To determine Determine the resulting acceleration of the unloaded rack. Explanation Show the free body diagram of the gear A as in Figure (1). Since the gear B and gear C is keyed to the same shaft, the angular acceleration of the gear B and gear C is equal. α B = α C = α B C Here, the angular acceleration of gears B and C is α B and α C respectively and the angular acceleration of the gear B and gear C is α B C . Write the relation between angular acceleration of the gear A and angular acceleration of the gear B and C . d A 2 α A = − d B 2 α B C α B C = − d A α A d B Here, the angular acceleration of the gear A is α A , pitch diameter of the gear B is d B and pitch diameter of gear A is d A . Express the acceleration relation for rack. a = d C 2 α B C (I) Here, the resulting acceleration of the unloaded rack is a and pitch diameter of gear C is d C . From the Figure (1), take moments about point A . ∑ M A = I A α A M − d A 2 F A B = I A α A M − I A α A = d A 2 F A B F A B = 2 d A ( M − I A α A ) Here, the input torque is M , the centroidal mass moment of inertia of the gear A is I A and the force acting on the gear A and gear B is F A B . Show the free body diagram of the gear B and C as in Figure (2). From the Figure (2), take moments about the center of the gear. ∑ M B C = ( I B + I C ) α B C − d B 2 F A B + d C 2 F C D = ( I B + I C ) α B C (II) Here, the centroidal mass moment of inertia of the gear B is I B , the centroidal mass moment of inertia of the gear C is I C , the force acting on the gear C and rack D is F C D and pitch diameter of gear C is d C . Show the free body diagram of the rack D as in Figure (3). From the Figure (3), resolve the forces in the vertical direction. ∑ F y = m D a − F C D = m D a (III) Here, the mass of the rack D is m D . Conclusion: Substitute − d A α A d B for α B C in Equation (I). a = d C 2 [ − d A α A d B ] α A = − 2 a d B d C d A Substitute 2 d A ( M − I A α A ) for F A B in Equation (II)

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 6.4, Problem 63P

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Determine the resulting acceleration of the unloaded rack.

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Chapter 6.4, Problem 62P

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