Solution to Hydrostatic Bearing Integration Task

[mathi85]

Hi everyone!
I would like to ask you for help with one of the tasks from my assignment. The rest of the assignment is done including some simple integration but I struggle with this one:

Task
"The total load capacity for a circular hydrostatic bearing is given as

$W=\int_0^{R_o} p_r(2πr dr) + \int_{R_o}^R p(2πr dr)$

By expressing the radial pressure in terms of the recess pressure, and by step by step argument, show that:

$W={\frac{π}{2}}{\frac{R^2-R_o^2}{2ln(R/R_o)}}p_r$ "

I think that radial pressure in terms of recess pressure is:

$p=p_r{\frac{ln(R/r)}{ln(R/R_o)}}$

I really cannot get my head around it. Shall I just substitute above equation for 'p'? Then I would get:

$W=\int_0^{R_o} p_r(2πr dr) + \int_{R_o}^R{\frac{p_r2πrdrln(R/r)}{ln(R/R_o)}}$

Do I have to then sort both integrals and just add them up together?

 

[Ray Vickson]

Hi everyone!
I would like to ask you for help with one of the tasks from my assignment. The rest of the assignment is done including some simple integration but I struggle with this one:

Task
"The total load capacity for a circular hydrostatic bearing is given as

$W=\int_0^{R_o} p_r(2πr dr) + \int_{R_o}^R p(2πr dr)$

By expressing the radial pressure in terms of the recess pressure, and by step by step argument, show that:

$W={\frac{π}{2}}{\frac{R^2-R_o^2}{2ln(R/R_o)}}p_r$ "

I think that radial pressure in terms of recess pressure is:

$p=p_r{\frac{ln(R/r)}{ln(R/R_o)}}$

I really cannot get my head around it. Shall I just substitute above equation for 'p'? Then I would get:

$W=\int_0^{R_o} p_r(2πr dr) + \int_{R_o}^R{\frac{p_r2πrdrln(R/r)}{ln(R/R_o)}}$

Do I have to then sort both integrals and just add them up together?

Yes. That is exactly what the formula says.

BTW: I think the given answer is too small by a factor of 2.

 

[mathi85]

Here is first part:

$\int_R_o^0$

 

[Ray Vickson]

Here is first part:

$\int_R_o^0$

Are you saying that your equation in the original post is wrong?

 

[Mark44]

Here is first part:

$\int_R_o^0$

Is this what you meant to write?
$$\int_{R_0}^0$$
The LaTeX script for the above is \int_{R_0}^0. If a limit of integration is more than one character, you need to put it in braces - { }.