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Roller Coaster

Example – Mechanics

Last edited: 19th of November 2017


Introduction

In a series of laboratory sessions in a physics course at NTNU, the students are studying the rolling motion of objects on a curved track. Due to the curvature, the velocity and acceleration will vary. The students uses a high speed camera to record the motion and compare the results with numerical simulations.

In this notebook, we will simulate a rolling ball on some arbitrary track in two dimensions by solving Newton's second law. The track is made by performing cubic spline interpolation on a set of points. The setup will be based on the setup used in the aforementioned laboratory sessions. For more information, see [1].

First, we import neccesary library packages.

In [1]:
import numpy as np
import scipy.interpolate as interp
import matplotlib.pyplot as plt
plt.style.use('bmh')    # Nicer looking plots

Theory

Equation of Motion

An object, such as a ball, is rolling on a curved track. The rotation axis passes thorugh the center of mass. Let $m$ be the mass, $r$ be the radius of the ball and $I_n$ the moment of inertia. The forces on the center of mass is described by Newton's second law

\begin{equation} \mathbf F = m\mathbf a. \end{equation}

The forces on the ball can be decomposed into a component parallel to the track and a component normal to the track, as shown in figure 1. Let the local slope of the track be described by an angle $\theta(x)$. The gravitational force acts on the center of mass vertically downwards. This amounts to a component parallel to the track $mg\sin \theta(x)$ and a normal component $mg\cos \theta(x)$. The forces from the track on the ball have a normal component $N$ (normal force) and the parallel component $f$ (friction).

Figure 1: A cylinder or sphere is rolling on curve $y(x)$. The forces acting on the object are indicated on the figure. At each point $x$, the slope of the curve is defined by an angle $\theta(x)$.

Due to the curvature of the track, the center of mass has a centripetal acceleration normal to the track. The total acceleration is thus given by

\begin{equation} \mathbf a = \dot v \mathbf{e}_\parallel + \frac{v^2}{R(x)} \mathbf{e}_\perp, \end{equation}

where $R(x)$ is the radius of the curvature and $v$ is the speed (along the track). Thus, we obtain the two equations

\begin{equation} mg \sin \theta(x)-f=m\dot v, \label{eq:parallel} \end{equation} \begin{equation} N-mg\cos\theta(x) = \frac{mv^2}{R(x)}. \label{eq:normal} \end{equation}

The first equation describes the motion parallel to the track, while the second equation yields, with $N\geq 0$, a condition for when the ball loses contact with the surface.

An expression for the friction is found by using Newton's second law of rotation,

\begin{equation} \tau = I_0\omega. \end{equation}

The friction $f$ is the only force that do not pass through the rotation axis, and is thus the only force that contribute to the total torque $\tau$. The friction acts on the object a distance $r$ from the rotation axis (and at an angle $\pi/2$), so that $fr = I_0 \omega$. The ball is assumed to roll without any gliding. By using the rolling condition $v=\omega r$, equation \eqref{eq:parallel} can be written as

\begin{equation} \dot v = \frac{g\sin (\theta(x))}{1 + I_0/mr^2}. \label{eq:vdot} \end{equation}

A complete discussion on rotational dynamics can be found in e.g. [chap. 9-10, 2] and [chap. 6, 3].

Curvature

Consider the curve from $y(x)$ from $A$ to $B$ as shown in figure 2. The curve from $A$ to $B$, with an arclenth $\Delta s$, can be considered as a small circle sector with angle $\Delta \theta$. The circle has a centre at $C$ and a radius $R\approx R_A\approx R_B$. The radius of the curvature thus becomes $R=\Delta s/\Delta \theta$. The curvature is in turn defined as $\kappa= 1/R$.

Figure 2: Sketch used to describe curvature and the radius of the curvature. (Taken from [1])

Consider now a infinitesimal arclenth $\Delta s \to\text{d} s$. From the figure one can see that

\begin{equation} \text{d}y/\text{d} x = \tan\theta, \label{eq:theta} \end{equation}

which gives

$$\frac{\text{d} \theta}{\text{d}x} = \frac{\text{d}}{\text{d}x}\arctan\left(\frac{\text{d}y}{\text{d}x}\right)=\frac{1}{1+\left(\text d y/\text dx\right)^2}\frac{\text d^2 y}{\text dx^2}.$$

The differential $\text d s$ is given by

$$(\text{d}s)^2 = (\text d x)^2 + (\text d y)^2 \quad \Longrightarrow \quad \text d s=\text d x\sqrt{1 + \left(\text d y/dx\right)^2}.$$

The curvature can thus be written as

\begin{equation} \kappa =\frac{\text{d}\theta}{\text{d}s} = \frac{\text{d}^2 y/\text{d}x^2}{\left(1 +\left(\text d y/\text d x\right)^2\right)^{3/2}}, \end{equation}

and the radius of the curvature becomes

\begin{equation} R(x) = \frac{\text{d}s}{\text{d}\theta} = \frac{\left(1 +\left(\text d y/\text d x\right)^2\right)^{3/2}}{\text{d}^2 y/\text{d}x^2}. \label{eq:R} \end{equation}

Note that $\kappa$ is always finite as long as $\text d y/\text d x$ is continuous, while $R\to \infty$ if $\text d^2 y/\text d x^2\to 0$.

Curvature is discussed in more detail in e.g. [chap. 11, 4] and [chap. 13, 5].

Setup

We will be considering a solid sphere (ball), which has a moment of inertia $I_0= 2mr^2/5$. The moment of inertia is easily computed for such objects, but this will not be discussed here. See e.g. [chap. 9, 2] for a general discussion and [6] for a list of some typical values for $I_0$. Assume that the ball has a radius $r=1\,\text{cm}$ and is made of iron with a density $\rho=7850\,\text{kg/m}^3$ (density found in [7]). The mass of the ball is in this case $m= 32.9\,\text{g}$ and the moment of inertia is thereby $I_0=13.2\,\mathrm{g\,cm^2}$.

In [2]:
# Properties of the rolling object
r = 0.01                 # m      (radius)
rho = 7850               # kg/m^3 (density)
g = 9.81                 # m/s^2  (gravitational acceleration)

m = (4/3)*np.pi*r**3*rho # kg     (mass)
c = 2/5
I0 = c*m*r**2            # kg m^2 (moment of inertia)

Figure 3: The actual setup used in the laboratory frame is shown to the left. A sketch is shown to the right. (Right image taken from [1])

The track is made by a plastic bar which is attached to a solid frame at $N=8$ different mounts, as shown in figure 3. Let $(x_i, y_i)$ be the position of the $i$th mount. The $x$ positions are fixed and uniformly distributed across the frame. We will assume that the frame has a length $L=1.4\,\text{m}$, such that the distance between the mounts is $l = 20\,\text{cm}$. The $y$ positions can be changed.

In [3]:
# Properties of the frame
L = 1.4                   # m (length)
yi = [.5, .3, .25, .3,
     .35, .34, .25, .15]  # m (y-positions)

N = len(yi)               #   (# of mounts)
xi = np.linspace(0, L, N) # m (x-positions)

The track can be described by a cubic spline that interpolates $(x_i, y_i)$. The cubic spline consists of a set of cubic polynomials that has continuous first and second derivaties at the interpolation points. See our notebook on Cubic Splines for a general discussion. We will be using the function CubicSpline from scipy.interpolate to perform the interpolation. This creates a callable class, which we can treat as a function $y(x)$ that describes the track.

In [4]:
# Callable class for the track curve
get_y = interp.CubicSpline(xi, yi, bc_type="natural")

As we saw earlier, $\theta$ and $R$ depends on $\text dy/\text dx$ and $\text d^2y/\text dx^2$. The class CubicSpline has a function called derivative which returns a class PPoly, which essentially is the derivative of the spline. PPoly has also a function derivative. We can thus easily compute $\theta(x)$ and $R(x)$ by using equations \eqref{eq:theta} and \eqref{eq:R}.

In [5]:
get_dydx = get_y.derivative()      # Callable derivative of track
get_d2ydx2 = get_dydx.derivative() # Callable double derivative of track

def get_theta(x):
    """ Returns the angle of the track. """
    return -np.arctan(get_dydx(x))
    
def get_R(x):
    """ Returns the radius of the curvature. """
    return (1 + (get_dydx(x))**2)**1.5/get_d2ydx2(x)

def get_curvature(x):
    """ Returns the curvature (1/R). """
    return get_d2ydx2(x)/(1 + (get_dydx(x))**2)**1.5

Let's plot the track!

In [6]:
x = np.linspace(xi[0], xi[-1], 200)

# Create figure
fig, axarr = plt.subplots(3, 1, sharex=True, figsize=(12, 9), dpi=600)
fig.subplots_adjust(hspace=0.02)

# Axes 1:
axarr[0].plot(x, get_y(x), 'C0', label=r"$y(x)$")
axarr[0].plot(xi, yi, 'C1o', label="Mounts")
axarr[0].set_ylabel(r"$y(x)$, [m]", size='15')
#axarr[0].set_aspect('equal')

# Axes 2: 
axarr[1].plot(x, get_theta(x), 'C0')
axarr[1].set_ylabel(r"$\theta(x)$, [rad]", size='15')

# Axes 2: 
axarr[2].plot(x, get_curvature(x), 'C0')
axarr[2].set_ylabel(r"$\kappa(x)$, [1/m]", size='15')

plt.show()

We are assuming that there is no loss of mechanical energy. Thus, if the highest point of the track is $y(x=0)$, then the ball will fall of to the right. If this is not the case, the ball will roll back and forth.

Numerics

We start by assuming that the ball is always in contact with the track. Equation \eqref{eq:vdot} yields a coupled set of ordinary differential equations (ODE)

$$\frac{\text dv}{\text dt} = \frac{g\sin (\theta(t))}{1 + I_0/mr^2},$$

where $v$ is the momentary velocity along the track. The $x$ position is in turn given by

$$ \text dx = \text ds \cos(\theta)\: \Longrightarrow \: \frac{\text dx}{\text dt} = v\cos(\theta).$$

In [7]:
def get_vdot(theta):
    """ Returns the time derivative of the (total) velocity. """
    return g*np.sin(theta)/(1 + c)

def RHS(z):
    """ Evaluates the right hand side of the two coupled 
    ODEs given in the text.
    
    Parameters:
        z : array-like, len(2), float. [v, x]
            The first element is the velocity, the second is the x-position.
        
    Returns:
        array-like, len(2), float. [a, v]
        The first element is the time derivative of the velocity (acceleration),
        the second is the time derivative of the position (velocity).
    """
    # z = [x, v]
    # w = [vx, a]
    w = np.zeros(2)
    theta = get_theta(z[0])
    w[0] = z[1]*1/np.sqrt(1+np.tan(theta)**2)
    w[1] = get_vdot(theta)
    return w

$v(t)$ and $x(t)$ can be found by applying a method for solving ODEs, such as an Euler method, a Runge-Kutta method or a more advanced addative method. We refer you to the respective modules or an example that solves an ODE at numfys.net. In this notebook we will be using the 4th order Runge-Kutta method.

In [8]:
def rk4step(f, y, h):
    """ Performs one step of the 4th order Runge-Kutta method.
    
    Parameters:
        f : Callable function with one input parameter.
            The right hand side of the ODE. Note that the
            RHS is in our case not a function of time.
        y : array-like, float. Current position.
        h : float. Time step.
    """
    s1 = f(y)
    s2 = f(y + h*s1/2.0)
    s3 = f(y + h*s2/2.0)
    s4 = f(y + h*s3)
    
    return y + h/6.0*(s1 + 2.0*s2 + 2.0*s3 + s4)

We choose $x=0$ and $v=0$ as initial conditions, and a time-step $\Delta t = 10^{-3}\, \text{s}$. We have assumed that the ball do not loose any mechanical energy. One method of checking whether the numerical computation gives realistic results is to check if the total mechanical energy

\begin{equation} E = \frac{1}{2}mv^2 + mgh + \frac{1}{2}I_0 \omega^2 = \frac{1}{2}m(1 + c)v^2 + mgh, \end{equation}

is constant.

In [9]:
dt = 1e-3 # s
tstop = 5 # s. If the ball has not reached the end within 5 seconds, we stop.
x0 = 0    # m. Initial x-position
v0 = 0    # m/s. Inital velocity

def get_K(v):
    """ Returns the kinetic energy. """
    return .5*m*(1 + c)*v**2

def get_U(h):
    """ Returns the potential energy. """
    return m*g*h

Everything is now set to roll the ball down the track.

In [10]:
# Set initial values
x = [x0]        # x-position
v = [v0]        # velocity
h = get_y(x0)   # height
K = [get_K(v0)] # kinetic energy
U = [get_U(h)]  # potential energy

it = 0 # Iterator
itmax = tstop/dt # Maximum number og iterations
while x0 <= L and it < itmax:
    # Perform one step of the Runge-Kutta method
    [x0, v0] = rk4step(RHS, [x0, v0], dt)
    # Append different values to their arrays
    x = np.append(x, x0)
    v = np.append(v, v0)
    h = get_y(x0)
    K = np.append(K, get_K(v0))
    U = np.append(U, get_U(h))
    it += 1

print("Iterations: %i"%(it))
print("Final time: %.2f s"%(it*dt))
dE = (K[0] - K[-1] + U[0] - U[-1])/(K[0] + U[0])
print("Relative change in mechanical energy: %.2e"%(dE))
Iterations: 1075
Final time: 1.07 s
Relative change in mechanical energy: -1.08e-09

The relative change in the mechanical energy was minimal, which means that the time step was more than small enough. The ball used $1.07\,\text{s}$ to reach the end of the track.

Let's plot the computed quantities!

In [11]:
t = np.linspace(0, it*dt, it + 1)

# Create figure
fig, axarr = plt.subplots(3, 1, sharex=True, figsize=(10, 8), dpi=400)
fig.subplots_adjust(hspace=0.02)

# Axes 1:
axarr[0].plot(t, x)
axarr[0].set_ylabel(r"$x(t)$, [m]")

# Axes 2: 
axarr[1].plot(t, v)
axarr[1].set_ylabel(r"$v(t)$, [m/s]")

# Axes 2: 
axarr[2].plot(t, U, label="Potential energy, $U$")
axarr[2].plot(t, K, label="Kinetic energy, $K$")
axarr[2].plot(t, K + U, label="Mechanical energy, $E=U+K$")
axarr[2].set_ylabel(r"Energy, [J]")
axarr[2].set_xlabel(r'$t$, [s]')
axarr[2].legend()

plt.show()