a LECTURE 6-IV Kinematic Analysis of Mechanisms 溷 6 6 SHANG 1日gG ERSITY
LECTURE 6-IV Kinematic Analysis of Mechanisms Kinematic Analysis of Mechanisms
OUTLINE Some important definitions Simple cases stud小y Coriolis acceleration Vector graphical analysis method Complex vector analytical method ME357 Design Manufacturing ll
OUTLINE Some important definitions Simple cases study Coriolis acceleration V hi l l i h d Vector graphical analysis met h o d Com p y lex vector anal ytical method ME357 Design & Manufacturing II
Some important definitions Displacement R=Reio Linear displacement:All particles of a body move in parallel planes and travel by same distance is known as linear displacement Angular displacement:A body rotating about a fixed point in such a way that all particles move in circular path is known as angular displacement y AImaginary axis Real axis ME357 Design Manufacturing ll
Some important definitions Displacement θ Linear displacement: All particles of a body move in parallel planes d t l b di t i k li di l t j R e θ R = an d travel by same di s tance is known as linear displacemen t Angular displacement: A body rotating about a fixed point in such a way that all particles move in circular path is known as angular way that all particles move in circular path is known as angular displacement ME357 Design & Manufacturing II
Some important definitions Velocity -Rate of change of displacement is velocity.Velocity can be linear velocity of angular velocity. First order: 出Re=e+Ree+ie do Angular Velocity:@ dt yAImaginary axis dR Linear Velocity:V= R dt Real axis R
Some important definitions Velocity - Rate of change of displacement is velocity. Velocity can be linear velocity of angular velocity can be linear velocity of angular velocity. First order: First order: ( ) ( )= d jj j j j R e Re R e j Re R j e dt θ θθ θ θ =+ + θ θ dt A l V l it dθ Angu lar V e locity: dt ω = Linear Velocity: d dt V = R
Some important definitions Acceleration-Rate of change of velocity Second order: 品Re-e+Rje-(a+1e+me =ReB+20Rie°+dRe°-R0ea Angular Velocity:== dw dt yAImaginary axis Linear Velocity:a= dv dt Real axis R
Some important definitions Acceleration- Rate of change of velocity 2 Second order: ( ) ( )( ) ( ) d jj j j j R e Re R j e R R j e R je j θθ θ θ θ = + ++ + θ θθ θ θ 2 2 ( ) ( )( ) ( ) = 2 j j jj R e Re R j e R R j e R je j dt Re Rj e Rj e R e θ θ θθ θ θθ θ θ θθ θ = + ++ + + +− A l V l it d ω θ A ngu lar V e locity: dt α θ = = Linear Velocity: d a R dt = = v
Simple cases study A link in pure rotation ©When point A is moving ME357 Design Manufacturing ll
Simple cases study A link in pure rotation Wh i t A i i When poin t A is moving ME357 Design & Manufacturing II
Simple cases study @/ A link in pure rotation Displac- ement R PA=pero RPA 2 Velocity A Vra=poje 十2 02 APA Acceleration A=pajere-pae 2 VPA =APA+A”PA ME357 Design Manufacturing Il
Simple cases study A link in pure rotation Displac - ement j PA p e θ R = Velocit y j PA p j e θ = ω V Acceleration j j θ θ 2 APA PA PA p je p e θ θ = − α ω = + t n A A A ME357 Design & Manufacturing II
Simple cases study When point A is moving Displac- 2 ement Rp=R+RPA RPA NPA NP Velocity 。=4+4 X =Va+pe(io） Accelerati Graphical solution: on VA Vp
Simple cases study When point A is moving Displac - ement RRR P = +A PA Velocit y G G G ( ω) θ V pe i V V V i P A PA = + = + G G G G Accelerati V pe ( i ω) = A + Graphical solution: on
Simple cases study +2 When point A is moving Displac- 2 ement Rp=R+RPA APA APA Velocity 。=4+pM 3 AA =V+pe(io) X Accelerati APA on A。=A+Ap4 APA AP APA -A-0'pe+iapei AA
Simple cases study When point A is moving Displac - ement RRR P = +A PA Velocit y G G G ( ω) θ V pe i V V V i P A PA = + = + G G G G Accelerati V pe ( i ω) = A + AAA GGG on 2 P A PA i i A AAA A pe i pe θ θ ω α = + =− + G
Coriolis Acceleration Position of slider 一02 AP脚 Rp=pe 02 Velocity of slider ,-pei Rp Transmission Slip velocity velocity 02 02 Acceleration: A,=peio+pe”(io）'+peia+ie”+pei0 Combining terms: -[(p-poi)+i(rg+2p Coriolis acc.occurs when a body has vslip and w Slip Normal Tangential Coriolis
Coriolis Acceleration i . R p p e θ = G Position of slider p p Velocity of slider i i θ θ G V pe i pe p = + ω Transmission Slip velocity velocity Acceleration: ( ) 2 i i i ii A pe i pe i pe i pe pe i p θ θ θ θθ = + + ++ ω ωα ω G ( ) ( ) 2 2 i A p p ip p e θ = ⎡ ⎤ − ω αω + + ⎣ ⎦ G Combining terms: Coriolis acc. occurs when A p p ip p e p ( ω αω ) ( 2 ) = ++ ⎡ ⎤ ⎣ ⎦ Slip Normal Tangential Coriolis a body has vslip and w