Modeling of Physical Systems: HW 7–due 11/8/12 Page 1
Problem 1: The child’s toy shown below is referred to as “magic man” because the body of
the clown straddles the rolling ball and is stabilized by a counter-mass, ms. The conceptual
schematic provided illustrates how this system may be realized. For the purposes of this
exercise, assume all motion is constrained to the x−zplane, even though in reality there is
usually some yaw motion (rotation about z). The angular motion of the body about the y
axis is quantified by the angle θ.
When the ball rolls on the ground, there is an induced rolling resistance modeled by the
relation, Fr=frW, where fris a dimensionless rolling resistance coefficient and Wis the
effective contact force. This force is applied in the xdirection and always opposes motion of
the body. It is also necessary to model sliding between the ball and the ground, since there
may either be pure rolling or sliding depending on the particular conditions. In addition,
the bearing that supports the body on the ball also introduces friction between these two
bodies.
(a) Develop a bond graph of this system. Note that this system does not have any inputs.
Motion is usually induced by giving the whole system an initial translational velocity. For
this reason, your model should not have any input effort or flow sources.
(b) Assign causality to your system, identify state variables, and derive state equations for
this system.
(c) Simulation problem: The total mass of this system is 0.19 kg, and the ball radius is
3.5 inches. Assume the ball mass alone is 0.1 kg. Estimate the other parameters needed to
simulate the behavior of your system model. For example, you need to estimate the value
of Jbfor the ball (it is a hollow sphere). Assume that the friction coefficient between the
ball and ground is about 0.3. At t= 0, the system is released moving in the xdirection at
a known velocity, 0.25 m/sec. The body pitch angle is zero. Assume that the ball is not
initially rotating (ωb(t= 0) = 0). Use your simulation model to: i) fine tune the unknown
parameters, keeping them realistic, and ii) solve for the response of the system from the time
the ball touches the ground to the point that all motion ceases.
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
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