CHAPTER 3 MESB 374 System Modeling and Analysis ME375 Handouts - Spring 2002 CHAPTER 3 MESB 374 System Modeling and Analysis Rotational Mechanical Systems

Rotational Mechanical Systems ME375 Handouts - Spring 2002 Rotational Mechanical Systems Variables Basic Modeling Elements Interconnection Laws Derive Equation of Motion (EOM)

Variables q : angular displacement [rad] ME375 Handouts - Spring 2002 Variables q t (t) J K B q : angular displacement [rad] w : angular velocity [rad/sec] a : angular acceleration [rad/sec2] t : torque [Nm] p : power [Nm/sec] w : work ( energy ) [Nm] 1 [Nm] = 1 [J] (Joule)

Basic Rotational Modeling Elements ME375 Handouts - Spring 2002 Basic Rotational Modeling Elements Spring Stiffness Element Analogous to Translational Spring. Stores Potential Energy. E.g. shafts Damper Friction Element Analogous to Translational Damper. Dissipate Energy. E.g. bearings, bushings, ... tD tD tS B tS q1 q2 t = B q & & t = K q - q - q = B w - w D 2 1 2 1 S 2 1

Basic Rotational Modeling Elements ME375 Handouts - Spring 2002 Basic Rotational Modeling Elements Moment of Inertia Inertia Element Analogous to Mass in Translational Motion. Stores Kinetic Energy. Parallel Axis Theorem Ex: d

Massless or small acceleration ME375 Handouts - Spring 2002 Interconnection Laws Newton’s Second Law Newton’s Third Law Action & Reaction Torque Lever (Motion Transformer Element) d J w = J q & & å = t dt 1 2 3 EXTi i Angular Momentum Massless or small acceleration

Motion Transformer Elements ME375 Handouts - Spring 2002 Motion Transformer Elements Rack and Pinion Gears Ideal case Gear 1 Gear 2

Example Derive a model (EOM) for the following system: ME375 Handouts - Spring 2002 Example Derive a model (EOM) for the following system: FBD: (Use Right-Hand Rule to determine direction) q t (t) J K B J K B

Energy Distribution J q & & + B q & + K q = t ( t ) J q & & × q & + B ME375 Handouts - Spring 2002 Energy Distribution EOM of a simple Mass-Spring-Damper System We want to look at the energy distribution of the system. How should we start ? Multiply the above equation by angular velocity term w : Ü What have we done ? Integrate the second equation w.r.t. time: Ü What are we doing now ? J q & & + B q & + K q = t ( t ) Contributi on Contributi on Contributi on Total of Inertia of the Dam per of the Spr ing Applied To rque J q & & × q & + B q & × q & + K q × q & = t t × w

ME375 Handouts - Spring 2002 Energy Method Change of energy in system equals to net work done by external inputs Change of energy in system per unit time equals to net power exerted by external inputs q t (t) J K B EOM of a simple Mass-Spring-Damper System

Example Rolling without slipping FBD: Elemental Laws: ME375 Handouts - Spring 2002 Example Rolling without slipping FBD: Elemental Laws: Coefficient of friction m x q K R f(t) B J, M Kinematic Constraints: (no slipping) f(t) fs fd ff Mg N q x

ME375 Handouts - Spring 2002 Example (cont.) Q: How would you decide whether or not the disk will slip? I/O Model (Input: f, Output: x): If Slipping happens. Q: How will the model be different if the disk rolls and slips ? does not hold any more

Example (Energy Method) ME375 Handouts - Spring 2002 Example (Energy Method) Rolling without slipping Power exerted by non-conservative inputs Coefficient of friction m x q K R f(t) B Energy equation J, M Kinetic Energy Kinematic Constraints: (no slipping) Potential Energy EOM

Example (coupled translation-rotation) ME375 Handouts - Spring 2002 Example (coupled translation-rotation) Real world Think about performance of lever and gear train in real life