Power decreases as the diameter increases

Question

I have a question about the use of a servo motor.

I need to lift a load of $130\ kg$ with two servo motors with $13\ Nm$.

The diameter of the shaft is $Ø40\ mm$, ie, $r=\ 20mm$.

$1\ N = 1\ Kg\ *\ 1\ m/s^2$

$1\ Nm = 1\ Kg\ *\ 1\ m/s^2*\ 1\ m$

Therefore:

$25.506\ Nm = 130\ Kg\ *\ 9.81\ m/s^2*\ 0.02\ m$

By my reasoning, two servo motors could raise this load if the point of application of the load is exactly the diameter of the shaft.

If I use a cable that every time its diameter increases inside a pulley, at some point I will not be able to lift that load anymore?

Answer 1

In addition to what Jonathan has said, there is several other practical things to consider:

1. not only torque is limited, but power also; since power is torque times angular velocity (P=T*omega), your motor specification will limit the maximum speed you will be able to lift the load at
2. depending on your application, you will not necessarily have to apply all the weight force by your motors; in typical elevators there is a counterweight (usually the size of the average load) which balances the weight force, so the motor will only have to apply the inertial force (of load + counterweight, plus any deviations from the average weight)
3. so, if you are building a kind of elevator, you don't have to wind the cable around a pulley either; because the counterweight pulls at the cable, you only need to take care that there is enough traction at the cable sheave
4. if you really want to lift all the weight by the motors, you could also use a gear or a block and tackle mechanism, which can translate torque into a more manageable range (without resorting to ridiculously small shaft diameters)

So your pulley diameter of 20mm is not set in stone after all. But remember: if you use a mechanical advantage of some sort (like in bullet 4), you will only change force/torque, whereas power stays the same (or almost, considering friction effects). So the power limitation of your motors is usually the more severe one, and this depends on what ultimate speed you want to move the load at.

Answer 2

The downwards tension force in a cable with a $130\text{kg}$ mass hanging under gravity is:

$$F=mg=130*9.81=1275.3\text{N}$$

The upwards force provided by a servo with arm radius $0.02\text{m}$ and torque $13\text{Nm}$ is:

$$F=\frac{T}{r}=\frac{13}{0.02}=650N$$

With two servos in your proposed system, the torque would be doubled, and so your theoretical lifting force would be 1300N.

According to the maths, as long as the radius remains under $\frac{13}{1275.3\ / \ 2}=0.02039\text{m}$, then you will be able to lift the mass.

Your problem, however, comes at the speed of lifting. At your original radius, the resultant force (lift minus weight) is only $1300-1275.3=24.7\text{N}$. This means a maximum acceleration of:

$$a=\frac{F}{m}=\frac{24.7}{130}=0.19\text{m/s}^2$$

This is really very slow, and will only go down as the cable winds in. For any practical application you will find that the performance of the servos may not match the theoretical peak values from the data sheets. I'd be extremely surprised if the servo motors that you have in mind are able to lift the $130\text{kg}$ mass at all, let alone at a speed that is useful.