Virtual work, principle

Simo Vu-Quoc (1986), which is another way of writing the virtual work principle,... 

Let B be the virtual angular displacement of the plastic mechanism (Fig. 8.6). In this case the virtual work principle gives the following plastic equilibrium equation ... 

We propose a simple method for the linearization of the equations, which are established in our case, based on the virtual work principle. The kinematic relations between the interconnected bodies are represented by the recursive equations. Under the small deformation assumption, the system generalized variables used in the equations are the relative joint coordinates at the connections and the deformation modal coordinates of the flexible bodies. In the linearization process, the differentiation of the kinematic terms with respect to the generalized variables must be performed. In our method, these partial derivatives are attained through the first and second order time differentiations of the body absolute angular velocities and through the first, second, and third order time differentiations of the mass center coordinates. This is the essential idea behind our method. The partial differentiation of the mechanical terms, for example, of the inertial tensors will also be presented. We have developed specific operators to perform the time differentiations. This method makes both the theoretical formulation and the programming task relatively simple, and allows fast computation. 

The virtual work principle [4, 5] states that the sum of the virtual works of elastic forces, active forces and inertia forces is equal to zero, all of which are acting on the system. Note that the non-working interconnected constraint forces between the bodies are eliminated when using this principle. The principle can be expressed as ... 

We have proposed a linearization procedure for the nonlinear dynamic equations which have been established in our case based on the virtual work principle. We have shown that the partial differentiation of the kinematic terms with respect to the generalized coordinates and velocities could be fulfilled through the first and second order time differentiation of the body absolute angular velocities and through the first, second, and third order time differentiation of the mass center vectors. The advantage of our method lies in the... 

That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

As before, we obtain the expression for the mass law by application of the principle of virtual work to the system in equilibrium. If sc) be the concentration of hydroxyl ions corresponding to a concentration X of hydrogen, viz. ... 

The generalized solution of this problem is found by the variational method of virtual displacements.140 According to this principle, the sum of the virtual work done in traversing possible displacement path is equal to zero ... 

A quasistatic finite strain analysis is based on the rate form of the principle of virtual work,... 

In principle, eqn (11.59) is all that we need to go about computing the interaction force between an obstacle and a dislocation. However, we will find it convenient to rewrite this equation in a more transparent form for the present discussion. Using the principle of virtual work, we may rewrite the interaction energy as... 

This is known as the chemical application of the Principle of Virtual Work For an account of Virtual Work m Mechanics, for example, see A, W Porter s book, Intermediate Mechanics... 

Note on Virtual Work —It is essential to be clear as to the sign of each virtual work term in the correct application of the principle It is therefore necessary to be able to visualise as far as possible the mechanism of the process A work term must be reckoned as essentially positive if the force (which does the work) assists the motion of the mass moved, or is m the same direction as the motion When... 

According to Newton, equilibrium is established when the resultant forces on the masses, i.e., in the direction of the respective planes, are equal. There is another formulation of equilibrium, however, due to the principle of virtual work. Equilibrium is established in this way, if a small, allowed displacement of the masses will not cause a change in energy. 

In order to move the atoms depicted in Fig. 2.3, work has to be performed on the system. For example, if one atom is moved by an amount 8m, the work done is / 8m, using the principle of virtual work. In doing this work, the bond energy has changed by an amount A=(f) u+bu)—(u). As indicated earlier, the equilibrium condition for the deformation is given by f—dapproaches zero, A =fhu. Thus, for small displacements, the work done on the atoms is equivalent to the change in bond energy. It is now possible to define a... 

Fig. 7 Schematic diagram of the working principle of easy confocal microscopy using a virtual pinhole, by controlling the slit width and the acquisition pixel of the CCD detector.

The stress formula (3.134) can be put in the form equivalent to the principle of virtual work. Consider a virtual deformation which displaces the point on the material to Tq. -I- de firp in a very short time dt. In the limit of dt- 0, the velocity gradient x p = dE pldt becomes very large, so that the time evolution of V in the time interval 6t is dominated by ticpA... 

The actual formula for the stress tensor is complicated. However, a neat expression is obtained in the form of the principle of virtual work. By a straightforward calculation, it can be shown ... 

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