Drillstring vibration: stick-slip, whirl and dogleg fatigue

Axial and torsional modes are solved as the same class of problem: a mass-spring chain discretized by finite elements, with element stiffness E·A/L on the axial axis and G·J/L on the torsional axis, lumped mass at each node, and fixed-free boundary conditions — top held at the rotary table, bit end free. The generalized eigenproblem K·φ = ω²·M·φ returns the system's natural frequencies fₙ.

Each natural frequency turns into an RPM band to avoid: RPM = 60·fₙ/order, for each relevant harmonic order. Running sustained inside one of those bands is the mechanical equivalent of striking the same note on a tuning fork — amplitude grows cycle after cycle until something gives, typically a connection or the BHA.

The Richard-Germay-Detournay (RGD) model couples two degrees of freedom — axial and torsional — through a single channel: the Detournay-Defourny bit-rock law, which splits weight and torque on bit into a cutting component (proportional to depth of cut, hence rate-independent) and a friction component acting on the bit's wearflat.

The source of self-excitation is a state delay: cutting depth at time t depends on the bit's angular position one revolution earlier, t−tₙ. That regenerative delay is what turns a cutting process that, on its own, does not depend on speed into a response that looks velocity-weakening — the bit slows down, nearly stops, then releases with a jump in angular velocity, the classic stick-slip pattern seen in surface torque.

Loss of bit-rock contact during the acceleration phase (bit-bounce) and mechanical specific energy (Teale's MSE) round out the picture: high MSE with torque oscillating at a low-frequency cycle is the signature of severe stick-slip, distinct from steady high MSE, which points to a worn bit instead.

Lateral vibration models the BHA as an Euler-Bernoulli beam with two degrees of freedom per node (deflection and rotation), using consistent mass instead of lumped mass — the difference matters because geometric stiffness induced by weight on bit compresses the lower BHA and lowers lateral natural frequency and critical rotary speed. Lateral supports sit at the bit, at stabilizers and at the top of the analyzed span, and the resulting side force — whose direction shows whether the tendency is build (fulcrum, pushing the high side) or drop (pendulum, pushing the low side) — feeds Warren's lateral-penetration model to estimate instantaneous dogleg severity.

That same side force couples direction and vibration: a BHA that builds angle faster is, typically, also the one that accumulates more bending fatigue at curvature transitions, because that is exactly where geometric stiffness changes fastest.

Fatigue combines the von Mises stress history from axial and torsional load (counted by rainflow, corrected by Goodman) with rotating bending induced by the dogleg — Lubinski's mechanism, in which the pipe bends around the dogleg and every surface revolution applies a full tension-compression cycle to the outer fiber. Damage accumulates through Palmgren-Miner up to D = 1, and it is this dogleg component, added to static axial tension, that typically dominates drill pipe fatigue life in curved sections — not torsion.