INSTANTANEOUS MOTION- INSTANTANEOUS CENTER (IC) AND VELOCITY ANALYSIS USING IC

Instantaneous Motion -Four bar link with the initial position A0A1B1B0, is displaced to the position A0A2B2B0, after a certain length of time. -Instantaneous motions of cranks A0A1 and B0B1 are rotations about respective fixed pivots. -The instantaneous motion of connecting rod A1B1 will be examined now: The displacement of A1 and B1 during this interval may be accepted as lines A1A2 and B1B2. The perpendicular bisectors of these two lines pass through A0 and B0 and intersect at P12. The triangles A1P12B1 and A2P12B2 are identical (A1P12=A2P12, B1P12=B2P12, A1B1=A2B2) If link A1B1 were considered as a free body, its motion would be a rotation about P12 (pole).

-If displacements A1A2 and B1B2 are infinitely small, then instantaneous motion of link A1B1 is a rotation about point P12. -It may be concluded that the motion of a link at any instant, regardless of the complexity of the motion during a longer interval, is a rotation about a certain point called the instantaneous center (IC) or centro. Properties of IC (centro): It is a common point to two links about which one is rotating relative to the other (instantaneous motion). It is a common point to two links which have no relative motion at that point. Since it is a common point to two links, it may have identical velocity if it is considered as a point on each of the two links.

Absolute center of rotations may be fixed or variable. Types of IC (Cento) ICs of moving links with other moving links are called relative center of rotation. ICs of moving links with fixed link are called absolute center of rotation. Absolute center of rotations may be fixed or variable. ICs are classified also as primary (directly visible) and secondary (obtained by solution procedure). Number of ICs in a mechanism. n is the number of link.

ICs are the center of radii in sliding pairs  23 is secondary IC INSTANTANEOUS CENTER OF ROTATION OF MECHANISMS ICs are the joints in rotating pairs and are the contact points in rolling pairs ICs are the center of radii in sliding pairs  23 is secondary IC for the cam pair

Determination of secondary instantaneous center of rotations 13 12 23 14 34 1 24 12 14 23 34 2 4 3 34 23 4 2 3 12 1 14 1

13 12 23 14 34 13 24 12 14 23 34 3 34 23 24 4 2 12 1 14 1

KENNEDY’S THEOREM OF THREE CENTERS - If there are 3 bodies moving relative to each other, there should be 3 ICs. - 12 and 13 are primary ICs. 23 (point C) should be at such place that bodies 2 and 3 have no relative velocity (VC3=VC2) Since VC3=VC2, 3 (13-23)= 2 (12-23) - This is possible only if C is on the line 12-13 Here 12-23 and 13-23 are directional distances Kennedy’s Theorem of Three Centers Therefore, ICs are used for velocity analysis of mechanisms. Angular velocities of any two bodies (2 and 3) relative to the third body (1) are inversely proportional to the directional distances from their respective ICs (12 and 13) to their relative IC (23)

VELOCITY ANALYSIS VIA INSTANTANEOUS CENTER OF ROTATIONS EXAMPLE 1: Vp is known. Find VC and 4 13 -------- 23 14 34 13 Line 14-34 Line 12-23 C’ VC C VC’ V34 D’ 3 B 34 4=V34/DE D 23 V34’ V23 P 4 2 Vp 12 14 E

EXAMPLE 2: 2 is known. Find VE and 6 15 To 15 6=VE/EF 12 34 4 23 35 3 16 56 E D 14  5 2 6 VE D’ F 13 -------- 23 14 34 15 -------- 35 16 56 13

6=VG/GH EXAMPLE 3: VB is known. Find VG and 6 VB VG 45 5 56 V56 E G 25 -------- 12 15 24 45 6=VG/GH 15 -------- 14 45 16 56 45 5 56 V56 E VG G 4 6 VB B 23 D H 24 -------- 12 14 23 34 16 3 25 VB’ 2 V25=V56 C 24 12 A 14 34  15

EXAMPLE 4: VA is known. Find 3 and 4 34  VA(V23) (34) VC 23 VC’ 12 14 C’ 3=VC/(C-13) 4=VC/(C-14) 13