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Unformatted text preview: 88 Chapter 3. Kinetics of Particles Question 312 A particle of mass m is attached to a linear spring with spring constant K and un stretched length r as shown in Fig. P312. The spring is attached at its other end at point P to the free end of a rigid massless arm of length l . The arm is hinged at its other end and rotates in a circular path at a constant angular rate . Knowing that the angle is measured from the downward direction and assuming no friction, determine a system of two differential equations of motion for the particle in terms of r and . t l m r K O P Figure P312 Solution to Question 312 Kinematics First, let F be a fixed reference frame. Then, choose the following coordinate system fixed in reference frame F : Origin at O E x = Along OP When t = E z = Out of Page E y = E z E x Next, let A be a reference frame fixed to the arm. Then, choose the following coordinate system fixed in reference frame A : Origin at O e x = Along OP e z = Out of Page ( = E z ) e y = e z e x Finally, let B be a reference frame fixed to the direction along which the spring lies (i.e., the direction Pm ). Then, choose the following coordinate system fixed in reference 89 frame B : Origin at O u r = Along Pm u z = Out of Page ( = E z = e z ) u = u z u r The geometry of the bases { E x , E y , E z } , { e x , e y , e z } , and { u r , u , u z } is shown in Fig. 3 10. Using Fig. 310, we have the following relationship between the basis { e x , e y , e z } e t t u r u e y u z , e z , E z E x E y x Figure 310 Geometry of Bases { E x , E y , E z } , { e x , e y , e z } , and { u r , u , u z } for Question 312 . and the basis { u r , u , u z } : e x = cos ( t) u r sin ( t) u e y = sin ( t) u r + cos ( t) u (3.319) Next, observing that the basis { e x , e y , e z } rotates with angular rate relative to the basis { E x , E y , E z } , the angular velocity of reference frame A in reference frame F is given as...
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This note was uploaded on 04/03/2012 for the course AERO 310 taught by Professor Chakravorty during the Spring '07 term at Texas A&M.
 Spring '07
 Chakravorty
 Dynamics

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