n
(4) Higher pair (rolling & sliding pair) Attention: Instant centre is not located at the point of contact or infinity！
V A1A2 1 n
2 A
P12 n
3.2.4 Theorem(定理) of Three Centres (Aronhold-Kennedy Theorem)
V A1A2 1 n
2 n A
(4) Higher pair (rolling & sliding pair) The direction of relative velocities, VA1A2 and VA2A1, between A1 and A2 must be along the common tangent(切线). Otherwise, the two links will separate(分离) or interfere(干涉). So the instant centre P12 or P21 must lie V A1A2 somewhere on the 2 P12 common normal n-n through the point A of n 1 contact. A
Let us consider any point, e.g. point C, outside the line P12P13.
C 2 A(P12 ) 1
3 B ( P13 )
V C 2C1 AC.
VC2C1 C 2 A(P12 ) 1 B ( P13 ) 3
V C 2C1 AC.
Since V C 2 V C1 V C 2C1 ,
V C 3C1 BC.
3.1.2 Methods： （1）Graphical method(图解法) (a) Geometrical(几何学的) method for position (b) Instant(瞬时的) centre method for velocity (c) Vector(矢量) equation method* （2）Analytical(解析的) method (a) Closed-loop method* (b) Assur group method （3）Experimental(实验) method*
1
(4) Higher pair (rolling & sliding pair)
2 1 A
(4) Higher pair (rolling & sliding pair) The direction of relative velocities, VA1A2 and VA2A1, between A1 and A2 must be along the common tangent(切线). Otherwise, the two links will separate(分离) or interfere(干涉).
Since P12 is the instantaneous(瞬时的) centre of relative rotation, VA2A1AP.
A 2 1
V A2A1
P12
Suppose that the positions of points A and B, the directions of VA2A1 and VB2B1 are known. The position of instant centre P12 V A2A1 is to be located. A
3.2.3 Location of the Instant Centre of Two Links Connected by a Kinematic Pair (1) Revolute pair If two links 1 and 2 are connected by a revolute pair, the centre of the revolute pair is obviously(明显地) the instant centre P12 or P21.
3 2 1
3.2.4 Theorem(定理) of Three Centres (Aronhold-Kennedy Theorem) Suppose that A is the instant centre P12 of the links 1 and 2. B is the instant centre P13 of the links 1 and 3. Where is P23? Any three links have three instant centres: P12, P13, and P23. They must 3 lie on a straight 2 A(P12 ) B ( P13 ) line. 1
V C 3C1 BC.
V C 2 C 1 V C 2 V C1 . Similarly, V C 3C1 V C 3 V C1. Obviously, for any point C outside the line P12P13, V C 2C1 V C 3C1 . Therefore, V C 2 V C 3 .
B 2 1 V B2B1
VA2A1AP. VB2B1BP.
A 2 1
V A2A1 B V B2B1 P12
3.2.2 Number N of instant centres: N=K(K-1)/2 Note: The frame is included in the number K. Classification(分类) of instant centres： (1) Absolute (绝对) instant centre: one of links is the frame. Its velocity is zero, but its acceleration may not be zero. (2) Relative(相对) instant centre: both links are moving links, velocity of which may not be zero.
2 A 1 P12 1 P12 2
(3) Sliding pair Relative translation(平移) is equivalent (等价 于) to relative rotation about a point located at infinity(无穷远) in either direction perpendicular(垂直的) to the guideway(导路). Therefore, their instant centre lies at infinity in either direction perpendicular to the guideway. P12 Attention：The common normal(公法线) may pass 2 through any point !!
C B A
2 3
D E
ω1 θ 1
5
F
4
6
3.2 Velocity Analysis by the Method of Instant Centres(瞬心) 3.2.1 Definition of the Instant Centre(瞬心) 1. a pair of coincident(重合) points, the absolute(绝对) velocities of which are the same, in both magnitude(大小) and direction. 2. relative velocity is zero. V P2 V P1 3. Instantaneous(瞬时的) 2 centre of relative rotation, or more briefly the instant P12 1 centre, denoted(标为) as P12 or P21.
C B A
2 3
D E
ω1 θ 1
5
F
4
6
(4) Locate point E according to LDE and CDE.
C B A
2 3
D E
ω1 θ 1
5
F
4
6
(5)Draw an arc with the point E as the centre and LEF as the radius(半径). The intersection of the arc and the horizontal pathway(导路) is point F.
V C 3C1 BC.
V C 2 C 1 V C 2 V C1 . Similarly, V C 3C1 V C 3 V C1.
VC2C1 V C3C1 C 2 A(P12 ) 1 B ( P13 ) 3
V C 2C1 AC.
Since V C 2 V C1 V C 2C1 ,
C B A
2 3
D E
ω1 θ 1
5
F
4
6
(2)Locate B cording to LAB and 1
C B A
2 3
D E
ω1 θ 1
5
F
4
6
(3)Draw two arcs with the points B and D as the centres and LBC and LDC as the radii(半径复数). The intersection(交点) of the two arcs is point C.
All kinematics dimensions(LAD、LAB、 LBC、LDC、LDE、CDE、LEF) are known. Draw the kinematic diagram of the mechanism when 1=80.