The differential equations for a rotating shaft system are derived variationally by the use of the Lagrange equations, where the kinetic and potential energy terms are obtained by integration of differential volume expressions which are in turn derived from system displacements and strains. Included, as a result, are the effects of initial and dynamic shaft curvature, gyroscopic moments, Coriolis forces, unbalances, rotatory inertia, static weight, and varying shaft cross-sections. Shown to be insignificant are the interacting effects of transverse shear, extensional displacements (and therefore axial constraint), torsion, product of inertia (for most shaft elements). Bearing forces are included as generalized forces on the shaft system. A sample solution of the equations uses the short bearing approximation to model bearing forces and derives expressions for shaft critical speed, equivalent bearing stiffness, and damping constants as functions of bearing eccentricity.

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