Modeling of Physical Systems: HW 2–due 9/13/12 Page 1
Problem 1: A motor-driven fan (or blower) is mounted on a steel frame which is rigidly
attached to the floor as shown in Figure 1. The rotor is unbalanced, and forced rotation at
angular velocity, ω, induces a dynamic force on the main shaft. Assume you are given net
weight/mass of the blower, M, and you can measure static deflection of the frame when the
fan is mounted on it. You might also be given or can determine the ‘eccentric’ mass on the
rotor, say m, as well its radial eccentricity, e, from the shaft center.
Figure 1: Unbalanced fan/blower
a. Draw a schematic that represents how you
would model this problem; i.e., using ideal
model elements (masses, spring elements, damp-
ing, etc.). List the specific constitutive relations
for all model elements. Include an input forcing
due to the unbalanced rotor rotation. Discuss
assumptions you would make, and list informa-
tion you would need to have. Justify all physical
effects you include in your model.
b. Prove that you can model the force applied
by the unbalanced rotor by a one-dimensional
dynamic force in the vertical direction (call it
z). Also show that this force can be quantified by, F(t) = Fosin(ωt), and determine how
Foand ωare related to the physical parameters of the problem (parameters for your model,
speed of rotation, etc.).
c. The most fundamental model of the vertical motion of the total fan mass can be repre-
sented by a second order differential equation. Derive this mathematical model.
Figure 2: Basic belt drive
Problem 2: Consider the system shown in Fig-
ure 2. A stepper motor drives pulley A, which
has moment of inertia JA, the timing belt has
total mass m, and pulley B has moment of in-
ertia, JB. There is static friction acting in the
rotational elements (which we can assume move
together) that has been measured at the input
shaft (at A) as, Tf. Assume that the shafts and
the belts are very stiff (negligible compliance).
a. To simplify the model, develop an expression
for the effective rotational inertia, Jeff, seen at
the motor shaft. Use energy and speed relations,
and assume that the two pulleys have equal di-
ameters.
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
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