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| {{Classical mechanics|cTopic=Fundamental concepts}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| '''Virtual work''' arises in the application of the ''[[principle of least action]]'' to the study of forces and movement of a [[mechanical system]]. The [[Work (physics)|work]] of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all of the possible displacements that a particle may follow, called [[virtual displacement]]s, one will minimize the action, and, therefore, is the one followed by the particle by the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work. | |
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| Historically, virtual work and the associated [[calculus of variations]] were formulated to analyze systems of rigid bodies,<ref name=Lanczos>[http://books.google.com/books?id=ZWoYYr8wk2IC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false C. Lánczos, The Variational Principles of Mechanics, 4th Ed., General Publishing Co., Canada, 1970]</ref> but they have also been developed for the study of the mechanics of deformable bodies.<ref>Dym, C. L. and I. H. Shames, ''Solid Mechanics: A Variational Approach'', McGraw-Hill, 1973.</ref>
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| == History ==
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| The introduction of virtual work and the principle of least action was guided by the view that the actual movement of a body is the one in a set of "tentative" realities that minimizes a particular quantity. This idea that nature minimizes is a version of the "simplicity hypothesis" that can be traced to Aristotle.<ref name=Yourgrau>[http://books.google.com/books?id=OwTyrJJXZbYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory, 3rd Ed., General Publishing Co., Canada, 1968]</ref> Another form of this hypothesis is [[Occam's razor]] which states that "it is futile to employ many principles when it is possible to employ fewer." These ideas illustrate a view of physics that nature optimizes in some way.
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| [[Gottfried Leibniz]] formulated [[Newton's laws of motion]] in terms of work and kinetic energy, or ''vis viva'' (living force), which are minimized as a system moves.<ref name=Lanczos/><ref name=Yourgrau/> [[Maupertuis]] adapted Leibniz's ideas as the ''principle of least action'' that nature minimizes action. But it was Euler and Lagrange who provided the mathematical foundation of the calculus of variations and applied it to the study of the statics and dynamics of mechanical systems.
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| | :<math forcemathmode="mathml">E=mc^2</math> |
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| Hamilton's reformulation of the principle of least action and Lagrange's equations yielded a theory of dynamics that is the foundation for modern physics and quantum mechanics.
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| | :<math forcemathmode="png">E=mc^2</math> |
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| ==Overview==
| | '''source''' |
| If a force acts on a particle as it moves from point ''A'' to point ''B'', then, for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path. The ''principle of virtual work'', which is the form of the principle of least action applied to these systems, states that the path actually followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero. The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, and is termed ''the calculus of variations''.
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| Let the function '''x'''(''t'') define the path followed by a point. A nearby path can then be defined by adding the function ''δ'''''x'''(''t'') to the original path, so that the new path is given by '''x'''(''t'')+''δ'''''x'''(''t''). The function ''δ'''''x'''(''t'') is called the ''variation'' of the original path, and each of the components of ''δ'''''x'''=(''δx'', ''δy'', ''δz'') is called a ''virtual displacement''. This can be generalized to an arbitrary mechanical system defined by the [[generalized coordinates]] ''q''<sub>''i'' </sub>, ''i'' = 1, ..., ''n''. In which case, the variation of the trajectory ''q''<sub>''i'' </sub>(''t'') is defined by the virtual displacements ''δq''<sub>''i'' </sub>, ''i'' = 1, ..., ''n''.
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| Virtual work can now be described as the work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements. When considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero.
| | ==Demos== |
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| '''Virtual work''' on a [[physical system|system]] is the [[mechanical work|work]] resulting from either virtual forces acting through a real [[Displacement (vector)|displacement]] or real [[forces]] acting through a [[virtual displacement]]. In this discussion, the term ''displacement'' may refer to a translation or a rotation, and the term ''force'' to a force or a moment. When the virtual quantities are [[independent variable]]s, they are also ''arbitrary''. Being arbitrary is an essential characteristic that enables one to draw important conclusions from mathematical relations. For example, in the matrix relation
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| :<math>\mathbf{R}^{*T} \mathbf{r} = \mathbf{R}^{*T} \mathbf{B}^{T} \mathbf{q}</math>, | | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| if <math>\mathbf{R}^{*}</math> is an arbitrary vector, then one can conclude that <math> \mathbf{r} = \mathbf{B}^{T} \mathbf{q} </math>. In this way, the arbitrary quantities disappear from the final useful results.
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| == Introduction ==
| | * accessibility: |
| In this introduction basic definitions are presented that will assist in understanding later sections.
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| Consider a particle ''P'' that moves along a trajectory '''r'''(''t'') from a point ''A'' to a point ''B'', while a force '''F''' is applied to it. Then the work done by the force is given by the integral
| | ==Test pages == |
| :<math> W = \int_{\mathbf{r}(t_0)=A}^{\mathbf{r}(t_1)=B}\mathbf{F}\cdot d\mathbf{r} = \int_{t_0}^{t_1}\mathbf{F}\cdot \mathbf{v}dt ,</math>
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| where ''d'''''r''' is the differential element along the curve that is the trajectory of ''P'', and '''v''' is its velocity. It is important to notice that the value of the work ''W'' depends on the trajectory '''r'''(''t'').
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| Now consider the work done by the same force on the same particle ''P'' again moving from point ''A'' to point ''B'', but this time moving along the nearby trajectory that differs from '''r'''(''t'') by the variation ''δ'''''r'''(''t'')=''ε'''''h'''(''t''), where ''ε'' is a scaling constant that can be made as small as desired and '''h'''(''t'') is an arbitrary function that satisfies '''h'''(''t''<sub>0</sub>) = '''h'''(''t''<sub>1</sub>) = 0,
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| :<math>\bar{W} = \int_{A}^{B}\mathbf{F}\cdot d(\mathbf{r}+\epsilon \mathbf{h})=\int_{t_0}^{t_1}\mathbf{F}\cdot (\mathbf{v}+\epsilon \dot{\mathbf{h}})dt .</math>
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| The variation of the work ''δW'' associated with this nearby path, known as the ''virtual work'', can be computed to be
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| :<math> \delta W = \bar{W}-W = \int_{t_0}^{t_1}(\mathbf{F}\cdot \epsilon\dot{\mathbf{h}})dt.</math> | | *[[Styling]] |
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| | *[[Unique Ids]] |
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| Now assume that '''r'''(''t'') and '''h'''(''t'') depend on the generalized coordinates ''q''<sub>''i'' </sub>, ''i'' = 1, ..., ''n'', then the derivative of the variation ''δ'''''r'''=''ε'''''h'''(''t'') is given by
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| :<math> \frac{d}{dt}\delta \mathbf{r} = \epsilon\dot{\mathbf{h}} = \epsilon \left(\frac{\partial \mathbf{h}}{\partial q_1} \dot{q}_1 + \ldots + \frac{\partial \mathbf{h}}{\partial q_n} \dot{q}_n\right),</math>
| | *[[Url2Image|Url2Image (private Wikis only)]] |
| then we have
| | ==Bug reporting== |
| :<math> \delta W = \int_{t_0}^{t_1}\left(\mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_1} \epsilon\dot{q}_1 + \ldots + \mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_n} \epsilon\dot{q}_n\right)dt =
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| \int_{t_0}^{t_1}\left(\mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_1}\right) \epsilon\dot{q}_1 dt + \ldots + \int_{t_0}^{t_1}\left(\mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_n}\right) \epsilon\dot{q}_n dt.</math>
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| The requirement that the virtual work be zero for an arbitrary variation ''δ'''''r'''(''t'')=ε'''h'''(''t'') is equivalent to the set of requirements
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| :<math> F_i = \mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_i} = 0, \quad i=1, \ldots, n.</math>
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| The terms ''F<sub>i</sub>'' are called the ''generalized forces'' associated with the virtual displacement δ'''r'''.
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| == Static equilibrium ==
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| Static equilibrium is the condition in which the applied forces and constraint forces on a mechanical system balance such that the system does not move. The ''principle of virtual work'' states that the virtual work of the applied forces is zero for all virtual movements of the system from static equilibrium, that is, ''δW'' = 0 for any variation ''δ'''''r'''.<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref> This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is ''F''<sub>''i''</sub> = 0.
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| Let the forces on the system be '''F'''<sub>''j'' </sub>, ''j'' = 1, ..., ''m'' and let the virtual displacement of each point of application of these forces be ''δ'''''r'''<sub>''j'' </sub>, ''j'' = 1, ..., ''m'', then the virtual work generated by a virtual displacement of these forces from the equilibrium position is given by
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| :<math>\delta W = \sum_{j=1}^m \mathbf{F}_j\cdot \delta\mathbf{r}_j.</math>
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| Now assume that each ''δ'''''r'''<sub>''j''</sub> depends on the generalized coordinates ''q''<sub>''i'' </sub>, ''i'' = 1, ..., ''n'', then
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| :<math> \delta \mathbf{r}_j = \frac{\partial \mathbf{r}_j}{\partial q_1} \delta{q}_1 + \ldots + \frac{\partial \mathbf{r}_j}{\partial q_n} \delta{q}_n,</math>
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| :<math> \delta W = \left(\sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_1}\right) \delta{q}_1 + \ldots + \left(\sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_n}\right) \delta{q}_n. </math>
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| The ''n'' terms
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| :<math> F_i = \sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_i},\quad i=1,\ldots, n,</math>
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| are the generalized forces acting on the system. Kane<ref>T. R. Kane and D. A. Levinson, Dynamics: theory and applications, McGraw-Hill, New York, 1985</ref> shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
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| :<math> F_i = \sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{v}_j}{\partial \dot{q}_i},\quad i=1,\ldots, n,</math>
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| where '''v'''<sub>''j''</sub> is the velocity of the point of application of the force '''F'''<sub>''j'' </sub>.
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| In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
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| :<math> \delta W = 0 \quad \Rightarrow \quad F_i =0, i=1,\ldots, n.</math>
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| === Constraint forces ===
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| An important benefit of the principle of virtual work is that only forces that do work as the system moves through a virtual displacement are needed to determine the mechanics of the system. There are many forces in a mechanical system that do no work during a virtual displacement, which means that they need not be considered in this analysis. The two important examples are (i) the internal forces in a rigid body, and (ii) the constraint forces at an ideal joint.
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| Lanczos<ref name=Lanczos/> presents this as the postulate: "The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints." The argument is as follows. The principle of virtual work states that in equilibrium the virtual work of the forces applied to a system is zero. Newton's laws state that at equilibrium the applied forces are equal and opposite to the reaction, or constraint, forces. This means the virtual work of the constraint forces must be zero as well.
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| === Couples ===
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| A pair of forces acting on a rigid body can form a couple defined by the moment vector '''M'''. The virtual work of a moment vector is obtained from the virtual rotation of the rigid body.
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| For planar movement, the moment acts perpendicular to the plane with magnitude ''M'' and the virtual work due to this moment is
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| :<math> \delta W = M\delta \phi,\!</math>
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| where ''δφ'' is the virtual rotation angle of the body.
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| In order to extend this to three dimensional rotations, use the angular velocity vector '''ω''' of the body to obtain the virtual work as
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| :<math> \delta W = \left(\mathbf{M}\cdot \frac{\partial\vec{\omega}}{\partial\dot{\phi}}\right) \delta\phi.</math>
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| Now consider the moments '''M'''<sub>''j''</sub> acting on ''m'' rigid bodies in a mechanical system. Let the angular velocity vectors '''ω'''<sub>''j'' </sub>, ''j'' = 1, ..., ''m'' of each body depend on the ''n'' generalized coordinates ''q''<sub>''j'' </sub>, ''i'' = 1, ..., ''n''. Then the virtual work obtained from these moments for a virtual displacement from equilibrium is given by
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| :<math> \delta W = \sum_{j=1}^m \mathbf{M}_j\cdot \left(\frac{\partial\vec{\omega}_j}{\partial\dot{q}_1} \delta q_1 + \ldots +\frac{\partial\vec{\omega}_j}{\partial\dot{q}_n} \delta q_n\right) .</math>
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| Collect the coefficients of the virtual displacements ''δq<sub>i</sub>'' to obtain
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| :<math> \delta W = \left(\sum_{j=1}^m \mathbf{M}_j\cdot \frac{\partial\vec{\omega}_j}{\partial\dot{q}_1}\right) \delta q_1 + \ldots + \left(\sum_{j=1}^m \mathbf{M}_j\cdot\frac{\partial\vec{\omega}_j}{\partial\dot{q}_n} \right) \delta q_n .</math>
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| === Forces and moments ===
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| Combine the virtual work above for couples with the virtual work of forces in order to obtain the virtual work of a system of forces and moments acting on system of rigid bodies displaced from equilibrium as
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| :<math> \delta W = \left(\sum_{j=1}^m \mathbf{F}_j \cdot \frac{\partial\mathbf{v}_j}{\partial\dot{q}_1} + \sum_{j=1}^m \mathbf{M}_j\cdot \frac{\partial\vec{\omega}_j}{\partial\dot{q}_1}\right) \delta q_1 + \ldots + \left(\sum_{j=1}^m \mathbf{F}_j \cdot \frac{\partial\mathbf{v}_j}{\partial\dot{q}_n} + \sum_{j=1}^m \mathbf{M}_j\cdot\frac{\partial\vec{\omega}_j}{\partial\dot{q}_n} \right) \delta q_n ,</math>
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| where the generalized forces are now defined to be
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| :<math> F_i = \sum_{j=1}^m \left(\mathbf{F}_j \cdot \frac{\partial\mathbf{v}_j}{\partial\dot{q}_i} + \mathbf{M}_j\cdot \frac{\partial\vec{\omega}_j}{\partial\dot{q}_i}\right), \quad i=1,\ldots, n.</math>
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| The principle of virtual work requires that a system of rigid bodies acted on by the forces and moments '''F'''<sub>''j''</sub> and '''M'''<sub>''j''</sub> is in equilibrium if the generalized forces ''F''<sub>''i''</sub> are zero, that is
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| :<math> \delta W=0\quad \Rightarrow \quad F_i = 0, i=1,\ldots, n.</math>
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| <!-- I rewrote this. However, please restore anything that you feel that I did not present clearly
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| Consider a system of particles, i, in static equilibrium. The total force on each particle is<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|263}}
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| :<math>\mathbf {F}_{i}^{(T)} = 0</math>.
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| Summing the work exerted by the force on each particle that acts through an arbitrary virtual displacement, <math>\delta \mathbf r_i</math>, of the system leads to an expression for the virtual work that must be zero since the forces are zero:<ref name="Torby1984"/>{{rp|263}}
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| :<math>\delta W = \sum_{i} \mathbf {F}_{i}^{(T)} \cdot \delta \mathbf r_i = 0</math>.
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| The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary virtual displacements. Separating the forces into applied forces, <math>\mathbf F_i</math>, and constraint forces, <math>\mathbf C_i</math>, yields<ref name="Torby1984"/>{{rp|263}}
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| :<math>\delta W = \sum_{i} \mathbf {F}_{i} \cdot \delta \mathbf r_i + \sum_{i} \mathbf {C}_{i} \cdot \delta \mathbf r_i = 0</math>.
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| If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be ''[[consistent]]'' with the constraints.<ref name="laa">{{cite web |url=http://comp.uark.edu/~icjong/docu/05Portland.pdf |title=Teaching Students Work and Virtual Work Method in Statics:A Guiding Strategy with Illustrative Examples |accessdate=2007-09-24 |author=Ing-Chang Jong |year=2005 |format=PDF |work=Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition |publisher=American Society for Engineering Education }}</ref> This leads to the formulation of the ''principle of virtual work for applied forces'', which states that forces applied to a static system do no virtual work:<ref name="Torby1984"/>{{rp|263}}
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| :<math>\delta W = \sum_{i} \mathbf {F}_{i} \cdot \delta \mathbf r_i = 0</math>.
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| There is also a corresponding principle for accelerating systems called [[D'Alembert's principle]], which forms a theoretical basis for [[Lagrangian mechanics]].
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| == One degree-of-freedom mechanisms ==
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| In this section, the principle of virtual work is used for the static analysis of one degree-of-freedom mechanical devices. Specifically, we analyze the lever, a pulley system, a gear train, and a four-bar linkage. Each of these devices moves in the plane, therefore a force '''F'''=(''f<sub>x</sub>'', ''f<sub>y</sub>'') has two components and acts on a point with coordinates '''r''' = (''r<sub>x</sub>'', ''r<sub>y</sub>'') and velocity '''v''' = (''v<sub>x</sub>'', ''v<sub>y</sub>''). A moment, also called a [[torque]], ''T'' acting on a body that moves in the plane has one component as does the angular velocity ''ω'' of the body.
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| Assume the bodies in the mechanism are rigid and the joints are ideal so that the only change in virtual work is associated with the movement of the input and output forces and torques.
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| === Applied Forces ===
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| Consider a mechanism such as a lever that operates so that an input force generates an output force. Let ''A'' be the point where the input force '''F'''<sub>''A''</sub> is applied, and let ''B'' be the point where the output force '''F'''<sub>''B''</sub> is exerted. Define the position and velocity of ''A'' and ''B'' by the vectors '''r'''<sub>''A''</sub>, '''v'''<sub>''A''</sub> and '''r'''<sub>''B''</sub>, '''v'''<sub>''B''</sub>, respectively.
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| Because the mechanism has one degree-of-freedom, there is a single generalized coordinate ''q'' that defines the position vectors '''r'''<sub>''A''</sub>(''q'') and '''r'''<sub>''B''</sub>(''q'') of the input and output points in the system. The principle of virtual work requires that the generalized force associated with this coordinate be zero, thus
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| :<math> F_q = \mathbf{F}_A \cdot \frac{\partial\mathbf{v}_A}{\partial\dot{q}} - \mathbf{F}_B \cdot \frac{\partial\mathbf{v}_B}{\partial\dot{q}}=0.</math>
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| The negative sign on the output force '''F'''<sub>''B''</sub> arises because the convention of virtual work assumes the forces are applied to the device.
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| === Applied Torque ===
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| Consider a mechanism such as a gear train that operates so that an input torque generates an output torque. Let body ''E''<sub>''A''</sub> have the input moment ''T''<sub>''A''</sub> applied to it, and let body ''E''<sub>''B''</sub> exert the output torque ''T''<sub>''B''</sub>. Define the angular position and velocity of ''E''<sub>''A''</sub> and ''E''<sub>''B''</sub> by ''θ''<sub>''A''</sub>, ''ω''<sub>''A''</sub> and ''θ''<sub>''B''</sub>, ''ω''<sub>''B''</sub>, respectively.
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| Because the mechanism has one degree-of-freedom, there is a single generalized coordinate ''q'' that defines the angles ''θ<sub>A</sub>''(''q'') and ''θ<sub>B</sub>''(''q'') of the input and output of the system. The principle of virtual work requires that the generalized force associated with this coordinate be zero, thus
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| :<math> F_q = T_A \frac{\partial\mathbf{\omega}_A}{\partial\dot{q}} - T_B \frac{\partial\mathbf{\omega}_B}{\partial\dot{q}}=0.</math>
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| The negative sign on the output torque ''T''<sub>''B''</sub> arises because the convention of virtual work assumes the torques are applied to the device.
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| == Law of the Lever ==
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| A [[lever]] is modeled as a rigid bar connected to a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force '''F'''<sub>''A''</sub> at a point ''A'' located by the coordinate vector '''r'''<sub>''A''</sub> on the bar. The lever then exerts an output force '''F'''<sub>''B''</sub> at the point ''B'' located by '''r'''<sub>''B''</sub>. The rotation of the lever about the fulcrum ''P'' is defined by the rotation angle ''θ''.
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| [[Image:Archimedes lever (Small).jpg|thumb|right|300px|This is an engraving from ''Mechanics Magazine'' published in London in 1824.]] | |
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| Let the coordinate vector of the point ''P'' that defines the fulcrum be '''r'''<sub>''P''</sub>, and introduce the lengths
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| :<math> a = |\mathbf{r}_A - \mathbf{r}_P|, \quad b = |\mathbf{r}_B - \mathbf{r}_P|, </math>
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| which are the distances from the fulcrum to the input point ''A'' and to the output point ''B'', respectively.
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| Now introduce the unit vectors '''e'''<sub>''A''</sub> and '''e'''<sub>''B''</sub> from the fulcrum to the point ''A'' and ''B'', so
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| :<math> \mathbf{r}_A - \mathbf{r}_P = a\mathbf{e}_A, \quad \mathbf{r}_B - \mathbf{r}_P = b\mathbf{e}_B.</math>
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| This notation allows us to define the velocity of the points ''A'' and ''B'' as
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| :<math> \mathbf{v}_A = \dot{\theta} a \mathbf{e}_A^\perp, \quad \mathbf{v}_B = \dot{\theta} b \mathbf{e}_B^\perp,</math>
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| where '''e'''<sub>''A''</sub><sup>⊥</sup> and '''e'''<sub>''B''</sub><sup>⊥</sup> are unit vectors perpendicular to '''e'''<sub>''A''</sub> and '''e'''<sub>''B''</sub>, respectively.
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| The angle ''θ'' is the generalized coordinate that defines the configuration of the lever, therefore using the formula above for forces applied to a one degree-of-freedom mechanism, the generalized force is given by
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| :<math> F_\theta = \mathbf{F}_A \cdot \frac{\partial\mathbf{v}_A}{\partial\dot{\theta}} - \mathbf{F}_B \cdot \frac{\partial\mathbf{v}_B}{\partial\dot{\theta}}= a(\mathbf{F}_A \cdot \mathbf{e}_A^\perp) - b(\mathbf{F}_B \cdot \mathbf{e}_B^\perp).</math>
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| Now, denote as ''F''<sub>''A''</sub> and ''F''<sub>''B''</sub> the components of the forces that are perpendicular to the radial segments ''PA'' and ''PB''. These forces are given by
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| :<math> F_A = \mathbf{F}_A \cdot \mathbf{e}_A^\perp, \quad F_B = \mathbf{F}_B \cdot \mathbf{e}_B^\perp.</math>
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| This notation and the principle of virtual work yield the formula for the generalized force as
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| :<math> F_\theta = a F_A - b F_B = 0. \,\!</math>
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| The ratio of the output force ''F''<sub>''B''</sub> to the input force ''F''<sub>''A''</sub> is the [[mechanical advantage]] of the lever, and is obtained from the principle of virtual work as
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| :<math> MA = \frac{F_B}{F_A} = \frac{a}{b}.</math>
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| This equation shows that if the distance ''a'' from the fulcrum to the point ''A'' where the input force is applied is greater than the distance ''b'' from fulcrum to the point ''B'' where the output force is applied, then the lever amplifies the input force. If the opposite is true that the distance from the fulcrum to the input point ''A'' is less than from the fulcrum to the output point ''B'', then the lever reduces the magnitude of the input force.
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| This is the ''law of the lever'', which was proven by [[Archimedes]] using geometric reasoning.<ref name="Usher1954">{{cite book|author=Usher, A. P.|authorlink=Abbott Payson Usher|title=A History of Mechanical Inventions|url=http://books.google.com/books?id=Zt4Aw9wKjm8C&pg=PA94|page=94|accessdate=7 April 2013|year=1929|publisher=Harvard University Press (reprinted by Dover Publications 1988)|isbn=978-0-486-14359-0|oclc=514178}}</ref>
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| == Gear train ==
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| A gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next. For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.
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| [[Image:Transmission of motion by compund gear train (Army Service Corps Training, Mechanical Transport, 1911).jpg|thumb|right|300px|Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train]]
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| The size of the gears and the sequence in which they engage define the ratio of the angular velocity ''ω<sub>A</sub>'' of the input gear to the angular velocity ''ω<sub>B</sub>'' of the output gear, known as the speed ratio, or [[gear ratio]], of the gear train. Let ''R'' be the speed ratio, then
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| :<math> \frac{\omega_A}{\omega_B} = R.</math>
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| The input torque ''T''<sub>''A''</sub> acting on the input gear ''G''<sub>''A''</sub> is transformed by the gear train into the output torque ''T''<sub>''B''</sub> exerted by the output gear ''G''<sub>''B''</sub>. If we assume, that the gears are rigid and that there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train.
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| Let the angle ''θ'' of the input gear be the generalized coordinate of the gear train, then the speed ratio ''R'' of the gear train defines the angular velocity of the output gear in terms of the input gear, that is
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| :<math> \omega_A = \omega, \quad \omega_B = \omega/R.\!</math>
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| The formula above for the principle of virtual work with applied torques yields the generalized force
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| :<math> F_\theta = T_A \frac{\partial\omega_A}{\partial\omega} - T_B \frac{\partial \omega_B}{\partial\omega}= T_A - T_B/R = 0.</math>
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| The [[mechanical advantage]] of the gear train is the ratio of the output torque ''T''<sub>''B''</sub> to the input torque ''T''<sub>''A''</sub>, and the above equation yields
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| :<math> MA = \frac{T_B}{T_A} = R.</math>
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| Thus, the speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque.
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| ==Virtual work and rigid body dynamics==
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| If the principle of virtual work for applied forces is used on individual particles of a [[rigid body]], the principle can be generalized for a rigid body: ''When a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero; and conversely, if the total virtual work of all external forces acting on a rigid body is zero then the body is in equilibrium''.
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| If a system is not in static equilibrium, D'Alembert showed that by introducing the acceleration terms of Newton's laws as inertia forces, this approach is generalized to define dynamic equilibrium. The result is D'Alembert's form of the principle of virtual work, which is used to derive the equations of motion for a mechanical system of rigid bodies.
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| The expression ''compatible displacements'' means that the particles remain in contact and displace together so that the work done by pairs of action/reaction inter-particle forces cancel out. Various forms of this principle have been credited to [[Johann Bernoulli|Johann (Jean) Bernoulli]] (1667–1748) and [[Daniel Bernoulli]] (1700–1782).
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| ===Generalized active forces===
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| The static equilibrium of a mechanical system of rigid bodies is defined by the condition that the virtual work of the applied forces is zero for any virtual displacement of the system. This is known as the ''principle of virtual work.''<ref name="Torby1984"/> This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is ''Q''<sub>''i''</sub> = 0.
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| Let a mechanical system be constructed from ''n'' rigid bodies, ''B''<sub>''i'' </sub>, ''i'' = 1, ..., ''n'', and let the resultant of the applied forces on each body be the force–torque pairs, '''F'''<sub>''i''</sub> and '''T'''<sub>''i'' </sub>, ''i'' = 1, ..., ''n''. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity '''V'''<sub>''i''</sub> and angular velocities '''ω'''<sub>''i'' </sub>, ''i'' = 1, ..., ''n'', for each rigid body, are defined by a single generalized coordinate ''q''. Such a system of rigid bodies is said to have one [[degree of freedom (mechanics)|degree of freedom]].
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| The virtual work of the forces and torques, '''F'''<sub>''i''</sub> and '''T'''<sub>''i'' </sub>, applied to this one degree of freedom system is given by
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| :<math> \delta W = \sum_{i=1}^n \left(\mathbf{F}_i\cdot \frac{\partial \mathbf{V}_i}{\partial \dot{q}} + \mathbf{T}_i\cdot\frac{\partial \vec{\omega}_i}{\partial \dot{q}}\right)\delta q = Q\,\delta q,</math>
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| where
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| :<math> Q = \sum_{i=1}^n \left(\mathbf{F}_i\cdot \frac{\partial \mathbf{V}_i}{\partial \dot{q}} + \mathbf{T}_i\cdot\frac{\partial \vec{\omega}_i}{\partial \dot{q}}\right),</math>
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| is the generalized force acting on this one degree of freedom system.
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| If the mechanical system is defined by ''m'' generalized coordinates, ''q''<sub>''j'' </sub>, ''j'' = 1, ..., ''m'', then the system has ''m'' degrees of freedom and the virtual work is given by,
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| :<math> \delta W = \sum_{j=1}^m Q_j\,\delta q_j,</math>
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| where
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| :<math> Q_j = \sum_{i=1}^n \left(\mathbf{F}_i\cdot \frac{\partial \mathbf{V}_i}{\partial \dot{q}_j} + \mathbf{T}_i\cdot\frac{\partial \vec{\omega}_i}{\partial \dot{q}_j}\right),\quad j=1, \ldots, m.</math>
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| is the generalized force associated with the generalized coordinate ''q''<sub>''j''</sub>. The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is
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| :<math> Q_j = 0,\quad j=1, \ldots, m.</math>
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| These ''m'' equations define the static equilibrium of the system of rigid bodies.
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| ===Generalized inertia forces===
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| Let a mechanical system be constructed from n rigid bodies, B<sub>i</sub>, i=1,...,n, and let the resultant of the applied forces on each body be the force-torque pairs, '''F'''<sub>i</sub> and '''T'''<sub>i</sub>, i=1,...,n. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity '''V'''<sub>i</sub> and angular velocities '''ω'''<sub>i</sub>, i=,1...,n, for each rigid body, are defined by a single generalized coordinate q. Such a system of rigid bodies is said to have one [[degree of freedom (mechanics)|degree of freedom]].
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| Consider a single rigid body which moves under the action of a resultant for '''F''' and torque '''T''', with one degree of freedom defined by the generalized coordinate q. Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force Q* associated with the generalized coordinate q is given by
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| :<math> Q^* = -(M\mathbf{A})\cdot \frac{\partial \mathbf{V}}{\partial \dot{q}} - ([I_R]\alpha+ \omega\times[I_R]\omega)\cdot \frac{\partial \vec{\omega}}{\partial \dot{q}}.</math>
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| This inertia force can be computed from the kinetic energy of the rigid body,
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| :<math> T = \frac{1}{2}M\mathbf{V}\cdot\mathbf{V} + \frac{1}{2}\vec{\omega}\cdot [I_R]\vec{\omega},</math>
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| by using the formula
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| :<math> Q^* = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}} -\frac{\partial T}{\partial q}\right).</math>
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| A system of n rigid bodies with m generalized coordinates has the kinetic energy
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| :<math>T = \sum_{i=1}^n(\frac{1}{2}M\mathbf{V}_i\cdot\mathbf{V}_i + \frac{1}{2}\vec{\omega}_i\cdot [I_R]\vec{\omega}_i),</math>
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| which can be used to calculate the m generalized inertia forces<ref>T. R. Kane and D. A. Levinson, [http://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460 Dynamics, Theory and Applications], McGraw-Hill, NY, 2005.</ref>
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| :<math> Q^*_j = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j}\right),\quad j=1, \ldots, m.</math>
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| ===D'Alembert's form of the principle of virtual work===
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| D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that
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| :<math> \delta W = (Q_1 + Q^*_1)\delta q_1 + \ldots + (Q_m + Q^*_m)\delta q_m = 0,</math>
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| for any set of virtual displacements δq<sub>j</sub>. This condition yields m equations,
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| :<math> Q_j + Q^*_j = 0, \quad j=1, \ldots, m,</math>
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| which can also be written as
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| :<math> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = Q_j, \quad j=1,\ldots,m.</math>
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| The result is a set of m equations of motion that define the dynamics of the rigid body system.
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| If the generalized forces Q<sub>j</sub> are derivable from a potential energy V(q<sub>1</sub>,...,q<sub>m</sub>), then these equations of motion take the form
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| :<math> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = -\frac{\partial V}{\partial q_j}, \quad j=1,\ldots,m.</math>
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| In this case, introduce the [[Lagrangian]], L=T-V, so these equations of motion become
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| :<math> \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j} -\frac{\partial L}{\partial q_j} =0 \quad j=1,\ldots,m.</math>
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| These are known as [[Lagrangian mechanics|Lagrange's equations of motion]].
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| ==Virtual work principle for a deformable body==
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| Consider now the [[free body diagram]] of a [[deformable body]], which is composed of an infinite number of differential cubes. Let's define two unrelated states for the body:
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| * The <math> \boldsymbol{\sigma} </math>-State (Fig.a): This shows external surface forces '''T''', body forces '''f''', and internal stresses <math> \boldsymbol{\sigma} </math> in equilibrium.
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| * The <math> \boldsymbol{\epsilon} </math>-State (Fig.b): This shows continuous displacements <math> \mathbf {u}^* </math> and consistent strains <math> \boldsymbol{\epsilon}^* </math>.
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| The superscript * emphasizes that the two states are unrelated. Other than the above stated conditions, there is no need to specify if any of the states are real or virtual.
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| Imagine now that the forces and stresses in the <math> \boldsymbol{\sigma} </math>-State undergo the [[displacement (vector)|displacement]]s and [[deformation (engineering)|deformation]]s in the <math> \boldsymbol{\epsilon} </math>-State: We can compute the total virtual (imaginary) work done by '''''all forces acting on the faces of all cubes''''' in two different ways:
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| * First, by summing the work done by forces such as <math> F_A </math> which act on individual common faces (Fig.c): Since the material experiences compatible [[displacement (vector)|displacement]]s, such work cancels out, leaving only the virtual work done by the surface forces '''T''' (which are equal to stresses on the cubes' faces, by equilibrium).
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| * Second, by computing the net work done by stresses or forces such as <math> F_A </math>, <math> F_B </math> which act on an individual cube, e.g. for the one-dimensional case in Fig.(c):
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| :<math> F_B \big ( u^* + \frac{ \partial u^*}{\partial x} dx \big ) - F_A u^* \approx \frac{ \partial u^* }{\partial x}
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| \sigma dV + u^* \frac{ \partial \sigma }{\partial x} dV = \epsilon^* \sigma dV - u^* f dV </math>
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| :where the equilibrium relation <math> \frac{ \partial \sigma }{\partial x}+f=0 </math> has been used and the second order term has been neglected.
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| :Integrating over the whole body gives:
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| :<math>\int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} \, dV </math> – Work done by the body forces '''f'''.
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| Equating the two results leads to the principle of virtual work for a deformable body:
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| :<math>\mbox{Total external virtual work} = \int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} dV \qquad \mathrm{(d)} </math>
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| where the total external virtual work is done by '''T''' and '''f'''. Thus,
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| :<math> \int_{S} \mathbf{u}^{*T} \mathbf{T} dS + \int_{V} \mathbf{u}^{*T} \mathbf{f} dV = \int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} dV \qquad \mathrm{(e)} </math>
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| The right-hand-side of (d,e) is often called the internal virtual work. The principle of virtual work then states: ''External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains''. It includes the principle of virtual work for rigid bodies as a special case where the internal virtual work is zero.
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| ===Proof of Equivalence between the Principle of Virtual Work and the Equilibrium Equation===
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| We start by looking at the total work done by surface traction on the body going through the specified deformation:
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| :<math> \int_{S} \mathbf{u \cdot T} dS = \int_{S} \mathbf{u \cdot \boldsymbol {\sigma} \cdot n} dS </math>
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| Applying divergence theorem to the right hand side yields:
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| :<math> \int_{S} \mathbf{u \cdot \boldsymbol \sigma \cdot n} dS = \int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol {\sigma} \right) dV </math>
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| Now switch to indicial notation for the ease of derivation.
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| :<math>
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| \begin{align}
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| \int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol {\sigma} \right) dV
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| &= \int_V \frac{\partial}{\partial x_j} \left( u_i \sigma_{ij} \right) dV \\
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| &= \int_V \frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j} dV
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| \end{align}
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| </math>
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| To continue our derivation, we substitute in the equilibrium equation <math> \frac{\partial \sigma_{ij}}{\partial x_j} + f_i = 0 </math>. Then
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| :<math>
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| \int_V \frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j} dV
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| = \int_V \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i dV
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| </math>
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| The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows:
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| :<math>
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| \begin{align}
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| \int_V \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i dV
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| &= \int_V \frac12 \left[ \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)
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| + \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i dV \\
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| &= \int_V \left[ \epsilon_{ij}
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| + \frac12 \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i dV \\
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| &= \int_V \epsilon_{ij} \sigma_{ij} - u_i f_i dV\\
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| &= \int_V \boldsymbol\epsilon : \boldsymbol\sigma - \mathbf u \cdot \mathbf f dV
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| \end{align}
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| </math>
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| where <math> \boldsymbol\epsilon </math> is the strain that is consistent with the specified displacement field. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero.
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| Now recap. We have shown through the above derivation that
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| :<math> \int_{S} \mathbf{u \cdot T} dS = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV - \int_V \mathbf u \cdot \mathbf f dV </math>
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| Move the 2nd term on the right hand side of the equation to the left:
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| :<math> \int_{S} \mathbf{u \cdot T} dS + \int_V \mathbf u \cdot \mathbf f dV = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV </math>
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| The physical interpretation of the above equation is, ''the External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains''.
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| For practical applications:
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| * In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.
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| * In order to impose consistent displacements and strains, we use equilibriated virtual stresses and forces in the virtual work equation.
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| These two general scenarios give rise to two often stated variational principles. They are valid irrespective of material behaviour.
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| ===Principle of virtual displacements===
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| Depending on the purpose, we may specialize the virtual work equation. For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify:
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| * Virtual displacements and strains as variations of the real displacements and strains using variational notation such as <math> \delta\ \mathbf {u} \equiv \mathbf{u}^* </math> and <math> \delta\ \boldsymbol {\epsilon} \equiv \boldsymbol {\epsilon}^* </math>
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| * Virtual displacements be zero on the part of the surface that has prescribed displacements, and thus the work done by the reactions is zero. There remains only external surface forces on the part <math> S_t </math> that do work.
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| The virtual work equation then becomes the principle of virtual displacements:
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| :<math> \int_{S_t} \delta\ \mathbf{u}^T \mathbf{T} dS + \int_{V} \delta\ \mathbf{u}^T \mathbf{f} dV = \int_{V}\delta\boldsymbol{\epsilon}^T \boldsymbol{\sigma} dV \qquad \mathrm{(f)} </math>
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| This relation is equivalent to the set of equilibrium equations written for a differential element in the deformable body as well as of the stress boundary conditions on the part <math> S_t </math> of the surface. Conversely, (f) can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on <math> S_t </math>, and proceeding in the manner similar to (a) and (b).
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| Since virtual displacements are automatically compatible when they are expressed in terms of [[Continuous function|continuous]], [[single-valued function]]s, we often mention only the need for consistency between strains and displacements. The virtual work principle is also valid for large real displacements; however, Eq.(f) would then be written using more complex measures of stresses and strains.
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| ===Principle of virtual forces===
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| Here, we specify:
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| * Virtual forces and stresses as variations of the real forces and stresses.
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| * Virtual forces be zero on the part <math> S_t </math> of the surface that has prescribed forces, and thus only surface (reaction) forces on <math> S_u </math> (where displacements are prescribed) would do work.
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| The virtual work equation becomes the principle of virtual forces:
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| :<math> \int_{S_u} \mathbf{u}^T \delta\ \mathbf{T} dS + \int_{V} \mathbf{u}^T \delta\ \mathbf{f} dV = \int_{V} \boldsymbol{\epsilon}^T \delta \boldsymbol{\sigma} dV \qquad \mathrm{(g)} </math>
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| This relation is equivalent to the set of strain-compatibility equations as well as of the displacement boundary conditions on the part <math> S_u </math>. It has another name: the principle of complementary virtual work.
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| ==Alternative forms==
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| A specialization of the principle of virtual forces is the [[unit dummy force method]], which is very useful for computing displacements in structural systems. According to [[D'Alembert's principle]], inclusion of inertial forces as additional body forces will give the virtual work equation applicable to dynamical systems. More generalized principles can be derived by:
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| * allowing variations of all quantities.
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| * using [[Lagrange multipliers]] to impose boundary conditions and/or to relax the conditions specified in the two states.
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| These are described in some of the references.
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| Among the many [[energy principles in structural mechanics]], the virtual work principle deserves a special place due to its generality that leads to powerful applications in [[structural analysis]], [[solid mechanics]], and [[finite element method in structural mechanics]].
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| ==See also==
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| *[[Flexibility method]]
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| *[[Unit dummy force method]]
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| *[[Finite element method in structural mechanics]]
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| *[[Calculus of variations]]
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| *[[Lagrangian mechanics]]
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| *[[Müller-Breslau's principle]]
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| ==References==
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| {{reflist}}
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| ==Bibliography==
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| *[[Klaus-Jürgen Bathe|Bathe, K.J.]] "Finite Element Procedures", Prentice Hall, 1996. ISBN 0-13-301458-4
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| *Charlton, T.M. ''Energy Principles in Theory of Structures'', Oxford University Press, 1973. ISBN 0-19-714102-1
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| *Dym, C. L. and I. H. Shames, ''Solid Mechanics: A Variational Approach'', McGraw-Hill, 1973.
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| *Greenwood, Donald T. ''Classical Dynamics'', Dover Publications Inc., 1977, ISBN 0-486-69690-1
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| *Hu, H. ''Variational Principles of Theory of Elasticity With Applications'', Taylor & Francis, 1984. ISBN 0-677-31330-6
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| *Langhaar, H. L. ''Energy Methods in Applied Mechanics'', Krieger, 1989.
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| *[[J. N. Reddy|Reddy, J.N.]] ''Energy Principles and Variational Methods in Applied Mechanics'', John Wiley, 2002. ISBN 0-471-17985-X
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| *Shames, I. H. and Dym, C. L. ''Energy and Finite Element Methods in Structural Mechanics'', Taylor & Francis, 1995, ISBN 0-89116-942-3
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| *Tauchert, T.R. ''Energy Principles in Structural Mechanics'', McGraw-Hill, 1974. ISBN 0-07-062925-0
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| *Washizu, K. ''Variational Methods in Elasticity and Plasticity'', Pergamon Pr, 1982. ISBN 0-08-026723-8
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| *Wunderlich, W. ''Mechanics of Structures: Variational and Computational Methods'', CRC, 2002. ISBN 0-8493-0700-7
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| [[Category:Mechanics]]
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| [[Category:Dynamical systems]]
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| [[Category:Structural analysis]]
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| [[Category:Mechanical engineering]]
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| [[Category:Machines]]
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| [[Category:Kinematics]]
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| [[Category:Mechanisms (engineering)]]
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| [[Category:Linkages (mechanical)]]
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