530_343lecture04

530_343lecture04 - ME 530.343 Design and Analysis of...

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Unformatted text preview: ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 4 - Fundamentals of Translational and Rotational Mechanical Systems Friday, February 6, 2009 1 Today’s Objectives • Elements of translational mechanical systems • Interconnection laws/Free body diagram for translational systems • Elements of rotational mechanical systems • Interconnection laws/Free body diagram for rotational systems Handout: Problem Set #2 For example problems, see web postings. Reading: Palm 4.1–4.2, 4.4 2 Translational Mechanical Systems 2.1 Basic Variables x , displacement (m) v = ˙ x = dx dt ,velocity ( m s ) a = ˙ v = ¨ x = d 2 x dt 2 ,acceleration ( m s 2 ) f , force (N) these variables are functions of time, t. 2.2 Elements 1. Mass (kg): 1 Newton’s 2 nd Law: sum of forces action on a body = time rate of change of momentum: f = d dt ( mv ) If dm dt = 0 then f = m dv dt f = ma Assumptions: • motions defined with respect to an inertial reference frame • scalar quantities (1 degree of freedom) Energy: • kinetic T = 1 2 mv 2 • potential U = mgh Initial conditions: v ,h . 2. Damping (Ns/m): Also known as viscous friction or linear friction f = B Δ v where Δ v = v 2- v 1 above left: b ∝ contact area & viscosity of oil, 1 α thickness of film above right: b small enough to be neglected (note that this is always an approximation!) Damping is used to model a dashpot, e.g. shock absorbers on car. 2 3. Friction (N) Dry Friction (Coulomb): f =- A for Δ v < A for Δ v > Drag Friction: f = D | Δ v | Δ v Power dissipated by friction = f Δ v This is converted to heat (not stored as potential) 4. Stiffness (N/m) Stiffness element: only element that undergoes a change in shape when subjected to a force, provided that an algebraic relationship exists between elongation and force Most common: ideal spring! d = length of spring when no force is applied x = elongation caused by f d ( t ) = d + x ( t ) f = kx → k Δ x = k ( x 2- x 1 ) (Note that the d ’s cancel out.) The linear spring is an approximation of something like: 3 Multiple applied forces: Structural shaft: k ∝ cross-sectional area and Young’s modulus However, k 1 ∝ length Potential energy: U = 1 2 k (Δ x ) 2 Need to know initial condition x 2.3 Interconnection Laws 1. Newton’s 21....
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This note was uploaded on 03/02/2010 for the course MECH 530.343 taught by Professor Sun during the Spring '08 term at Johns Hopkins.

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530_343lecture04 - ME 530.343 Design and Analysis of...

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