Coulomb damping
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Coulomb damping is a type of constant mechanical
damping Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
in which the system's
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
is absorbed via
sliding friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
(the friction generated by the relative motion of two surfaces that press against each other). Coulomb damping is a common damping mechanism that occurs in machinery.


History

Coulomb damping was so named because
Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...
carried on research in mechanics. He later published a work on
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
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 Coulombian friction

Coulomb damping absorbs energy with friction, which converts that kinetic energy into thermal energy, i.e. heat. Coulomb friction considers this under two distinct modes: either static, or kinetic.
Static friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
occurs when two objects are not in relative motion, e.g. if both are stationary. The force exerted between the objects does exceed—in magnitude—the product of the
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance '' normal'' is used in the geometric sense and means perpendicular, as oppos ...
and the ''coefficient of static friction'' : :, F_, < \mu_ N.
Kinetic friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
on the other hand, occurs when two objects are undergoing relative motion, as they slide against each other. The force exerted between the moving objects is equal in magnitude to the product of the normal force and the ''coefficient of kinetic friction'' : :, F_, = \mu_ N. Regardless of the mode, friction always acts to oppose the objects' relative motion. The normal force is taken perpendicularly to the direction of relative motion; under the influence of gravity, and in the common case of an object supported by a horizontal surface, the normal force is just the weight of the object itself. As there is no relative motion under static friction, no work is done, and hence no energy can be dissipated. An oscillating system is (by definition) only dampened via kinetic friction.


Illustration

Consider a block of mass m that slides over a rough horizontal surface under the restraint of a spring with a
spring constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
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 :F = k x, where x is the horizontal displacement of the block from when the spring is unstretched. On a horizontal surface, the normal force is constant and equal to the weight of the block by
Newton's third law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
, i.e. :N = mg. As stated earlier, F_ acts to opposite the motion of the block. Once in motion, the block will oscillate horizontally back and forth around the equilibrium.
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
states that the equation of motion of the block is :m \ddot x \ = -F - (\sgn) F_k = -k x - (\sgn) \mu_mg. Above, \dot x and \ddot x respectively denote the velocity and acceleration of the block. Note that the sign of the kinetic friction term depends on \sgn—the ''direction'' the block is travelling in—but not the ''speed''. 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\pm 2\mu mg\omega_/(k\pi) where ''ω''n is the
natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
. 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 Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
''τ'' is the amount of time between the repetition of phases during
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
. 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.


See also

* Dry friction *
Viscous damping In continuum mechanics, viscous damping is a formulation of the damping phenomena, in which the source of damping force is modeled as a function of the volume, shape, and velocity of an object traversing through a real fluid with viscosity. Typic ...


References

* * * {{cite book , last = Walshaw , first = A.C. , title = Mechanical Vibrations with Applications , edition = 1st , publisher = Ellis Horwood Limited , year = 1984 , isbn = 0-85312-593-7


External links


Friction
2009-10-31) - Microsoft Encarta Online Encyclopedia 2006

- Science and Engineering Encyclopedia Mechanical vibrations