Class 11: System dynamics models: Damped oscillator and bungee jumping I¶

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Load packages¶

In [1]:

%matplotlib inline

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

Model of an undamped spring¶

Constants¶

In [2]:

sim_time = 3.0  # s
delta_t = 0.00001 # s
sim_steps = int(sim_time / delta_t)

mass = 0.2  # kg
spring_constant = 10  # N/m
gravity_acceleration = -9.81  # m/s^2
weight = mass * gravity_acceleration  # N

unweighted_length = 1.000  # m
weight_displacement = 1.962 / 10 # m
weighted_length = unweighted_length + weight_displacement  # m
init_displacement = 0.3  # m

length = weighted_length + init_displacement  # m
velocity = 0  # m/s

Simulation history¶

In [3]:

trace = [[0, length, 0]]

Algorithm¶

In [4]:

for step_index in range(1, sim_steps + 1):
    restoring_spring_force = spring_constant * (length - unweighted_length)
    total_force = restoring_spring_force + weight
    acceleration = total_force / mass

    length += -velocity * delta_t
    velocity += delta_t * acceleration

    trace.append([step_index, length, velocity])

Convert results to data frame¶

In [5]:

simulation_df = pd.DataFrame({
    "time": [x[0] * delta_t for x in trace],
    "length": [x[1] for x in trace],
    "velocity": [x[2] for x in trace],
})

Visualization¶

In [6]:

figwidth = 5  # inches
figheight = 0.618 * figwidth  # golden ratio
figsize = (figwidth, 1.5 * figheight)

In [7]:

fig, ax = plt.subplots(nrows=2, ncols=1, figsize=figsize, dpi=120, sharex=True)
ax[0].plot(simulation_df["time"], simulation_df["length"], "-")
ax[1].plot(simulation_df["time"], simulation_df["velocity"], "-")
ax[0].set_xlabel("time (s)")
ax[0].set_ylabel("length (m)")
ax[1].set_xlabel("time (s)")
ax[1].set_ylabel("velocity (m/s)");