Example of selecting a gearbox for a 3DOF platform

Hello everyone!
I want to show how I calculate the torque for selecting a gear motor. My calculation is very approximate and does not take into account many factors, such as the moments of inertia of the drive shafts of the motor and gearbox, friction coefficients in the bearings, the gearbox operating factor, etc., so I roughly round all the calculated values up.
I will be glad if you supplement or point out errors in my calculation.

For example, let's take a RIG: 3DOF with rotating cranks and try to select a worm gear.
As a motor I will use 80ST-M02430 + A1-SVD15 (750 W, 2.39 Nm, 3000 rpm).



m = 200 kg
"m" is the mass that we are going to "swing". It includes the mass of the equipped cockpit, the mass of the pilot, the mass of the upper platform and the mass of the rods attached to the cranks and the upper platform.

m_crank = 5 kg is the mass of the crank.
l = 0.1 m is the length of the crank.

n = 3 is the number of gearboxes.

g = 9.81 m/s^2 is the acceleration of gravity
a = 2 m/s^2 is the acceleration with which we want to move the platform with the load.

1. Let's define the force (F) acting on the edge of the crank. We will assume that the mass (m) is uniformly distributed over 3 cranks:
F = (m * (g + a)) / n ;
F = (200 * (9.81 + 2)) / 3 = 787.33 N ≈ 788 N;

2. Determine the torque acting on the gearbox shaft from the applied force (M_F):
M_F = F * l;
М_F = 788 * 0.1 = 78.8 Nm ≈ 79 Nm;

3. Determine the torque acting on the gearbox shaft from the moment of inertia of the rotating crank (M_I):
M_I = I * ε,
"I" is the moment of inertia of the crank rotating around one of the ends;
I = (m_crank * l^2)/3;
I = (5 * 0.1^2)/3 = 0.0167 kg/m^2;

"ε" is the angular acceleration of the crank;
ε = a / l;
ε = 2 / 0.1 = 20 rad/s^2;

As a result, we calculate M_I:
M_I = 0.0167 * 20 = 0.33 Nm ≈ 0.4 Nm;

4. Determine the total torque acting on the gearbox shaft (M):
M = M_F + M_I;
M = 79 + 0.4 = 79.4 Nm ≈ 80 Nm;

5. I decided that the angular velocity at the gearbox output ω = 60 rpm would be sufficient. In this case, with a crank length of l = 0.1 m, we obtain the linear velocity:
V = (2 * π * ω * l) / 60;
V = (2 * 3.14 * 60 * 0.1) / 60 = 0.628 m/s.

6. Let's determine the gearbox gear ratio:
i = 3000 / ω; (3000 is the angular velocity of the 80th servomotor).
i = 3000 / 60 = 50;

From the documentation for NMRV worm gearboxes, we find that with an electric motor power of P = 750 W and the required gear ratio of i = 50, a gearbox of size 063 (NMRV063-50) is used.

The maximum torque that the NMRV063-50 gearbox design can withstand is M_gmax = 124 Nm (from the documentation).

The required torque we calculated earlier (M = 80 Nm) does not exceed the maximum torque of the gearbox (M_gmax = 124 Nm). From this we can conclude that the gearbox will not fail under our loads.

7. Now let's check what torque (M_mg) the gearmotor can develop, taking into account the engine speed (ω_m = 3000 rpm), engine power (P = 0.75 kW), gear ratio of the gearbox (i = 50) and the efficiency of the worm gearbox (η = 0.75):
M_mg = (9550 * P * i * η) / ω_m;
M_mg = (9550 * 0.75 * 50 * 0.75) / 3000 = 89.53 Nm.

Almost forgot!
The magnitude of the gearmotor torque (M_mg) can also be estimated using the motor torque (M_m = 2.39 Nm):
M_mg = M_m * i * η;
M_mg = 2.39 * 50 * 0.75 = 89.625 Nm.
This method is rougher, so for further steps we will take M_mg = 89.53 Nm.

From this calculation, we see that the torque (M_mg = 89.53 Nm) that the selected gearmotor (NMRV063-50 + 80st) can provide is slightly higher than the calculated torque (M = 80 Nm) and does not exceed the maximum torque (M_gmax = 124 Nm) that the gearbox design can withstand:

M < M_mg < M_gmax;
80 Nm < 89.53 Nm < 124 Nm.

I repeat once again - this calculation is only an approximate guide when choosing a gearbox. This is exactly the calculation I used when choosing gearboxes for my 3DOF platform, which works properly.

Note that this is a 3 axis rig so the motors have to support the entire load. For a 2 DOF rig with a Ujoint most of the weight is on the Ujoint. My 2 DOF rig does fine with 36nM of torque measured at the motor. Sorry, didn't do the math.

Are you happy with the worm gears? IIRC you have reported problems with the backlash causing noises when reversing direction. I think there are planetary gearboxes with very low backlash available at stepperonline. They are, well, not cheap but affordable (€216 for 7 arcmin, €126 for 15arcmin). How do they compare to your worm gearboxes?

In my personal experience the gearbox (looks like his picture) has never been an issue for backlash. Certainly don't notice anything changing directions. The issues I have had are from the coupling of the gearbox to the lever arm. This particular gearbox at least mine has a 6mm key. This allows the shaft to have a solid minimum backlash connection to the shaft. If I had to do it over (or should I say when?) I will weld the lever arm to the shaft. Also, the motor and shaft need a super solid connection to the stationary part of the rig. These issues could have been reported as backlash.

The key really has to be oversized. What you get from Amazon and the like will be all over the place but mostly under. McMaster Carr has oversized keys which are 100% better. Do you have a hammer? I have a reverse one now too.

https://www.mcmaster.com/catalog/98870A351

After more than 2 years of struggling I don't have any "slop" in the drive after 130 hours of operation.

I spoke with the service department of the supplier of my gearboxes. I was advised to add adjusting rings before the worm shaft bearing (immediately after the servo motor flange).

Unfortunately, this did not help reduce the noise.

I was advised to use bevel-helical gearboxes - I will use them in the next project. For now, I am looking at Tramec TC100C.

The problem of those cheap worm gearboxes is they are designed for drive applications like conveyors, not precise and fast positioning with quick direction reversals. But their advantage is they have output shaft ends on both sides which is good for your design with the two lever arms per motor.