Alternating frequency time domain approach to calculate the forced response of drill strings

A method for drilling a borehole penetrating an earth formation, the method comprising:

• drilling a borehole with a drill rig that operates a drill string having a drill bit;
• obtaining borehole geometry data;
• calculating a first displacement of the drill string in a frequency domain for a first excitation force frequency using an equation of motion of the drill string that is solved by a processor, the equation of motion having a static force component, an excitation force component, and a non-linear force component with respect to at least one of a deflection and a derivative of the deflection of the drill string;
• transforming the first displacement from the frequency domain into a time domain using the processor;
• calculating a non-linear force in the time domain based on the borehole geometry data and at least one of the calculated displacement and a derivative of the calculated displacement using the processor;
• calculating a frequency domain coefficient derived from the calculated non-linear force in the time domain using the processor;
• and
• calculating a second displacement of the drill string in the frequency domain using the equation of motion and the frequency domain coefficient using the processor;
• and
• transmitting a control signal from the processor to the drill rig to control a drilling parameter, the processor being configured to execute a control algorithm having the second displacement as an input.

An apparatus for drilling a borehole penetrating an earth formation using a drill rig configured to operate a drill string having a drill bit, the apparatus comprising:

• a borehole caliper tool disposed at the drill string and configured to provide borehole geometry data;
• a processor configured to receive the borehole geometry data and to implement a method comprising:
• calculating a first displacement of the drill string in a frequency domain for a first excitation force frequency using an equation of motion of the drill string, the equation of motion having a static force component, an excitation force component, and a non-linear force component with respect to at least one of a deflection and a derivative of the deflection of the drill string;
• transforming the first displacement from the frequency domain into a time domain;
• calculating a non-linear force in the time domain based on the borehole geometry data and at least one of the calculated displacement and a derivative of the calculated displacement;
• calculating a frequency domain coefficient derived from the calculated non-linear force in the time domain;
• and
• calculating a second displacement of the drill string in the frequency domain using the equation of motion and the frequency domain coefficient;
• a controller configured to receive the second displacement and to transmit a control signal to the drill rig to control a drilling parameter, the controller being configured to execute a control algorithm having the second displacement as an input.