Virtual work

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.

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. ==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 (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 \mathbf{r}(t) from point A , where \mathbf{r}(t)=t_0 , to point B , where \mathbf{r}(t)=t_1 . It is possible that the particle moves from A to B along a nearby path described by \mathbf{r}(t) + \delta \mathbf{r}(t) , where \delta \mathbf{r}(t) is called the variation of \mathbf{r}(t) . The variation \delta \mathbf{r}(t) satisfies the requirement \delta \mathbf{r}(t_0) = \delta \mathbf{r}(t_1) = 0 . The scalar components of the variation \delta r_1(t) , \delta r_2(t) and \delta r_3(t) are called virtual displacements. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates q_i , i = 1,2,...,n . In which case, the variation of the trajectory q_i(t) is defined by the virtual displacements \delta q_i , 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. 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 is the differential element along the curve that is the trajectory of P, and is its velocity. It is important to notice that the value of the work depends on the trajectory . 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 by the variation , where is a scaling constant that can be made as small as desired and is an arbitrary function that satisfies . Suppose the force is the same as . The work done by the force is given by the integral \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 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 \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 . If there are ( ) constraint forces, then parameters are needed. Hence, we can define generalized coordinates ( ), and express and in terms of the generalized coordinates. That is, \mathbf{r}(t) = \mathbf{r}(q_1,q_2,\dots,q_n;t), \mathbf{h}(t) = \mathbf{h}(q_1,q_2,\dots,q_n;t). Then, the derivative of the variation is given by \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 \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 is equivalent to the set of requirements Q_i = \mathbf{F} \cdot \frac{\partial \mathbf{h}}{\partial q_i} = 0, \quad i=1, \ldots, n. The terms are called the generalized forces associated with the virtual displacement . \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 F i , i = 1, 2, ..., m and M j , j = 1, 2, ..., n are the applied forces and applied moments, respectively, and δr i , i = 1, 2, ..., m and δφ j , 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 q k , k = 1, 2, ..., f are defined such that the virtual movements can be expressed in terms of these generalized coordinates. That is, \delta \mathbf{r}_i (q_1, q_2, \dots, q_f; t), \quad i = 1, 2, \dots, m ; \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: \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 Q k are defined as 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 = \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 F i and moments M j vanishes if it is in equilibrium. Therefore, the generalized forces Q k are zero, that is \delta W=0 \quad \Rightarrow \quad Q_k = 0 \quad k =1, 2, \dots, f . 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 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 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 Q = a F_A - b F_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. \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 Q = 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. ===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 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 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}}. 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} \boldsymbol{\omega} \cdot [I_R] \boldsymbol{\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 \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), 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 + \dots + (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, known as Lagrange's equations or the generalized equations of motion. If the generalized forces Q j are derivable from a potential energy V(q 1 ,...,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, , 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 the Euler-Lagrange equations for a system with m degrees of freedom, or Lagrange's equations of the second kind. ==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 : This shows external surface forces T, body forces f, and internal stresses \boldsymbol{\sigma} in equilibrium. The \boldsymbol{\epsilon} -State : 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 \left( u^* + \frac{ \partial u^*}{\partial x} dx \right ) - F_A u^* \approx \frac{ \partial u^* }{\partial x} \sigma dV + u^* \frac{ \partial \sigma }{\partial x} dV = \epsilon^* \sigma dV - u^* f dV 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: where the total external virtual work is done by T and f. Thus, The right-hand-side of ( , ) 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 \mathbf T dS = \int_{S} \mathbf u \cdot \boldsymbol \sigma \cdot \mathbf 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{align} \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 \left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j}\right) dV \end{align} To continue our derivation, we substitute in the equilibrium equation \frac{\partial \sigma_{ij}}{\partial x_j} + f_i = 0 . Then \int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j}\right) dV = \int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows: \begin{align} \int_V\left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV &= \int_V\left( \frac12 \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 \right) dV \\ &= \int_V \left( \left[ \epsilon_{ij} + \frac12 \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i\right) dV \\ &= \int_V\left( \epsilon_{ij} \sigma_{ij} - u_i f_i \right) dV\\ &= \int_V \left( \boldsymbol\epsilon : \boldsymbol\sigma - \mathbf u \cdot \mathbf f \right) dV \end{align} 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: 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, ( ) 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 ( ) and ( ). 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.( ) 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: 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 D'Alembert's principle ==References== ==Bibliography== Bathe, K.J. "Finite Element Procedures", Prentice Hall, 1996. Charlton, T.M. Energy Principles in Theory of Structures, Oxford University Press, 1973. Dym, C. L. and I. H. Shames, Solid Mechanics: A Variational Approach, McGraw-Hill, 1973. Greenwood, Donald T. Classical Dynamics, Dover Publications Inc., 1977, Hu, H. Variational Principles of Theory of Elasticity With Applications, Taylor & Francis, 1984. Langhaar, H. L. Energy Methods in Applied Mechanics, Krieger, 1989. Reddy, J.N. Energy Principles and Variational Methods in Applied Mechanics, John Wiley, 2002. Shames, I. H. and Dym, C. L. Energy and Finite Element Methods in Structural Mechanics, Taylor & Francis, 1995, Tauchert, T.R. Energy Principles in Structural Mechanics, McGraw-Hill, 1974. Washizu, K. Variational Methods in Elasticity and Plasticity, Pergamon Pr, 1982. Wunderlich, W. Mechanics of Structures: Variational and Computational Methods, CRC, 2002.

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