Dampers for mitigation of downhole tool vibrations

1. A system for damping torsional oscillations of downhole systems, the system comprising:

a damping system configured on the downhole system, the damping system comprising:

a first element; and

a second element in frictional contact with the first element,

wherein the second element moves relative to the first element with a velocity that is a sum of a periodic velocity fluctuation having an amplitude and a mean velocity, wherein the mean velocity is lower than the amplitude of the periodic velocity fluctuation.

2. The system of claim 1 , further comprising an adjusting element arranged to adjust a force between the first element and the second element.

3. The system of claim 2 , wherein the adjustment is based on a threshold of at least one of the amplitude and a frequency of the torsional oscillations.

4. The system of claim 1 , wherein the first element comprises a first portion that is fixedly attached to the second element, such that the first portion does not move relative to the second element.

5. The system of claim 1 , wherein the torsional oscillations comprise a first oscillation mode and a second oscillation mode.

6. The system of claim 5 , wherein the second element comprises a first body and a second body, wherein the first body moves relative to the first element with a velocity that is a first sum of a first periodic velocity fluctuation having a first amplitude and a first mean velocity and the second body moves relative to the first element with a velocity that is a second sum of a second periodic velocity fluctuation having a second amplitude and a second mean velocity, wherein the first mean velocity is lower than the first amplitude of the first periodic velocity fluctuation and the second mean velocity is lower than the second amplitude of the second periodic velocity fluctuation, wherein the first body is selected to damp the first oscillation mode and the second body is selected to damp the second oscillation mode.

7. The system of claim 6 , wherein the downhole system rotates about a rotation axis and wherein the first body and the second body are positioned at different locations along the rotation axis.

8. The system of claim 5 , further comprising a processor configured to calculate a mode shape of at least one of the first oscillation mode and the second oscillation mode and wherein at least one of the first element and second element is located in the damping system based on the calculation.

9. The system of claim 5 , wherein at least one of the first oscillation mode and the second oscillation mode has a shape comprising a maximum and a minimum and the length of at least one of the first element and the second element is a tenth of the distance between the maximum and the minimum.

10. The system of claim 1 , wherein the frictional contact switches from a static friction to a dynamic friction during each period of the periodic velocity fluctuation.

11. A method of damping torsional oscillations of a downhole system in a borehole, the method comprising:

installing a damping system on a downhole system, the damping system comprising:

a first element; and

a second element in frictional contact with the first element,

wherein the second element moves relative to the first element with a velocity that is a sum of a periodic velocity fluctuation having an amplitude and a mean velocity, wherein the mean velocity is lower than the amplitude of the periodic velocity fluctuation.

12. The method of claim 11 , further comprising adjusting, with an adjusting element, a force between the first element and the second element.

13. The method of claim 12 , wherein adjusting is based on a threshold of at least one of the amplitude and a frequency of the torsional oscillations.

14. The method of claim 11 , wherein the first element comprises a first portion that is fixedly attached to the second element such that the first portion does not move relative to the second element.

15. The method of claim 11 , wherein the torsional oscillations comprise a first oscillation mode and a second oscillation mode.

16. The method of claim 15 , wherein the second element comprises a first body and a second body, wherein the first body moves relative to the first element with a velocity that is a first sum of a first periodic velocity fluctuation having a first amplitude and a first mean velocity and the second body moves relative to the first element with a velocity that is a second sum of a second periodic velocity fluctuation having a second amplitude and a second mean velocity, wherein the first mean velocity is lower than the first amplitude of the first periodic velocity fluctuation and the second mean velocity is lower than the second amplitude of the second periodic velocity fluctuation, wherein the first body is selected to damp the first oscillation mode and the second body is selected to damp the second oscillation mode.

17. The method of claim 16 , further comprising rotating the downhole system about a rotation axis, wherein the first body and the second body are positioned at different locations along the rotation axis.

18. The method of claim 15 , further comprising calculating, with a computer, a mode shape of at least one of the first oscillation mode and second oscillation mode and placing at least one of the first element and the second element based on the calculation.

19. The method of claim 15 , wherein at least one of the first oscillation mode and the second oscillation mode has a shape comprising a maximum and a minimum and the length of at least one of the first element and second element is a tenth of the distance between the maximum and the minimum.

20. The method of claim 11 , wherein the frictional contact switches from a static friction to a dynamic friction during each period of the periodic velocity fluctuation.