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Module 1. Introduction to Theory of Machine
Module 2. Planar Mechanism
Module 3. Velocity and Acceleration Analysis
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Lesson 7.
7.1 ARONHOLDKENNEDY THEOREM
It states that "if three bodies, having relative motion with respect to each other will have three Icentres all of which lies on the same line". It is very useful in the mechanism where there are three links and will have three Icentres, if two them are known then third will lie on the line joining the two Icentres.
Fig. 3.11 Aronhold –Kennedy’s theorem
as shown in the above figure 3.11, the Icentre I_{13} and I_{12} are located by observation , then the third Icentre I_{23} will lie on the line extended from I_{13 }& I_{12 }and cutting the line perpendicular to the velocity .
7.2 METHOD FOR LOCATING ICENTRE
Step 1. The number of Icentre in a four bar mechanism is
\[n = \frac{{n(n  1)}}{2}\]
Where n = No. of links
The number of Icentre in a four bar mechanism will be
\[\frac{{{\text{4}} \times \left( {{\text{4}}  {\text{1}}} \right)}}{{\text{2}}} = {\text{6}}\] .
So there are 6 Icentre in a four bar mechanism
Step 2. These Icentre’s is shown below
Links 
1 
2 
3 
4 
Icentre’s 
I_{12} 
I_{23} 
I_{34} 
 
I_{13} 
I_{24} 
 
 

I_{14} 
 
 
 
Fig. 3.12
Step 3. I_{12 }and I_{14} are found to be fixed Icentre because they will change their position during the rotation of crank or at any position of crank. The I_{23} and I_{34} are permanent types of Icentre as they will change during crank movement but will remain at the joint of link 2 & 3.
Step 4. I_{13} and I_{24} are secondary Icentre and are located by circle diagram given below.
Fig. 3.13 circle diagram
Take four points 1, 2, 3 and 4 on the circle equal to the number of links in the mechanism. Join these point by solid lines 12,23,34 and 41.These lines will be specify the Icentre’s I_{12},I_{23,}I_{34} and I_{14}.Two remaining Icentre’s I_{24} and I_{13 }are located by joining points 1 to 3 and 2 to 4 showed by dotted lines.
7.3 TYPES OF ICENTRE
There are basically three types of Icentre Fixed, Permanent, Neither fixed nor permanent
The fixed Icentre means its location will remain same during the relative motion between the links because one of the links involve is fixed to the ground as Icentres I_{12} & I_{14} in the figure 3.12. The permanent Icentres are those whose location will change depending on the position of links but they will remain at permanent joining points of the two links as Icentres I_{23} & I_{34} in figure 3.12. The remaining two Icentres I_{24} & I_{13} are neither fixed nor permanent types of Icentres as it will change continuously depending on the location of other Icentres.
Example: 3.2 Slidercrank mechanism has a crank length of 125 mm which is rotating at 600 r.p.m. (clockwise direction). Find the velocity of slider and angular velocity of connecting rod if the length of connecting rod is 500 mm and crank makes an angle of 45^{0} with the inner dead centre.
Solution:
Fig.3.14 slider crank mechanism
Find the number and location of all the Icenters by following the method as describe in above article. Then, Locate the fixed and permanent Icentre I_{12,}I_{34} and I_{ 23} by inspection. As slider (link 4) moves in a straight line so Icentre I_{14 }will lie at infinity. Locate remaining two Icentre’s I_{24} and I_{13} by using Kennedy theorem.
By measuring from the above diagram we can have
I_{13}A = 581 mm
I_{13}B = 690 mm
Velocity of slider A
\[\omega\times {\text{OB}} = \frac{{{\text{2}}\pi\times {\text{6}}00}}{{{\text{6}}0}}{\text{ = 62}}.{\text{9}}0{\text{rad/s}}\]
\[{{\text{V}}_B}{\text{=}}{\omega _{{\text{OB}}}} \times {\text{OB = 62}}.{\text{9}}0 \times 0.{\text{125 = 7}}.{\text{86 m}}/{\text{s}}\]
\[\frac{{{{\text{V}}_{\text{A}}}}}{{{{\text{I}}_{{\text{13}}}}{\text{A}}}}= \frac{{{{\text{V}}_{\text{B}}}}}{{{{\text{I}}_{{\text{13}}}}{\text{B}}}},{V_A} = 0.{\text{581}} \times {\text{7}}.{\text{86}}/0.{\text{69}} = {\text{6}}.{\text{61m/sec}}\]
Angular velocity of connecting rod
\[{\omega_{AB}}{\text{=}}\frac{{{{\text{V}}_{\text{A}}}}}{{{{\text{I}}_{{\text{13}}}}{\text{A}}}}= \frac{{{\text{6}}.{\text{61}}}}{{0.{\text{581}}}} = {\text{11}}.{\text{37rad}}/{\text{s}}\]
Example 3.3: Find the linear velocities of the points B, C & D by using Icentre method and also find the angular velocities of the links AB, BC and CD for the diagram shown below.
Fig: 3.15
Solution:
Step 1. Write down all the Icenter’s in the table
I_{12} 
I_{23} 
I_{34} 
I_{45} 
I_{56} 
I_{13} 
I_{24} 
I_{35} 
I_{46} 

I_{14} 
I_{25} 
I_{36} 


I_{15} 
I_{26} 



I_{16} 




Step 2. Locate all the fixed and permanent Icentre by inspection
I_{12},I_{23,} I_{34},I_{45,} I_{56},I_{16,} I_{14}
Step 3. Locate Icentre I_{15} by using Kennedy theorem as it will be used in finding the slider velocity
Step 4. Locate I_{13 }by Kennedy theorem as it is common Icenter of points A and B.
Step 5. find V_{A}= ω × OA
Step 6. considering Icenter I_{13} because point A and point B rotate about this Icenter and have zero velocity at this point. So use
eq. \[\frac{{{V_A}}}{{{I_{13}}A}} = \frac{{{V_B}}}{{{I_{13}}B}}\] and find V_{B }
Step 7. similarly, considering Icenter I_{14}, use eq. \[\frac{{{V_B}}}{{{I_{14}}B}} = \frac{{{V_C}}}{{{I_{14}}C}}\] and find V_{C}
Step 8. similarly, considering Icenter I_{15}, use eq. \[\frac{{{V_C}}}{{{I_{15}}C}} = \frac{{{V_D}}}{{{I_{15}}D}}\] and find V_{D}
Angular velocity of links AB can be found by using eq. \[{\omega _{AB}} = \frac{{{V_A}}}{{A{I_{13}}}}\]
Similarly find velocities for links BC and CD