Method to predict, illustrate, and select drilling parameters to avoid severe lateral vibrations

An apparatus for drilling a borehole in an earth material, the apparatus comprising:

• a drill string coupled to a drill bit configured to drill the borehole;
• a mud motor disposed at the drill string and configured to rotate the drill bit;
• a drill rig in operable communication with the drill string and configured to operate the drill string to drill the borehole, the drill rig being receptive to adjustable rotational speed of the drill string and adjustable rotational speed of the mud motor;
• a processor configured to:
• receive a mathematical model of a system comprising the drill sting, the mud motor, and a geometry of the borehole, the model comprising dimensions, mass distribution, material density, and material stiffness using the processor;
• calculate a mud motor lateral excitation force imposed on the drill string by the mud motor for one or more of combinations of drilling parameters;
• calculate lateral motion of the drill string and a force imposed on the drill string at a plurality of positions along the drill string for the one or more combinations of drilling parameters using the mathematical model and the mud motor lateral excitation force;
• select a range of combinations of drilling parameters that result in the force imposed upon the drill string being less than a threshold value;
• provide the range of combinations to a display;
• a display configured to receive the range of combinations from the processor and to display the range of combinations to a user.