Editing Virtual work (section)

== Dynamic equilibrium for rigid bodies ==
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''.

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.

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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totally the same as in static equilibrium
===Generalized active forces===
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">{{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 ''Q''<sub>''i''</sub>&nbsp;=&nbsp;0.

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]].

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
:<math display="block"> \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>
where 
:<math display="block"> 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>
is the generalized force acting on this one degree of freedom system.

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,
:<math display="block"> \delta W = \sum_{j=1}^m Q_j\,\delta q_j,</math>
where
:<math display="block"> 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>
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
:<math display="block"> Q_j = 0,\quad j=1, \ldots, m.</math>
These ''m'' equations define the static equilibrium of the system of rigid bodies.
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===Generalized inertia forces===
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]].

Consider a single rigid body which moves under the action of a resultant force '''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
<math display="block"> Q^* = -(M\mathbf{A}) \cdot  \frac{\partial \mathbf{V}}{\partial \dot{q}} - ([I_R]\alpha+ \omega\times[I_R]\omega) \cdot \frac{\partial \boldsymbol{\omega}}{\partial \dot{q}}.</math>
This inertia force can be computed from the kinetic energy of the rigid body,
<math display="block"> T = \frac{1}{2} M \mathbf{V} \cdot \mathbf{V} + \frac{1}{2} \boldsymbol{\omega} \cdot [I_R] \boldsymbol{\omega},</math>
by using the formula
<math display="block"> Q^* = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}} -\frac{\partial T}{\partial q}\right).</math>

A system of n rigid bodies with m generalized coordinates has the kinetic energy
<math display="block">T = \sum_{i=1}^n \left(\frac{1}{2} M \mathbf{V}_i \cdot \mathbf{V}_i + \frac{1}{2} \boldsymbol{\omega}_i \cdot [I_R] \boldsymbol{\omega}_i\right),</math>
which can be used to calculate the m generalized inertia forces<ref>T. R. Kane and D. A. Levinson, [https://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460 Dynamics, Theory and Applications], McGraw-Hill, NY, 2005.</ref>
<math display="block"> 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>

===D'Alembert's form of the principle of virtual work===
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
<math display="block"> \delta W = (Q_1 + Q^*_1)\delta q_1 + \dots + (Q_m + Q^*_m)\delta q_m = 0,</math>
for any set of virtual displacements ''δq<sub>j</sub>''.  This condition yields ''m'' equations,
<math display="block"> Q_j + Q^*_j = 0, \quad j=1, \ldots, m,</math>
which can also be written as
<math display="block"> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = Q_j, \quad j=1,\ldots,m.</math>
The result is a set of m equations of motion that define the dynamics of the rigid body system, known as [[Lagrangian mechanics|Lagrange's equations]] or the '''generalized equations of motion'''.

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
<math display="block"> \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>

In this case, introduce the [[Lagrangian mechanics|Lagrangian]], {{math|1=''L'' = ''T'' − ''V''}}, so these equations of motion become
<math display="block"> \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j} = 0 \quad j=1,\ldots,m.</math>
These are known as the [[Euler–Lagrange equation|Euler-Lagrange equations]] for a system with m degrees of freedom, or '''Lagrange's equations of the second kind'''.