Virtual work

Virtual work is a concept of analytical mechanics or technical mechanics and describes both the work that a force does on a system during a virtual displacement and the work that a virtual force does on a real displacement. A virtual shift is understood to be a change in the shape or position of the system that is compatible with the bonds (e.g. bearings) and "instantaneous", but otherwise arbitrary and also infinitesimally small. The principle of virtual work results from the principle of virtual performance and is also used to calculate the equilibrium in statics and to set up equations of motion ( d'Alembert's principle ).

description

Virtual displacement, virtual work

In the following an N-particle system is considered which is restricted by constraints .

A virtual displacement is a fictitious infinitesimal displacement of the -th particle that is compatible with the constraints. The dependence on time is not considered. ${\displaystyle \delta \mathbf {x} _{i}}$ ${\displaystyle i}$

The holonomic constraints,,   are fulfilled by using so-called generalized coordinates : ${\displaystyle s}$ ${\displaystyle f_{l}\,(\mathbf {x} _{1},\dots ,\mathbf {x} _{N},\,t)=0\,\,,\quad l=1,\dots ,s\,\,}$ ${\displaystyle n=3N-s}$ ${\displaystyle \,q_{k}}$

${\displaystyle \delta \mathbf {x} _{i}=\sum _{k=1}^{n}{\frac {\partial \mathbf {x} _{i}}{\partial q_{k}}}\delta q_{k}}$

(The holonomic constraints are therefore explicitly eliminated by selecting and reducing the generalized coordinates accordingly.)

To fulfill the anholonomic constraints, the   other conditions are subject to , e.g. B. differential non-integrable equations: ${\displaystyle \delta q_{k}}$ ${\displaystyle r}$

${\displaystyle \,\sum _{k}a_{k}^{(l)}\delta q_{k}=0\ ,\quad l=1,\ldots ,r}$

The virtual work that the force would do with virtual displacement on the -th particle is: ${\displaystyle \mathbf {F} _{i}}$ ${\displaystyle \delta \mathbf {x} _{i}}$ ${\displaystyle i}$

${\displaystyle \delta W_{i}=\mathbf {F} _{i}\cdot \delta \mathbf {x} _{i}}$

System in balance

If the -particle system is in equilibrium, the acceleration for each particle is zero: ${\displaystyle N}$

${\displaystyle {\ddot {\mathbf {x} }}_{i}=0}$

Therefore the resulting force on each particle must be zero:

${\displaystyle \mathbf {F} _{i}=m_{i}{\ddot {\mathbf {x} }}_{i}=0}$

If the system is in equilibrium, the virtual work of the force in the case of displacement is zero, since the force itself disappears: ${\displaystyle \mathbf {F} _{i}}$ ${\displaystyle \delta \mathbf {x} _{i}}$

${\displaystyle \delta W_{i}=\mathbf {F} _{i}\cdot \delta \mathbf {x} _{i}=0}$

Thus, the sum of the work done by all forces with virtual displacements is zero:

${\displaystyle \sum _{i=1}^{N}\mathbf {F} _{i}\cdot \delta \mathbf {x} _{i}=0}$

The resulting forces can be put together from the applied forces and constraining forces : ${\displaystyle \mathbf {F} _{i}}$ ${\displaystyle \mathbf {F} _{i}^{e}}$ ${\displaystyle \mathbf {F} _{i}^{z}}$

${\displaystyle \mathbf {F} _{i}=\mathbf {F} _{i}^{e}+\mathbf {F} _{i}^{z}}$

Inserted in the above relationship:

${\displaystyle \sum _{i=1}^{N}\mathbf {F} _{i}^{e}\cdot \delta \mathbf {x} _{i}+\sum _{i=1}^{N}\mathbf {F} _{i}^{z}\cdot \delta \mathbf {x} _{i}=0}$

Principle of virtual work

Usually the constraining force is perpendicular to the virtual displacement , so that applies. This is e.g. B. the case when movement is limited to curves or surfaces. ${\displaystyle \mathbf {F} _{i}^{z}}$ ${\displaystyle \delta \mathbf {x} _{i}}$ ${\displaystyle \mathbf {F} _{i}^{z}\cdot \delta \mathbf {x} _{i}=0}$

However, there are systems in which individual coercive forces do work . ${\displaystyle \mathbf {F} _{i}^{z}\cdot \delta \mathbf {x} _{i}\neq 0}$

The principle of virtual work now requires that the sum of all virtual work carried out by the constraining forces vanishes in a system in equilibrium:

${\displaystyle \sum _{i=1}^{N}\mathbf {F} _{i}^{z}\cdot \delta \mathbf {x} _{i}=0}$

For the impressed forces, the principle of virtual work means:

${\displaystyle \sum _{i=1}^{N}\mathbf {F} _{i}^{e}\cdot \delta \mathbf {x} _{i}=0}$

Note that the principle of virtual work is only an equilibrium principle of statics. The D'Alembert principle provides the extension to dynamics .

Principle of virtual work in conservative systems

In conservative systems, all applied forces can be derived from a potential : ${\displaystyle V}$

${\displaystyle \mathbf {F} _{i}^{e}=-\nabla _{\mathbf {x} _{i}}V=-{\frac {\partial V}{\partial \mathbf {x} _{i}}}}$

In this case, the principle of virtual work

${\displaystyle \sum _{i=1}^{N}\mathbf {F} _{i}^{e}\cdot \delta \mathbf {x} _{i}=-\sum _{i=1}^{N}{\frac {\partial V}{\partial \mathbf {x} _{i}}}\cdot \delta \mathbf {x} _{i}=0}$

in the shape

${\displaystyle \delta \,V=0}$

represent. The symbol is to be understood as a variation symbol in the sense of the calculation of variations . thus means the first variation of the potential energy . ${\displaystyle \delta }$ ${\displaystyle \delta V=0}$

example

Articulated angle lever, the virtual displacement is characterized by the angle of rotation δΦ.

At an angle lever, which is freely rotatably mounted on an axis, 2 applied forces and act . The virtual displacements of the force application points are and . The virtual work of the impressed forces is thus ${\displaystyle \mathbf {F} _{1}}$ ${\displaystyle \mathbf {F} _{2}}$ ${\displaystyle \delta \mathbf {x} _{1}}$ ${\displaystyle \delta \mathbf {x} _{2}}$

${\displaystyle \mathbf {F} _{1}\delta \mathbf {x} _{1}-\mathbf {F} _{2}\delta \mathbf {x} _{2}=0}$

Because the bell crank is considered to be rigid, the sizes and are not independent of each other. Their dependence can be expressed by varying the generalized coordinate : ${\displaystyle \delta \mathbf {x} _{1}}$ ${\displaystyle \delta \mathbf {x} _{2}}$ ${\displaystyle \delta \Phi }$ ${\displaystyle \Phi }$

${\displaystyle \delta \mathbf {x} _{1}=a_{1}\delta \Phi \quad {\text{und}}\quad \delta \mathbf {x} _{2}=a_{2}\delta \Phi }$

This makes virtual work:

${\displaystyle (\mathbf {F} _{1}a_{1}-\mathbf {F} _{2}a_{2})\delta \Phi =0}$

Since the equation holds for any , the expression in brackets must be identical to 0: ${\displaystyle \delta \Phi }$

${\displaystyle \mathbf {F} _{1}a_{1}=\mathbf {F} _{2}a_{2}}$

So the system remains in equilibrium, i.e. H. it does not tilt either to the right or to the left if the forces multiplied by their axis distance are equal.

Principle of virtual work for dynamic systems

→ Main article : D'Alembert's principle

The virtual work of the constraining forces or moments is zero in dynamic systems. If one expresses the virtual displacements in the generalized coordinates, equations of motion for large multibody systems can be set up using the principle of virtual work.

Alternatives

In addition to the principle of virtual work, the principle of virtual performance is also used. The main difference in this principle is that instead of virtual shifts, virtual speed variations are used here. This principle is seldom used in statics, but its extension to dynamic systems, the so-called Jourdain principle , proves to be advantageous, since nonholonomic bonds can be taken into account very elegantly .

Remarks

1. ↑ The  desired virtual change arises from the total differential of a function , i.e. an expression of the form . The term "instantaneous" is thereby mathematized. ${\displaystyle g(q_{1},\dots ,q_{n},t)}$ ${\displaystyle \mathrm {d} g=\sum _{i=1}^{n}{\frac {\partial g}{\partial q_{i}}}\,\mathrm {d} q_{i}+{\frac {\partial g}{\partial t}}\,\mathrm {d} t}$ ${\displaystyle \delta g=\sum _{i=1}^{n}{\frac {\partial g}{\partial q_{i}}}\,\delta q_{i}}$
2. ↑ The generalized coordinates can depend on the time, although again this is not included, since only the current value is required.

literature

• Herbert Goldstein , Charles P. Poole, John L. Safko: Classical Mechanics . 3. Edition. Wiley-VCH, 2006, ISBN 978-3-527-40589-3 .
• Danilo Capecchi: History of Virtual Work Laws. A History of Mechanics Prospective . Birkhäuser, 2012 Milan, ISBN 978-88-470-2055-9 .
• Karl-Eugen Kurrer : The History of the Theory of Structures. Searching for Equilibrium , Ernst and Sohn, Berlin 2018, pp. 27–31, pp. 476–481, pp. 811–814, pp. 821–824 and pp. 929–931, ISBN 978-3-433-03229- 9 .

Individual evidence

1. ^ Rolf Mahnken: Textbook of Technical Mechanics - Statics: Basics and Applications . Springer, ISBN 978-3-642-21710-4 ( limited preview in Google book search).