1 / 14

CHAPTER 3 MESB 374 System Modeling and Analysis

CHAPTER 3 MESB 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]

reese
Download Presentation

CHAPTER 3 MESB 374 System Modeling and Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CHAPTER 3MESB 374 System Modeling and Analysis Rotational Mechanical Systems

  2. Rotational Mechanical Systems • Variables • Basic Modeling Elements • Interconnection Laws • Derive Equation of Motion (EOM)

  3. 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)

  4. 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

  5. d Basic Rotational Modeling Elements • Parallel Axis Theorem • Ex: • Moment of Inertia • Inertia Element • Analogous to Mass in Translational Motion. • Stores Kinetic Energy.

  6. 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

  7. Motion Transformer Elements • Rack and Pinion • Gears Ideal case Gear 1 Gear 2

  8. 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

  9. 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

  10. 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

  11. 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)

  12. 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

  13. 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

  14. Example (coupled translation-rotation) Real world Think about performance of lever and gear train in real life

More Related