2D Motion of Rigid Bodies: Falling Stick Example, Work-Energy Principle

Transcription

1 Example: Fallng Stck 1.003J/1.053J Dynamcs and Control I, Sprng 007 Professor Thomas Peacock 3/1/007 ecture 10 D Moton of Rgd Bodes: Fallng Stck Example, Work-Energy Prncple Example: Fallng Stck Fgure 1: Fallng Stck. Stck has mass, M and ength,. Fgure by MIT OCW. What s the equaton of moton f the stck sldes wthout frcton along flat a surface? Applcaton: Jont slppng and sldng. Unform Rod: Center of Mass C. Knematcs Constrant: y B = generalzed coordnates So we choose x C, φ to be the generalzed coordnates because body s rgd. x B s related to x C. Forces not asked for n problem. r C = x C ê x + sn φê y Cte as: Thomas Peacock and Ncolas Hadjconstantnou, course materals for.003j/1.053j Dynamcs and Control I, Sprng 007. MIT OpenCourseWare Massachusetts Insttute of Technology.

2 Example: Fallng Stck Knetcs Free Body Dagram ṙ C = ẋ C ê x + φ cos φê y r C = ẍ C ê x + φ cos φ φ sn φê y Fgure : Free Body Dagram of Fallng Stck. Fgure by MIT OCW. Whch prncples to apply? Want to avod usng N 1 Angular momentum about B near momentum n x No forces n x-drecton Could use work-energy prncple, because N does no work. near Momentum n x-drecton F = x d P dt x 0 = mẍ c Angular Momentum about B d τ B = H B + v B P dt Cte as: Thomas Peacock and Ncolas Hadjconstantnou, course materals for.003j/1.053j Dynamcs and Control I, Sprng 007. MIT OpenCourseWare Massachusetts Insttute of Technology.

3 Example: Fallng Stck 3 d d v B = x B ê x = x C cos φê x dt dt [ ] v B = ẋ C + φ sn φ ê x H B = H C + r C P = I C φ ê z + cos φêx + sn φê y m ẋ c ê x + φ cos φê y m = I C φ ê z + m cos φφ ê z ẋ c sn φê z [ 4 ] m = I C + m cos φ φ ẋ c sn φ ê z 4 m = I C + cos φ φê z 4 Note: We cannot wrte n ths problem that I B ω = H B. Pont B s movng, not statonary. [ Ḣ B = I C + m cos φ φ m cos φ sn φφ m ẍ C sn φ m ] ẋ C φ cos φ 4 4 v B P = ẋ C + φ sn φ ê x m ẋ C ê x + φ cos φê y m v B P = ẋ c m φ cos φ + sn φ cos φφ 4 τ B = mg cos φê z Notce all terms are n the ê z or ê z drecton. ê z ê z Cte as: Thomas Peacock and Ncolas Hadjconstantnou, course materals for.003j/1.053j Dynamcs and Control I, Sprng 007. MIT OpenCourseWare Massachusetts Insttute of Technology.

4 Example: Fallng Stck 4 Fgure 3: Free Body Dagram of Fallng Stck. Fgure by MIT OCW. Therefore: m m m mg cos φ = I C + cos φ φ cos φ sn φφ ẋ C φ cos φ 4 m m + ẋ C φ cos φ + sn φ cos φφ 4 m m = I C + cos φ φ cos φ sn φφ 4 4 m m mg cos φ = I C + cos φ φ cos φ sn φφ = mẍ C Alternatve Approach: Apply Angular Momentum About Pont C, The Center of Mass We should get the same answer by applyng angular momentum about C. d d τ C = H C = Iω = I C φê z dt dt N cos φê z = I C φê z I C φ = N cos φ Cte as: Thomas Peacock and Ncolas Hadjconstantnou, course materals for.003j/1.053j Dynamcs and Control I, Sprng 007. MIT OpenCourseWare Massachusetts Insttute of Technology.

5 Work-Energy Prncple for Rgd Bodes 5 Fgure 4: Free Body Dagram of Fallng Stck. The approach employed here s one that uses angular momentum about C. Fgure by MIT OCW. Smpler expresson but must fnd N. Use lnear momentum n y-drecton. From above: N mg = m φ cos φ φ sn φ I C φ N = cos φ I C φ + mg cos φ = ml cos φ φ cos φ φ sn φ m m mg cos φ = I C + cos φ φ cos φφ 4 4 Notce the equatons are the same. Work-Energy Prncple for Rgd Bodes We need a knetc energy expresson. If you can show that all non-conservatve external forces do no work, V + T = Constant. V s potental energy defned based on center of mass. Cte as: Thomas Peacock and Ncolas Hadjconstantnou, course materals for.003j/1.053j Dynamcs and Control I, Sprng 007. MIT OpenCourseWare Massachusetts Insttute of Technology.

6 Work-Energy Prncple for Rgd Bodes 6 Fgure 5: Rgd Body. Fgure by MIT OCW. 1 T = m v v 1 = m v C + ω ρ v C + ω ρ [ ] 1 = m v C v C + v C ω ρ + ω ρ ω ρ 1 1 = Mv c + v C ω m ρ + m ρ ω ω m ρ = 0 because of center of mass. 1 1 T = MvC + I C ω Cte as: Thomas Peacock and Ncolas Hadjconstantnou, course materals for.003j/1.053j Dynamcs and Control I, Sprng 007. MIT OpenCourseWare Massachusetts Insttute of Technology.