This paper presents the results of an analytical frequency study which illustrates the effect of a select set of variables on the first natural frequency of a producing well. Specifically, axial force effects, fluid flow effects, and hydrodynamic effects, by means of a hydrodynamic function, are investigated. Due to the nature of the hydrodynamic function, the model is derived in the frequency domain and solved using the spectral element method.
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Eric Kjolsing, Michael Todd, "A frequency study of a clamped-clamped pipe immersed in a viscous fluid conveying internal steady flow for use in energy harvester development as applied to hydrocarbon production wells," Proc. SPIE 9435, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2015, 943505 (3 April 2015);