Assuming a linearly changing gap eg h h0 hL h0xL solve th

Assuming a linearly changing gap, e.g. h = h_0 + (h_L - h_0)x/L, solve the 2D Reynold\'s Lubrication Equation and show that [White, equation 3-240)]: p - p_infinity/mu UL/h_0^2 = 6(x/L)(1 - x/L)(1 - h_L/h_0)/(1 + h_L/h_0)[1 - (1 -h_L/h_0)(x/L)]^2

Solution

solution:

1) reynolds equation in differential form is given as

d/dx(h3dp/dx)=6*mu*U*dh/dx

as there is no motion along y and z direction hence motion along only x direction is consider

where dh/dx=hl-ho/L

on putting value and on integrating we get

dp/6*mu*U=(hl-ho/L)dx/h3

where

dx=(L/(hl-ho))*dh

putting value we get

dp/6*mu*U=dh/h3

on integrating

p from P to Pinfinity and h from ho to h

hence we get

Pinf-P/6*mu*U=(-.5)(1/h2-1/ho2)

P-Pinf/(6*mu*U/ho2)=.5*(1/((1-(1-(hl/ho)x/L))2-1)

on taking positive term out fromdenometer and and taking negative reciprocal to numerator and on solving finally get given equation as

P-Pinf/(6*mu*U/ho2)=6*(x/L)(1-x/L)(1-hl/ho)/(1+hl/ho)((1-(1-(hl/ho)x/L))2)

hence equation is prove for bearing