This work describes an alternative formulation of a system of nonlinear state-dependent delay differential equations (SDDDEs) that governs the coupled axial and torsional vibrations of a 2 DOF drillstring model considering a Polycrystalline Diamond Compact (PDC) bit with realistic cutter layout. Such considerations result in up to 100 state-dependent delays due to the regenerative effect of the drilling process, which renders the computational efficiency of conventional solution strategies unacceptable. The regeneration of the rock surface, associated with the bit motion history, can be described using the bit trajectory function, the evolution of which is governed by a partial differential equation (PDE). Thus the original system of SDDDEs can be replaced by a nonlinear coupled system of a PDE and ordinary differential equations (ODEs). Via the application of the Galerkin method, this system of PDE-ODEs is transformed into a system of coupled ODEs, which can be readily solved. The algorithm is further extended to a linear stability analysis for the bit dynamics. The resulting stability boundaries are verified with time-domain simulations. The reported algorithm could, in principle, be applied to a more realistic drillstring model, which may lead to an in-depth understanding of the mitigation of self-excited vibrations through PDC bit designs.

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