# Modeling and Simulation in Python

Case study

Copyright 2017 Allen Downey

License: [Creative Commons Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0)



```python
# 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 modsim.py module
from modsim import *
```

## Yo-yo

Suppose you are holding a yo-yo with a length of string wound around its axle, and you drop it while holding the end of the string stationary.  As gravity accelerates the yo-yo downward, tension in the string exerts a force upward.  Since this force acts on a point offset from the center of mass, it exerts a torque that causes the yo-yo to spin.



This figure shows the forces on the yo-yo and the resulting torque.  The outer shaded area shows the body of the yo-yo.  The inner shaded area shows the rolled up string, the radius of which changes as the yo-yo unrolls.

In this model, we can't figure out the linear and angular acceleration independently; we have to solve a system of equations:

$\sum F = m a $

$\sum \tau = I \alpha$

where the summations indicate that we are adding up forces and torques.

As in the previous examples, linear and angular velocity are related because of the way the string unrolls:

$\frac{dy}{dt} = -r \frac{d \theta}{dt} $

In this example, the linear and angular accelerations have opposite sign.  As the yo-yo rotates counter-clockwise, $\theta$ increases and $y$, which is the length of the rolled part of the string, decreases.

Taking the derivative of both sides yields a similar relationship between linear and angular acceleration:

$\frac{d^2 y}{dt^2} = -r \frac{d^2 \theta}{dt^2} $

Which we can write more concisely:

$ a = -r \alpha $

This relationship is not a general law of nature; it is specific to scenarios like this where there is rolling without stretching or slipping.

Because of the way we've set up the problem, $y$ actually has two meanings: it represents the length of the rolled string and the height of the yo-yo, which decreases as the yo-yo falls.  Similarly, $a$ represents acceleration in the length of the rolled string and the height of the yo-yo.

We can compute the acceleration of the yo-yo by adding up the linear forces:

$\sum F = T - mg = ma $

Where $T$ is positive because the tension force points up, and $mg$ is negative because gravity points down.

Because gravity acts on the center of mass, it creates no torque, so the only torque is due to tension:

$\sum \tau = T r = I \alpha $

Positive (upward) tension yields positive (counter-clockwise) angular acceleration.

Now we have three equations in three unknowns, $T$, $a$, and $\alpha$, with $I$, $m$, $g$, and $r$ as known parameters.  It is simple enough to solve these equations by hand, but we can also get SymPy to do it for us.




```python
from sympy import init_printing, symbols, Eq, solve

init_printing()
```


```python
T, a, alpha, I, m, g, r = symbols('T a alpha I m g r')
```


```python
eq1 = Eq(a, -r * alpha)
```


```python
eq2 = Eq(T - m * g, m * a)
```


```python
eq3 = Eq(T * r, I * alpha)
```


```python
soln = solve([eq1, eq2, eq3], [T, a, alpha])
```


```python
soln[T]
```


```python
soln[a]
```


```python
soln[alpha]
```


The results are

$T      = m g I / I^*   $

$a      = -m g r^2 / I^* $

$\alpha = m g r / I^*    $

where $I^*$ is the augmented moment of inertia, $I + m r^2$.

You can also see [the derivation of these equations in this video](https://www.youtube.com/watch?v=chC7xVDKl4Q).

To simulate the system, we don't really need $T$; we can plug $a$ and $\alpha$ directly into the slope function.


```python
radian = UNITS.radian
m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram
N = UNITS.newton
```

**Exercise:**  Simulate the descent of a yo-yo.  How long does it take to reach the end of the string?

I provide a `Params` object with the system parameters:

* `Rmin` is the radius of the axle.  `Rmax` is the radius of the axle plus rolled string.

* `Rout` is the radius of the yo-yo body.  `mass` is the total mass of the yo-yo, ignoring the string.  

* `L` is the length of the string.

* `g` is the acceleration of gravity.


```python
params = Params(Rmin = 8e-3 * m,
                Rmax = 16e-3 * m,
                Rout = 35e-3 * m,
                mass = 50e-3 * kg,
                L = 1 * m,
                g = 9.8 * m / s**2,
                t_end = 1 * s)
```

Here's a `make_system` function that computes `I` and `k` based on the system parameters.

I estimated `I` by modeling the yo-yo as a solid cylinder with uniform density ([see here](https://en.wikipedia.org/wiki/List_of_moments_of_inertia)).

In reality, the distribution of weight in a yo-yo is often designed to achieve desired effects.  But we'll keep it simple.


```python
def make_system(params):
    """Make a system object.
    
    params: Params with Rmin, Rmax, Rout, 
                              mass, L, g, t_end
    
    returns: System with init, k, Rmin, Rmax, mass,
                         I, g, ts
    """
    unpack(params)
    
    init = State(theta = 0 * radian,
                 omega = 0 * radian/s,
                 y = L,
                 v = 0 * m / s)
    
    I = mass * Rout**2 / 2
    k = (Rmax**2 - Rmin**2) / 2 / L / radian    
    
    return System(init=init, k=k,
                  Rmin=Rmin, Rmax=Rmax,
                  mass=mass, I=I, g=g,
                  t_end=t_end)
```

Testing `make_system`


```python
system = make_system(params)
```


```python
system.init
```

Write a slope function for this system, using these results from the book:

$ r = \sqrt{2 k y + R_{min}^2} $ 

$ T      = m g I / I^*  $

$ a      = -m g r^2 / I^* $

$ \alpha  = m g r / I^*  $

where $I^*$ is the augmented moment of inertia, $I + m r^2$.



```python
# Solution goes here
```

Test your slope function with the initial paramss.


```python
# Solution goes here
```

Write an event function that will stop the simulation when `y` is 0.


```python
# Solution goes here
```

Test your event function:


```python
# Solution goes here
```

Then run the simulation.


```python
# Solution goes here
```

Check the final state.  If things have gone according to plan, the final value of `y` should be close to 0.


```python
# Solution goes here
```

Plot the results.

`theta` should increase and accelerate.


```python
def plot_theta(results):
    plot(results.theta, color='C0', label='theta')
    decorate(xlabel='Time (s)',
             ylabel='Angle (rad)')
plot_theta(results)
```

`y` should decrease and accelerate down.


```python
def plot_y(results):
    plot(results.y, color='C1', label='y')

    decorate(xlabel='Time (s)',
             ylabel='Length (m)')
    
plot_y(results)
```

Plot velocity as a function of time; is the yo-yo accelerating?


```python
# Solution goes here
```

Use `gradient` to estimate the derivative of `v`.  How goes the acceleration of the yo-yo compare to `g`?


```python
# Solution goes here
```


```python

```

b9650019cb423883e9c2cb63835c7b54e3076c44
ipynb
Jupyter Notebook
code/yoyo.ipynb
leilamerz/ModSimPy
b877b6053a461c179643f55d0f1c1d929af0aef8
MIT
2021-08-05T14:06:08.000Z
2021-08-05T14:06:08.000Z
code/yoyo.ipynb
leilamerz/ModSimPy
b877b6053a461c179643f55d0f1c1d929af0aef8
MIT
code/yoyo.ipynb
leilamerz/ModSimPy
b877b6053a461c179643f55d0f1c1d929af0aef8
MIT
Qwen/Qwen-72B
1. YES
2. YES