AV_modelling_example

AV_modelling_example - The system shown in the figure below...

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Unformatted text preview: The system shown in the figure below consists of a disk of radius r , mass 1 m and moment of inertia G I which is rolled on the ground with no slip. A flexible pendulum of mass 2 m and spring k , of free length L , is attached to the center of the disk as shown in the figure. Denote the length of the spring by ( ) t ρ , and assume that the mass of the spring is negligible. Numerical data: m 4 = R , m 1 = r , kg 120 1 = m , kg 10 2 = m , 2 m kg 60 ⋅ = G I , N/m 1000 = k , m 3 = L (a) Determine the equations of motion by both Newton’s and energy methods (Present your solution in a symbolic form) (b) If for a particular instant rad 1 . 1 = θ , rad/s 3 . 1 = θ & , rad 15 . 2 = θ , rad/s 5 . 2 = θ & m 2 . 3 = ρ , and m/s 5 − = ρ & , what are 1 θ & & , 2 θ & & , and ρ & & for this instant of time? (c) Linearize the equations of motion obtained in (a) (d) What are the poles of the linear system? What are its natural frequencies? R r 1 θ 2 θ ρ g O R r 1 θ 2 θ ρ g O Solution (a) Equations of motion via the energy method To determine the kinetic energy we need to find 2 B v . From the velocity diagram we have: ( ) ( ) 2 1 2 1 cos θ ρ θ θ θ & & + − − = r R v x B ( ) ( ) ρ θ θ θ & & + − − = 1 2 1 sin r R v y B ⇒ ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 2 1 2 1 2 sin cos ρ θ θ θ θ ρ θ θ θ & & & & + − − + + − − = r R r R v B . So that the kinetic energy is ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 sin cos 2 1 2 1 ρ θ θ θ θ ρ θ θ θ θ & & & & & + − − + + − − + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = r R r R m r R r I m T G and the potential energy is ( ) ( ) ( ) ( ) 2 2 2 2 1 2 1 2 1 cos cos 1 L k k g m L g m r R g m m V − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + − − + = ρ θ ρ θ . Equation of motion for 1 θ ( ) 1 θ & r R − ρ & 2 θ ρ & B v 2 x 2 y 1 2 θ θ − ( ) 1 θ & r R − ρ & 2 θ ρ & B v 2 x 2 y 1 2 θ θ − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 2 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 cos sin sin cos sin sin cos sin θ θ θ ρ θ θ θ θ ρ θ θ θ θ ρ θ θ θ θ θ θ θ ρ θ θ θ θ θ − − + − − − − + = − − + − − + − − + − − − − + = ∂ − ∂ & & & & & & & & & & r R m r R m r R g m m r R r R m r R r R m r R g m m T V ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 2 2 2 1 2 2 1 2 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 1 sin cos sin sin cos cos θ θ ρ θ θ θ ρ θ θ θ θ ρ θ θ θ θ θ θ ρ θ θ θ θ θ − − − − − − − − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = − − + − − − − − + − − − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = ∂ − ∂ r R m r R m r R m r R r I m r R r R m r R r R m r R r I m T V G G & & & & & & & & & & ( )...
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This note was uploaded on 12/02/2011 for the course ME 7153 taught by Professor Ram,y during the Fall '08 term at LSU.

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AV_modelling_example - The system shown in the figure below...

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