Mechanical design engineers who have to make a machine or modify the machine usually deal with designing timing diagram of related movements between each part in the machine. Timing diagram is a useful tool for the designer not only to see how each part of the machine works together, but also to see the opportunity to improve the machine movement through “overlap” motion. If we design the timing diagram using the old-fashioned robot style as you see in the movies, it will waste the time in waiting for another part to finish its movement first. If we think about human’s hand movement in transferring the object, we will see that it will not act like the robot. We can see some overlaps. For example, if we transfer a rod from right hand to left hand, we will see that the left hand close a little bit already when right hand move the rod to left hand. Left hand will not open widely and wait until the rod reach it then close, because it’s not natural. If we use the same approach for design the machine timing diagram, we will get smoother motion of relevant parts.
Why overlap motion gives smoother movement of the parts?
Assume we use cycloidal motion cam profile to move the part. So first we have to get the formula to calculate the maximum acceleration of cycloid cam profile. If the machine speed is N (pcs/h) and the indexing angle (degree) is Bm, the indexing time (second) tm can be calculated as follows.
Cycle time (sec) = 3600/N
Indexing time tm (sec) = (Bm/360) x Cycle time = (Bm/360) x (3600/N)
Hence, Indexing time tm (sec) = 10Bm/N
Cycloidal motion cam profile has the equation of displacement as follows.
h = hm x [t/tm- 1/(2 x 3.141592654) x sin(2 x 3.141592654 x t/tm)]
where: hm = Maximum displacement (m) and tm = Indexing time (s)
We can get velocity equation by differentiation.
v = dh/dt = hm x [1/tm – (2 x 3.141592654)/(2 x 3.141592654 x tm) x cos(2 x 3.141592654 x t/tm)]
v = hm/tm x [1 – cos(2 x 3.141592654 x t/tm)]
Then, the acceleration is as follows.
a = dv/dt = hm/tm x [0 – (-2 x 3.141592654/tm) x sin(2 x 3.141592654 x t/tm)]
a = 2 x 3.141592654 x hm/tm^2 x sin(2 x 3.141592654 x t/tm)
The maximum acceleration (amplitude) occurs when sin(2 x 3.141592654 x t/tm) = 1 or -1. Therefore the amplitude of of maximum acceleration is as follows. amax = 2 x 3.141592654 x hm/tm^2
We can clearly see from the above derivations that the acceleration is inversely proportional to the square of indexing time. Since the indexing time ™ is proportional to the indexing angle (Bm), then the maximum acceleration is also inversely proportional to the square of the indexing angle. That means if we can increase the indexing angle by a factor of two, the maximum acceleration will reduce by a factor of four!! And we can do this by putting more overlapped motion in the timing diagram design. Read more details about how to design smarter timing diagram at http://mechanical-design-handbook.blogspot.com/.