·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