D'Alembert's principle

The d' Alembert principle allowed ( Jean Baptiste le Rond d' Alembert ) of classical mechanics the equations of motion of a mechanical system with constraints. The principle is based on the proposition that the constraint forces or moments afford virtual work.

Introduction

The equation of motion for a system of N point masses is given by Newton's second law as

By rearranging results in

The resultant external force on the mass point i It is the sum of the impressed force and coercive force.

Thus is the Newtonian equation of motion:

It forms the dot product with the virtual displacements [Note 1]. If the principle of virtual work, the constraint forces do no virtual work itself, the scalar product of coercive forces and virtual displacement vanishes. This gives the d' Alembert's principle ( in the formulation of Lagrange ):

In the equation, the constraint forces do no longer occur - only the active forces. The constraints still hiding in the virtual displacements, because there are only those permits that are consistent with the constraints.

To gain from the equations of motion, one goes in ( holonomic ) constraints on ( at N mass ) independent coordinates ( degrees of freedom ) over ( " generalized coordinates "). The virtual displacements are thereby expressed by these new position coordinates

This is inserted into the previous equation:

Since the new coordinates are chosen already taking into account the constraints, they can be independently varied (eg all shifts except one equal to zero). Therefore, in the above equation vanish all the coefficients of the individual, resulting in second order differential equations arise.

For holonomic constraints and conservative forces ( which can be derived from a potential function ) the d' Alembert principle is equivalent then to the Lagrange equations of the first kind ( see there).

Occasionally the outset reproduced simple conversion of the Newtonian equations of motion is already known as the d' Alembert's principle. But this overlooks the important conclusions (such as the elimination of coercive forces that make no virtual work ) and comes in the words of Georg Hamel almost an insult by d' Alembert same. It is also to note that the principle of virtual work used does not follow from Newton's axioms, but is a different basic postulate.

Extension to multi-body systems

In the general case of multi-body systems is considered that the virtual work of the constraint moments disappears on the virtual distortions. To calculate the forced moments Euler's equation is used.

With N bodies and k bonds to degrees of freedom. The virtual rotations is obtained analogously to the shifts of the partial derivatives with respect to the generalized coordinates:

The accelerations can be divided into a part which depends only on the second derivatives of the generalized coordinates, and a residual term disassemble:

This means that the second-order differential equation system in matrix form can be presented.

Where:

The elements of the mass matrix are calculated as follows:

For the components of the generalized forces or torques yields:

The calculation of the mass matrix and the generalized forces and moments can be performed numerically in the computer. The system of differential equations can also be solved numerically with popular programs. The treatment of large multibody systems with kinematic bonds is possible.

Example Pendulum

The flat pendulum connected to the ground, the angle at which the yarn is deflected from the rest position, is selected as a degree of freedom. The constant thread length represents a scleronomic constraint dar. position, velocity and acceleration of the mass can therefore be expressed as a function of this angle:

The virtual displacement is given by:

As an active force acts the weight:

The equation of motion follows from the condition that the virtual work of the constraint forces vanishes.

By evaluating the scalar products is finally obtained:

Mass and thread length can be shortened, so that you the well-known differential equation:

Receives.

The procedure appears to be very awkward at this simple example. However, since only scalar products to be evaluated, it may be automated in large-scale systems and can be numerically implemented in the computer. This will facilitate the preparation of the equations of motion considerably.