In 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 is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to 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 principle of virtual work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, and Renaissance Italians as "the law of lever". The idea of virtual work was invoked by many notable physicists of the 17th century, such as Galileo, Descartes, Torricelli, Wallis, and Huygens, in varying degrees of generality, when solving problems in statics. Working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both rigid bodies as well as fluids. Bernoulli's version of virtual work law appeared in his letter to Pierre Varignon in 1715, which was later published in Varignon's second volume of Nouvelle mécanique ou Statique in 1725. This formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. In 1743 D'Alembert published his Traité de Dynamique where he applied the principle of virtual work, based on Bernoulli's work, to solve various problems in dynamics. His idea was to convert a dynamical problem into static problem by introducing inertial force. In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved. A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic, essentially D'Alembert's principle, was given in his Mécanique Analytique of 1788. Although Lagrange had presented his version of least action principle prior to this work, he recognized the virtual work principle to be more fundamental mainly because it could be assumed alone as the foundation for all mechanics, unlike the modern understanding that least action does not account for non-conservative forces.
Overview If a force acts on a particle as it moves from point A {\displaystyle A} to point B {\displaystyle 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 (to the first order). 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. Consider a point particle that moves along a path which is described by a function r ( t ) {\displaystyle \mathbf {r} (t)} from point A {\displaystyle A} , where r ( t = t 0 ) {\displaystyle \mathbf {r} (t=t_{0})} , to point B {\displaystyle B} , where r ( t = t 1 ) {\displaystyle \mathbf {r} (t=t_{1})} . It is possible that the particle moves from A {\displaystyle A} to B {\displaystyle B} along a nearby path described by r ( t ) + δ r ( t ) {\displaystyle \mathbf {r} (t)+\delta \mathbf {r} (t)} , where δ r ( t ) {\displaystyle \delta \mathbf {r} (t)} is called the variation of r ( t ) {\displaystyle \mathbf {r} (t)} . The variation δ r ( t ) {\displaystyle \delta \mathbf {r} (t)} satisfies the requirement δ r ( t 0 ) = δ r ( t 1 ) = 0 {\displaystyle \delta \mathbf {r} (t_{0})=\delta \mathbf {r} (t_{1})=0} . The scalar components of the variation δ r 1 ( t ) {\displaystyle \delta r_{1}(t)} , δ r 2 ( t ) {\displaystyle \delta r_{2}(t)} and δ r 3 ( t ) {\displaystyle \delta r_{3}(t)} are called virtual displacements. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates q i {\displaystyle q_{i}} , i = 1 , 2 , . . . , n {\displaystyle i=1,2,...,n} . In which case, the variation of the trajectory q i ( t ) {\displaystyle q_{i}(t)} is defined by the virtual displacements δ q i {\displaystyle \delta q_{i}} , i = 1 , 2 , . . . , n {\displaystyle i=1,2,...,n} . Virtual work is the total 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.
Mathematical treatment Consider a particle P that moves from a point A to a point B along a trajectory r(t), while a force F(r(t)) is applied to it. The work done by the force F is given by the integral
W = ∫ r ( t 0 ) = A r ( t 1 ) = B F ⋅ d r = ∫ t 0 t 1 F ⋅ d r d t d t = ∫ t 0 t 1 F ⋅ v d t , {\displaystyle 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 {\frac {d\mathbf {r} }{dt}}~dt=\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 particle P that moves from point A to point B again, but this time it moves 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. Suppose the force F(r(t) + εh(t)) is the same as F(r(t)). The work done by the force is given by the integral
W ¯ = ∫ r ( t 0 ) = A r ( t 1 ) = B F ⋅ d ( r + ε h ) = ∫ t 0 t 1 F ⋅ d ( r ( t ) + ε h ( t ) ) d t d t = ∫ t 0 t 1 F ⋅ ( v + ε h ˙ ) d t . {\displaystyle {\bar {W}}=\int _{\mathbf {r} (t_{0})=A}^{\mathbf {r} (t_{1})=B}\mathbf {F} \cdot d(\mathbf {r} +\varepsilon \mathbf {h} )=\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot {\frac {d(\mathbf {r} (t)+\varepsilon \mathbf {h} (t))}{dt}}~dt=\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot (\mathbf {v} +\varepsilon {\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
δ W = W ¯ − W = ∫ t 0 t 1 ( F ⋅ ε h ˙ ) d t . {\displaystyle \delta W={\bar {W}}-W=\int _{t_{0}}^{t_{1}}(\mathbf {F} \cdot \varepsilon {\dot {\mathbf {h} }})~dt.}
If there are no constraints on the motion of P, then 3 parameters are needed to completely describe P's position at any time t. If there are k (k ≤ 3) constraint forces, then n = (3 − k) parameters are needed. Hence, we can define n generalized coordinates qi (t) (i = 1,...,n), and express r(t) and δr = εh(t) in terms of the generalized coordinates. That is,
r ( t ) = r ( q 1 , q 2 , … , q n ; t ) , {\displaystyle \mathbf {r} (t)=\mathbf {r} (q_{1},q_{2},\dots ,q_{n};t),}
h ( t ) = h ( q 1 , q 2 , … , q n ; t ) . {\displaystyle \mathbf {h} (t)=\mathbf {h} (q_{1},q_{2},\dots ,q_{n};t).}
Then, the derivative of the variation δr = εh(t) is given by
d d t δ r = d d t ε h = ∑ i = 1 n ∂ h ∂ q i ε q ˙ i , {\displaystyle {\frac {d}{dt}}\delta \mathbf {r} ={\frac {d}{dt}}\varepsilon \mathbf {h} =\sum _{i=1}^{n}{\frac {\partial \mathbf {h} }{\partial q_{i}}}\varepsilon {\dot {q}}_{i},}
then we have
δ W = ∫ t 0 t 1 ( ∑ i = 1 n F ⋅ ∂ h ∂ q i ε q ˙ i ) d t = ∑ i = 1 n ( ∫ t 0 t 1 F ⋅ ∂ h ∂ q i ε q ˙ i d t ) . {\displaystyle \delta W=\int _{t_{0}}^{t_{1}}\left(\sum _{i=1}^{n}\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{i}}}\varepsilon {\dot {q}}_{i}\right)dt=\sum _{i=1}^{n}\left(\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{i}}}\varepsilon {\dot {q}}_{i}~dt\right).}
The requirement that the virtual work be zero for an arbitrary variation δr(t) = εh(t) is equivalent to the set of requirements
Q i = F ⋅ ∂ h ∂ q i = 0 , i = 1 , … , n . {\displaystyle Q_{i}=\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{i}}}=0,\quad i=1,\ldots ,n.}
The terms Qi are called the generalized forces associated with the virtual displacement δr.
Static equilibrium Static equilibrium is a state in which the net force and net torque acted upon the system is zero. In other words, both linear momentum and angular momentum of the system are conserved. 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. This principle can be generalized such that three dimensional rotations are included: the virtual work of the applied forces and applied moments is zero for all virtual movements of the system from static equilibrium. That is
δ W = ∑ i = 1 m F i ⋅ δ r i + ∑ j = 1 n M j ⋅ δ φ j = 0 , {\displaystyle \delta W=\sum _{i=1}^{m}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}+\sum _{j=1}^{n}\mathbf {M} _{j}\cdot \delta \mathbf {\varphi } _{j}=0,}
where Fi , i = 1, 2, ..., m and Mj , j = 1, 2, ..., n are the applied forces and applied moments, respectively, and δri , i = 1, 2, ..., m and δφj, j = 1, 2, ..., n are the virtual displacements and virtual rotations, respectively. Suppose the system consists of N particles, and it has f (f ≤ 6N) degrees of freedom. It is sufficient to use only f coordinates to give a complete description of the motion of the system, so f generalized coordinates qk , k = 1, 2, ..., f are defined such that the virtual movements can be expressed in terms of these generalized coordinates. That is,
δ r i ( q 1 , q 2 , … , q f ; t ) , i = 1 , 2 , … , m ; {\displaystyle \delta \mathbf {r} _{i}(q_{1},q_{2},\dots ,q_{f};t),\quad i=1,2,\dots ,m;}
δ ϕ j ( q 1 , q 2 , … , q f ; t ) , j = 1 , 2 , … , n . {\displaystyle \delta \phi _{j}(q_{1},q_{2},\dots ,q_{f};t),\quad j=1,2,\dots ,n.}
The virtual work can then be reparametrized by the generalized coordinates:
δ W = ∑ k = 1 f [ ( ∑ i = 1 m F i ⋅ ∂ r i ∂ q k + ∑ j = 1 n M j ⋅ ∂ ϕ j ∂ q k ) δ q k ] = ∑ k = 1 f Q k δ q k , {\displaystyle \delta W=\sum _{k=1}^{f}\left[\left(\sum _{i=1}^{m}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{k}}}+\sum _{j=1}^{n}\mathbf {M} _{j}\cdot {\frac {\partial \mathbf {\phi } _{j}}{\partial q_{k}}}\right)\delta q_{k}\right]=\sum _{k=1}^{f}Q_{k}\delta q_{k},}
where the generalized forces Qk are defined as
Q k = ∑ i = 1 m F i ⋅ ∂ r i ∂ q k + ∑ j = 1 n M j ⋅ ∂ ϕ j ∂ q k , k = 1 , 2 , … , f . {\displaystyle Q_{k}=\sum _{i=1}^{m}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{k}}}+\sum _{j=1}^{n}\mathbf {M} _{j}\cdot {\frac {\partial \mathbf {\phi } _{j}}{\partial q_{k}}},\quad k=1,2,\dots ,f.}
Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives. That is,
Q k = ∑ i = 1 m F i ⋅ ∂ v i ∂ q ˙ k + ∑ j = 1 n M j ⋅ ∂ ω j ∂ q ˙ k , k = 1 , 2 , … , f . {\displaystyle Q_{k}=\sum _{i=1}^{m}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {v} _{i}}{\partial {\dot {q}}_{k}}}+\sum _{j=1}^{n}\mathbf {M} _{j}\cdot {\frac {\partial \mathbf {\omega } _{j}}{\partial {\dot {q}}_{k}}},\quad k=1,2,\dots ,f.}
The principle of virtual work requires that the virtual work done on a system by the forces Fi and moments Mj vanishes if it is in equilibrium. Therefore, the generalized forces Qk are zero, that is
δ W = 0 ⇒ Q k = 0 k = 1 , 2 , … , f . {\displaystyle \delta W=0\quad \Rightarrow \quad Q_{k}=0\quad k=1,2,\dots ,f.}
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.
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 FA at a point A located by the coordinate vector rA on the bar. The lever then exerts an output force FB at the point B located by rB. The rotation of the lever about the fulcrum P is defined by the rotation angle θ.
Let the coordinate vector of the point P that defines the fulcrum be rP, and introduce the lengths
a = | r A − r P | , b = | r B − r P | , {\displaystyle 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 eA and eB from the fulcrum to the point A and B, so
r A − r P = a e A , r B − r P = b e B . {\displaystyle \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
v A = θ ˙ a e A ⊥ , v B = θ ˙ b e B ⊥ , {\displaystyle \mathbf {v} _{A}={\dot {\theta }}a\mathbf {e} _{A}^{\perp },\quad \mathbf {v} _{B}={\dot {\theta }}b\mathbf {e} _{B}^{\perp },}
where eA⊥ and eB⊥ are unit vectors perpendicular to eA and eB, 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
Q = F A ⋅ ∂ v A ∂ θ ˙ − F B ⋅ ∂ v B ∂ θ ˙ = a ( F A ⋅ e A ⊥ ) − b ( F B ⋅ e B ⊥ ) . {\displaystyle Q=\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 FA and FB the components of the forces that are perpendicular to the radial segments PA and PB. These forces are given by
F A = F A ⋅ e A ⊥ , F B = F B ⋅ e B ⊥ . {\displaystyle 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
Q = a F A − b F B = 0. {\displaystyle Q=aF_{A}-bF_{B}=0.}
The ratio of the output force FB to the input force FA is the mechanical advantage of the lever, and is obtained from the principle of virtual work as
M A = F B F A = a b . {\displaystyle 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.
The size of the gears and the sequence in which they engage define the ratio of the angular velocity ωA of the input gear to the angular velocity ωB of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio, then
ω A