# Physics:Coulomb damping

Coulomb damping is a type of constant mechanical damping in which energy is absorbed via sliding friction. The friction generated by the relative motion of the two surfaces that press against each other is a source of energy dissipation. In general, damping is the dissipation of energy from a vibrating system where the kinetic energy is converted into heat by the friction. Coulomb damping is a common damping mechanism that occurs in machinery.

## History

Coulomb damping was so named because Charles-Augustin de Coulomb carried on research in mechanics. He later published a work on friction in 1781 entitled "Theory of Simple Machines" for an Academy of Sciences contest. Coulomb then gained much fame for his work with electricity and magnetism.

## Modes of Coulomb damping

Coulomb damping absorbs energy with friction, which converts that kinetic energy into thermal energy or heat. The Coulomb friction law is associated with two aspects. Static and kinetic frictions occur in a vibrating system undergoing Coulomb damping. Static friction occurs when the two objects are stationary or undergoing no relative motion. For static friction, the friction force F exerted between the surfaces having no relative motion cannot exceed a value that is proportional to the product of the normal force N and the coefficient of static friction μs:

$\displaystyle{ F_s = \mu_s N. }$

Kinetic friction occurs when the two objects are undergoing relative motion and they are sliding against each other. The friction force F exerted between the moving surfaces is equal to a value that is proportional to the product of the normal force N and the coefficient of kinetic friction μk:

$\displaystyle{ F_k = \mu_k N. }$

In both of these cases, the frictional force always opposes the direction of motion of the object. The normal force is perpendicular to the direction of motion of the object and equal to the weight of the object sliding.

## Example

For a simple example, a block of mass $\displaystyle{ m }$ slides over a rough horizontal surface under the restraint of a spring with a spring constant $\displaystyle{ k }$. The spring is attached to the block and mounted to an immobile object on the other end allowing the block to be moved by the force of the spring

$\displaystyle{ F = k x. }$

Because the surface is horizontal, the normal force is constant and equal to the weight of the block, or $\displaystyle{ N = mg }$. This can be determined by summing the forces in the vertical direction. A position $\displaystyle{ x }$ is then measured horizontally to the right from the location of the block when the spring is unstretched. As stated earlier, the friction force acts in a direction opposite the motion of the block. Once put into motion the block will oscillate back and forth around the equilibrium position. Newton's Second Law states that the equation of motion of the block is $\displaystyle{ m \ddot x \ = -k x - F }$ or $\displaystyle{ m \ddot x \ = -k x + F }$ depending on the direction of motion of the block. In this equation $\displaystyle{ \ddot x \ }$ is the acceleration of the block and $\displaystyle{ x }$ is the position of the block. A real-life example of Coulomb damping occurs in large structures with non-welded joints such as airplane wings.

## Theory

Coulomb damping dissipates energy constantly because of sliding friction. The magnitude of sliding friction is a constant value; independent of surface area, displacement or position, and velocity. The system undergoing Coulomb damping is periodic or oscillating and restrained by the sliding friction. Essentially, the object in the system is vibrating back and forth around an equilibrium point. A system being acted upon by Coulomb damping is nonlinear because the frictional force always opposes the direction of motion of the system as stated earlier. And because there is friction present, the amplitude of the motion decreases or decays with time. Under the influence of Coulomb damping, the amplitude decays linearly with a slope of ±((2μmgωn)/(πk)) where ωn is the natural frequency. The natural frequency is the number of times the system oscillates between a fixed time interval in an undamped system. It should also be known that the frequency and the period of vibration do not change when the damping is constant, as in the case of Coulomb damping. The period τ is the amount of time between the repetition of phases during vibration. As time progresses, the object sliding slows and the distance it travels during these oscillations becomes smaller until it reaches zero, the equilibrium point. The position where the object stops, or its equilibrium position, could potentially be at a completely different position than when initially at rest because the system is nonlinear. Linear systems have only a single equilibrium point.

## References

• Ginsberg, Jerry (2001). Mechanical and Structural Vibrations: Theory and Applications (1st ed.). John Wiley & Sons, Inc.. ISBN 0-471-37084-3.
• Inman, Daniel (2001). Engineering Vibration (2nd ed.). Prentice Hall. ISBN 0-13-726142-X.
• Walshaw, A.C. (1984). Mechanical Vibrations with Applications (1st ed.). Ellis Horwood Limited. ISBN 0-85312-593-7.