Modeling and Simulation in Python

Chapter 21

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International

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# 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 *

With air resistance

Next we'll add air resistance using the drag equation

I'll start by getting the units we'll need from Pint.

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m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram

Now I'll create a Params object to contain the quantities we need. Using a Params object is convenient for grouping the system parameters in a way that's easy to read (and double-check).

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params = Params(height = 381 * m,
                v_init = 0 * m / s,
                g = 9.8 * m/s**2,
                mass = 2.5e-3 * kg,
                diameter = 19e-3 * m,
                rho = 1.2 * kg/m**3,
                v_term = 18 * m / s)

Now we can pass the Params object make_system which computes some additional parameters and defines init.

make_system uses the given radius to compute area and the given v_term to compute the drag coefficient C_d.

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def make_system(params):
    """Makes a System object for the given conditions.
    params: Params object
    returns: System object
    diameter, mass = params.diameter, params.mass
    g, rho = params.g, params.rho, 
    v_init, v_term = params.v_init, params.v_term
    height = params.height
    area = np.pi * (diameter/2)**2
    C_d = 2 * mass * g / (rho * area * v_term**2)
    init = State(y=height, v=v_init)
    t_end = 30 * s
    dt = t_end / 100
    return System(params, area=area, C_d=C_d, 
                  init=init, t_end=t_end, dt=dt)

Let's make a System

In [5]:
system = make_system(params)

Here's the slope function, including acceleration due to gravity and drag.

In [6]:
def slope_func(state, t, system):
    """Compute derivatives of the state.
    state: position, velocity
    t: time
    system: System object
    returns: derivatives of y and v
    y, v = state
    rho, C_d, g = system.rho, system.C_d, system.g
    area, mass = system.area, system.mass
    f_drag = rho * v**2 * C_d * area / 2
    a_drag = f_drag / mass
    dydt = v
    dvdt = -g + a_drag
    return dydt, dvdt

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

In [7]:
slope_func(system.init, 0, system)

We can use the same event function as in the previous chapter.

In [8]:
def event_func(state, t, system):
    """Return the height of the penny above the sidewalk.
    y, v = state
    return y

And then run the simulation.

In [9]:
results, details = run_ode_solver(system, slope_func, events=event_func)

Here are the results.

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The final height is close to 0, as expected.

Interestingly, the final velocity is not exactly terminal velocity, which suggests that there are some numerical errors.

We can get the flight time from results.

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t_sidewalk = get_last_label(results) * s

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

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

And velocity as a function of time:

In [14]:
def plot_velocity(results):
    plot(results.v, color='C1', label='v')
    decorate(xlabel='Time (s)',
             ylabel='Velocity (m/s)')

From an initial velocity of 0, the penny accelerates downward until it reaches terminal velocity; after that, velocity is constant.

Exercise: Run the simulation with an initial velocity, downward, that exceeds the penny's terminal velocity. Hint: You can create a new Params object based on an existing one, like this:

params2 = Params(params, v_init=-30 * m/s)

What do you expect to happen? Plot velocity and position as a function of time, and see if they are consistent with your prediction.

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Exercise: Suppose we drop a quarter from the Empire State Building and find that its flight time is 19.1 seconds. Use this measurement to estimate the terminal velocity.

  1. You can get the relevant dimensions of a quarter from

  2. Create a Params object with the system parameters. We don't know v_term, so we'll start with the inital guess v_term = 18 * m / s.

  3. Use make_system to create a System object.

  4. Call run_ode_solver to simulate the system. How does the flight time of the simulation compare to the measurement?

  5. Try a few different values of t_term and see if you can get the simulated flight time close to 19.1 seconds.

  6. Optionally, write an error function and use root_scalar to improve your estimate.

  7. Use your best estimate of v_term to compute C_d.

Note: I fabricated the observed flight time, so don't take the results of this exercise too seriously.

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