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# note3 - MEEN 364 The following problems will be covered 1 2...

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MEEN 364 Notes from Session on September 22, 2004 304 Fermier Hall, 6:00~7:00 pm The following problems will be covered. 1. The major mistakes in HW2. 2. Several examples of deriving the governing equation of motion. 3. “Rolling without slipping”. 4. Questions? 1

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MEEN 364 Notes from Session on September 22, 2004 304 Fermier Hall, 6:00~7:00 pm 1) Derive the governing differential equations of the following system: b 2 b 4 2
MEEN 364 Notes from Session on September 22, 2004 304 Fermier Hall, 6:00~7:00 pm Solution: There are basically two steps to obtain the set of differential equations that represent a system: 1. The Kinematics Stage . In this stage, the position, velocity and the acceleration of all the rigid bodies are defined. It can be seen that there are three rigid bodies. The total number of degrees of freedom of the system is three. The degrees of freedom of the system are defined as the horizontal displacements of the three bodies of mass m 1 , m 2 , and m 3 . Let the displacement of the bodies of mass m 1 , m 2 , and m 3 be equal to x 1 , x 2 , and x 3 , respectively. The velocity and acceleration of the body with mass m 1 is given by x ˙ 1 and x ˙˙ 1 respectively; for body m 2 , they are x ˙ 2 and x ˙˙ 2 respectively; and the velocity and the acceleration of the body with mass m 3 , they are x ˙ 3 and x ˙˙ 3 , respectively. This completes the kinematics stage. 2. The Kinetics Stage . In this stage, the Newton’s second law of motion is used to obtain the final governing differential equations of motion. That law, either for translational or rotational motion, is applied separately to each rigid body as if it were free in the space, considering links to other bodies as external forces, in addition to any other force that could be acting on it. To ease this analysis,

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