This paper outlines the theoretical description of the vibratory portion of the rotary-vibratory drilling process. A multiple mechanical element model is used to describe the drill string and rock-rock bit assembly. The drill string model has continuously distributed properties of mass, stiffness, and external drilling-mud damping. A closed form solution is developed using boundary condition matching at the end of each mechanical element. The solution is used to compute the power input to the system by a vibratory unit, the power delivered to the rock, and the power lost to the drilling mud through vibratory losses. From these data, efficiencies are computed. The analytical solution has been checked in parallel with a transfer matrix computer solution. The results are identical within computer precision. The analytical model is then applied to the study of the Drilling Research Incorporated (DRI) prototype drilling system and its test drilling parameters for the 1957 test drilling. Explanations of the limits of the increase in drilling rates to 2:1 are explored. The results are explored relative to the potential for increasing the drill penetration rates by system redesign. Conclusions are drawn concerning the most productive routes to be taken for rotary-vibratory drilling systems and for the vibratory driver. It has become clear that successful future downhole rotary-vibratory drilling rigs will require a complete system understanding, a complete system design, and a new concept in vibratory driver.

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