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Unformatted text preview: AE 352: Aerospace Dynamics II, Fall 2008 Homework 7 Due Friday, October 24 Problem 1. Write down the constraint equations for each of the following descrip- tions. Note that you must first identify the generalized coordinates. In your answer, also specify the number of degrees of freedom before and after the constraint, and whether the constraints are holonomic or nonholonomic. a) A particle is constrained to lie in a horizontal plane. b) A particle is attached to a massless inextensible rod in a vertical plane. The end of the rod opposite the mass is constrained to be attached to a pivot located at a particular x,y coordinate. c) A particle moving in a vertical plane is constrained to be a distance l from the origin. d) A unicycle (thin disc of mass m ) is constrained to roll without slipping on a horizontal surface (ignore the z-direction, i.e. assume the disc is always in contact with a horizontal plane). Problem 2. Use Lagrange multipliers to find the tension in the simple pendulum. Problem 3. Greenwood 6-14. Solution. Part (a): To begin, we note that our generalized coordinates are r, . And that we have one holonomic constraint: r = 0. Thus we have one DOF: . In this part, we just care about equations of motion, so we will use the standard form of Lagranges equations (no multipliers), and we will make use of the holonomic constraints by considering r = 0. First, we must find the kinetic and potential energy of the particle. Lets define a coordinate system e r , e which has its origin at O , and is rotating such that e r points towards the particle. Lets also define a coordinate system e 1 , e 2 with origin at O where e 1 points towards O . Then we can write down an expression for the position of the particle: r = r O /O + r O /P = r 3 e 1 + r e r And the velocity of the particle is:...
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This note was uploaded on 09/16/2009 for the course AE ae352 taught by Professor Sri during the Spring '09 term at University of Illinois at Urbana–Champaign.
- Spring '09