AV_modelling_example

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 & & & & & & & & & & ( )...
View Full Document

## 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.

### Page1 / 10

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online