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This chapter provides an in-depth exploration of modeling and analysis techniques for rotational mechanical systems. It covers essential variables such as angular displacement, velocity, acceleration, torque, power, and work. Key modeling elements including dampers, springs, and inertia are discussed, along with the application of Newton’s laws and energy distribution methods. By deriving example equations of motion for various systems and analyzing energy changes, this guide equips readers with the necessary tools to understand and effectively model rotational dynamics.
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CHAPTER 3MESB 374 System Modeling and Analysis Rotational Mechanical Systems
Rotational Mechanical Systems • Variables • Basic Modeling Elements • Interconnection Laws • Derive Equation of Motion (EOM)
J q K t (t) B Variables • 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 • Damper • Friction Element • Analogous to Translational Damper. • Dissipate Energy. • E.g. bearings, bushings, ... • Spring • Stiffness Element • Analogous to Translational Spring. • Stores Potential Energy. • E.g. shafts tD tD tS B tS q1 q2 & & t = q - q = w - w B B t = q - q K D 2 1 2 1 S 2 1
d Basic Rotational Modeling Elements • Parallel Axis Theorem • Ex: • Moment of Inertia • Inertia Element • Analogous to Mass in Translational Motion. • Stores Kinetic Energy.
Interconnection Laws • Newton’s Second Law • Newton’s Third Law • Action & Reaction Torque • Lever (Motion Transformer Element) d å & & w = q = t J J EXTi 1 2 3 dt i Angular Momentum Massless or small acceleration
Motion Transformer Elements • Rack and Pinion • Gears Ideal case Gear 1 Gear 2
J q K t (t) B Example Derive a model (EOM) for the following system: FBD: (Use Right-Hand Rule to determine direction) J K B
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 termw : Ü What have we done ? • Integrate the second equation w.r.t. time:Ü What are we doing now ? & & & q + q + q = t J B K ( t ) Total Contributi on Contributi on Contributi on Applied To rque of Inertia of the Dam per of the Spr ing & & & & & & q × q + q × q + q × q = t × w J B K t
J q K t (t) B 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 • EOM of a simple Mass-Spring-Damper System
Rolling without slipping FBD: Elemental Laws: x x q q fs f(t) fd Mg ff N Example Coefficient of friction m K R B f(t) J, M Kinematic Constraints: (no slipping)
Example (cont.) I/O Model (Input: f, Output: x): Q: How would you decide whether or not the disk will slip? If Slipping happens. Q: How will the model be different if the disk rolls and slips ? does not hold any more
Rolling without slipping Kinetic Energy x q Example (Energy Method) Power exerted by non-conservative inputs Coefficient of friction m K R B f(t) Energy equation J, M Kinematic Constraints: (no slipping) Potential Energy EOM
Example (coupled translation-rotation) Real world Think about performance of lever and gear train in real life
