Principle of virtual performance

The principle of virtual power , also Jourdain's principle according to Philip Jourdain , is used in classical mechanics to set up the equations of motion of mechanical systems with constraints . In contrast to the principle of virtual work , it can also be used if the speeds are included in the constraints.

Formal representation

For the sake of simplicity, the principle is only presented here for a system of  point masses . It is assumed that the locations and speeds , which are summarized in the following in matrices and , a constraint: ${\displaystyle n}$ ${\displaystyle {\vec {x}}_{1},\ldots ,{\vec {x}}_{n}}$ ${\displaystyle {\vec {v}}_{1}={\dot {{\vec {x}}_{1}}},\ldots ,{\vec {v}}_{n}={\dot {{\vec {x}}_{n}}}}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {v} }$

${\displaystyle f(t,\mathbf {x} ,\mathbf {v} )=0}$

suffice.

The mechanical system then moves in such a way that the virtual power balance for all virtual speeds compatible with the constraints ${\displaystyle \delta {\vec {v}}}$

${\displaystyle \delta P:=\sum _{k=1}^{n}(m_{k}{\dot {\vec {v}}}_{k}-{\vec {F}}_{\mathrm {e} k}(\mathbf {x} ,\mathbf {v} ))\cdot \delta {\vec {v}}_{k}=0}$

is fulfilled, where stands  for the impressed force acting on the  -th point mass (without constraining force ). ${\displaystyle {\vec {F}}_{\mathrm {e} k}}$ ${\displaystyle k}$

If the constraint is free of hidden constraints , the virtual speeds compatible with it are  described by the following equation: ${\displaystyle \delta \mathbf {v} }$

${\displaystyle \partial _{\mathbf {v} }f(t,\mathbf {x} ,\mathbf {v} )\cdot \delta \mathbf {v} =0}$

By reducing the geometric index of the algebro differential equation system

${\displaystyle {\begin{matrix}f(t,\mathbf {x} ,\mathbf {v} )&=&0\\\mathbf {v} &=&{\dot {\mathbf {x} }}\end{matrix}}}$

down to zero one can (normally) eliminate any hidden constraints that may occur.

Applications

The Jourdain principle is used, for example, when setting up the equations of motion for multi-body systems . For the rotational movements occurring there , the virtual angular velocities can be represented more simply than the virtual rotations.

The principle of virtual performance, which was only demonstrated here for a point mass system, is also applied in practice to mechanical systems with distributed parameters.

For example, the principle is used to partially discretize the equations of motion of flexible bodies. In this case, the approach space for the solutions of these equations is restricted to a finite-dimensional subspace . This restriction of the movement possibilities of the system is then interpreted as a constraint. Polynomial spaces or spaces of a finite selection are used as starting spaces for the problem of particularly interesting intrinsic movements of the elastic body.

literature

1. Jean-Claude Samin and Paul Fisette: Symbolic modeling of multibody systems. Kluwer Academic Press, 2003.