Gearbox sizing

SevenToFive
If I have a 7.5HP motor running at 1750rpm with a pulley and v-belt and connect it to a gearbox, where the right angle worm gearbox sees an input speed of 580rpm, how would I go about sizing the gearbox?

Am I correct to say that I would size the gearbox for the lower input speed and more torque, and lower output rpm? So I would then calculate the torque generated with a 580rpm input speed, and verify that the gearing is capable of handling the increased torque.

I don't think the gearbox is going to be really seeing the 7.5HP, and if the gearbox becomes overloaded, the belt would probably slip on the pulleys connecting the motor and gearbox correct?

Just looking to confirm that I am on the correct path of thought here. Thanks to everyone who replies.

If I have a 7.5HP motor running at 1750rpm with a pulley and v-belt and connect it to a gearbox, where the right angle worm gearbox sees an input speed of 580rpm, how would I go about sizing the gearbox?
The gearbox will need to handle 7.5 HP at 580 RPM. Check the duty cycle, lubrication and cooling of the gearbox.

Am I correct to say that I would size the gearbox for the lower input speed and more torque, and lower output rpm?
Yes, the slower the output shaft, the heavier it will need to be to handle the torque when transmitting 7.5 HP. But you know the input RPM and HP so you can base selection on input shaft torque.

I don't think the gearbox is going to be really seeing the 7.5HP, and if the gearbox becomes overloaded, the belt would probably slip on the pulleys connecting the motor and gearbox correct?
Belts will not slip if they are correctly fitted. If they do slip they will creep during normal operation and so will wear the belts. What section belt and how many will you need for the input torque? Use bigger V-belt pulleys to reduce belt tension or number of belts.

Do you have a link to a list of worm boxes rated at 7.5 HP or more?
What reduction ratio or output RPM is your target ?

SevenToFive and PaulK2
SevenToFive
Baluncore, as always thanks for your help.