# Converting belt tension to radial compression force

• MechEgr
In summary, by considering the tension T applied to a belt and its dimensions, one can calculate the pressure it applies to one's waist. This can be done by considering the force applied to a semicircular segment of the belt and using the length and width of the belt to determine the pressure per unit area.

#### MechEgr

Imagine you are getting dressed in the morning and fastening a belt around your waste. You pull on the loose end of the belt, developing tension T throughout the length of the belt. Using this information, how can you determine the amount of compression the belt is applying to your waste?

I apologize if this topic has already been covered. I could not find it in the forums, but if it has, please direct me to the correct thread.

Thanks.

Suppose the belt is circular and it is pulled by force T.

One can calculate the force perpendicular to the belt element delimited by angle sector of $\Delta \varphi$ radians (draw a picture).

There are two forces pushing the band of the belt towards the belly, each acting on one end of the band and having equal magnitude $T\Delta \varphi/2$. Their sum is $T\Delta \varphi$.

This is the total force pushing the band of the belt towards the belly.

With information on the radius and width of the belt, one can then calculate the pressure.

Rather than compute the tension on a small belt element of size Δ, one can consider the tension on the front 180 degrees.

The net force on this semicircular segment is 2T. If you consider a vertical slice through yourself from hip to hip, the total force applied across this slice must therefore be 2T. The length of this slice is the diameter D of the circle formed by your belt.

Accordingly, the pressure within the bulk of your body must be given by 2T/D. This is the same as the pressure exerted by the belt.

[This "pressure" is per unit length, not per unit area]

Divide by the width of the belt to get pressure per unit area.

On the [questionable] assumption that your waist is circular and that the belt has length L and width W and just barely reaches around, this would mean that the pressure exerted by the belt on your body is given by 2piT/(L*W)

Jbriggs,

I follow your logic all the way until your very last point. I don't quite understand where the pi term came from in your equation. I understand that the force exerted over half of the cross-sectional area should be 2T/(L*W). It seems that somehow when you convert that to pressure applied circumferentially, you get an extra "pi" term from somewhere?

The pressure per unit distance at the centerline of your body must be given by 2T/D, you agree?

Given that an equilibrium exists, the pressure per unit distance around the circumference of your body must therefore also be given by 2T/D, you agree?

The length L of the belt is given by pi * D. It follows that D = L/pi.

This means that 2T/D can be rewrtten as 2T/(L/pi) = 2piT/L

Factoring in the width of the belt, the pressure per unit area is 2piT/(L*W)

## 1. How do you convert belt tension to radial compression force?

To convert belt tension to radial compression force, you can use the formula F = T/r, where F is the radial compression force, T is the belt tension, and r is the radius of the pulley. This formula assumes that the belt is in pure rolling contact with the pulley.

## 2. What factors affect the conversion of belt tension to radial compression force?

The factors that affect the conversion of belt tension to radial compression force include the material properties of the belt, the geometry and surface roughness of the pulley, and the tension in the belt. The coefficient of friction between the belt and pulley can also affect the conversion.

## 3. Can the conversion of belt tension to radial compression force be affected by misalignment?

Yes, misalignment between the belt and pulley can affect the conversion of belt tension to radial compression force. This is because misalignment can cause the belt to slip or experience uneven tension, which can affect the calculation of the radial compression force.

## 4. How accurate is the conversion of belt tension to radial compression force?

The accuracy of the conversion depends on a variety of factors, such as the accuracy of the measurement of belt tension, the accuracy of the measurement of the pulley radius, and the accuracy of the assumptions made in the calculation. Generally, the conversion can be accurate within a certain range, but it is always best to verify the results through experimentation.

## 5. Are there any software or tools available for converting belt tension to radial compression force?

Yes, there are various software and tools available that can help with the conversion of belt tension to radial compression force. These tools use different formulas and algorithms to calculate the force and may also consider other factors such as misalignment and belt slip. It is important to choose a reliable and accurate tool for your specific application.