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Unformatted text preview: ASE 330M ‐ Fall 2008: Homework #2 Due Monday, September 29th The diagram below illustrates a particle P of mass m that is constrained to move radially along a recessed channel on a spinning turntable. The motion of the mass is subject to some stiffness and frictional losses, each represented by a linear spring of constant k and a linear damper of constant c, respectively. The unstretched length of the linear spring is roughly 0.10 m. In the diagram, the unit ˆ
vectors ui are fixed on the spinning turntable so that u1 is aligned with the channel, u3 is normal to the ˆ
plane of motion, and u 2 u3 u1 . An inertial coordinate system is defined through unit vectors ei . The relative orientation of the turntable with respect to the inertial frame is given by the angle t such ˆ
that when t 0 , the u i and ei unit vectors are aligned. The orientation illustrated below is a top view of the turntable such that gravity acts into the page (i.e. down). (a) In the 2DOF case, when both r t and t are free to vary, the inertial acceleration of the particle is given by, d e dr OP 2 ˆ ˆ
r r u1 r 2r u2 . dt dt e Suppose that a motor forces the table to spin at a constant rate instead such t Where is a constant parameter. Derive the EOM of particle P relative to O. (b) Select three different initial configurations and numerically simulate the nonlinear behavior of the system using MATLAB. Assume m = 2kg, , k / m 2 2 , c / m 4 / 3 . For example, choose various combinations of small radial displacement and radial rate relative to the equilibrium position. (c) Plot r t and r t vs. time over a 20 second time interval. Comment on the numerically predicted behavior. Are the results consistent with the initial conditions? Are they consistent with what you would expect the behavior of the mass to be? (d) Find a general equation that describes the relation between the spin rate and the dynamic equilibrium position of the mass and linearize about this equilibrium point using any of the methods described in class. (e) Identify the linear state space model associated with this equilibrium point and numerically simulate the dynamics in MATLAB. Compare the difference between the nonlinear perturbed state and that obtained from the linearized approximation. Comment on the accuracy of the results. (f) Repeat parts (a) through (e) when constant . Does the linear model, in this case, accurately represent the true evolution of the nonlinear system. Justify your answers. Problem #2: Find the equilibrium solutions for the following systems x1 x2 x2 sin x1 6 x1 x2 x2 x1 x13 x1 x2 x2 0 (a) (b) (c) ...
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This note was uploaded on 09/25/2011 for the course ASE 18510 taught by Professor Jeannefalcon during the Spring '10 term at University of Texas at Austin.
- Spring '10