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Unformatted text preview: B.J. Fregly March 20, 2009 PENDULUM DYNAMICS EXAMPLES OVERVIEW Below are three pendulum dynamics problems that demonstrate concepts related to formulation of dynamics equations using the linear and angular moment principles. Each subsequent problem grows in complexity compared to the previous problem. The final equations demonstrate how the selected equations (linear or angular momentum) and directions in which dot products are taken affects the unknowns that appear in the resulting scalar equations. In each of the three examples, the final vector equations and associated scalar equations produced by the linear and angular momentum principles are presented. You should re-derive these equations on your own to verify that you understand how to derive them. EXAMPLE 1: ONE DEGREE OF FREEDOM PENDULUM PARTICLE P N O L n x n y B N b x b y T mg Complete Model Free Body For this first example, particle P is taken as the free body. The possible unknowns that we could find are and T (we know the direction of T since the string from which particle P is suspended is assumed to be massless). Application of the linear momentum principle to particle P produces the following vector equation: [ ] [ ] 2 sin( ) cos( ) x y x y mg T mg mL mL + = + b b b b (1) Dotting Eq. (1) in the x b direction produces sin( ) mg mL = (2) while dotting this equation in the y b direction yields 2 cos( ) T mg mL = (3) Alternatively, dotting Eq. (1) in the x n direction produces 2 sin( ) ( ) sin( ) T m L c o s m L = (4) 2 B.J. Fregly March 20, 2009 and dotting in the y n direction yields 2 cos( ) sin( ) cos( ) T m g m L m L = + (5) Application of the angular momentum principle to particle P about point P produces the following vector equation: z z = b b (6) which is not useful. Alternatively, application of the angular momentum principle to particle P about point O N produces the following vector equation: [ ] 2 sin( ) z z mgL mL = b b (7) Dotting Eq. (7) in the z b direction produces 2 sin( ) mgL mL =...
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This note was uploaded on 06/13/2011 for the course EGM 3400 taught by Professor Matthews during the Spring '08 term at University of Florida.
- Spring '08