# Arduino Forum

## Using Arduino => Motors, Mechanics, and Power => Topic started by: DryRun on May 18, 2019, 11:46 pm

Title: Derive DC motor formula with drum driving a conveyor belt
Post by: DryRun on May 18, 2019, 11:46 pm
The setup is a DC motor directly connected via its shaft to a drum which drives a horizontal conveyor belt. There is a load of mass, M, on the conveyor belt.
I am stuck on trying to figure out what 'D' in the equation below means.

Title: Pulley, Conveyor Belt & Load system

Description: Load-Inertia to the motor consists of pulley (of radius r) and mass of load (M) on conveyor belt. The load-torque, to be delivered by the motor is given as:

where,

M is the mass of load on the conveyor belt.

ω = angular velocity

But i don't know what D means and i cannot figure out how that equation was derived in the first place. Any help would be much appreciated.
Title: Re: Derive DC motor formula with drum driving a conveyor belt
Post by: jremington on May 19, 2019, 02:44 am
Quote
But i don't know what D means and i cannot figure out how that equation was derived in the first place.
Where did you find that equation and why do you think it applies to your problem?

Please state the mechanical situation as clearly as you can. There is an enormous difference between a situation where the mass is being conveyed at constant velocity along the track or a situation where the mass must be accelerated from standstill (i.e. when it is dropped on to the belt).

If the latter, D*omega  probably represents the rate of change of angular velocity (the angular acceleration), because M*r^2 is the moment of inertia of the mass.
Title: Re: Derive DC motor formula with drum driving a conveyor belt
Post by: DryRun on May 19, 2019, 08:13 am
I searched for the equation and you're right - D*omega is indeed the derivative of angular velocity. I was thinking the wrong way, searching for the meaning of a variable 'D'.

It is from my notes but obviously, it's not fully proofed, and it also seems to be missing some steps. I'm just trying to piece it all together so it makes sense. It is about deriving the equation for a DC motor driving a conveyor belt with a load already on it (and it is, as you suggested, the load is not moving at constant velocity but it is being accelerated from standstill so that it moves only a certain distance along the conveyor belt) and finally, i have to write the transfer function for it. It is already given in Equation 3 below but i cannot figure out how it reached that point, since in Equation 2, there is a 'D' when i think it should be an 's' after doing Laplace transform, and then in Equation 3, there is an extra 's' in the numerator. I might have made a mistake somewhere... but no clue what i did wrong.

Here is the complete slide:

Title: Re: Derive DC motor formula with drum driving a conveyor belt
Post by: Robin2 on May 19, 2019, 09:46 am
Diameter?

...R
Title: Re: Derive DC motor formula with drum driving a conveyor belt
Post by: Paul_KD7HB on May 19, 2019, 04:49 pm
After all that work you have still not accounted for the stiffness of the belt, nor the loss in driving the belt.

Paul
Title: Re: Derive DC motor formula with drum driving a conveyor belt
Post by: jremington on May 19, 2019, 05:33 pm
To increase the angular velocity W of a point mass M moving in a circle of radius R requires torque

Tau = M*R^2*dW/dt

If the mass is initially dropped onto the belt, then dW/dt is approximately V/(R*dt), where V is the final linear velocity of the belt and dt is the time it takes to get the mass velocity up to V.

As Paul notes above, this torque is required in addition to torque required to maintain the motion of the unloaded belt and pulleys.