Virtual Work Principle

Virtual work principle

The principle stating that the total virtual work done by all the forces acting on a system in static equilibrium is zero for a set of infinitesimal virtual displacements from equilibrium. The infinitesimal displacements are called virtual because they need not be obtained by a displacement that actually occurs in the system. The virtual work is the work done by the virtual displacements, which can be arbitrary, provided they are consistent with the constraints of the system. See Constraint

The principle of virtual work is equivalent to the conditions for static equilibrium of a rigid body expressed in terms of the total forces and torques. That is, the principle of virtual work can be derived from these conditions, and conversely. See Equilibrium of forces, Statics

One advantage of the principle of virtual work is that it can serve as a basis for all of statics. In the solution of problems the principle of virtual work is often useful for eliminating the need for consideration of the forces of constraint, since these forces often are perpendicular to the virtual displacements and consequently do no work.

Virtual Work Principle

one of the variational principles of mechanics determining the general condition of equilibrium of a mechanical system. According to the virtual work principle, for the equilibrium of a mechanical system with ideal constraints it is necessary and sufficient that the sum of works δA_i due to all active forces applied to the system in any virtual displacement be equal to zero. Mathematically, the virtual work principle is expressed by the equation

(1) Σδ·A_i≡ΣF_iδs_i cos α=0

where the F_i are the operating active forces, the δs_i are the magnitudes of virtual displacements at the points of application of these forces, and the α_i are the angles between the directions of forces and the virtual displacements. For systems with several degrees of freedom, equation (1) must be formulated separately for each independent dis-placement.

Thus, the virtual work principle makes it possible to find the conditions of equilibrium of a system without introducing unknown constraint forces, which significantly simplifies the solution and expands the class of problems that can be solved. For example, it is easy to find the conditions of equilibrium of a lifting device whose parts are hidden in box K of Figure 1 using this principle. From equation (1) we obtain

Pδs_b—Qδ s_D=0

where P and Q are the acting active forces. For a definitive calculation it is necessary to determine the relation between displacements δs_B and δs_D. If a screw is raised by the

Figure 1

amount h by a single turn of handle AB, then this relation will be found from the proportion δs_B:δs_D = 2πa:h, where a is the length of the handle. Finally, equation (2) gives the following condition of equilibrium: P = Qh/2πa. This problem cannot, in general, be solved by the methods of geometrical statics (if the parts of the mechanism hidden in the box are unknown).

S. M. TARG

virtual work principle

[′vər·chə·wəl ¦wərk ‚prin·sə·pəl] (mechanics) principle of virtual work

单词	virtual work principle
释义	Virtual Work Principle Virtual work principle The principle stating that the total virtual work done by all the forces acting on a system in static equilibrium is zero for a set of infinitesimal virtual displacements from equilibrium. The infinitesimal displacements are called virtual because they need not be obtained by a displacement that actually occurs in the system. The virtual work is the work done by the virtual displacements, which can be arbitrary, provided they are consistent with the constraints of the system. See Constraint The principle of virtual work is equivalent to the conditions for static equilibrium of a rigid body expressed in terms of the total forces and torques. That is, the principle of virtual work can be derived from these conditions, and conversely. See Equilibrium of forces, Statics One advantage of the principle of virtual work is that it can serve as a basis for all of statics. In the solution of problems the principle of virtual work is often useful for eliminating the need for consideration of the forces of constraint, since these forces often are perpendicular to the virtual displacements and consequently do no work. Virtual Work Principle one of the variational principles of mechanics determining the general condition of equilibrium of a mechanical system. According to the virtual work principle, for the equilibrium of a mechanical system with ideal constraints it is necessary and sufficient that the sum of works δA_i due to all active forces applied to the system in any virtual displacement be equal to zero. Mathematically, the virtual work principle is expressed by the equation (1) Σδ·A_i≡ΣF_iδs_i cos α=0 where the F_i are the operating active forces, the δs_i are the magnitudes of virtual displacements at the points of application of these forces, and the α_i are the angles between the directions of forces and the virtual displacements. For systems with several degrees of freedom, equation (1) must be formulated separately for each independent dis-placement. Thus, the virtual work principle makes it possible to find the conditions of equilibrium of a system without introducing unknown constraint forces, which significantly simplifies the solution and expands the class of problems that can be solved. For example, it is easy to find the conditions of equilibrium of a lifting device whose parts are hidden in box K of Figure 1 using this principle. From equation (1) we obtain Pδs_b—Qδ s_D=0 where P and Q are the acting active forces. For a definitive calculation it is necessary to determine the relation between displacements δs_B and δs_D. If a screw is raised by the Figure 1 amount h by a single turn of handle AB, then this relation will be found from the proportion δs_B:δs_D = 2πa:h, where a is the length of the handle. Finally, equation (2) gives the following condition of equilibrium: P = Qh/2πa. This problem cannot, in general, be solved by the methods of geometrical statics (if the parts of the mechanism hidden in the box are unknown). S. M. TARG virtual work principle [′vər·chə·wəl ¦wərk ‚prin·sə·pəl] (mechanics) principle of virtual work
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