exam2_recon - 10 pts 3 A uniform rod of length L and mass m...

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2.003 Monday 30 April 2006 10 pts 1. A mass m slides without friction along a horizontal guide as shown in figure 1. Motion is restrained by a linear spring of stiffness k and unstretched length L/ 2. Determine the equation of motion for the mass using its horizontal displacement x as a generalized coordinate. Figure 1: 1 Cite as: Thomas Peacock, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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10 pts 2. A uniform rod of length L and mass m 2 is attached by way of a frictionless pin to a sliding collar of mass m 1 (figure 2). The collar slides without friction on a horizontal guide and is further restrained by two identical springs of stiffness k/ 2 ( x is the extension/compression of the left/right springs respectively). A force of magnitude F is applied to the rod in such a way that it is always perpendicular to the rod. Derive the equations of motion. Figure 2: 2 Cite as: Thomas Peacock, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Unformatted text preview: 10 pts 3. A uniform rod of length L and mass m 1 is attached to a cart of mass m 2 by means of a spring with spring constant k (figure 3). There is a constant force F that is always applied perpendicular to the free end. The nonlinear equations of motion for this system in terms of the generalized coordinates θ and x are: ¨ L L L θ : I θ + m 1 g sinθ − k cosθ ( x − sinθ ) = FL (7) 2 2 2 L x : m 2 x ¨ + k ( x − sinθ ) = 0 (8) 2 Identify the equilibria and derive the linearized equations for small perturbations about the equilibria (you do not have to solve these equations). Figure 3: Cite as: Thomas Peacock, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]....
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