= Full-film journal-bearing whirl
A freely translating inner axle with prescribed positive <angular velocity> $\Omega$ and downward load $F>0$ has <full-film journal bearing> <force balance> equations
$$
R_r\dot r=-F\sin\phi,\qquad R_t r(\dot\phi-\Omega/2)=-F\cos\phi,
$$
where $R_r,R_t$ are the <squeeze resistance of an eccentric journal bearing>. Rotation supplies tangential fluid <force> $R_t\Omega r/2$. The equilibrium lies horizontally to the right and satisfies $F=R_t\Omega r/2$. <Linearization> there has imaginary <eigenvalues>: the prescribed rotating axle continuously supplies energy, so viscous dissipation does not imply settling. For an initially concentric axle, putting $e=r/\Delta$ and $K=6\pi\mu\Omega a^3/\Delta^2$ gives the closed orbit
$$
\frac x\Delta=\frac{2K}{5F}\left[(1-e^2)^{-1/2}-(1-e^2)^{3/4}\right],\qquad z^2=\Delta^2e^2-x^2.
$$
This follows by eliminating time and integrating $x_e+3ex/[2(1-e^2)]=\Omega\Delta^2R_0e/[2F(1-e^2)^{3/2}]$. It applies to the leading inertia-free full-film model; adding <cavitation>, constraints or finite-length leakage changes the prediction.
Back to article page
