lec12 (1)

# lec12 (1) - Example Rolling Cylinder Inside A Fixed Tube 1...

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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, 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]. 1 Example: Rolling Cylinder Inside A Fixed Tube 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 3/19/2007 Lecture 12 2D Motion of Rigid Bodies: Rolling Cylinder and Rocker Examples Example: Rolling Cylinder Inside A Fixed Tube Initial Conﬁguration: Figure 1: Initial conﬁguration of rolling cylinder inside ﬁxed tube. Figure by MIT OCW. Derive the equations of motion. Assume no slip. Displaced conﬁguration: Figure 2: Displaced conﬁguration of rolling cylinder inside ﬁxed tube. Figure by MIT OCW.

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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, 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]. 2 Example: Rolling Cylinder Inside A Fixed Tube Kinematics May not know everything but at least choose generalized coordinates. How many generalized coordinates? 3 coordinates initially. 2 constraints. 1: Rolling on inside of cylinder. 2: No slip. Only need 1 generalized coordinate: either φ or θ . We will choose θ . Recognize angular velocity ω = φ ˙ e ˆ z . Must express φ in terms of θ . No-slip condition: Figure 3: Kinematic diagram of rolling cylinder inside ﬁxed tube. Figure by MIT OCW. ( R r ) ˙ ( R r ) θ ˙ = r ( θ + φ ) φ = r θ ; φ = r Kinetics 3 diﬀerent methods of solution. Angular momentum about B Conservation of energy Angular momentum about C
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, 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]. Figure 4: Kinematic diagram of cylinder in ﬁxed tube. Point B is an imaginary particle marking the point of contact. Point B moves. B on cylinder. B on tube. If you choose B , at a later point in time B would have moved away from the contact marker B. Likewise B . Figure by MIT OCW. ± ² ³ 3 Example: Rolling Cylinder Inside A Fixed Tube Angular Momentum about B d τ B = H B + v B × P dt v B = ˙ e ˆ θ mv c ˙ e ˆ θ = m ( R r ) θ ˙ e ˆ θ Therefore: v B × P = 0 H B : H C + r BC × P r BC × P : r BC is perpendicular to P . rm

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## This note was uploaded on 11/29/2011 for the course CIVIL 1.018j taught by Professor Markusbuehler during the Fall '08 term at MIT.

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lec12 (1) - Example Rolling Cylinder Inside A Fixed Tube 1...

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