Energy Losses in the Hydrodynamic Lubrication of Bearings

To calculate friction from the Reynolds equation, we begin by differentiating Eqn 2-8 with respect to t/, keeping in mind the restrictions imposed on p which make dp/dx a simple derivative  

In the plane slider bearing the shear stress on the surface y = h is on the surface y = 0, (Tj ). These conditions, together with the convention used for the direction of the velocity give the expression below for the frictional force in the oil film between two surfaces  

On introducing Eqn 2-23 for the relation between h and x into Eqn 2-28 and then integrating each term by parts. 

The first term is the horizontal component of the vertically integrated pressure acting on the shoe. The second term is the viscous frictional shear in the oil film and is the one that is dominant in lubricated sliding systems which can be treated by the Reynolds equation. 

Another way to treat the loss problem is to compute the power which is lost as the product of friction force and the shearing velocity at the moving boundary. If the force is in newtons and the velocity in meters per second, the power loss is in watts. In English engineering units, if the force is in lbs and the sliding speed in inches per second, the loss expressed as horsepower is  

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