300)  # Increased to 300 for smoother animation

We can then use scipy to solve the initial value problem (IVP) and store the solutions in a pandas dataframe.

import pandas as pd
from scipy.integrate import solve_ivp

# Solve the ODE
sol = solve_ivp(mass_spring_damper, t_span, y0, t_eval=t_eval)

# Create a DataFrame
df = pd.DataFrame({'time': sol.t, 'displacement': sol.y[0]})

Using plotly we can animate the trajectory and display it in a Javascript player.

import plotly.graph_objects as go
animation_time = 10

# Create animation frames
frames = [
    go.Frame(data=[
        go.Scatter(x=df['time'][:i+1], y=df['displacement'][:i+1], mode='lines', name='trajectory'),  # Line showing trajectory
        go.Scatter(x=[df['time'][i]], y=[df['displacement'][i]], mode='markers', marker=dict(size=10, color='red'), name='current point')  # Marker for current point
    ],
    name=str(df['time'][i]))
    for i in range(len(df))
]

# Create the figure
fig = go.Figure(
    data=[
        go.Scatter(x=[df['time'][0]], y=[df['displacement'][0]], mode='lines', name='trajectory'),  # Initial empty line
        go.Scatter(x=[df['time'][0]], y=[df['displacement'][0]], mode='markers', marker=dict(size=10, color='red'), name='current point') # Initial marker
    ],
    layout=go.Layout(
        xaxis=dict(range=t_span, autorange=False, title='Time'),
        yaxis=dict(range=[-1,1], autorange=False, title='Displacement'),
        title="Mass-Spring-Damper System Animation",
        updatemenus=[dict(type="buttons",
                        buttons=[dict(label="Play",
                                        method="animate",
                                        args=[None, {"frame": {"duration": animation_time, "redraw": False},
                                                    "fromcurrent": True}])])]),
    frames=frames
)

fig.show()
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