Editing Simple machine

{{short description|Mechanical device that changes the direction or magnitude of a force}}
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{{Use mdy dates|date=November 2022}}[[File:Six Mechanical Powers.png|thumb|250x250px|The six classical simple machines]]
{{History of technology sidebar}}
A '''simple machine''' is a [[machine|mechanical device]] that changes the [[Direction (geometry) |direction]] or [[Magnitude_(mathematics) |magnitude]] of a [[force]].<ref name="Paul-Roy-Mukherjee-2005">{{Citation |last1=Paul |first1=Akshoy |last2=Roy |first2=Pijush |last3=Mukherjee |first3=Sanchayan |title=Mechanical sciences: engineering mechanics and strength of materials |year=2005 |publisher=Prentice Hall of India |isbn=978-81-203-2611-8 |page=215 |postscript=.}}</ref> In general, they can be defined as the simplest [[Mechanism (engineering) |mechanism]]s that use [[mechanical advantage]] (also called leverage) to multiply force.<ref name="Asimov1988">{{Citation |last=Asimov |first=Isaac |title=Understanding Physics |year=1988 |publisher=Barnes & Noble |location=New York |isbn=978-0-88029-251-1 |url=https://books.google.com/books?id=pSKvaLV6zkcC&q=Asimov+simple+machine&pg=PA88 |page=88 |postscript=.}}</ref> Usually the term refers to the six classical simple machines that were defined by [[Renaissance]] scientists:<ref name="Anderson">{{cite book
  |last=Anderson
  |first=William Ballantyne
  |title=Physics for Technical Students: Mechanics and Heat
  |year=1914
  |publisher=McGraw Hill
  |location=New York
  |url=https://archive.org/details/bub_gb_Pa0IAAAAIAAJ/page/n131
  |access-date=2008-05-11
  |pages=112}}</ref><ref name="Britannica1773">{{cite encyclopedia
  | title = Mechanics
  | encyclopedia = Encyclopædia Britannica
  | volume = 3
  | pages = 44
  | publisher = John Donaldson
  | date =1773
  | url = https://books.google.com/books?id=Ow8UAAAAQAAJ&q=%22simple+machine%22+%22mechanical+powers%22+lever+screw+inclined+plane+wedge+wheel+pulley&pg=PA44
  | access-date = 5 April 2020}}</ref><ref name="Morris">{{cite book
 | last1  = Morris
 | first1 = Christopher G.
 | title  = Academic Press Dictionary of Science and Technology
 | publisher = Gulf Professional Publishing
 | date   = 1992
 | pages  = 1993
 | url    = https://books.google.com/books?id=nauWlPTBcjIC&q=%22simple+machine%22&pg=PA1993
 | isbn   = 978-0122004001
 }}</ref>
* [[Lever]]
* [[Wheel and axle]]
* [[Pulley]]
* [[Inclined plane]]
* [[Wedge (mechanical device)|Wedge]]
* [[Screw (simple machine)|Screw]]
A simple machine uses a single applied force to do [[Mechanical work|work]] against a single load force. Ignoring [[friction]] losses, the work done on the load is equal to the work done by the applied force. The machine can increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the applied force is called the ''mechanical advantage''.

Simple machines can be regarded as the elementary "building blocks" of which all more complicated [[machine]]s (sometimes called "compound machines"<ref name="U_Virginia_elementary_curriculum">{{Citation|url=http://galileo.phys.virginia.edu/outreach/8thgradesol/compoundmachine.htm | title=Compound machines|publisher=University of Virginia Physics Department|access-date=2010-06-11|postscript=.}}</ref><ref name="Usher">{{cite book|last=Usher|first=Abbott Payson |title=A History of Mechanical Inventions|publisher=Courier Dover Publications|year=1988|location=US|pages=98 | url=https://books.google.com/books?id=xuDDqqa8FlwC&q=wedge+and+screw&pg=PA196 |isbn=978-0-486-25593-4}}</ref>) are composed.<ref name="Asimov1988"/><ref>{{cite conference  |last=Wallenstein  |first=Andrew  |title=Foundations of cognitive support: Toward abstract patterns of usefulness  |date=June 2002  |book-title=Proceedings of the 9th Annual Workshop on the Design, Specification, and Verification of Interactive Systems  |publisher=Springer  |url=https://books.google.com/books?id=G9sZf7D24a8C&q=simple+machines&pg=PA136  |access-date=2008-05-21  |page=136|isbn=978-3540002666 }}</ref> For example, wheels, levers, and pulleys are all used in the mechanism of a [[bicycle]].<ref name="Prater1994">{{Citation|last=Prater|first=Edward L.|year=1994|title=Basic machines | url=http://www.constructionknowledge.net/public_domain_documents/Div_1_General/Basic_Skills/Basic%20Machines%20NAVEDTRA%2014037%201994.pdf |publisher=U.S. Navy Naval Education and Training Professional Development and Technology Center, NAVEDTRA 14037|postscript=.}}</ref><ref name="USBureauNavalPersonnel1971">{{Citation|author=U.S. Navy Bureau of Naval Personnel|year=1971| url=http://www.webpal.org/SAFE/aaarecovery/5_simple_technology/basic_machines.pdf | title=Basic machines and how they work|publisher=Dover Publications|postscript=.}}</ref> The mechanical advantage of a compound machine is just the product of the mechanical advantages of the simple machines of which it is composed.

Although they continue to be of great importance in mechanics and applied science, modern mechanics has moved beyond the view of the simple machines as the ultimate building blocks of which all [[machine (mechanical)|machines]] are composed, which arose in the Renaissance as a [[neoclassicism|neoclassical]] amplification of [[ancient Greece|ancient Greek]] texts. The great variety and sophistication of modern machine linkages, which arose during the [[Industrial Revolution]], is inadequately described by these six simple categories. Various post-Renaissance authors have compiled expanded lists of "simple machines", often using terms like ''basic machines'',<ref name="Prater1994"/> ''compound machines'',<ref name="U_Virginia_elementary_curriculum"/> or ''machine elements'' to distinguish them from the classical simple machines above. By the late 1800s, [[Franz Reuleaux]]<ref name="Reuleaux1876">{{Citation |last=Reuleaux |first=F. |orig-year=1876 |year=1963 |title=The kinematics of machinery (translated and annotated by A.B.W. Kennedy) |publisher=reprinted by Dover |location=New York |postscript=.}}</ref> had identified hundreds of machine elements, calling them ''simple machines''.<ref name="KMODDL">{{Citation |author=Cornell University |author-link=Cornell University |title=Reuleaux Collection of Mechanisms and Machines at Cornell University |url=http://kmoddl.library.cornell.edu/rx_collection.php |publisher=Cornell University |postscript=.}}</ref> Modern machine theory analyzes machines as [[kinematic chain]]s composed of elementary linkages called [[kinematic pair]]s.

==History==
[[File:Archimedes lever.png|thumb|upright=1.4|Engraving from an 1824 mechanics magazine illustrating Archimedes's statement that given a place to stand, with a lever a person could move the Earth]] 
The idea of a simple machine originated with the Greek philosopher [[Archimedes]] around the 3rd century BC, who studied the [[Aristotelian physics|Archimedean]] simple machines: lever, pulley, and [[Screw (simple machine)|screw]].<ref name="Asimov1988"/><ref name="Chiu">{{Citation
  | last = Chiu
  | first = Y. C.
  | title = An introduction to the History of Project Management
  | publisher = Eburon Academic Publishers
  | year = 2010
  | location = Delft
  | pages = 42
  | url = https://books.google.com/books?id=osNrPO3ivZoC&q=%22heron+of+alexandria%22++load+motion&pg=PA42
  | isbn = 978-90-5972-437-2}}</ref> He discovered the principle of [[mechanical advantage]] in the lever.<ref>{{cite book
  |last1=Ostdiek
  |first1=Vern
  |last2=Bord
  |first2=Donald
  |title=Inquiry into Physics
  |year=2005
  |publisher=Thompson Brooks/Cole
  |isbn=978-0-534-49168-0
  |url=https://books.google.com/books?id=7kz2pd14hPUC&pg=PA123
  |access-date=2008-05-22
  |page=123}}</ref> Archimedes' famous remark with regard to the lever: "Give me a place to stand on, and I will move the Earth," ({{langx|el|δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω}})<ref>Quoted by [[Pappus of Alexandria]] in ''Synagoge'', Book VIII</ref><ref name="Dupac">{{cite book
  | last1 = Dupac
  | first1 = Mihai
  | last2  = Marghitu
  | first2 = Dan B.
  | title = Engineering Applications: Analytical and Numerical Calculation with MATLAB
  | publisher = John Wiley and Sons
  | series = 
  | volume = 
  | edition = 
  | date = 2021
  | location = 
  | pages = 295
  | language = 
  | url = https://books.google.com/books?id=13whEAAAQBAJ&pg=PA295
  | archive-url= 
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  | doi = 
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  | isbn = 9781119093633
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  | jfm =}}</ref><ref name="Dijksterhuis">{{cite book
  | last = Dijksterhuis
  | first = Eduard Jan 
  | title = Archimedes
  | publisher = Princeton University Press
  | date = 2014
  | location = 
  | pages = 15
  | language = 
  | url = https://books.google.com/books?id=Vvj_AwAAQBAJ&q=Archimedes+%22Give+me+a+place+to+stand+on,+and+I+will+move+the+Earth%22
  | archive-url= 
  | archive-date=
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  | jfm =}}</ref> expresses his realization that there was no limit to the amount of force amplification that could be achieved by using mechanical advantage. Later Greek philosophers defined the classic five simple machines (excluding the [[inclined plane]]) and were able to calculate their (ideal) mechanical advantage.<ref name="Usher"/> For example, [[Heron of Alexandria]] ({{circa|10}}–75 AD) in his work ''Mechanics'' lists five mechanisms that can "set a load in motion": [[lever]], [[windlass]], [[pulley]], [[wedge (mechanical device)|wedge]], and [[screw]],<ref name="Chiu" /> and describes their fabrication and uses.<ref>{{cite conference
  | first = Viktor
  | last = Strizhak
  |author2=Igor Penkov |author3=Toivo Pappel
   | title = Evolution of design, use, and strength calculations of screw threads and threaded joints
  | book-title = HMM2004 International Symposium on History of Machines and Mechanisms
  | publisher = Kluwer Academic
  | year = 2004
  | url = https://books.google.com/books?id=FqZvlMnjqY0C&q=%22archimedean+simple+machine%22
  | isbn = 1-4020-2203-4
  | access-date = 2008-05-21
  |page=245}}</ref> However the Greeks' understanding was limited to the [[statics]] of simple machines (the balance of forces), and did not include [[Dynamics (mechanics)|dynamics]], the tradeoff between force and distance, or the concept of [[Work (physics)|work]].

During the [[Renaissance]] the dynamics of the ''mechanical powers'', as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. In 1586 Flemish engineer [[Simon Stevin]] derived the mechanical advantage of the inclined plane, and it was included with the other simple machines. The complete dynamic theory of simple machines was worked out by Italian scientist [[Galileo Galilei]] in 1600 in {{lang|it|Le Meccaniche}} (''On Mechanics''), in which he showed the underlying mathematical similarity of the machines as force amplifiers.<ref name="Krebs">{{cite book
  |last=Krebs
  |first=Robert E.
  |title=Groundbreaking Experiments, Inventions, and Discoveries of the Middle Ages
  |year=2004
  |publisher=Greenwood
  |isbn=978-0-313-32433-8
  |url=https://books.google.com/books?id=MTXdplfiz-cC&q=%22mechanics+Galileo+analyzed%22&pg=PA163
  |access-date=2008-05-21
  |page=163}}</ref><ref name="Stephen">{{cite book
  | last = Stephen
  | first = Donald
  |author2=Lowell Cardwell
  | title = Wheels, clocks, and rockets: a history of technology
  | publisher = W. W. Norton & Company
  | year = 2001
  | location = US
  | pages = 85–87
  | url = https://books.google.com/books?id=BSfpFLV1nkAC&q=%22simple+machine%22+galileo&pg=PA86
  | isbn = 978-0-393-32175-3}}</ref> He was the first to explain that simple machines do not create [[energy]], only transform it.<ref name="Krebs" />

The classic rules of sliding [[friction]] in machines were discovered by [[Leonardo da Vinci]] (1452–1519), but were unpublished and merely documented in his notebooks, and were based on pre-Newtonian science such as believing friction was an [[Aether (classical element)|ethereal]] fluid. They were rediscovered by [[Guillaume Amontons]] (1699) and were further developed by [[Charles-Augustin de Coulomb]] (1785).<ref>{{cite book
  | last = Armstrong-Hélouvry
  | first = Brian
  | title = Control of machines with friction
  | publisher = Springer
  | year = 1991
  | pages = 10
  | url = https://books.google.com/books?id=0zk_zI3xACgC&q=friction+leonardo+da+vinci+amontons+coulomb&pg=PA10
  | isbn = 978-0-7923-9133-3}}</ref>
<!-- I am commenting this section out because the references are not substantial and there is no question about what the Renaissance scientists considered to be simple machines. I think this section is confusing to readers.
==Alternate definitions==
Any list of simple machines is somewhat arbitrary; the central idea is that every mechanism that manipulates force should be able to be understood as a combination of devices on the list.  Some variations that have been proposed to the classical list of six simple machines:
* Some exclude the wedge from the list of simple machines, as it is a moving inclined plane.<ref name="Asimov1988"/>
* The screw, being a [[helical]] inclined plane,<ref>{{cite web|url=http://cnx.org/content/m13594/latest/|title=Simple Machine Elements|website=cnx.org}}</ref> is sometimes also excluded.<ref>{{cite book
  |last=Carhart
  |first=Henry S.
  |last2=Chute
  |first2=Horatio N.
  |title=Physics with Applications
  |year=1917
  |publisher=Allyn & Bacom
  |pages=[https://archive.org/details/physicswithappl00chutgoog/page/n188 159]–60
  |url=https://archive.org/details/physicswithappl00chutgoog
  |access-date=2008-05-20}}</ref> This position is less accepted because a screw converts a rotational force ([[torque]]) to a linear force.
* It has been said that the pulley and the wheel and axle can be viewed as unique forms of levers, leaving only the lever and the inclined plane as simple machines from which all others can be derived.<ref>{{cite web
  |last=Isbell
  |first=Pam
  |title=Simple machines, or are they?
  |year=2001
  |work=Grade 5–7 lesson plan
  |publisher=2001 National Teacher Training Institute
  |url=http://www.myetv.org/education/ntti/lessons/2001_lessons/simplemachines.cfm
  |access-date=2008-05-13}}</ref><ref name="Clute">{{cite book
  |last=Clute
  |first=Willard N.
  |title=Experimental General Science
  |year=1917
  |publisher=P. Blakiston's Son & Co.
  |location=Philadelphia, Pennsylvania
  |pages=[https://archive.org/details/experimentalgen00clutgoog/page/n208 188]
  |url=https://archive.org/details/experimentalgen00clutgoog
  |access-date=2008-05-20}}</ref><ref name="BNET">{{cite web
  |title=Mechanical Advantage and Simple Machines
  |year=2002
  |work=BNET Business Network
  |publisher=CNET
  |url=http://findarticles.com/p/articles/mi_gx5226/is_2002/ai_n19143765/pg_1
  |access-date=2008-05-21}}</ref><ref name="Beiser">{{cite book
  |last=Beiser
  |first=Arthur
  |year=2004
  |title=Schaum's Outline of Applied Physics
  |publisher=McGraw-Hill
  |url=https://books.google.com/books?id=soKguvJDgmsC&dq=Hydraulic+%22simple+machines%22&cad=0
  |isbn=0-07-142611-6
  |access-date=2008-05-21
  |page=145}}</ref>
* [[Hydraulic]] systems can also provide amplification of force, so some say they should be added to the list.<ref name="BNET"/><ref>This was first suggested by [[Blaise Pascal]] in the 17th century: {{cite book
  |last=Meli
  |first=Domenico Bertolini
  |title=Thinking with Objects:The Transformation of Mechanics in the 17th Century
  |year=2006
  |publisher=JHU Press
  |isbn=0-8018-8427-6
  |url=https://books.google.com/books?id=qbS_0qAB3_cC&dq=Hydraulic+%22simple+machines%22&cad=0|page=175}}</ref><ref>{{cite web
  |title=Mechanical Advantage – Simple Machines
  |work=MCAT Exam preparation
  |date=January 7, 2008
  |publisher=Eduturca
  |url=http://www.eduturca.com/mcat-exam/mechanical-advantage-simple-machines-mcat.html
  |access-date=2008-05-21}}</ref>
-->

==Ideal simple machine==
If a simple machine does not dissipate energy through friction, wear or deformation, then energy is conserved and it is called an ideal simple machine. In this case, the power into the machine equals the power out, and the mechanical advantage can be calculated from its geometric dimensions.

Although each machine works differently mechanically, the way they function is similar mathematically.<ref>This fundamental insight was the subject of Galileo Galilei's 1600 work {{lang|it|Le Meccaniche}} (''On Mechanics'').</ref> In each machine, a force <math>F_\text{in}</math> is applied to the device at one point, and it does [[Mechanical work|work]] moving a load <math>F_\text{out}</math> at another point.<ref name="Bhatnagar">{{cite book
  | last = Bhatnagar
  | first = V. P.
  | title = A Complete Course in Certificate Physics
  | publisher = Pitambar
  | year = 1996
  | location = India
  | pages = 28–30
  | url = https://books.google.com/books?id=pb45yhuNltEC&q=%22simple+machine%22+%22&pg=PA28
  | isbn = 978-81-209-0868-0}}</ref> Although some machines only change the direction of the force, such as a stationary pulley, most machines multiply the magnitude of the force by a factor, the [[mechanical advantage]]

<math display="block">\mathrm{MA} = {F_\text{out} \over F_\text{in}}</math>

that can be calculated from the machine's geometry and friction.

Simple machines do not contain a source of [[energy]],<ref name="Simmons">{{cite book
  | last1 = Simmons
  | first1 = Ron
  | last2 = Cindy
  | first2 = Barden
   | title = Discover! Work & Machines
  | publisher = Milliken
  | year = 2008
  | location = US
  | pages = 29
  | url = https://books.google.com/books?id=geddKUGjI3cC&q=%22simple+machine%22+%22mechanical+advantage%22&pg=PA30
  | isbn = 978-1-4291-0947-5}}</ref> so they cannot do more [[Mechanical work|work]] than they receive from the input force.<ref name="Bhatnagar" /> A simple machine with no [[friction]] or [[Elasticity (physics)|elasticity]] is called an ''ideal machine''.<ref name="Gujral">{{cite book
  | last = Gujral
  | first = I. S.
  | title = Engineering Mechanics
  | publisher = Firewall Media
  | year = 2005
  | pages = 378–380
  | url = https://books.google.com/books?id=JM0OG-XUyu0C&q=%22simple+machine%22+self-locking&pg=PA382
  | isbn = 978-81-7008-636-9}}</ref><ref name="Uicker2003">{{Citation | first1=John J. Jr. | last1=Uicker |first2=Gordon R. |last2=Pennock |first3=Joseph E. |last3=Shigley |year=2003 |title=Theory of Machines and Mechanisms |edition=third |publisher=Oxford University Press |location=New York |isbn=978-0-19-515598-3 }}</ref><ref>{{cite book |first=Burton |last=Paul |year=1979 |title=Kinematics and Dynamics of Planar Machinery |publisher=Prentice Hall |isbn=978-0-13-516062-6 }}</ref> Due to [[conservation of energy]], in an ideal simple machine, the power output (rate of energy output) at any time <math>P_\text{out}</math> is equal to the power input <math>P_\text{in}</math>

<math display="block">P_\text{out} =  P_\text{in}\!</math>

The power output equals the velocity of the load <math>v_\text{out}\,</math> multiplied by the load force <math>P_\text{out} =  F_\text{out} v_\text{out}\,</math>. Similarly the power input from the applied force is equal to the velocity of the input point <math>v_\text{in}\,</math> multiplied by the applied force <math>P_\text{in} =  F_\text{in} v_\text{in}\!</math>. Therefore,

<math display="block">F_\text{out}v_\text{out} = F_\text{in}v_\text{in}\,</math>

So the mechanical advantage of an ideal machine <math>\mathrm{MA}_\text{ideal}\,</math> is equal to the ''velocity ratio'', the ratio of input velocity to output velocity

<math display="block">\mathrm{MA}_\text{ideal} = {F_\text{out} \over F_\text{in}} = {v_\text{in} \over v_\text{out}}\,</math>

The ''velocity ratio'' is also equal to the ratio of the distances covered in any given period of time<ref name="Rao1">{{cite book
 | last1  = Rao
 | first1 = S.
 | last2  = Durgaiah
 | first2 = R.
 | title  = Engineering Mechanics
 | publisher = Universities Press
 | date   = 2005
 | pages  = 80
 | url    = https://books.google.com/books?id=vRR4FKAkJl4C&q=%22simple+machine%22+%22velocity+ratio%22+distance&pg=PA80
 | isbn   = 978-81-7371-543-3
 }}</ref><ref name="Goyal1">{{cite book
 | last1  = Goyal
 | first1 = M. C.
 | last2  = Raghuvanshee
 | first2 = G. S.
 | title  = Engineering Mechanics
 | publisher = PHI Learning
 | date   = 2011
 | page  = 212
 | url    = https://books.google.com/books?id=qNPE9RVkSTUC&q=%22simple+machine%22+%22velocity+ratio%22+distance&pg=PA212
 | isbn   = 978-81-203-4327-6
 }}</ref><ref name="Avison">{{cite book
 | last1  = Avison
 | first1 = John
 | title  = The World of Physics
 | publisher = Nelson Thornes
 | date   = 2014
 | page  = 110
 | url    = https://books.google.com/books?id=DojwZzKAvN8C&q=machine+%22velocity+ratio%22+%22distance+ratio%22&pg=PA110
 | isbn   = 978-0-17-438733-6
 }}</ref>

<math display="block">{v_\text{out} \over v_\text{in}} = {d_\text{out} \over d_\text{in}}</math>

Therefore, the mechanical advantage of an ideal machine is also equal to the ''distance ratio'', the ratio of input distance moved to output distance moved
{{Equation box 1
|indent =:
|equation = <math>\mathrm{MA}_\text{ideal} = {F_\text{out} \over F_\text{in}} = {d_\text{in} \over d_\text{out}}\,</math>
|cellpadding = 0
|border = 1
|border colour = black
|background colour = transparent
}}
This can be calculated from the geometry of the machine. For example, the mechanical advantage and distance ratio of the [[lever]] is equal to the ratio of its [[lever arm]]s.

The mechanical advantage can be greater or less than one:
* If <math>\mathrm{MA} > 1\,</math>, the output force is greater than the input, the machine acts as a force amplifier, but the distance moved by the load <math>d_\text{out}</math> is less than the distance moved by the input force <math>d_\text{in}\,</math>.
* If <math>\mathrm{MA} < 1\,</math>, the output force is less than the input, but the distance moved by the load is greater than the distance moved by the input force.

In the [[screw (simple machine)|screw]], which uses rotational motion, the input force should be replaced by the [[torque]], and the velocity by the [[angular velocity]] the shaft is turned.

==Friction and efficiency==
All real machines have friction, which causes some of the input power to be dissipated as heat. If <math>P_\text{fric}\,</math> is the power lost to friction, from conservation of energy

<math display="block">P_\text{in} = P_\text{out} + P_\text{fric}</math>

The [[mechanical efficiency]] <math>\eta</math> of a machine (where <math> 0 < \eta \ < 1</math>) is defined as the ratio of power out to the power in, and is a measure of the frictional energy losses

<math display="block">\begin{align}
\eta & \equiv {P_\text{out} \over P_\text{in}} \\
P_\text{out} & = \eta P_\text{in}
\end{align}</math>

As above, the power is equal to the product of force and velocity, so

<math display="block">F_\text{out} v_\text{out}  =  \eta F_\text{in} v_\text{in}</math>

Therefore,
{{Equation box 1  |cellpadding = 0 |border = 1 |border colour = black |background colour = transparent  |indent =:
|equation = <math>\mathrm{MA} = {F_\text{out} \over F_\text{in}} = \eta {v_\text{in} \over v_\text{out}}</math>
}}
So in non-ideal machines, the mechanical advantage is always less than the velocity ratio by the product with the efficiency <math>\eta</math>. So a machine that includes friction will not be able to move as large a load as a corresponding ideal machine using the same input force.

==Compound machines==
A ''compound machine'' is a [[machine (mechanical)|machine]] formed from a set of simple machines connected in series with the output force of one providing the input force to the next. For example, a [[Vise|bench vise]] consists of a lever (the vise's handle) in series with a screw, and a simple [[gear train]] consists of a number of [[gear]]s ([[Wheel and axle|wheels and axles]]) connected in series.

The mechanical advantage of a compound machine is the ratio of the output force exerted by the last machine in the series divided by the input force applied to the first machine, that is

<math display="block">\mathrm{MA}_\text{compound}   =  {F_{\text{out}N} \over F_\text{in1}}</math>

Because the output force of each machine is the input of the next, <math>F_\text{out1} = F_\text{in2}, \; F_\text{out2} = F_\text{in3},\, \ldots \; F_{\text{out}K} = F_{\text{in}K+1}</math>, this mechanical advantage is also given by

<math display="block">\mathrm{MA}_\text{compound}  =  {F_\text{out1} \over F_\text{in1}} {F_\text{out2} \over F_\text{in2}} {F_\text{out3} \over F_\text{in3}}\ldots  {F_{\text{out}N} \over F_{\text{in}N}}  \,</math>

Thus, the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of the series of simple machines that form it

<math display="block">\mathrm{MA}_\text{compound} = \mathrm{MA}_1 \mathrm{MA}_2 \ldots  \mathrm{MA}_N</math>

Similarly, the efficiency of a compound machine is also the product of the efficiencies of the series of simple machines that form it

<math display="block">\eta_\text{compound} = \eta_1  \eta_2 \ldots \; \eta_N.</math>

==Self-locking machines==
[[Image:BOLT SCREW UBT 199.JPG|thumb|The [[screw (simple machine)|screw]]'s self-locking property is the reason for its wide use in [[threaded fastener]]s like [[bolt (fastener)|bolts]] and [[wood screw]]s ]]

In many simple machines, if the load force <math>F_{\textrm{out}}</math> on the machine is high enough in relation to the input force <math>F_{\textrm{in}}</math>, the machine will move backwards, with the load force doing work on the input force.<ref name="Gujral2">{{cite book
  | last = Gujral
  | first = I. S.
  | title = Engineering Mechanics
  | publisher = Firewall Media
  | year = 2005
  | page = 382
  | url = https://books.google.com/books?id=JM0OG-XUyu0C&q=%22simple+machine%22+self-locking&pg=PA382
  | isbn = 978-81-7008-636-9}}</ref> So these machines can be used in either direction, with the driving force applied to either input point. For example, if the load force on a lever is high enough, the lever will move backwards, moving the input arm backwards against the input force. These are called ''reversible'', ''non-locking'' or ''overhauling'' machines, and the backward motion is called ''overhauling''.

However, in some machines, if the frictional forces are high enough, no amount of load force can move it backwards, even if the input force is zero. This is called a ''self-locking'', ''nonreversible'', or ''non-overhauling'' machine.<ref name="Gujral2" /> These machines can only be set in motion by a force at the input, and when the input force is removed will remain motionless, "locked" by friction at whatever position they were left.

Self-locking occurs mainly in those machines with large areas of sliding contact between moving parts: the [[screw (simple machine)|screw]], [[inclined plane]], and [[wedge (mechanical device)|wedge]]:
* The most common example is a screw. In most screws, one can move the screw forward or backward by turning it, and one can move the nut along the shaft by turning it, but no amount of pushing the screw or the nut will cause either of them to turn.
* On an inclined plane, a load can be pulled up the plane by a sideways input force, but if the plane is not too steep and there is enough friction between load and plane, when the input force is removed the load will remain motionless and will not slide down the plane, regardless of its weight.
* A wedge can be driven into a block of wood by force on the end, such as from hitting it with a sledge hammer, forcing the sides apart, but no amount of [[compression (physics)|compression]] force from the wood walls will cause it to pop back out of the block.

A machine will be self-locking if and only if its efficiency <math>\eta</math> is below 50%:<ref name="Gujral2" />

<math display="block">\eta \equiv \frac {F_\text{out}/F_\text{in} }{d_\text{in}/d_\text{out} } < 0.5</math>

Whether a machine is self-locking depends on both the friction forces ([[Coefficient of friction|coefficient of static friction]]) between its parts, and the distance ratio <math>d_{\textrm{in}}/d_{\textrm{out}}</math> (ideal mechanical advantage). If both the friction and ideal mechanical advantage are high enough, it will self-lock.

===Proof===
When a machine moves in the forward direction from point 1 to point 2, with the input force doing work on a load force, from conservation of energy<ref name="Rao">{{cite book
  | last1 = Rao
  | first1 = S.
  | first2 = R.
  | last2 = Durgaiah
  | title = Engineering Mechanics
  | publisher = Universities Press
  | year = 2005
  | pages = 82
  | url = https://books.google.com/books?id=vRR4FKAkJl4C&q=%22simple+machine%22+%22&pg=PA80
  | isbn = 978-81-7371-543-3}}</ref><ref name="Goyal">{{cite book
  | last1 = Goyal
  | first1 = M. C.
  | first2 = G. S.
  | last2 = Raghuvanshi
   | title = Engineering Mechanics
  | publisher = PHI Learning Private Ltd.
  | year = 2009
  | location = New Delhi
  | pages = 202
  | url = https://books.google.com/books?id=vRR4FKAkJl4C&pg=PA82
  | isbn = 978-81-203-3789-3}}</ref> the input work <math>W_\text{1,2}</math> is equal to the sum of the work done on the load force <math>W_\text{load}</math> and the work lost to friction <math>W_\text{fric} </math>
{{NumBlk2|:|<math>W_\text{1,2} = W_\text{load} + W_\text{fric}</math>|Eq. 1}}

If the efficiency is below 50% {{nowrap|(<math>\eta = W_\text{load}/W_\text{1,2} < 0.5</math>):}}

<math display="block">2W_\text{load} < W_\text{1,2} \,</math>

From {{EquationNote|Eq. 1}}
<math display="block">\begin{align}
2W_\text{load} & < W_\text{load} + W_\text{fric} \\
W_\text{load} & < W_\text{fric}
\end{align}</math>

When the machine moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to friction <math>W_\text{fric}</math> is the same

<math display="block">W_\text{load} = W_\text{2,1} + W_\text{fric}</math>

So the output work is
<math display="block">W_\text{2,1} = W_\text{load} - W_\text{fric} < 0</math>

Thus the machine self-locks, because the work dissipated in friction is greater than the work done by the load force moving it backwards even with no input force.

==Modern machine theory==
Machines are studied as mechanical systems consisting of [[actuator]]s and [[mechanism (engineering)|mechanism]]s that transmit forces and movement, monitored by sensors and controllers. The components of actuators and mechanisms consist of links and joints that form kinematic chains.

===Kinematic chains===
[[File:Kinematics of Machinery - Figure 21.jpg|thumb|right|200px|alt=Illustration of a Four-bar linkage from Kinematics of Machinery, 1876|Illustration of a four-bar linkage from [[s:The Kinematics of Machinery|Kinematics of Machinery, 1876]]]]Simple machines are elementary examples of [[kinematic chain]]s that are used to model [[mechanical systems]] ranging from the steam engine to robot manipulators. The bearings that form the fulcrum of a lever and that allow the wheel and axle and pulleys to rotate are examples of a [[kinematic pair]] called a hinged joint. Similarly, the flat surface of an inclined plane and wedge are examples of the kinematic pair called a sliding joint. The screw is usually identified as its own kinematic pair called a helical joint.

Two levers, or cranks, are combined into a planar [[four-bar linkage]] by attaching a link that connects the output of one crank to the input of another. Additional links can be attached to form a [[six-bar linkage]] or in series to form a robot.<ref name="Uicker2003"/>

===Classification of machines===
The identification of simple machines arises from a desire for a systematic method to invent new machines. Therefore, an important concern is how simple machines are combined to make more complex machines. One approach is to attach simple machines in series to obtain compound machines.

However, a more successful strategy was identified by [[Franz Reuleaux]], who collected and studied over 800 elementary machines. He realized that a lever, pulley, and wheel and axle are in essence the same device: a body rotating about a hinge. Similarly, an inclined plane, wedge, and screw are a block sliding on a flat surface.<ref>Hartenberg, R.S. & J. Denavit (1964) [http://kmoddl.library.cornell.edu/bib.php?m=23 Kinematic synthesis of linkages], New York: McGraw-Hill, online link from [[Cornell University]].</ref>

This realization shows that it is the joints, or the connections that provide movement, that are the primary elements of a machine. Starting with four types of joints, the [[revolute joint]], [[prismatic joint|sliding joint]], [[Cam (mechanism)|cam joint]] and [[gear train|gear joint]], and related connections such as cables and belts, it is possible to understand a machine as an assembly of solid parts that connect these joints.<ref name="Uicker2003"/>

===Kinematic synthesis===
The design of mechanisms to perform required movement and force transmission is known as [[kinematic synthesis]]. This is a collection of geometric techniques for the mechanical design of [[linkage (mechanical)|linkage]]s, [[Cam (mechanism)|cam and follower mechanisms]] and [[Gear train|gears and gear trains]].

==See also==
* [[Linkage (mechanical)]]
* [[Cam (mechanism)|Cam and follower mechanisms]]
* [[Gear train|Gears and gear trains]]
* [[Mechanism (engineering)]]
* [[Rolamite]], the only elementary machine discovered in the 20th century

==References==
{{Reflist}}

{{Machines}}

{{Authority control}}

{{DEFAULTSORT:Simple Machine}}
[[Category:Mechanical engineering]]
[[Category:Simple machines| ]]