The rotor is supported on journal bearings. Its model would include the support forces which are usually modeled by linearized stiffness and damping matrices or full non-linear forces and viscosity appears therein. The expression for linearized matrices changes depending upon what kind of a bearing is used. For short bearings, you can find these expressions derived in "Holmes, R., 1972, “The effect of sleeve bearings on the vibration of rotating shafts, Tribology, 161-168.". The matrix form is given in many references. The code for full non-linear force components computation (Px and Py) for a short bearing can be given as follows where inputs are X and Y direction displacements from bearing center, clearance (C), Velocities in X and Y directions (DX and DY), Viscosity (MU), radius (R), bearing length (B), and angular speed of journal (OMEGA). Note that all data must be supplied in SI units only, pow is a function for exponent, sqrt is for square root, and atan2 is signed arctan function. In the code, EPS is eccentricity ratio, SI is attitude angle measured from vertical down position, TH1 and TH2 define fluid film extent (positive pressure zone). In this model, initial forces are computed in r-s frame (PR and PS, along attitude line and normal to it) and then transformed to X-Y reference frame. This is a validated code taken from the sub-routine library of Symbols bond graph modelling software. Same solution is given in "A. Mukherjee, Effect of Biphase Lubricants on Dynamics of Rigid Rotors, J. of Lubrication Tech 105(1), 29-38 (Jan 01, 1983) (10 pages) doi:10.1115/1.3254539".

EPS=sqrt(X*X+Y*Y)/C;

DEPS=(X*DX+Y*DY)/(C*C*EPS);

DSI=(-DY*X+DX*Y)/pow(EPS*C,2.0);

COFF=MU*R*B*B*B*0.5/pow(C,2.0);

TH1=atan2(2*DEPS, EPS*(OMEGA-2*DSI)); if (TH1

Thanks a lot Dr. Arun Kumar Samantaray, for the above adequate answer.

Dr. Arun, could you please explain how can I estimate the steady state eccentricity ratio ?please.

It comes from load bearing capacity equation which gives static equilibrium position (EPS). The reference is O. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, Mcgraw-Hill Book Co. Inc., New York, 1961. The formula is

P or W (load on bearing) =

MU*OMEGA*R*pow(B,3)*EPS/XXX

where XXX= 4*C*C*(1-EPS*EPS)*(1-EPS*EPS)*YYY

and YYY= sqrt(PI*PI*(1-EPS*EPS)+16*EPS*EPS)

Please recheck the formula from the reference. Once you have this relation and you know the static load on the bearing then the above equation needs to be solved to obtain EPS. Here, MU is viscosity, C is clearance, B is length, EPS is eccentricity ratio, R is radius, OMEGA is speed and PI is atan(1.0).

Thanks Dr., but for the above formula is it for finite length bearing or short?

I need the steady state eccentricity ratio for finite length bearing, please.

also the reference you mentioned I search for it but I didn't get it, could you please send it to me.

thanks again Dr. Kumar for your reply.

Bearing length B is a parameter. So, it is for short bearing. I had the book of Pinkus, but have misplaced it and unable to trace it. Please see if you can find it from somewhere. It is a very nice book and will give you a lot of details.

Thank you Dr. Kumar, I get the book.

I will read it and hope to find how to calculate the eccentricity ratio for finite length bearing.