Automated drilling methods and systems using real-time analysis of drill string dynamics

A drilling system comprising:

• a plurality of sensors mounted to a drill string at a plurality of sensor locations, the sensors configured to sense a plurality of acceleration measurements corresponding to a plurality of locations in a first two-dimensional reference frame, the first two-dimensional reference frame coincident with a plane orthogonal to a rotational axis of the drilling system and based on a position of the drilling system in the plane, the plurality of acceleration measurements comprising measurements in a plurality of non-parallel directions lying in the plane;
• a processor in communication with the plurality of sensors, the processor configured to:
• receive the plurality of acceleration measurements;
• determine a rotational position of the drilling system in the first two-dimensional reference frame based on the plurality of acceleration measurements, the rotational position describing rotation of the drilling system in the plane about the rotational axis;
• convert a first location of the plurality of locations to a corresponding second location of the drilling system in a second two-dimensional reference frame based on the plurality of acceleration measurements, the second two-dimensional reference frame coincident with the plane and invariant with the position of the drilling system in the plane;
• determine a revolution position of the drilling system in the second two-dimensional reference frame based on the rotational position and the second location, the revolution position describing revolution of the drilling system in the plane about a revolution axis;
• and
• identify a whirl dynamic of the drilling system based on the revolution position;
• wherein the processor being configured to determine a rotational position comprises the processor being configured to:
• determine a rotational velocity of the drilling system in the first two-dimensional reference frame based on the plurality of acceleration measurements;
• and
• determine the rotational position based on the rotational velocity;
• wherein the first location comprises a first sensor location of the first sensor, a first measurement subset of the plurality of acceleration measurements are sensed by the first sensor, the first sensor location spaced apart from the rotational axis by a first radius RA;
• and
• wherein a second measurement subset of the plurality of acceleration measurements are sensed by a second sensor, the second sensor having a second sensor location spaced apart from the rotational axis by a second radius RB and spaced apart from the first sensor location.