Large-amplitude vibrations in drilling often occur at frequencies near multiples of the rotation frequency, even when these are much lower than the system’s first natural frequency. These vibrations are responsible for out-of-round, “lobed” holes. A simplified model of the mechanics of this phenomenon is presented in this paper. The model includes cutting and “rubbing” forces on the drill, but inertia and damping of the tool are neglected at low speeds. This quasi-static model remains dynamic because of the regenerative nature of cutting; the force on each cutting element depends on both the tool’s current position and its position at the time of the previous tooth passage. Characteristic solutions, including unstable retrograde “whirling” modes, are found in terms of eigenvalues and eigenvectors of a discrete state-transition matrix. These unstable modes correspond closely to behavior observed in drilling tests.

[1] In two recent papers, C. L. Rino, H. Ngo and V. Kruger (RNK) proposed a novel scheme for tropospheric propagation over one-dimensional (1D) terrain or ocean surfaces. This scheme is based on a boundary integral equation (BIE) formalism, the iterative solution of which is carried out efficiently via the method of ordered multiple interactions (MOMI). An important merit of the RNK scheme is the fact that, being a BIE-based model, it rigorously accounts for the effect of the multiple-scale ocean surface roughness on the propagated field. In their most recent paper, C. L. Rino and V. Kruger suggested that the accelerated version of MOMI proposed by H.-T. Chou and J. T. Johnson (MOMI/ CJ) could substantially enhance the efficiency of their scheme. Motivated by this suggestion, this paper proposes a combined RNK/CJ algorithm for propagation over finitely conducting (FC) ocean surfaces. While maintaining its accuracy, the proposed scheme enhances the computational efficiency of the rigorous RNK model by reducing the number of mathematical operations needed in the iterative solution of the governing BIE.INDEX TERMS: 6964 Radio Science: Radio wave propagation; 0659 Electromagnetics: Random media and rough surfaces; 0644 Electromagnetics: Numerical methods; KEYWORDS: wave propagation, rough surface scattering, integral equation methods, fast numerical algorithms Citation: Awadallah, R. S., M. T. Lamar, and J. R. Kuttler, An accelerated boundary integral equation scheme for propagation over the ocean surface,

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Large-amplitude vibrations in drilling often occur near multiples of the rotation frequency, even when these frequencies are much lower than the system’s first natural frequency. These vibrations are responsible for out-of-round, “lobed” holes. A simplified model of the mechanics of this phenomenon is presented in this paper. The model includes cutting and “rubbing” forces on the drill, but inertia and damping of the tool are neglected at low speeds. This quasi-static model remains dynamic because of the regenerative nature of cutting; the force on each cutting element depends on both the tool’s current position and its position at the time of the previous tooth passage. Characteristic solutions, including unstable retrograde “whirling” modes, are found in terms of eigenvalues and eigenvectors of a discrete state-transition matrix. These unstable modes correspond closely to behavior observed in drilling tests.

No abstract