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Template:Classical mechanics 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 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 displacements, 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.

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.

History

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. 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.

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. 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.

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.

Overview

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.

Let the function x(t) define the path followed by a point. A nearby path can then be defined by adding the function δx(t) to the original path, so that the new path is given by x(t)+δx(t). The function δx(t) is called the variation of the original path, and each of the components of δx=(δx, δy, δz) is called a virtual displacement. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates q_i , i = 1, ..., n. In which case, the variation of the trajectory q_i (t) is defined by the virtual displacements δq_i , i = 1, ..., n.

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.

Introduction

In this introduction basic definitions are presented that will assist in understanding later sections.

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

$W=\int _{\mathbf {r} (t_{0})=A}^{\mathbf {r} (t_{1})=B}\mathbf {F} \cdot d\mathbf {r} =\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot \mathbf {v} dt,$

where dr 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).

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 δr(t)=εh(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(t₀) = h(t₁) = 0,

${\bar {W}}=\int _{A}^{B}\mathbf {F} \cdot d(\mathbf {r} +\epsilon \mathbf {h} )=\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot (\mathbf {v} +\epsilon {\dot {\mathbf {h} }})dt.$

The variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be

$\delta W={\bar {W}}-W=\int _{t_{0}}^{t_{1}}(\mathbf {F} \cdot \epsilon {\dot {\mathbf {h} }})dt.$

Now assume that r(t) and h(t) depend on the generalized coordinates q_i , i = 1, ..., n, then the derivative of the variation δr=εh(t) is given by

${\frac {d}{dt}}\delta \mathbf {r} =\epsilon {\dot {\mathbf {h} }}=\epsilon \left({\frac {\partial \mathbf {h} }{\partial q_{1}}}{\dot {q}}_{1}+\ldots +{\frac {\partial \mathbf {h} }{\partial q_{n}}}{\dot {q}}_{n}\right),$

then we have

$\delta W=\int _{t_{0}}^{t_{1}}\left(\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{1}}}\epsilon {\dot {q}}_{1}+\ldots +\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{n}}}\epsilon {\dot {q}}_{n}\right)dt=\int _{t_{0}}^{t_{1}}\left(\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{1}}}\right)\epsilon {\dot {q}}_{1}dt+\ldots +\int _{t_{0}}^{t_{1}}\left(\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{n}}}\right)\epsilon {\dot {q}}_{n}dt.$

The requirement that the virtual work be zero for an arbitrary variation δr(t)=εh(t) is equivalent to the set of requirements

$F_{i}=\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{i}}}=0,\quad i=1,\ldots ,n.$

The terms F_i are called the generalized forces associated with the virtual displacement δr.

Static equilibrium

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 δr. This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F_i = 0.

Let the forces on the system be F_j , j = 1, ..., m and let the virtual displacement of each point of application of these forces be δr_j , j = 1, ..., m, then the virtual work generated by a virtual displacement of these forces from the equilibrium position is given by

$\delta W=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot \delta \mathbf {r} _{j}.$

Now assume that each δr_j depends on the generalized coordinates q_i , i = 1, ..., n, then

$\delta \mathbf {r} _{j}={\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\delta {q}_{1}+\ldots +{\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\delta {q}_{n},$

and

$\delta W=\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\right)\delta {q}_{1}+\ldots +\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\right)\delta {q}_{n}.$

The n terms

$F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{i}}},\quad i=1,\ldots ,n,$

are the generalized forces acting on the system. Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,

$F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}},\quad i=1,\ldots ,n,$

where v_j is the velocity of the point of application of the force F_j .

In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is

$\delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.$

Constraint forces

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.

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.

Couples

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.

For planar movement, the moment acts perpendicular to the plane with magnitude M and the virtual work due to this moment is

$\delta W=M\delta \phi ,\!$

where δφ is the virtual rotation angle of the body.

In order to extend this to three dimensional rotations, use the angular velocity vector ω of the body to obtain the virtual work as

$\delta W=\left(\mathbf {M} \cdot {\frac {\partial {\vec {\omega }}}{\partial {\dot {\phi }}}}\right)\delta \phi .$

Now consider the moments M_j acting on m rigid bodies in a mechanical system. Let the angular velocity vectors ω_j , j = 1, ..., m of each body depend on the n generalized coordinates q_j , i = 1, ..., n. Then the virtual work obtained from these moments for a virtual displacement from equilibrium is given by

$\delta W=\sum _{j=1}^{m}\mathbf {M} _{j}\cdot \left({\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{1}}}\delta q_{1}+\ldots +{\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{n}}}\delta q_{n}\right).$

Collect the coefficients of the virtual displacements δq_i to obtain

$\delta W=\left(\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{1}}}\right)\delta q_{1}+\ldots +\left(\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{n}}}\right)\delta q_{n}.$

Forces and moments

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

$\delta W=\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{1}}}+\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{1}}}\right)\delta q_{1}+\ldots +\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{n}}}+\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{n}}}\right)\delta q_{n},$

where the generalized forces are now defined to be

$F_{i}=\sum _{j=1}^{m}\left(\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}}+\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{i}}}\right),\quad i=1,\ldots ,n.$

The principle of virtual work requires that a system of rigid bodies acted on by the forces and moments F_j and M_j is in equilibrium if the generalized forces F_i are zero, that is

$\delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.$

One degree-of-freedom mechanisms

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_x, f_y) has two components and acts on a point with coordinates r = (r_x, r_y) and velocity v = (v_x, v_y). 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.

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.

Applied Forces

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_A is applied, and let B be the point where the output force F_B is exerted. Define the position and velocity of A and B by the vectors r_A, v_A and r_B, v_B, respectively.

Because the mechanism has one degree-of-freedom, there is a single generalized coordinate q that defines the position vectors r_A(q) and r_B(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

$F_{q}=\mathbf {F} _{A}\cdot {\frac {\partial \mathbf {v} _{A}}{\partial {\dot {q}}}}-\mathbf {F} _{B}\cdot {\frac {\partial \mathbf {v} _{B}}{\partial {\dot {q}}}}=0.$

The negative sign on the output force F_B arises because the convention of virtual work assumes the forces are applied to the device.

Applied Torque

Consider a mechanism such as a gear train that operates so that an input torque generates an output torque. Let body E_A have the input moment T_A applied to it, and let body E_B exert the output torque T_B. Define the angular position and velocity of E_A and E_B by θ_A, ω_A and θ_B, ω_B, respectively.

Because the mechanism has one degree-of-freedom, there is a single generalized coordinate q that defines the angles θ_A(q) and θ_B(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

$F_{q}=T_{A}{\frac {\partial \mathbf {\omega } _{A}}{\partial {\dot {q}}}}-T_{B}{\frac {\partial \mathbf {\omega } _{B}}{\partial {\dot {q}}}}=0.$

The negative sign on the output torque T_B arises because the convention of virtual work assumes the torques are applied to the device.

Law of the Lever

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_A at a point A located by the coordinate vector r_A on the bar. The lever then exerts an output force F_B at the point B located by r_B. The rotation of the lever about the fulcrum P is defined by the rotation angle θ.

This is an engraving from Mechanics Magazine published in London in 1824.

Let the coordinate vector of the point P that defines the fulcrum be r_P, and introduce the lengths

$a=|\mathbf {r} _{A}-\mathbf {r} _{P}|,\quad b=|\mathbf {r} _{B}-\mathbf {r} _{P}|,$

which are the distances from the fulcrum to the input point A and to the output point B, respectively.

Now introduce the unit vectors e_A and e_B from the fulcrum to the point A and B, so

$\mathbf {r} _{A}-\mathbf {r} _{P}=a\mathbf {e} _{A},\quad \mathbf {r} _{B}-\mathbf {r} _{P}=b\mathbf {e} _{B}.$

This notation allows us to define the velocity of the points A and B as

$\mathbf {v} _{A}={\dot {\theta }}a\mathbf {e} _{A}^{\perp },\quad \mathbf {v} _{B}={\dot {\theta }}b\mathbf {e} _{B}^{\perp },$

where e_A^⊥ and e_B^⊥ are unit vectors perpendicular to e_A and e_B, respectively.

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

$F_{\theta }=\mathbf {F} _{A}\cdot {\frac {\partial \mathbf {v} _{A}}{\partial {\dot {\theta }}}}-\mathbf {F} _{B}\cdot {\frac {\partial \mathbf {v} _{B}}{\partial {\dot {\theta }}}}=a(\mathbf {F} _{A}\cdot \mathbf {e} _{A}^{\perp })-b(\mathbf {F} _{B}\cdot \mathbf {e} _{B}^{\perp }).$

Now, denote as F_A and F_B the components of the forces that are perpendicular to the radial segments PA and PB. These forces are given by

$F_{A}=\mathbf {F} _{A}\cdot \mathbf {e} _{A}^{\perp },\quad F_{B}=\mathbf {F} _{B}\cdot \mathbf {e} _{B}^{\perp }.$

This notation and the principle of virtual work yield the formula for the generalized force as

$F_{\theta }=aF_{A}-bF_{B}=0.\,\!$

The ratio of the output force F_B to the input force F_A is the mechanical advantage of the lever, and is obtained from the principle of virtual work as

$MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}}.$

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.

This is the law of the lever, which was proven by Archimedes using geometric reasoning.

Gear train

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.

Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train

The size of the gears and the sequence in which they engage define the ratio of the angular velocity ω_A of the input gear to the angular velocity ω_B of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio, then

${\frac {\omega _{A}}{\omega _{B}}}=R.$

The input torque T_A acting on the input gear G_A is transformed by the gear train into the output torque T_B exerted by the output gear G_B. 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.

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

$\omega _{A}=\omega ,\quad \omega _{B}=\omega /R.\!$

The formula above for the principle of virtual work with applied torques yields the generalized force

$F_{\theta }=T_{A}{\frac {\partial \omega _{A}}{\partial \omega }}-T_{B}{\frac {\partial \omega _{B}}{\partial \omega }}=T_{A}-T_{B}/R=0.$

The mechanical advantage of the gear train is the ratio of the output torque T_B to the input torque T_A, and the above equation yields

$MA={\frac {T_{B}}{T_{A}}}=R.$

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.

Virtual work and rigid body dynamics

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 (Jean) Bernoulli (1667–1748) and Daniel Bernoulli (1700–1782).

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. This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is Q_i = 0.

Let a mechanical system be constructed from n rigid bodies, B_i , i = 1, ..., n, and let the resultant of the applied forces on each body be the force–torque pairs, F_i and T_i , 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_i and angular velocities ω_i , 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.

The virtual work of the forces and torques, F_i and T_i , applied to this one degree of freedom system is given by

$\delta W=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\vec {\omega }}_{i}}{\partial {\dot {q}}}}\right)\delta q=Q\,\delta q,$

where

$Q=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\vec {\omega }}_{i}}{\partial {\dot {q}}}}\right),$

is the generalized force acting on this one degree of freedom system.

If the mechanical system is defined by m generalized coordinates, q_j , j = 1, ..., m, then the system has m degrees of freedom and the virtual work is given by,

$\delta W=\sum _{j=1}^{m}Q_{j}\,\delta q_{j},$

where

$Q_{j}=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\vec {\omega }}_{i}}{\partial {\dot {q}}_{j}}}\right),\quad j=1,\ldots ,m.$

is the generalized force associated with the generalized coordinate q_j. The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is

$Q_{j}=0,\quad j=1,\ldots ,m.$

These m equations define the static equilibrium of the system of rigid bodies.

Generalized inertia forces

Let a mechanical system be constructed from n rigid bodies, B_i, i=1,...,n, and let the resultant of the applied forces on each body be the force-torque pairs, F_i and T_i, 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_i and angular velocities ω_i, 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.

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

$Q^{*}=-(M\mathbf {A} )\cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}-([I_{R}]\alpha +\omega \times [I_{R}]\omega )\cdot {\frac {\partial {\vec {\omega }}}{\partial {\dot {q}}}}.$

This inertia force can be computed from the kinetic energy of the rigid body,

$T={\frac {1}{2}}M\mathbf {V} \cdot \mathbf {V} +{\frac {1}{2}}{\vec {\omega }}\cdot [I_{R}]{\vec {\omega }},$

by using the formula

$Q^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}}}-{\frac {\partial T}{\partial q}}\right).$

A system of n rigid bodies with m generalized coordinates has the kinetic energy

$T=\sum _{i=1}^{n}({\frac {1}{2}}M\mathbf {V} _{i}\cdot \mathbf {V} _{i}+{\frac {1}{2}}{\vec {\omega }}_{i}\cdot [I_{R}]{\vec {\omega }}_{i}),$

which can be used to calculate the m generalized inertia forces

$Q_{j}^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right),\quad j=1,\ldots ,m.$

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

$\delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\ldots +(Q_{m}+Q_{m}^{*})\delta q_{m}=0,$

for any set of virtual displacements δq_j. This condition yields m equations,

$Q_{j}+Q_{j}^{*}=0,\quad j=1,\ldots ,m,$

which can also be written as

${\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=Q_{j},\quad j=1,\ldots ,m.$

The result is a set of m equations of motion that define the dynamics of the rigid body system.

If the generalized forces Q_j are derivable from a potential energy V(q₁,...,q_m), then these equations of motion take the form

${\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.$

In this case, introduce the Lagrangian, L=T-V, so these equations of motion become

${\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}-{\frac {\partial L}{\partial q_{j}}}=0\quad j=1,\ldots ,m.$

These are known as Lagrange's equations of motion.

Virtual work principle for a deformable body

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:

• The ${\boldsymbol {\sigma }}$-State (Fig.a): This shows external surface forces T, body forces f, and internal stresses ${\boldsymbol {\sigma }}$ in equilibrium.
• The ${\boldsymbol {\epsilon }}$-State (Fig.b): This shows continuous displacements $\mathbf {u} ^{*}$ and consistent strains ${\boldsymbol {\epsilon }}^{*}$.

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.

Imagine now that the forces and stresses in the ${\boldsymbol {\sigma }}$-State undergo the displacements and deformations in the ${\boldsymbol {\epsilon }}$-State: We can compute the total virtual (imaginary) work done by all forces acting on the faces of all cubes in two different ways:

• First, by summing the work done by forces such as $F_{A}$ which act on individual common faces (Fig.c): Since the material experiences compatible displacements, 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).

• Second, by computing the net work done by stresses or forces such as $F_{A}$, $F_{B}$ which act on an individual cube, e.g. for the one-dimensional case in Fig.(c):

$F_{B}{\big (}u^{*}+{\frac {\partial u^{*}}{\partial x}}dx{\big )}-F_{A}u^{*}\approx {\frac {\partial u^{*}}{\partial x}}\sigma dV+u^{*}{\frac {\partial \sigma }{\partial x}}dV=\epsilon ^{*}\sigma dV-u^{*}fdV$

where the equilibrium relation ${\frac {\partial \sigma }{\partial x}}+f=0$ has been used and the second order term has been neglected.

Integrating over the whole body gives:

$\int _{V}{\boldsymbol {\epsilon }}^{*T}{\boldsymbol {\sigma }}\,dV$ – Work done by the body forces f.

Equating the two results leads to the principle of virtual work for a deformable body:

${\mbox{Total external virtual work}}=\int _{V}{\boldsymbol {\epsilon }}^{*T}{\boldsymbol {\sigma }}dV\qquad \mathrm {(d)}$

where the total external virtual work is done by T and f. Thus,

$\int _{S}\mathbf {u} ^{*T}\mathbf {T} dS+\int _{V}\mathbf {u} ^{*T}\mathbf {f} dV=\int _{V}{\boldsymbol {\epsilon }}^{*T}{\boldsymbol {\sigma }}dV\qquad \mathrm {(e)}$

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.

Proof of Equivalence between the Principle of Virtual Work and the Equilibrium Equation

We start by looking at the total work done by surface traction on the body going through the specified deformation:

$\int _{S}\mathbf {u\cdot T} dS=\int _{S}\mathbf {u\cdot {\boldsymbol {\sigma }}\cdot n} dS$

Applying divergence theorem to the right hand side yields:

$\int _{S}\mathbf {u\cdot {\boldsymbol {\sigma }}\cdot n} dS=\int _{V}\nabla \cdot \left(\mathbf {u} \cdot {\boldsymbol {\sigma }}\right)dV$

Now switch to indicial notation for the ease of derivation.

${\begin{aligned}\int _{V}\nabla \cdot \left(\mathbf {u} \cdot {\boldsymbol {\sigma }}\right)dV&=\int _{V}{\frac {\partial }{\partial x_{j}}}\left(u_{i}\sigma _{ij}\right)dV\\&=\int _{V}{\frac {\partial u_{i}}{\partial x_{j}}}\sigma _{ij}+u_{i}{\frac {\partial \sigma _{ij}}{\partial x_{j}}}dV\end{aligned}}$

To continue our derivation, we substitute in the equilibrium equation ${\frac {\partial \sigma _{ij}}{\partial x_{j}}}+f_{i}=0$. Then

$\int _{V}{\frac {\partial u_{i}}{\partial x_{j}}}\sigma _{ij}+u_{i}{\frac {\partial \sigma _{ij}}{\partial x_{j}}}dV=\int _{V}{\frac {\partial u_{i}}{\partial x_{j}}}\sigma _{ij}-u_{i}f_{i}dV$

The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows:

${\begin{aligned}\int _{V}{\frac {\partial u_{i}}{\partial x_{j}}}\sigma _{ij}-u_{i}f_{i}dV&=\int _{V}{\frac {1}{2}}\left[\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)+\left({\frac {\partial u_{i}}{\partial x_{j}}}-{\frac {\partial u_{j}}{\partial x_{i}}}\right)\right]\sigma _{ij}-u_{i}f_{i}dV\\&=\int _{V}\left[\epsilon _{ij}+{\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}-{\frac {\partial u_{j}}{\partial x_{i}}}\right)\right]\sigma _{ij}-u_{i}f_{i}dV\\&=\int _{V}\epsilon _{ij}\sigma _{ij}-u_{i}f_{i}dV\\&=\int _{V}{\boldsymbol {\epsilon }}:{\boldsymbol {\sigma }}-\mathbf {u} \cdot \mathbf {f} dV\end{aligned}}$

where ${\boldsymbol {\epsilon }}$ 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.

Now recap. We have shown through the above derivation that

$\int _{S}\mathbf {u\cdot T} dS=\int _{V}{\boldsymbol {\epsilon }}:{\boldsymbol {\sigma }}dV-\int _{V}\mathbf {u} \cdot \mathbf {f} dV$

Move the 2nd term on the right hand side of the equation to the left:

$\int _{S}\mathbf {u\cdot T} dS+\int _{V}\mathbf {u} \cdot \mathbf {f} dV=\int _{V}{\boldsymbol {\epsilon }}:{\boldsymbol {\sigma }}dV$

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.

For practical applications:

• In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.

• In order to impose consistent displacements and strains, we use equilibriated virtual stresses and forces in the virtual work equation.

These two general scenarios give rise to two often stated variational principles. They are valid irrespective of material behaviour.

Principle of virtual displacements

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:

• Virtual displacements and strains as variations of the real displacements and strains using variational notation such as $\delta \ \mathbf {u} \equiv \mathbf {u} ^{*}$ and $\delta \ {\boldsymbol {\epsilon }}\equiv {\boldsymbol {\epsilon }}^{*}$

• 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 $S_{t}$ that do work.

The virtual work equation then becomes the principle of virtual displacements:

$\int _{S_{t}}\delta \ \mathbf {u} ^{T}\mathbf {T} dS+\int _{V}\delta \ \mathbf {u} ^{T}\mathbf {f} dV=\int _{V}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}dV\qquad \mathrm {(f)}$

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 $S_{t}$ 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 $S_{t}$, and proceeding in the manner similar to (a) and (b).

Since virtual displacements are automatically compatible when they are expressed in terms of continuous, single-valued functions, 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.

Principle of virtual forces

Here, we specify:

• Virtual forces and stresses as variations of the real forces and stresses.

• Virtual forces be zero on the part $S_{t}$ of the surface that has prescribed forces, and thus only surface (reaction) forces on $S_{u}$ (where displacements are prescribed) would do work.

The virtual work equation becomes the principle of virtual forces:

$\int _{S_{u}}\mathbf {u} ^{T}\delta \ \mathbf {T} dS+\int _{V}\mathbf {u} ^{T}\delta \ \mathbf {f} dV=\int _{V}{\boldsymbol {\epsilon }}^{T}\delta {\boldsymbol {\sigma }}dV\qquad \mathrm {(g)}$

This relation is equivalent to the set of strain-compatibility equations as well as of the displacement boundary conditions on the part $S_{u}$. It has another name: the principle of complementary virtual work.

Alternative forms

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:

• allowing variations of all quantities.
• using Lagrange multipliers to impose boundary conditions and/or to relax the conditions specified in the two states.

These are described in some of the references.

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.

See also

• Flexibility method
• Unit dummy force method
• Finite element method in structural mechanics
• Calculus of variations
• Lagrangian mechanics
• Müller-Breslau's principle

References

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