ase330m-hw2 - the true state of the particle using the...

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ASE 330M – Homework #2 Due: 9/21/2007 Problem #1: For problem #2 in homework 1, a) Find the equations of motion for particle P in terms of rotating unit vectors. b) Let t=0 denote the initial time and R the radius of the disk, where R = 36 inches. Suppose the particle is initially at rest relative to the turntable such that l(0)=0.5*R, and θ (0)=0 rad. Furthermore, the turntable is initially spinning at 1 rev/sec. Write the equations of motion in state space form and numerically integrate the nonlinear equations in MATLAB. Plot the radial position of the particle, l(t), as a function of time. Can you deduce, from this plot, if the particle remains on the turntable for all time? If not, at what point in time is the particle ejected from the turntable? c) Linearize the nonlinear equations of motion using a Taylor Series approximation; let the initial conditions in part(b) represent the reference solution. Perform a numerical integration of the linear equations in MATLAB to obtain the perturbed states as functions of time. Approximate
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Unformatted text preview: the true state of the particle using the reference state and the approximated state. Compare the approximated position of the particle to the true position computed in part(b). Comment on the accuracy of the nonlinear model. Problem #2: For Sample Problem 3.1 of Torby, write the equations of motion in state space form. Numerically integrate these equations in MATLAB using the following assumptions: a) Let the pendulum be initially oriented at a 30 degree angle from the “down” axis b) The spring is initially unstretched and k=10 N/m. c) L = 1 m and the mass is 1 kg Assume the pendulum’s rest position is the reference solution. Linearize the equations of motion about the reference. Define the initial perturbed state so that it corresponds to the nonlinear true state employed in part 2(a). Numerically integrate the linearized equations using these initial conditions. Compare the linear and nonlinear models and comment on the accuracy of the linearized equations....
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