Kaplansky density theorem - Revision history

en>Brirush at 13:04, 25 November 2014

2014-11-25T13:04:40Z

← Older revision		Revision as of 15:04, 25 November 2014
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	~~'''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.		Friends call her Claude Gulledge. One of his preferred hobbies is playing crochet but he hasn't produced a dime with it. Years in the past we moved to Arizona but my wife desires us to transfer. The occupation he's been occupying for years is a messenger.<br><br>Also visit my page [http://Www.Pietreta.com/UserProfile/tabid/42/userId/8740/language/en-US/Default.aspx car warranty]

	~~==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 μ<sub>s</sub>.
	~~:<math>F_s = \mu_s N \,</math>~~
	~~[[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 μ<sub>k</sub>.~~
	~~:<math>F_k = \mu_k N \,</math>~~
	~~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 m slides over a rough horizontal surface under the restraint of a spring with a [[spring constant]] 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.
	~~:<math>F = k x </math>~~
	Because the surface is horizontal, the normal force is constant and equal to the weight of the block, or N=mg. This can be determined by summing the forces in the vertical direction. A position 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 <math>m \ddot x \ = -k x - F</math> or <math>m \ddot x \ = -k x + F</math> depending on the direction of motion of the block. In this equation <math> \ddot x \ </math> is the acceleration of the block and 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 ±((2μmgω~~<~~sub~~>n<~~/sub~~>)/(πk)) where ω<sub>n</sub> 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 [[Wave period\|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 [[Mechanical equilibrium\|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==~~
	* {{cite book
	~~\| last = Ginsberg \| first = Jerry~~
	~~\| title = Mechanical and Structural Vibrations: Theory and Applications~~
	~~\| edition = 1st ed.~~
	~~\| publisher = John Wiley & Sons, Inc.~~
	~~\| year = 2001~~
	~~\| id = ISBN 0-471-37084-3~~
	}}
	* {{cite book
	~~\| last = Inman \| first = Daniel~~
	~~\| title = Engineering Vibration~~
	~~\| edition = 2nd ed.~~
	~~\| publisher = Prentice Hall~~
	~~\| year = 2001~~
	~~\| id = ISBN 0-13-726142-X~~
	}}
	* {{cite book
	~~\| last = Walshaw \| first = A.C.~~
	~~\| title = Mechanical Vibrations with Applications~~
	~~\| edition = 1st ed.~~
	~~\| publisher = Ellis Horwood Limited~~
	~~\| year = 1984~~
	~~\| id = ISBN 0-85312-593-7~~
	}}

	~~==External links==~~
	* [http://~~encarta~~.~~msn~~.com/~~encyclopedia_761569794/Friction.html Friction] ([http:/~~/~~www.webcitation.org~~/~~5kwaxHMbP Archived] 2009-10-31) - Microsoft Encarta Online Encyclopedia 2006~~
	* [http://~~www.diracdelta.co.uk~~/~~science~~/~~source~~/~~c/o/coulomb%20damping/source~~.~~html Coulomb Damping] - Science and Engineering Encyclopedia~~

	~~[[Category:Mechanical vibrations]~~]

en>Buspirtraz: edited Expand German tag to link it to corresponding article and put article in category "Science articles needing translation from German Wikipedia"

2013-07-26T14:51:25Z

edited Expand German tag to link it to corresponding article and put article in category "Science articles needing translation from German Wikipedia"

← Older revision		Revision as of 16:51, 26 July 2013
Line 1:		Line 1:
	~~Claude~~ is ~~her title~~ and ~~she totally digs~~ that ~~title~~. ~~What he really enjoys performing~~ is to ~~play handball but he~~ is ~~struggling~~ to ~~find time for it~~. ~~I am~~ a ~~production~~ and ~~distribution officer~~. ~~I presently live~~ in ~~Arizona but now I~~'m ~~considering other choices~~.<br><br>~~Visit my web site~~: [http://www.~~kindkids~~.~~net~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/61/~~UserId~~/~~260741~~/~~Default~~.~~aspx extended car warranty~~]		'''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 μ<sub>s</sub>.
			:<math>F_s = \mu_s N \,</math>
			[[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 μ<sub>k</sub>.
			:<math>F_k = \mu_k N \,</math>
			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 m slides over a rough horizontal surface under the restraint of a spring with a [[spring constant]] 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.
			:<math>F = k x </math>
			Because the surface is horizontal, the normal force is constant and equal to the weight of the block, or N=mg. This can be determined by summing the forces in the vertical direction. A position 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 <math>m \ddot x \ = -k x - F</math> or <math>m \ddot x \ = -k x + F</math> depending on the direction of motion of the block. In this equation <math> \ddot x \ </math> is the acceleration of the block and 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 ±((2μmgω<sub>n</sub>)/(πk)) where ω<sub>n</sub> 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 [[Wave period\|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 [[Mechanical equilibrium\|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==
			* {{cite book
			\| last = Ginsberg \| first = Jerry
			\| title = Mechanical and Structural Vibrations: Theory and Applications
			\| edition = 1st ed.
			\| publisher = John Wiley & Sons, Inc.
			\| year = 2001
			\| id = ISBN 0-471-37084-3
			}}
			* {{cite book
			\| last = Inman \| first = Daniel
			\| title = Engineering Vibration
			\| edition = 2nd ed.
			\| publisher = Prentice Hall
			\| year = 2001
			\| id = ISBN 0-13-726142-X
			}}
			* {{cite book
			\| last = Walshaw \| first = A.C.
			\| title = Mechanical Vibrations with Applications
			\| edition = 1st ed.
			\| publisher = Ellis Horwood Limited
			\| year = 1984
			\| id = ISBN 0-85312-593-7
			}}

			==External links==
			* [http://encarta.msn.com/encyclopedia_761569794/Friction.html Friction] ([http://www.webcitation.org/5kwaxHMbP Archived] 2009-10-31) - Microsoft Encarta Online Encyclopedia 2006
			* [http://www.diracdelta.co.uk/science/source/c/o/coulomb%20damping/source.html Coulomb Damping] - Science and Engineering Encyclopedia

			[[Category:Mechanical vibrations]]

en>Sodin: added Category:Theorems in functional analysis using HotCat

2011-10-28T00:15:48Z

added Category:Theorems in functional analysis using HotCat

New page

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