MATHEMATICAL METHODS FOR ENGINEERS

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MATHEMATICAL METHODS FOR ENGINEERS 1 (ENR 114)
Group Project, SP3 2020
See unit website for due date
Ground Rules for Students

  1. This is a group project, and each group has exactly two students.
  2. The two students must share the work equally between them.
  3. You have limited time to find a partner. If you do, inform the course coordinator by email or by a message through the web site. Students who do not find a partner will be
    randomly assigned to a group.
  4. Students may be given permission to do it individually. However, if a student elects to do
    the project individually, then that student takes full responsibility for the extra workload.
    The same applies to students who, through their own actions, end up doing the project on
    their own.
  5. Collaboration between partners will take place online, through email, OneDrive, and Lync
    (a Virtual Classroom tool).
  6. Troubleshooting. It sometimes happens that one partner feels that they are carrying
    the group, and that the second partner is not contributing. When that happens, the unit
    coordinator needs to be informed immediately.
  7. Troubleshooting. It sometimes happens that two partners do not get along, even though
    both are contributing to the project. If you feel uncomfortable in your group, inform the
    unit coordinator as soon as possible.
  8. Most importantly, one report (including mfile) per group must be submitted. The front
    page of the project must have the name of both partners.
    1
    A Bouncing Ball
    Consider a ball allowed to fall onto a hard surface and rebound, continuing to bounce until it is
    at rest. In the analysis undertaken here, the ball is dropped, i.e. its initial velocity is zero, and
    the surface it lands on is perpendicular to its flight. A ball with high resilience, for example,
    a new table tennis ball, retains most of its kinetic energy following the collision of ball and
    surface, and it rebounds to a high proportion of its original height.
    A simple model of the ball’s behaviour views the movement of the ball in one-dimension, and
    would ignore air resistance. There is loss of energy in the collision; some energy is transformed
    into sound, some into a slight rise in the heat of the ball following the deformation and compression it experiences during the impact, and some energy may impart a spin to the ball. The
    dissipation of energy can be accumulated into the term resilience, a characteristic of various
    types of balls. The main factors governing the height of the ball’s subsequent bounce are; its
    initial height (and thus velocity), the force of gravity, and the ball’s resilience. One measure of
    resilience is called the coefficient of restitution, commonly given the symbol e.
    2
    Task 1
    We assume that the potential energy of the ball at its initial position ( P.E. = mgh , where
    m is the mass of the ball, g is the gravitational constant (take g = 9.81 m/s2 here), and h is
    the height of the ball) is entirely converted to kinetic energy (K.E. =
    1
    2
    mv2
    , where v is the
    velocity of the ball just before it hits the surface).
    Note: it may be easier to take the positive direction as being downward for this problem.
    a) Show that the ball’s velocity just before it hits the surface is v =

    2gh .
    Denote the maximum height of the ball before each bounce with a subscript matching the
    number of the following bounce. Let the velocity of the ball just before its first bounce be
    v1 =

    2gh1 , then its velocity just before its second bounce is v2 =

    2gh2 . By symmetry,
    this has the same magnitude as the ball’s velocity just after its first bounce, and the coefficient
    of restitution is calculated as the ratio of these, i.e. e =
    |v2|
    |v1|
    . Note that 0 < e < 1 where
    e = 1 would be perfect elasticity.
    b) Show that e =
    r
    h2
    h1
    .
    Since we have constant acceleration while the ball either falls or bounces upward, the time
    for the ball to fall from its initial position to impact with the surface can be calculated as
    ∆t =
    vf − vi
    a
    , where vf is velocity at the end of the time interval, vi
    is velocity at the start
    of the time interval, and a is the acceleration applied.
    c) Show that the time for the ball to fall from its initial position until impact is t1 =
    r
    2h1
    g
    .
    d) If t2 is the time for the ball to fall from (or to reach) its subsequent maximum rebound height,
    show that t2 = et1 .
    Note that the flight time between the first and second bounces is 2t2 .
    e) Generalise the recurrence relation in part d) to show that the time until the ball comes to rest
    is
    T = t1 + 2t2 + 2t3 + … = t1 + 2et1
    X∞
    i=0
    e
    i =
    s
    2h1
    g
    
    1 + e
    1 − e
    
    .
    This equation for T says that although the model allows for an infinite number of bounces,
    they occur within finite time. And that time until the ball comes to rest depends on the square
    root of the ball’s initial height, which seems reasonable.
    3
    Task 2
    a) Demonstrate that the time when the ball attains maximum rebound height is
    t(hi) = X
    i−1
    j=1
    tj (1 + e)
    for t(h2), t(h3) and t(h4).
    I do not have experimental data to provide so the following plot only illustrates a theoretical
    model. A competition-approved table tennis ball (mass 2.7g, diameter 40mm) must have a
    coefficient of restitution (using a steel block as the surface) of 0.89 to 0.92, use a value of
    e = 0.9 here. Take initial height as 0.5 m.
    b) Construct a for loop in MATLAB to calculate the maximum rebound heights and the time
    they occur until the ball is at rest. Use the scatter command to plot this data using circles
    to indicate the position of the ball. Have the colour of the circles grade from yellow to red
    across the plot. Comment on the features of the graph.
    c) Does the time to rest calculated using the equation in part e)of Task 1, and estimated from
    the graph of part b) above agree? Comment on this.
    Task 3
    The relationship ti = eti−1 says that the duration of a rebound is proportional to the duration
    of the preceding rebound. This implies an exponential relationship (in this case, exponential
    decay) between successive rebound durations.
    a) Calculate the natural log of rebound durations (t1, t2, …) and produce a scatter plot in MATLAB of these values plotted against bounce number. Fit a line (use MATLAB ) to the plotted
    points and display the equation of the line on the plot. Comment on the result.
    b) Calculate the time between successive bounces and the time when the first of these bounces
    takes place. Produce a scatter plot of these data in MATLAB . Fit a line (use MATLAB )
    to the plotted points and display the equation of the line on the plot. The slope of this line
    should be e − 1.
    4

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