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| {{More footnotes|date=January 2009}}
| | Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br> |
| [[File:Capstan equation diagram.svg|thumb|right|220px|]]
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| [[File:Example, no knowledge of capstan equation.png|thumb|An example of when knowledge of the capstan equation might have been useful. The bent white tube contains a cord to raise and lower a curtain. The tube is bent some 40 degrees in two places. The blue line indicates a better design.]]
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| The '''capstan equation''' or '''belt friction equation''', also known as '''Eytelwein's formula''',<ref>http://www.atp.ruhr-uni-bochum.de/rt1/currentcourse/node57.html</ref><ref>http://www.jrre.org/att_frict.pdf</ref> relates the hold-force to the load-force if a flexible line is wound around a cylinder (a [[bollard]], a [[winch]] or a [[Capstan (nautical)|capstan]])<ref name=Johnson>{{cite book | last = Johnson | first = K.L. | title = Contact Mechanics | url = http://www.knovel.com/web/portal/browse/display?_EXT_KNOVEL_DISPLAY_bookid=2685 | format = PDF | accessdate = February 14, 2011 | year = 1985}}</ref>
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| .<ref name=Attaway>{{cite conference | last = Attaway | first = Stephen W. | title = The Mechanics of Friction in Rope Rescue | url = http://www.jrre.org/att_frict.pdf | format = PDF | accessdate = February 1, 2010 | conference = International Technical Rescue Symposium | year = 1999}}</ref> | |
| Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small ''holding'' force exerted on one side can carry a much larger ''loading'' force on the other side; this is the principle by which a capstan-type device operates. For instance in [[rock climbing]] with so-called [[top-roping]], a lighter person can hold (belay) a heavier person due to this effect.
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| The formula is:
| | Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. If you are a positive thinker businessman then today you have to put your business online. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. Here is more information on [http://roaaad.com/link//wordpress_dropbox_backup_4691962 wordpress backup plugin] look into our own web-page. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>This gives a clearer picture that online shoppers are familiar with the WP ecommerce system. Word - Press has different exciting features including a plug-in architecture with a templating system. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. Word - Press can also be quickly extended however improvement API is not as potent as Joomla's. with posts or testimonials updated as they are uploaded to a particular section of the website. Verify whether your company has a team of developers or programmers having hands-on experience and knowledge about all Word - Press concepts. |
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| :<math>T_\text{load} = T_\text{hold}\ e^{ \mu \phi} \,</math>
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| where <math>T_\text{load}</math> is the applied tension on the line, <math>T_\text{hold}</math> is the resulting force exerted at the other side of the capstan, <math>\mu</math> is the [[coefficient of friction]] between the rope and capstan materials, and <math>\phi</math> is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle <math>\phi =2\pi\,</math>).
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| Several assumptions must be true for the formula to be valid:
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| # The rope is on the verge of full sliding, i.e. <math>T_\text{load}</math> is the maximum load that one can hold. Smaller loads can be held as well, resulting in a smaller ''effective'' contact angle <math>\phi</math>.
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| # It is important that the line is not rigid, in which case significant force would be lost in the bending of the line tightly around the cylinder. (The equation must be modified for this case.) For instance a [[Bowden cable]] is to some extent rigid and doesn't obey the principles of the Capstan equation.
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| # The line is non-[[Elasticity (physics)|elastic]].
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| It can be observed that the force gain grows [[Exponential growth|exponentially]] with the coefficient of friction, the number of turns around the cylinder, and the angle of contact. Note that the radius of the cylinder has no influence on the force gain. The table below lists values of the factor <math>e^{ \mu \phi} \,</math> based on the number of turns and coefficient of friction ''μ''.
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| {| class="wikitable" style="text-align:right;"
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| ! rowspan="2" | Number<br/>of turns !! colspan="7" | Coefficient of friction ''μ''
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| ! scope="col" | 0.1
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| ! scope="col" | 0.2
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| ! scope="col" | 0.3
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| ! scope="col" | 0.4
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| ! scope="col" | 0.5
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| ! scope="col" | 0.6
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| ! scope="col" | 0.7
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| |-
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| ! scope="row" | 1
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| | 1.9 || 3.5 || 6.6 || 12 || 23 || 43 || 81
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| |-
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| ! scope="row" | 2
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| | 3.5 || 12 || 43 || 152 || 535 || {{gaps|1|881}} || {{gaps|6|661}}
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| |-
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| ! scope="row" | 3
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| | 6.6 || 43 || 286 || {{gaps|1|881}} || {{gaps|12|392}}|| {{gaps|81|612}}|| {{gaps|437|503}}
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| |-
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| ! scope="row" | 4
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| | 12 || 152 || {{gaps|1|881}} || {{gaps|23|228}} || {{gaps|286|751}} || {{gaps|3|540|026}} || {{gaps|43|702|631}}
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| |-
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| ! scope="row" | 5
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| | 23 || 535 || {{gaps|12|392}} || {{gaps|286|751}} || {{gaps|6|635|624}} || {{gaps|153|552|935}} || {{gaps|3|553|321|281}}
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| |}
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| From the table it is evident why one seldom sees a [[sheet (sailing)|sheet]] (a rope to the loose side of a sail) wound more than three turns around a winch. The force gain would be extreme besides being counter-productive since there is risk of a [[riding turn]], result being that the sheet will foul, form a knot and not run out when [[eased]] (by slacking grip on the [[tail]] (free end), or in land talk, one lets go of the hold end. It is both ancient and modern practice for anchor capstans and jib winches to be slightly flared out at the base, rather than cylindrical, to prevent the rope (anchor [[warp]] or sail sheet) from sliding down. The rope wound several times around the winch can slip upwards gradually, with little risk of a riding turn, provided it is [[tailed]] (loose end is pulled clear), by hand or a self-tailer.
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| For instance, the factor 153552935 means, in theory, that a newborn baby would be capable of holding the weight of two {{USS|Nimitz|CVN-68|6}} supercarriers (97 000 ton each, but for the baby it would be only a little more than 1 kg).
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| == Proof of the capstan equation ==
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| '''1. Circular coordinates''' | |
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| : <math>\varphi=\frac{s}{r}\qquad{d}\varphi=\frac{ds}{r}\qquad\frac{d\varphi}{ds}=\frac{1}{r}</math> (1), (2), (3)
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| <br />
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| [[File:Capstan eqn1.svg|right|thumb]]
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| Let <math>\mathbf{s}_{u}</math> and <math>\mathbf{n}_{u}</math> denote [[unit vectors]];
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| : <math>\mathbf{s}_{u}=-\sin\varphi\cdot{\mathbf{x}}_{u}+\cos\varphi\cdot{\mathbf{y}}_{u}</math> (4)
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| : <math>\mathbf{n}_{u}=\cos\varphi\cdot{\mathbf{x}}_{u}+\sin\varphi\cdot{\mathbf{y}}_{u}</math> (5)
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| Then from (5)
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| : <math>\frac{d}{d\varphi}{\mathbf{s}}_u = -\cos\varphi\cdot{\mathbf{x}}_u -\sin\varphi\cdot{\mathbf{y}}_u=-{\mathbf{n}}_u</math> (6)
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| : <math>\frac{d}{ds}=\frac{d}{d\varphi}\cdot\frac{d\varphi}{ds}=\frac{d}{d\varphi}\cdot\frac{1}{r}=\frac{1}{r}\cdot\frac{d}{d\varphi}</math> (7)
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| From (6) and (7), it follows that
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| : <math>\frac{d}{ds}{\mathbf{s}}_u=\frac{1}{r}\cdot\frac{d}{d\varphi}{\mathbf{s}}_u=-\frac{1}{r}\cdot{\mathbf{n}}_u.</math> '''(8)'''
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| '''2. Forces on cordage in general'''
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| Now, let's study a piece of cord in general, subject to an arbitrary force. Let <math>s\,</math> denote the length of the cord and let the force ''per unit length'' be <math>\mathbf{q}(s)</math>. Consider a short piece <math>\Delta{s}\,</math> of the cord and introduce the cross-sectional force <math>\mathbf{T}(s)\,</math>.
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| Balancing the forces, we get
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| : <math>\mathbf{T}(s+\Delta{s})-\mathbf{T}(s)+\mathbf{q}(s)\cdot\Delta{s}=0, </math> (9)
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| [[File:Capstan eqn2.png|right|thumb]]
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| : <math>\frac{\mathbf{T}(s+\Delta{s})-\mathbf{T}(s)}{\Delta{s}}=-\mathbf{q}(s). </math> (10)
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| Letting <math>\Delta{s}\rightarrow0</math>, we conclude that
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| : <math>\frac{d}{ds}\mathbf{T}(s)=-\mathbf{q}(s).</math> '''(11)'''
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| '''3. A line around a capstan'''
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| A line is wound around a cylinder(a bollard or a capstan). In this case the curvature of the line is circular which makes the problem easier. Let <math>s\,</math> be the length of the line from a point A where the line makes contact with the cylinder. At the point <math>s\,</math> on the short piece <math>\Delta{s}\,</math> of the line acts a force from the cylinder that can be subdivided into a ''tangential'' component <math>t\Delta{s}\,</math> (friction) and a ''normal'' component <math>n\Delta{s}\,</math>. That is to say that
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| : <math>\mathbf{q}(s)=t\cdot\mathbf{s}_u + n\cdot\mathbf{n}_u</math> (12)
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| [[File:Capstan eqn3.png|right|thumb]]
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| With the cross-sectional force <math>\mathbf{T}(s)\,</math> (which is tangential) we get
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| : <math>\mathbf{T}(s)=T(s)\cdot\mathbf{s}_u</math> (13)
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| From (11), (12) and (13), it follows that
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| : <math>\frac{d}{ds}T(s)\cdot\mathbf{s}_u = -t\cdot\mathbf{s}_u - n\cdot\mathbf{n}_u</math> '''(14)'''
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| Derivative of a product and (8) imply that
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| : <math>\frac{d}{ds}T(s)\cdot\mathbf{s}_{u}=\frac{dT(s)}{ds}\cdot\mathbf{s}_{u}+T(s)\cdot\frac{d}{ds}\mathbf{s}_u</math>
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| : <math>= \frac{dT(s)}{ds}\cdot\mathbf{s}_u-\frac{T(s)}{r}\cdot\mathbf{n}_u=-t\cdot\mathbf{s}_u-n\cdot\mathbf{n}_u</math> '''(15)'''
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| Identifying components in '''(15)''', we get
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| : <math>\frac{dT(s)}{ds}=-t</math> (16)
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| and | |
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| : <math>\frac{T(s)}{r}=n.</math> (17)
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| Dividing (16) by (17), we get
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| : <math>
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| \frac{dT(s)}{ds}/\frac{T(s)}{r}=-\frac{t}{n}</math> (18)
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| From (18) and reciprocal of (2), we get
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| : <math>\mathrm{LHS}=\frac{dT(s)}{T(s)}\cdot\frac{r}{ds}=\frac{dT(s)}{T(s)}\cdot\frac{1}{d\varphi}=\frac{1}{T(S)}\cdot\frac{dT(s)}{d\varphi}.</math> (19)
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| From (18) and (19) it follows that
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| : <math>\frac{1}{T(s)}\cdot\frac{dT(s)}{d\varphi}=-\frac{t}{n}.</math> (20)
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| Let <math>\mu=\frac{t}{n}</math> (21) be the ''coefficient of friction'' (no slip). Then
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| : <math>\frac{1}{T}\cdot\frac{dT}{d\varphi}=-\mu</math> (22) : <math>\Rightarrow</math>
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| : <math>\frac{1}{T}\cdot{dT}=-\mu\cdot{d}\varphi</math> '''(23)'''
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| Integration of '''(23)''' yields
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| : <math>\int_{T_\text{load}}^{T_\text{hold}} \frac{1}{T} \cdot{dT} = \int_0^\phi -\mu \cdot {d}\varphi</math> (24)
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| : <math>\ln T_\text{hold} - \ln T_\text{load} = \ln\frac{T_\text{hold}}{T_\text{load}} = -\mu\cdot\phi</math> (25) <math>\Rightarrow</math>
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| : <math>\frac{T_\text{hold}}{T_\text{load}}={e}^{-\mu\cdot\phi}</math> (26)
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| Finally,
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| : <math>T_\text{hold}=T_\text{load}\cdot{e}^{-\mu\cdot\phi}\quad\text{ or } \quad T_\text{load} = T_\text{hold}\cdot{e}^{\mu\cdot\phi} </math>
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| ==See also==
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| *[[Belt friction]]
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| *[[Frictional contact mechanics]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * Arne Kihlberg, Kompendium i Mekanik för E1, del II, Göteborg 1980, 60–62.
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| {{DEFAULTSORT:Capstan Equation}}
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| [[Category:Concepts in physics]]
| |
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