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I have seen the different definitions of the velocity ratio for Gears and Gear trains. I have referred a book called Theory of machines by S. S. Ratan. In that book the definitions are as following:

  1. For gears, velocity ratio is ratio of angular velocity of follower to that of the driving gear.
  2. For gear train, speed ratio is given as the ratio of speed of driving to that of the driven shaft.

Why are these two definitions different (inverse of each other) as I think Velocity ratio and speed ratio are the same thing. Also if there is a difference, it will be helpful if a bit explanation is provided.

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TL;DR IMHO, this is something that think has its roots to older times, where the strength design of gears required tables and charts

Like solarMike said, they are essentially the same thing. I don't think many people nowadays pay too much attention to this detail.

IMHO, the reason for the existence stems from the different need. Like you stated, (some authors) use (note that I'll start the opposite way):

  1. For gear train, speed ratio is given as the ratio of speed of driving to that of the driven shaft.

In that case, the big picture is important. I.e the entire drivetrain, and more specifically its kinematics. In that case the important is to define the angular velocities and torque on the various stages of the drivetrain (or maybe just at the end). Therefore, it that case, the speed ratio focuses on that.

  1. For gears, velocity ratio is ratio of angular velocity of follower to that of the driving gear.

In this particular case, the gears themselves are the object of study (to be honest I think the velocity ratio is a poor name, and I have seen it as gear ratio). In that case, except for the kinematic characteristics there are also concerns about strength of the gears. (I haven't seen S. S. Ratan's book but I bet that there is some calculation method for stress inside).

Therefore, in the case of the gears, you isolate two components and you try to design them so that they a) perform kinematically as they should and b) the components themselves don't fail.

The second part is what's important. I've found several different methodologies for calculating gears (US, European, German, British etc). Almost every manufacturer has one. However, in the majority of cases the gear ratio plays a significant role(even though in more recent cases they avoid it as much as possible). So you have graphs like the following (this is just one example, I am sure that Ratan's book will probably have something similar):

enter image description here

Figure : Hardness ratio factor $C_H$ (source: shigley Mechanical Engineering Design 8th edition)

On the x-axis, the reduction gear ratio $m_G$ is what you call velocity ratio. As you can see in this graph it is much easier to view the curves (rather than if they were between 0-1). So, because this is at component level (i.e. between the two gears), which is the driving and which is the driven doesn't really matter. Instead what it was important was to put a lot all the relevant information without duplicating it. Therefore, they opted to define another ratio which essentially it was always greater than zero, so that the tables could be used without unnecessary duplication.

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So basically this is the ratio of turns.

However for shafts that is sufficient while for gears the tooth speed becomes useful.

Just depends how you want to calculate the system.

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