The main goal of this research is to analyze the effects of drilling mud flow rate, drill string weight, weight on bit and angular velocity on stability and vibration of a drill string. To this end, kinetic and potential energies of a rotating drill string are written while axial and lateral vibrations are considered. The effects of the drill string’s weight, weight on bit and geometrical shortening are considered in the model. Drilling mud’s effects are modeled by the Paidoussis formulations. The finite element method is employed to discrete the formulations. The stabilizers are modeled by dropping the coincided nodes. Linear (Flutter method) and non-linear methods are employed to analyze a drill string’s stability for different weight on a bit, angular velocity, drilling mud flow rate and numbers and arrangements of stabilizers. These results represent the significant effects of non-linear terms. Also, the effects of drilling mud flow rate and weight on bit on the natural frequencies and time responses are illustrated. Increasing drilling mud flow rate causes decreasing of natural frequencies and vibrational amplitude. Furthermore, increasing weight on bit leads to decreasing natural frequencies and increasing vibrational amplitude. These formulations can be used to choose the safest working conditions in the drilling process.

The main aim of this article is to find the optimum positions of the stabilizers that reduces the vibration and leads to the largest weight on bit (WOB) in drill strings. In this work, the potential energy of drill strings has been derived by considering the drill string weight and WOB. The potential energy of this continuous system is considered as a multi-degree-of-freedom system by the mode summation method. The equilibrium position of the system and its stability is determined by finding the roots of the first derivative and the sign of the second derivative of the potential energy, respectively. Using this formulation, the best positions of stabilizers that lead to the largest WOB can be found for different numbers of stabilizers. The best arrangement of the stabilizer's position on a drill string with one, two, and three stabilizers is investigated.

In this research, the effects of drilling mud flow and WOB force on the lateral vibration of drill string are investigated. To this goal, the kinetic and potential energy of drill string for axial and lateral vibrations are written in an integral equation. In potential energy equation, the effect of geometrical shortening, which causes nonlinear coupling between axial and lateral vibration, is considered. Drilling mud forces are modeled by Paidoussis formulations. The works done by WOB force, weight of drill string and drilling mud forces are calculated. The mode summation method is employed to convert the continuous system to a discrete one. Dropping and considering third and fourth order tensor of potential energy lead to linear and nonlinear system, respectively. The effects of stabilizers are modeled by a linear stiff spring. The wall contact is modeled by Hertzian contact force. Lagrange equation is employed for finding the equations of motions. First and second natural frequencies of drill string are found for different WOB and drilling mud flow. Also the effects of drilling mud and nonlinear terms on lateral vibration of drill string are investigated. The effect of drilling mud on the post buckling vibration of drill string is also delivered. This formulation can be used for optimization of drilling mud flow, WOB and the number and positions of stabilizer so that the lateral vibration of drill string is minimized.

In this research, optimum arrangements of drill string’s stabilizers are investigated. The optimum arrangement leads to biggest WOB without buckling. So the penetration rate of the drill bit can be increases. To this goal, the potential energy of drill string for axial and lateral vibration is written in an integral equation. In this equation, the effect of geometrical shortening, which causes nonlinear coupling between axial and lateral vibration, is considered. The work done by WOB force and weight of drill string is calculated. The mode summation method is employed to convert the integral potential energy of the continuous system to a discrete one. The effects of stabilizers are modeled by linear springs with large stiffness constant. Dropping and considering third and fourth order tensor of potential energy lead to linear and nonlinear stability analysis, respectively. Taking first and second order derivatives of discrete potential energy, the stability of drill string can be analyzed. Repeating this procedure for different numbers and positions of stabilizers, the optimum arrangement of stabilizers can be found. This method is employed to find the best arrangement of one and two stabilizers, by linear and nonlinear method.

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