3.4 Gears and cam-followers 3.4.1 Gears Gear teeth on the periphery of the gears cause pitch circles to roll relative to each other without slip
3.4 Gears and cam-followers 3.4.1 Gears Gear teeth on the periphery of the gears cause pitch circles to roll relative to each other without slip
3.4.1 Gears 1.Outer meshing gear The arc lengths DO,and DO are equal. ↑9 1=0,Q,and Ocoincide a,R,=a R O,and are fixed points on y B;and Bi,0,e,are constant x+0=p+8 a,=4+0-0 R,(,+0,)+R+0)Rπ C%=-(9+0-0-π) R+R x,+8,+4=B+π am明一心1用一r收-=0-8 u Ru w-rk-r小m--0 Two constraint Φ》=(-y(G-）-(R+R,}=0 equations
Qi and Qj are fixed points on Bi and Bj, j j j 3.4.1 Gears The arc lengths DQi and DQj are equal. t = 0, Qi and Qj coincide ( ) j j j i i i sin cos 0 ( , ) P i P j P i P j P i P j T g i j u r r x x y y i j j i i j D Qi Qj u Pj u Pi 0 ( , ) 2 i j P i P j T P i P j rd i j r r r r R R i j i i i j j j j R R R R R Two constraint equations x y i x i y j x j y i j , are constant i i i iRi jRj u PiPj // u PiPj 0 P i P j T u r r u Ru sin cos u 1. Outer meshing gear
s》=ur(-）=-sine6-x）+cos6y-y）=0 R(,+0,)+R,,+日，-Rπ R,+R,, R+R R+R -[-l-8a. (Rg+R,) R+R 》=(-2）+w6-） R+BsB)=0 R,+R a”-w-ra y8)=0 eai-rg
sin cos 0 ( , ) P i P j P i P j P i P j T g i j u r r x x y y Since 0 ( , ) i P j i i i P j j j T i j i i j j P i P j T P i P j T P i P j T g i j R R R R u r B s r B s u r r u r r u r r i j i i j j R R R R u u u u u sin cos , cos sin , sin cos i j i i i j j j j R R R R R i j i i j j R R R R The velocity equation is The Jacobians of the constraint with respect to qi and qj are i j P i i P j P T i i T T g i j R R R i u u B s u r r q ( , ) i j P j i P j P T j j T T g i j R R R j u u B s u r r q ( , ) 0 g (i, j) v
W-R4+R)+r'6+B,s4-i-Bs”)=0 R+R -=i,+B功，-片-Bs4,B,=-中，A,B=-4A4 a=8]-加-）=6-y-）=0 un=-化-转年R）w6-k有+R⊙) R+Rj R+R +n'呢+B,y有-日-Bs)-2m6-A+R》 R+R +u（As+2AsP）=0 2m-R4+R）+u'6,24,s-2As）=0 R+R
0 ( , ) i P j i i i P j j j T i j i i j j P i P j T g i j R R R R u r B s r B s u r r Differentiating the velocity equation 0 2 2 2 ( , ) P i i i P j j j T i j i i j j P i P j T i P j i i i P j j j T i j i i j j P i P j T i j i i j j P i P j T g i j R R R R R R R R R R R R u A s As u r r u r B s r B s u r r u r r 0 2 ( , ) 2 2 P i i i P j j j T i j i i j j P i P j T g i j R R R R u A s A s u r r And using i j j j i i i P j i i i P j j j P i P r j r r B s r B s , B A , B A u u cos sin 0 P i P j T P i P j T u r r u r r
2.Inner meshing gear e. R>R The arc lengths DO;and DO,are equal. t=0,O;and coincide &R=&,R O;and Oare fixed points on Ri B;and Bi,0,,are constant a,+0=+8, %=4+0-0 -R(+0)+R(,+0,) ,+B=日，+9 %,=中，+8，-0 -R+R 吧一i1两一心-小-0[调 u=Ru n=r'6--m心-小-g)=0} Two constraint D》=(g-）Y(-)-(R,-R,)=0 equations
Qi and Qj are fixed points on Bi and Bj, j j j The arc lengths DQi and DQj are equal. t = 0, Qi and Qj coincide i i i j j j sin cos 0 ( , ) P i P j P i P j P i P j T g i j u r r x x y y i j i j j i D Qi Qj u u Pi Pj 2 ( , ) 0 T rd i j P P P P rj ri rj ri Ri Rj i i i j j j i j R R R R Two constraint equations x y i x i y j x j y i j , are constant i i i iRi jRj u PiPj // u PiPj 0 P i P j T u r r u Ru sin cos u 2. Inner meshing gear D i Ri Rj Rj Ri
·g -[]*4+4 Number of constraint s=2 9,=(9,9,=69,) q=ag）=(m,5p,) n=6 Degrees of freedom δ=n-S=4 Parameters of the constraint ,js”=(x”y）',s=(y),R,R,0,6
T T , , , , , , , P P P P P P i i i j j j i j i j i j s x y s x y R R s 2 Degrees of freedom n s 4 Number of constraint T T j j j q r Parameters of the constraint T T i i i q r T T T T T T i j i i j j q q q r r n 6 P P i P i P i i i i x y r r As ( , ) ( , ) 2 sin cos = P P P P g i j j i j i rd i j T P P P P j i j i i j x x y y R R r r r r 0 P P j P j P j j j j x y r r A s
Rack and pinion If the radius of the gear on body i becomes infinite,a straight gear profile is called a rack,and the gear on body j is called a pinion,and the gear pair is called a rack and pinion
Rack and pinion If the radius of the gear on body i becomes infinite, a straight gear profile is called a rack, and the gear on body j is called a pinion, and the gear pair is called a rack and pinion
Rack and pinion y 1.The pinion is above the rack This pair may be seen as a revolute- translational composite joint with the condition of arc DO=DO Ois fixed point on B,is constant v is fixed point on Bi,is constant P Xi The constraint equation of a revolute- translational composite joint is o=(g-）-R,=0 Two constraint equations are V. The equal arc length condition is s=(-）=Rg, ” where where 0=9+0,0,+a,=2 +0-9 3江+，+0，-4，-日， aj
This pair may be seen as a revolutetranslational composite joint with the condition of arc DQj = DQi i j j j xi yi xj yj Qj Pj Pi ( , ) 0 T rt i j i P P j i j i R v v r r T i P Q j i j j i s R v v r r i Qi D Rj The constraint equation of a revolutetranslational composite joint is i v i v Two constraint equations are s where 3 , 2 i i j j j ( , ) 0 rt i j T P P i j i i j v R v r r ( , ) 0 rp i j T P Q i j i i j j v r r v R The equal arc length condition is where j i i j j 2 3 P i P j r r Q i P j r r Rack and pinion 1. The pinion is above the rack Qj is fixed point on Bj, j is constant vi is fixed point on Bi, is constant i xj
2.The pinion is below the rack This pair may be seen as a revolute- translational composite joint with the condition of arc DO;=DO Ois fixed point on B,is constant v is fixed point on Bi,0 is constant The constraint equation of a revolute- translational composite joint is wn-(-）-R=0 Two constraint equations are The equal arc length condition is s=2(g-）=Ra where ） 0=项+0-元4+0，+a,=0+受 where a,=-7+项+0-4-0
i j j j xi xj yj Qj Pj i Qi D Rj i v i v s P i P j r r Q i P j r r yi Pi This pair may be seen as a revolutetranslational composite joint with the condition of arc DQj = DQi 2. The pinion is below the rack Qj is fixed point on Bj, j is constant vi is fixed point on Bi, is constant i The constraint equation of a revolutetranslational composite joint is ( , ) 0 T rt i j i P P j i j i R v v r r The equal arc length condition is T i P Q j i j j i s R v v r r where , 2 i i j j j Two constraint equations are ( , ) 0 rt i j T P P i j i i j v R v r r ( , ) 0 rp i j T P Q i j i i j j v r r v R where 2 j i i j j
o 10 =[ 0x,=c+4+8,-9-0 3π/2，The pinion is above the rack The pinion is below the rack Number of constraint S=2 9=a=(g,p, n=6 Degrees of freedom δ=n-s=4 Parameters of the constraint ij,s=(xy),s=(xy）,0=(xy）,R,8,c
T T T , , , , , , , , P P P P P P Q Q Q i i i j j j i i i j i j i j s x y s x y s x y R c s 2 Degrees of freedom n s 4 Number of constraint Parameters of the constraint T T T T T T i j i i j j q q q r r n 6 P P i P i P i i i i x y r r As ( , ) ( , ) = T P P rt i j i j i i j rp i j T T P Q i j i i j j v R v R v r r v r r 0 P P j P j P j j j j x y r r A s Q Q i Q i Q i i i i x y r r As i i i v Av j i i j j c cos sin i i i v 3 / 2, The pinion is above the rack / 2, The pinion is below the rack c
