Capstan equation - Revision history

en>Applejuicefool at 13:32, 8 January 2015

2015-01-08T13:32:49Z

← Older revision		Revision as of 15:32, 8 January 2015
Line 1:		Line 1:
	The '''Krylov–Bogolyubov averaging method''' ('''Krylov–Bogolyubov method of averaging''') is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics.<ref>[http://eom.springer.de/K/k055940.htm Krylov–Bogolyubov method of averaging] at Springer Encyclopaedia of Mathematics</ref> The method is based on the averaging principle when the exact differential equation of the motion is replaced by its averaged version. The method is named after [[Nikolay Mitrofanovich Krylov\|Nikolay Krylov]] and [[Nikolay Bogoliubov]].

	Various averaging schemes for studying problems of celestial mechanics were used since works of [[Carl Friedrich Gauss\|Gauss]], [[Pierre Fatou\|Fatou]], Delone, [[George William Hill\|Hill]]. The importance of the contribution of Krylov and Bogoliubov is that they developed a general averaging approach and proved that the solution of the averaged system approximates the exact dynamics.<ref>{{cite book \|title=Methodes approchees de la mecanique non-lineaire dans leurs application a l'Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s'y rapportant\|last=N. M. Krylov\|authorlink=\|coauthors=N. N. Bogolyubov\|year=1935\|publisher=Academie des Sciences d'Ukraine\|location=Kiev\|isbn=\|pages=\|language=French}}</ref><ref>{{cite book \|title=Introduction to non-linear mechanics\|last= N. M. Krylov\|authorlink= \|coauthors=N. N. Bogolyubov\|year=1937\|language=Russian \|publisher=Izd-vo AN SSSR\|location=Kiev\|isbn=\|pages=}}</ref><ref>{{cite book \|title=Introduction to non-linear mechanics\|last= N. M. Krylov\|authorlink= \|coauthors=N. N. Bogolyubov\|year=1947 \|publisher=Princeton Univ. Press\|location=Princeton\|isbn=9780691079851\|pages=\|language=English, partial translation from Russian}}</ref>

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	~~Krylov~~-~~Bogoliubov averaging~~ can ~~be used~~ to ~~approximate oscillatory problems when a classical perturbation expansion fails~~. ~~That is [[singular perturbation]] problems~~ of ~~oscillatory type, for example Einstein's correction~~ to ~~the~~ [~~[Two-body problem in general relativity\|perihelion precession of Mercury]]~~.~~<ref name~~=~~"Smith">{{cite book \| last = Smith \| first = Donald \| title = Singular~~-~~Perturbation Theory \| publisher = Cambridge University Press \| location = Cambridge \| year = 1985 \| isbn = 0~~-~~521-30042-8 }}</ref>~~

	~~==Derivation==~~
	~~The method deals with differential equations in~~ the ~~form~~

	~~:<math>~~
	~~\frac{d^2u}{dt^2} + k^2 u = a + \varepsilon f\left(u~~,~~\frac{du}{dt}\right)~~
	~~</math>~~
	~~for~~ a ~~smooth function ''f'' along with appropriate initial conditions~~. The ~~parameter ''ε''~~ is ~~assumed~~ to ~~satisfy~~

	~~:<math>~~
	~~0 < \varepsilon \ll k.~~
	~~</math>~~
	~~If ''ε'' = 0 then the equation becomes that of~~ the ~~simple harmonic oscillator with constant forcing~~, and ~~the general solution is~~

	:<~~math~~>
	~~u(t) = \frac{a}{k^2} + A \sin (kt + B),~~
	<~~/math~~>
	~~where ''A'' and ''B''~~ are ~~chosen~~ to ~~match~~ the ~~initial conditions~~. ~~The solution~~ to the ~~perturbed equation (when ''ε'' ≠ 0) is assumed to take~~ the ~~same~~
	~~form~~, ~~but now ''A'' and ''B'' are allowed to vary~~ with ~~''t'' (and ''ε'')~~. ~~If it~~ is ~~also assumed that~~
	:<~~math~~>
	~~\frac{du}{dt} = kA(t) \cos (kt + B(t)),~~
	<~~/math~~>
	~~then it~~ can ~~be shown that ''A''~~ and ~~''B'' satisfy~~ the ~~differential equation:<ref name="Smith" />~~
	~~:<math>~~
	~~\frac{d}{dt} \begin{bmatrix} A \\ B \end{bmatrix} = \frac{\varepsilon}{k} f\left( \frac{~~a~~}{k^2} + A \sin (\phi)~~, ~~kA \cos (\phi)\right) \begin{bmatrix} \cos(\phi) \\ - \frac{1}{A} \sin(\phi) \~~end~~{bmatrix}~~,
	~~</math>~~
	~~where~~ <~~math~~> ~~\phi = kt + B~~ <~~/math~~>~~. Note that this equation is still exact — no approximation~~ has ~~been made as yet. The method~~ of ~~Krylov~~ and ~~Bogolyubov is to note that~~ the ~~functions A~~ and ~~B vary slowly~~
	with ~~time~~ (in ~~proportion to ε~~), ~~so their dependence on φ can be (approximately) removed by averaging on~~ the ~~right hand side of the previous equation:~~
	~~:<math>~~
	~~\frac{d}{dt} \begin{bmatrix} A_0 \\ B_0 \end{bmatrix} = \frac{\varepsilon}{2\pi k} \int_0^{2 \pi} f( \frac{~~a~~}{k^2} + A \sin (\theta), kA \cos (\theta)) \begin{bmatrix} \cos(\theta) \\ - \frac{1}{A_0} \sin(\theta) \end{bmatrix} d\theta,~~
	~~</math>~~
	~~where <math>A_0</math>~~ and ~~<math>B_0</math> are held fixed during~~ the ~~integration~~. ~~After solving this (possibly) simpler set~~ of ~~differential equations, the Krylov–Bogolyubov averaged approximation~~ for ~~the original function is then given by~~
	:~~<math>~~
	~~u_0(t~~,~~\varepsilon) := \frac{a}{k^2} + A_0(t~~,~~\varepsilon) \sin (kt + B_0(t,\varepsilon))~~.
	<~~/math~~>
	~~This approximation has been shown to satisfy~~ <~~ref~~>~~{{cite book \| last = Bogolubov \| first = N. \| title = Asymptotic Methods in the Theory of Non-Linear Oscillations \| publisher = Gordon~~ & ~~Breach \| location = Paris \| year = 1961 \| isbn = 978-0-677-20050-7 }}</ref~~>
	~~:<math>~~
	~~\left\| u(t~~,~~\varepsilon)~~ - ~~u_0(t,\varepsilon) \right\| \le C_1 \varepsilon~~,
	~~</math>~~
	~~where t satisfies~~
	~~:<math>~~
	~~0 \le t \le \frac{C_2}{\varepsilon}~~
	~~</math>~~
	for ~~some constants <math>C_1</math>~~ and ~~<math>C_2</math>~~, ~~independent of ε~~.

	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Krylov-Bogoliubov averaging method}}~~
	~~[[Category:Dynamical systems]]~~

en>WPGA2345: template ship names, replaced: USS ''Nimitz'' → {{USS|Nimitz|CVN-68|6}} using AWB

2013-11-01T04:31:33Z

template ship names, replaced: USS ''Nimitz'' → {{USS|Nimitz|CVN-68|6}} using AWB

← Older revision		Revision as of 06:31, 1 November 2013
Line 1:		Line 1:
	~~They~~'~~re always ready to help, and they~~'~~re always making changes to the site to make sure you won~~'~~t have troubles in the first place. You may discover this probably the most time-consuming part~~ of ~~building~~ a ~~Word~~ - ~~Press MLM website~~. ~~Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs~~. ~~s ultimately easy to implement and virtually maintenance free~~. ~~It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed~~. <~~br><br~~>		The '''Krylov–Bogolyubov averaging method''' ('''Krylov–Bogolyubov method of averaging''') is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics.<ref>[http://eom.springer.de/K/k055940.htm Krylov–Bogolyubov method of averaging] at Springer Encyclopaedia of Mathematics</ref> The method is based on the averaging principle when the exact differential equation of the motion is replaced by its averaged version. The method is named after [[Nikolay Mitrofanovich Krylov\|Nikolay Krylov]] and [[Nikolay Bogoliubov]].

	~~These folders as well as files have to copied and the saved~~. The ~~higher your blog ranks on search engines,~~ the ~~more likely people will find your online marketing site. There are number~~ of ~~web services~~ that ~~offer Word press development across~~ the ~~world~~. ~~These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to~~ a ~~site easier to configure~~. ~~Once you~~'~~ve installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having~~ to ~~do all the coding yourself~~. <br><br>~~Saying that~~, ~~despite the launch of Wordpress Express many months ago, there has still been no sign of~~ a ~~Wordpress video tutorial on offer UNTIL NOW~~. As of ~~now~~, ~~Pin2Press is getting ready~~ to ~~hit~~ the ~~market~~. ~~This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words~~ in ~~their content to match~~ the ~~keywords entered by their web visitors in their search queries~~'~~a feat that they cannot easily achieve~~ with ~~older software~~. ~~After~~ that the ~~developer adds~~ the ~~unordered list for navigations.~~ <br><br>~~Word - Press installation is very easy~~ and ~~hassle free. And, make no mistake,India~~'~~s Fertility Clinics and IVF specialists~~ are ~~amongst~~ the ~~best in~~ the ~~world~~,and ~~have been for some time. Next you~~'~~ll go by way of~~ to ~~your simple Word - Press site~~. ~~There are many advantages of hiring Wordpress developers for Wordpress project development~~:~~. Word - Press offers constant updated services and products~~, that ~~too, absolutely free of cost.~~ <br><br>~~Instead~~, ~~you can easily just include it with our bodies integration field in e~~ - ~~Panel~~. ~~If you operate a website~~ that's been ~~built on HTML then you might have~~ to ~~witness traffic losses because such a site isn't competent enough~~ in ~~grabbing~~ the ~~attention~~ of ~~potential consumers. Useful Plugins~~ ~~Uber~~ - ~~Menu Top Megamenu~~ ~~Now it is~~ the ~~time~~ of ~~sticky Top navbar. Here~~ is ~~more in regards to [http~~:/~~/www~~.~~ruedalo.co/wordpress_backup_plugin_533377 wordpress backup] review~~ the ~~web~~-~~site. Word~~ - ~~Press is an open source content management system which is easy to use~~ and ~~offers many user friendly features. You can check out the statistics~~ of ~~page of views for your web pages using free tools that are available on the internet~~.		Various averaging schemes for studying problems of celestial mechanics were used since works of [[Carl Friedrich Gauss\|Gauss]], [[Pierre Fatou\|Fatou]], Delone, [[George William Hill\|Hill]]. The importance of the contribution of Krylov and Bogoliubov is that they developed a general averaging approach and proved that the solution of the averaged system approximates the exact dynamics.<ref>{{cite book \|title=Methodes approchees de la mecanique non-lineaire dans leurs application a l'Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s'y rapportant\|last=N. M. Krylov\|authorlink=\|coauthors=N. N. Bogolyubov\|year=1935\|publisher=Academie des Sciences d'Ukraine\|location=Kiev\|isbn=\|pages=\|language=French}}</ref><ref>{{cite book \|title=Introduction to non-linear mechanics\|last= N. M. Krylov\|authorlink= \|coauthors=N. N. Bogolyubov\|year=1937\|language=Russian \|publisher=Izd-vo AN SSSR\|location=Kiev\|isbn=\|pages=}}</ref><ref>{{cite book \|title=Introduction to non-linear mechanics\|last= N. M. Krylov\|authorlink= \|coauthors=N. N. Bogolyubov\|year=1947 \|publisher=Princeton Univ. Press\|location=Princeton\|isbn=9780691079851\|pages=\|language=English, partial translation from Russian}}</ref>

			==Background==
			Krylov-Bogoliubov averaging can be used to approximate oscillatory problems when a classical perturbation expansion fails. That is [[singular perturbation]] problems of oscillatory type, for example Einstein's correction to the [[Two-body problem in general relativity\|perihelion precession of Mercury]].<ref name="Smith">{{cite book \| last = Smith \| first = Donald \| title = Singular-Perturbation Theory \| publisher = Cambridge University Press \| location = Cambridge \| year = 1985 \| isbn = 0-521-30042-8 }}</ref>

			==Derivation==
			The method deals with differential equations in the form

			:<math>
			\frac{d^2u}{dt^2} + k^2 u = a + \varepsilon f\left(u,\frac{du}{dt}\right)
			</math>
			for a smooth function ''f'' along with appropriate initial conditions. The parameter ''ε'' is assumed to satisfy

			:<math>
			0 < \varepsilon \ll k.
			</math>
			If ''ε'' = 0 then the equation becomes that of the simple harmonic oscillator with constant forcing, and the general solution is

			:<math>
			u(t) = \frac{a}{k^2} + A \sin (kt + B),
			</math>
			where ''A'' and ''B'' are chosen to match the initial conditions. The solution to the perturbed equation (when ''ε'' ≠ 0) is assumed to take the same
			form, but now ''A'' and ''B'' are allowed to vary with ''t'' (and ''ε''). If it is also assumed that
			:<math>
			\frac{du}{dt} = kA(t) \cos (kt + B(t)),
			</math>
			then it can be shown that ''A'' and ''B'' satisfy the differential equation:<ref name="Smith" />
			:<math>
			\frac{d}{dt} \begin{bmatrix} A \\ B \end{bmatrix} = \frac{\varepsilon}{k} f\left( \frac{a}{k^2} + A \sin (\phi), kA \cos (\phi)\right) \begin{bmatrix} \cos(\phi) \\ - \frac{1}{A} \sin(\phi) \end{bmatrix},
			</math>
			where <math> \phi = kt + B </math>. Note that this equation is still exact — no approximation has been made as yet. The method of Krylov and Bogolyubov is to note that the functions A and B vary slowly
			with time (in proportion to ε), so their dependence on φ can be (approximately) removed by averaging on the right hand side of the previous equation:
			:<math>
			\frac{d}{dt} \begin{bmatrix} A_0 \\ B_0 \end{bmatrix} = \frac{\varepsilon}{2\pi k} \int_0^{2 \pi} f( \frac{a}{k^2} + A \sin (\theta), kA \cos (\theta)) \begin{bmatrix} \cos(\theta) \\ - \frac{1}{A_0} \sin(\theta) \end{bmatrix} d\theta,
			</math>
			where <math>A_0</math> and <math>B_0</math> are held fixed during the integration. After solving this (possibly) simpler set of differential equations, the Krylov–Bogolyubov averaged approximation for the original function is then given by
			:<math>
			u_0(t,\varepsilon) := \frac{a}{k^2} + A_0(t,\varepsilon) \sin (kt + B_0(t,\varepsilon)).
			</math>
			This approximation has been shown to satisfy <ref>{{cite book \| last = Bogolubov \| first = N. \| title = Asymptotic Methods in the Theory of Non-Linear Oscillations \| publisher = Gordon & Breach \| location = Paris \| year = 1961 \| isbn = 978-0-677-20050-7 }}</ref>
			:<math>
			\left\| u(t,\varepsilon) - u_0(t,\varepsilon) \right\| \le C_1 \varepsilon,
			</math>
			where t satisfies
			:<math>
			0 \le t \le \frac{C_2}{\varepsilon}
			</math>
			for some constants <math>C_1</math> and <math>C_2</math>, independent of ε.

			==References==
			{{reflist}}

			{{DEFAULTSORT:Krylov-Bogoliubov averaging method}}
			[[Category:Dynamical systems]]

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

2012-07-19T00:20:03Z

Robot - Moving category Fundamental physics concepts to Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

New page

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These folders as well as files have to copied and the saved. The higher your blog ranks on search engines, the more likely people will find your online marketing site. There are number of web services that offer Word press development across the world. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. As of now, Pin2Press is getting ready to hit the market. This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. After that the developer adds the unordered list for navigations. Word - Press installation is very easy and hassle free. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Next you'll go by way of to your simple Word - Press site. There are many advantages of hiring Wordpress developers for Wordpress project development:. Word - Press offers constant updated services and products, that too, absolutely free of cost. Instead, you can easily just include it with our bodies integration field in e - Panel. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. Here is more in regards to [http://www.ruedalo.co/wordpress_backup_plugin_533377 wordpress backup] review the web-site. Word - Press is an open source content management system which is easy to use and offers many user friendly features. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.