Download Modeling of Physical Systems: Homework 2 and more Exercises Mathematical Modeling and Simulation in PDF only on Docsity! 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 Fo and ω are 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 Docsity.com
