Modeling and Simulation in Python

Bungee jump case study

Copyright 2018 Allen Downey

License: Creative Commons Attribution 4.0 International

In [1]:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline

# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

# import functions from the module
from modsim import *

Bungee jumping

Suppose you want to set the world record for the highest "bungee dunk", as shown in this video. Since the record is 70 m, let's design a jump for 80 m.

We'll make the following modeling assumptions:

  1. Initially the bungee cord hangs from a crane with the attachment point 80 m above a cup of tea.

  2. Until the cord is fully extended, it applies no force to the jumper. It turns out this might not be a good assumption; we will revisit it.

  3. After the cord is fully extended, it obeys Hooke's Law; that is, it applies a force to the jumper proportional to the extension of the cord beyond its resting length.

  4. The jumper is subject to drag force proportional to the square of their velocity, in the opposite of their direction of motion.

Our objective is to choose the length of the cord, L, and its spring constant, k, so that the jumper falls all the way to the tea cup, but no farther!

First I'll create a Param object to contain the quantities we'll need:

  1. Let's assume that the jumper's mass is 75 kg.

  2. With a terminal velocity of 60 m/s.

  3. The length of the bungee cord is L = 40 m.

  4. The spring constant of the cord is k = 20 N / m when the cord is stretched, and 0 when it's compressed.

In [2]:
m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram
N = UNITS.newton


In [3]:
params = Params(y_attach = 80 * m,
                 v_init = 0 * m / s,
                 g = 9.8 * m/s**2,
                 mass = 75 * kg,
                 area = 1 * m**2,
                 rho = 1.2 * kg/m**3,
                 v_term = 60 * m / s,
                 L = 25 * m,
                 k = 40 * N / m,
                 zero_force = 0 * N)

y_attach 80 meter
v_init 0.0 meter / second
g 9.8 meter / second ** 2
mass 75 kilogram
area 1 meter ** 2
rho 1.2 kilogram / meter ** 3
v_term 60.0 meter / second
L 25 meter
k 40.0 newton / meter
zero_force 0 newton

Now here's a version of make_system that takes a Params object as a parameter.

make_system uses the given value of v_term to compute the drag coefficient C_d.

In [4]:
def make_system(params):
    """Makes a System object for the given params.
    params: Params object
    returns: System object
    area, mass = params.area, params.mass
    g, rho = params.g, params.rho
    v_init, v_term = params.v_init, params.v_term
    y_attach = params.y_attach
    C_d = 2 * mass * g / (rho * area * v_term**2)
    init = State(y=y_attach, v=v_init)
    t_end = 20 * s

    return System(params, C_d=C_d, 
                  init=init, t_end=t_end)

Let's make a System

In [5]:
system = make_system(params)

y_attach 80 meter
v_init 0.0 meter / second
g 9.8 meter / second ** 2
mass 75 kilogram
area 1 meter ** 2
rho 1.2 kilogram / meter ** 3
v_term 60.0 meter / second
L 25 meter
k 40.0 newton / meter
zero_force 0 newton
C_d 0.3402777777777778 dimensionless
init y 80 meter v 0.0 meter / secon...
t_end 20 second

spring_force computes the force of the cord on the jumper.

If the spring is not extended, it returns zero_force, which is either 0 Newtons or 0, depending on whether the System object has units. I did that so the slope function works correctly with and without units.

In [6]:
def spring_force(y, system):
    """Computes the force of the bungee cord on the jumper:
    y: height of the jumper
    Uses these variables from system|
    y_attach: height of the attachment point
    L: resting length of the cord
    k: spring constant of the cord
    returns: force in N
    y_attach, L, k = system.y_attach, system.L, system.k
    distance_fallen = y_attach - y
    if distance_fallen <= L:
        return system.zero_force
    extension = distance_fallen - L
    f_spring = k * extension
    return f_spring

The spring force is 0 until the cord is fully extended. When it is extended 1 m, the spring force is 40 N.

In [7]:
spring_force(80*m, system)

0 newton

In [8]:
spring_force(55*m, system)

0 newton

In [9]:
spring_force(54*m, system)

40.0 newton

drag_force computes drag as a function of velocity:

In [10]:
def drag_force(v, system):
    """Computes drag force in the opposite direction of `v`.
    v: velocity
    system: System object

    returns: drag force
    rho, C_d, area = system.rho, system.C_d, system.area
    f_drag = -np.sign(v) * rho * v**2 * C_d * area / 2
    return f_drag

Here's the drag force at 60 meters per second.

In [11]:
v = -60 * m/s
f_drag = drag_force(v, system)

735.0 kilogram meter/second2

Acceleration due to drag at 60 m/s is approximately g, which confirms that 60 m/s is terminal velocity.

In [12]:
a_drag = f_drag / system.mass

9.8 meter/second2

Now here's the slope function:

In [13]:
def slope_func(state, t, system):
    """Compute derivatives of the state.
    state: position, velocity
    t: time
    system: System object containing g, rho,
            C_d, area, and mass
    returns: derivatives of y and v
    y, v = state
    mass, g = system.mass, system.g
    a_drag = drag_force(v, system) / mass
    a_spring = spring_force(y, system) / mass
    dvdt = -g + a_drag + a_spring
    return v, dvdt

As always, let's test the slope function with the initial params.

In [14]:
slope_func(system.init, 0, system)

(0.0 <Unit('meter / second')>, -9.8 <Unit('meter / second ** 2')>)

And then run the simulation.

In [15]:
results, details = run_ode_solver(system, slope_func)

success True
message The solver successfully reached the end of the...

Here's the plot of position as a function of time.

In [16]:
def plot_position(results):
    decorate(xlabel='Time (s)',
             ylabel='Position (m)')

In [17]:

After reaching the lowest point, the jumper springs back almost to almost 70 m, and oscillates several times. That looks like more osciallation that we expect from an actual jump, which suggests that there some dissipation of energy in the real world that is not captured in our model. To improve the model, that might be a good thing to investigate.

But since we are primarily interested in the initial descent, the model might be good enough for now.

We can use min to find the lowest point:

In [18]:

5.413731910623594 meter

At the lowest point, the jumper is still too high, so we'll need to increase L or decrease k.

Here's velocity as a function of time:

In [19]:
def plot_velocity(results):
    plot(results.v, color='C1', label='v')
    decorate(xlabel='Time (s)',
             ylabel='Velocity (m/s)')

In [20]:

Although we compute acceleration inside the slope function, we don't get acceleration as a result from run_ode_solver.

We can approximate it by computing the numerical derivative of ys:

In [21]:
a = gradient(results.v)
decorate(xlabel='Time (s)',
         ylabel='Acceleration (m/$s^2$)')

And we can compute the maximum acceleration the jumper experiences:

In [22]:
max_acceleration = max(a) * m/s**2

16.64087716292061 meter/second2

Relative to the acceleration of gravity, the jumper "pulls" about "1.7 g's".

In [23]:
max_acceleration / system.g

1.6980486900939398 dimensionless

Under the hood

The gradient function in adapts the NumPy function of the same name so it works with Series objects.

In [24]:

def gradient(series, **options):
    """Computes the numerical derivative of a series.

    If the elements of series have units, they are dropped.

    series: Series object
    options: any legal options to np.gradient

    returns: Series, same subclass as series
    x = magnitudes(series.index)
    y = magnitudes(series.values)
    # units = get_units(series.values[0])

    a = np.gradient(y, x, **options)
    return series.__class__(a, series.index)

Solving for length

Assuming that k is fixed, let's find the length L that makes the minimum altitude of the jumper exactly 0.

The metric we are interested in is the lowest point of the first oscillation. For both efficiency and accuracy, it is better to stop the simulation when we reach this point, rather than run past it and the compute the minimum.

Here's an event function that stops the simulation when velocity is 0.

In [25]:
def event_func(state, t, system):
    """Return velocity.
    y, v = state
    return v

As usual, we should test it with the initial conditions.

In [26]:
event_func(system.init, 0, system)

0.0 meter/second

Now we can test it and confirm that it stops at the bottom of the jump.

In [27]:
results, details = run_ode_solver(system, slope_func, events=event_func)

In [28]:

5.448756012469637 meter

Exercise: Write an error function that takes L and params as arguments, simulates a bungee jump, and returns the lowest point.

Test the error function with a guess of 25 m and confirm that the return value is about 5 meters.

Use root_scalar with your error function to find the value of L that yields a perfect bungee dunk. Hint: before calling root_scalar, make a version of params with no dimensions.

Run a simulation with the result from root_scalar and confirm that it works.

In [29]:
# Solution

def error_func(L, params):
    """Minimum height as a function of length.
    L: length in m
    params: Params object
    returns: height in m
    params = Params(params, L=L)
    system = make_system(params)

    results, details = run_ode_solver(system, slope_func, events=event_func)
    min_height = get_last_value(results.y)
    return min_height

In [30]:
# Solution

guess1 = 25 * m
error_func(guess1, params)

5.448756012469637 meter

In [31]:
# Solution

guess2 = 30 * m
error_func(guess2, params)

-1.398018644734631 meter

In [32]:
# Solution

res = root_bisect(error_func, [guess1, guess2], params)

converged True
root 28.996669054031372 meter

In [33]:
# Solution

L = res.root
params_solution = Params(params, L=L)
system_solution = make_system(params_solution)

results, details = run_ode_solver(system_solution, slope_func, events=event_func)

success True
message A termination event occurred.

In [34]:
# Solution

min_height = get_last_value(results.y) * m

4.1424875280200724×10-7 meter2

In [ ]: