This is a long one, with a huge amount of material out there in white papers and on manufacturer’s websites. So see the Reference list below. I have tried to keep a hold of a lot of the things I have read over the years, but also I am not going to point you at an 800-page catalogue.

Defining the problem.

1. We have a mechanical system that needs an actuating force.

2. We have to get the system to meet some requirement of speed and positional accuracy.

Defining the system:

1. It has some physical characteristics that are a function of the materials and the shape of the materials that make them up. We can mathematically describe the system based on some rather basic shapes and then work on an assumption that we don’t need to get very very detailed unless we need to move very dynamically. Dynamically here means changing direction and speed very quickly.

2. The main things in the system are:

2.1 Mass of the load

2.2 damping effect of friction.

2.3 system stiffness

Now it gets counter-intuitive.

2.4 Mass of the actuator’s moving parts (ie the motor rotor and bearings)

2.5 The damping of the motor (ie the friction of the bearings)

A diagram of the system as described by a few system modelling textbooks is shown in Fig 1. This is pretty much as simple as you can go if you want to do the analysis properly.

**Equation 1: JL/JM **

where JL is the load inertia and JM is the motor inertia.

Using equation 1 and the system diagram above we can gain an understanding of how the system should behave based on the characteristics observed through mathematical modelling. A Bode plot of the system modelling is shown in Fig 2. Those peaks and troughs represent system resonance and anti-resonance spots in the frequency response of our system. You can see that they are amplified by the amount of the JL/JM ratio. So we ideally want to keep those as low as possible, right? Not quite. They aren’t a problem for you unless your system is operating within them. So we ‘tune’ the mechanical system by careful choices of gearbox and motor to adjust the JL/JM ratios in the frequency range our machine will operate within.

**Equation 2: Tm=Toutput*N **

where Tm is the Motor output torque and Toutput is the output torque from a gearbox with reduction ratio N.

**Equation 3: ωm=ωoutput/N**

where ωm is the output speed of the motor and ωoutput is the output speed of the gearbox which is experienced by the load and N is the reduction ratio of the gearbox.

**Equation 4: Jr = Jl/(N^2)**where Jr is the inertia load realised by the motor (the effective system inertia) once the gearbox of ratio N is applied to the inertia load Jl.

Now this is where the magic of gearboxes lies for system tuning the N^2 factor. We use the mechanical advantage of Equation 2 to get a smaller motor to do the same job with the trade off being speed as seen in Equation 3. BUT we use Equation 4 to adjust the Bode plot as seen in Fig 2 to fit the system requirements we need to gain a well-behaving system in the frequency band we will operate in. You will come across some textbooks/design guides that tell you to use a ratio of between 5 and 10 for JL/JM, they may be correct, but if you use this simple bit of a mental check and matching, you can come up with a much more informed decision for your application.

Next week I am thinking about doing a write-up on motor curves and torque response. Let me know what you think.

References

https://www.takealot.com/dynamic-modeling-and-control-of-engineering-systems/PLID34225275 Buy the textbook that I used at University.

If you are looking at very very good motors and a motor matching service, https://www.faulhaber.com/en/home/

Peter