# Pulley-to-Shaft-to-Pulley Calculation?

• playludesc
In summary, the conversation discusses a pulley configuration for a supercharger and how to calculate the RPM at the outer diameter of a blue pulley based on the outer diameter and RPM of a black pulley. The participants determine that the RPMs of both pulleys should be the same and the tangential velocities can be calculated using the ratio of the diameters. They also discuss the use of a jackshaft in this configuration.
playludesc
Hello all! I'm working with a particular pulley configuration and I realized after a few calculations that I don't have an accurate way to calculate one part of the set up.

Here's a quick MSPaint sketch showing what I need to solve for. If I know the outer diameter of both pulleys, and the RPM of the black pulley, how can I solve for the RPM at the outer diameter of the blue pulley? (If you care, this is regarding the jackshaft for my supercharger!)

Thanks very much for your help!

So I figured that this is probably a simple mechanical advantage equation, as the two pulleys simply work like levers on each other. Here's what I ended up with:

Blue pulley diameter = 1.835"
Black pulley diameter = 2.000"
Black pulley RPM = 22,023.5

1.835/2.000 = 0.9175

22,023.5*0.9175 = RPM for Blue pulley of 20,206.5

Can anyone confirm if I've done that correctly; or, more importantly, if mechanical advantage is the right way to calculate this?

If they're on the same shaft (as in the picture) then they would both rotate at the same rate so the speeds of the two belts (?) will be proportional to the diameters. If they are coupled by a belt then the peripheral speeds will be the same.

sophiecentaur said:
If they're on the same shaft (as in the picture) then they would both rotate at the same rate so the speeds of the two belts (?) will be proportional to the diameters. If they are coupled by a belt then the peripheral speeds will be the same.

Right, would the belts' speed be directly proportional, like in my calculation above? Or should I be using a different formula?

playludesc said:
Right, would the belts' speed be directly proportional, like in my calculation above? Or should I be using a different formula?

Which bit are you asking is right? In one bit you seem to imply that the RPMs would be different. How could that be if they are on the same shaft?
The belt speed 'out' will be belt speed 'in' times the ratio of diameters - it's that simple.

Hi playludesc! I'm not sure I understand your sketch (what does variable and constant mean in the sketch?). If two wheels are connected via a normal axle, they will have the same RPM (revolutions/minute), regardless of the diameter of the wheels (unless you mean some entirely different setup). You may be confusing rotational speed (revolutions/minute) with tangential speed (e.g. centimeters/second).

The circumferences c of the wheels are (d are diameters):

cblack = pi*dblack

cblue = pi*dblue

so the tangential velocities v at the circumference of the wheels will be

vblack = cblack/t = pi*dblack * RPM

vblue = cblue/t = pi*dblue * RPM

If you measure the diameters in cm, the velocities above will be cm/minute (and t is time in minutes).
If you measure the diameters in inches, the velocities above will be inches/minute.

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sophiecentaur said:
Which bit are you asking is right? In one bit you seem to imply that the RPMs would be different. How could that be if they are on the same shaft?
The belt speed 'out' will be belt speed 'in' times the ratio of diameters - it's that simple.

That's what I thought. Sorry if my phrasing was confusing.

I'm just asking if my method of calculating the "out" speed for the blue pulley is correct in my second post. Your posts seem to agree with my method, so we're good!

DennisN said:
Hi playludesc! I'm not sure I understand your sketch (what does variable and constant mean in the sketch?). If two wheels are connected via a normal axle, they will have the same RPM (revolutions/minute), regardless of the diameter of the wheels (unless you mean some entirely different setup). You may be confusing rotational speed (revolutions/minute) with tangential speed (e.g. centimeters/second).

The circumferences c of the wheels are (d are diameters):

cblack = pi*dblack

cblue = pi*dblue

so the tangential velocities v at the circumference of the wheels will be

vblack = cblack/t = pi*dblack * RPM

vblue = cblue/t = pi*dblue * RPM

If you measure the diameters in cm, the velocities above will be cm/minute (and t is time in minutes).
If you measure the diameters in inches, the velocities above will be inches/minute.

Thanks very much for taking the time to answer so thoroughly. I think sophie confirmed that I'm on track for my application. Even so, I'll wrap my English degree head around your post and get back to you in a bit!

A jackshaft, also called a countershaft, is a common mechanical design component used to transfer or synchronize rotational force in a machine. A jackshaft is often just a short stub with supporting bearings on the ends and two pulleys, gears, or cranks attached to it.
http://en.wikipedia.org/wiki/Jackshaft

Aha that solves the mystery. Sophie and dennis both picked up on it.

Is your blue pulley a variable diameter sheave?
What are you supercharging, just out of curiosity?

Nah, blue pulley will be a timing belt pulley, I'm just trying to figure out exactly what size to make it, which in the calculation/selection phase makes it the variable.

I've got a ported M62 going on the H22a4 in my fifth generation Prelude.

Hi again! I might as well also simplify my two velocity equations further. If the pullies have the same RPM (as I suppose), then the velocity relations can be divided to yield the following relation;

vblue/vblack = dblue/dblack

which means e.g.

dblue = (dblack*vblue)/vblack

(d=diameters, v=tangential speeds)
I don't know if it helps you, I'm not sure about the other stuff in your project . (Btw, the t I used before was the period, i.e. the time for one revolution.)

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## 1. What is a pulley-to-shaft-to-pulley calculation?

A pulley-to-shaft-to-pulley calculation is a mathematical formula used to determine the relationship between the size and speed of two pulleys connected by a shaft. This calculation is important in mechanical engineering and can be used to design systems such as conveyor belts, bicycles, and gearboxes.

## 2. How do you calculate the ratio of pulley-to-shaft-to-pulley?

The ratio of pulley-to-shaft-to-pulley is calculated by dividing the diameter of the driving pulley by the diameter of the driven pulley. This ratio determines the speed at which the driven pulley will rotate in relation to the driving pulley.

## 3. What are the factors that affect a pulley-to-shaft-to-pulley calculation?

The factors that affect a pulley-to-shaft-to-pulley calculation include the size and speed of the pulleys, the distance between them, and the material and coefficient of friction of the shaft. These factors can impact the efficiency and performance of a mechanical system.

## 4. How is a pulley-to-shaft-to-pulley calculation used in real-world applications?

A pulley-to-shaft-to-pulley calculation is used in a variety of real-world applications, such as in the design of machinery and equipment. For example, it is used to determine the appropriate gear ratios in a car's transmission or to calculate the speed and power of a conveyor belt system.

## 5. What are some common challenges when performing a pulley-to-shaft-to-pulley calculation?

Some common challenges when performing a pulley-to-shaft-to-pulley calculation include accounting for friction, accurately measuring the diameters of the pulleys, and considering the effects of wear and tear on the system over time. It is important to carefully consider all factors and make adjustments as needed to ensure accurate and efficient calculations.

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