D'Alembert's principle

The d'Alembert principle (after Jean-Baptiste le Rond d'Alembert ) of classical mechanics allows the establishment of the equations of motion of a mechanical system with constraints . The principle is based on the principle that the constraining forces or moments in a mechanical system do not perform any virtual work .

The name “d'Alembert's principle” is used by some authors for the dynamic equilibrium between external force and d'Alembert's inertial force, while other authors reject it with violent words as an inadmissible shortening.

Preliminary considerations

The equation of motion for a mass point is formulated in an inertial system. According to Newton's second law, it reads :

${\displaystyle m{\vec {a}}={\vec {F}}}$

This contains the mass, the absolute acceleration and the external force. This basic equation of mechanics can be expressed in the form: ${\displaystyle m}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {F}}}$

${\displaystyle {\vec {F}}-m{\vec {a}}={\vec {0}}}$

to be brought. The term is interpreted as a force and referred to as the d'Alembert inertial force . ${\displaystyle -m{\vec {a}}}$ ${\displaystyle {\vec {F}}_{T}}$

${\displaystyle {\vec {F}}+{\vec {F}}_{T}={\vec {0}}}$

The dynamic problem is due to an equilibrium problem in statics. The relationship is therefore also referred to as dynamic equilibrium . A problem of dynamics can thus also be dealt with with methods of statics, if inertial forces are taken into account. With the d'Alembert principle, the principle of virtual work is used in the following, which can be used in statics to calculate unknown bearing forces.

introduction

For a system of N mass points which is subject to constraints, the equation of motion for the mass is i

${\displaystyle m_{i}{\ddot {\vec {r}}}_{i}={\vec {F}}_{i}}$.

Here, the resulting external force to the mass point i . It is the sum of the applied force and constraining force . ${\displaystyle {\vec {F}}_{i}}$ ${\displaystyle {\vec {F_{i}^{e}}}}$ ${\displaystyle {\vec {F_{i}^{z}}}}$

${\displaystyle {\vec {F}}_{i}={\vec {F_{i}^{e}}}+{\vec {F_{i}^{z}}}\;.}$

Inserted into Newton's equation of motion:

${\displaystyle m_{i}{\ddot {\vec {r}}}_{i}={\vec {F_{i}^{e}}}+{\vec {F_{i}^{z}}}\;.}$

The constraining force is thus calculated

${\displaystyle {\vec {F}}_{i}^{z}=m_{i}{\ddot {\vec {r}}}_{i}-{\vec {F}}_{i}^{e}\;.}$

The scalar product of the constraining forces is formed with the virtual displacements . If, according to the principle of virtual work, the constraining forces do not perform any virtual work, the sum of the scalar products of constraining forces and virtual displacements vanishes: ${\displaystyle \delta {\vec {r}}_{i}}$

${\displaystyle \sum _{i=1}^{N}\left({\vec {F}}_{i}^{z}\cdot \delta {\vec {r}}_{i}\right)=0\;.}$

One obtains the d'Alembert principle (in Lagrange's formulation ):

${\displaystyle {\sum _{i=1}^{N}\left(m_{i}{\ddot {\vec {r}}}_{i}-{\vec {F}}_{i}^{e}\right)\cdot \delta {\vec {r}}_{i}=0}}$

The constraining forces no longer appear in the equation - only the applied forces. The constraints are still hidden in the virtual displacements, because only those are allowed that are compatible with the constraints.

In order to obtain equations of motion from this, one goes over to independent coordinates (degrees of freedom) with ( holonomic ) constraints and expresses position, speed, acceleration and virtual displacements of the N masses using these new position coordinates (" generalized coordinates "). ${\displaystyle \,k}$ ${\displaystyle f=3N-k}$ ${\displaystyle q=(q_{1}(t),...,q_{f}(t))}$

${\displaystyle {\vec {r}}_{i}={\vec {r}}_{i}(q)\;,\quad {\dot {\vec {r}}}_{i}={\dot {\vec {r}}}_{i}(q,{\dot {q}})\;,\quad {\ddot {\vec {r}}}_{i}={\ddot {\vec {r}}}_{i}(q,{\dot {q}},{\ddot {q}})\;,\quad \delta {{\vec {r}}_{i}}=\sum _{j=1}^{f}{\frac {\partial {\vec {r}}_{i}}{\partial q_{j}}}\,\delta q_{j}\,.}$

Since the new coordinates can be varied independently, there are differential equations of the second order that can be solved for. The concrete procedure for setting up the equations of motion can be found in the next section. ${\displaystyle f}$ ${\displaystyle {\ddot {q}}}$

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

Occasionally, the simple conversion of Newton's equation of motion described at the beginning is referred to as the d'Alembert principle. However, this overlooks important consequences such as the elimination of coercive forces that do not perform virtual work and, in the words of Georg Hamel, is almost an insult to d'Alembert . It should also be noted that the principle of virtual work used does not follow from Newton's axioms , but represents its own basic postulate.

Extension to multi-body systems

In the general case of multi-body systems , it is taken into account that the virtual work of the constraining torques on the virtual rotations also disappears. Euler's equation is used to calculate the constraining moments .

${\displaystyle {\sum _{i=1}^{N}\left(\left[m_{i}{\ddot {\vec {r}}}_{i}-{\vec {F}}_{i}^{e}\right]\delta {\vec {r}}_{i}^{\,T}+\left[I_{i}\,{\dot {\vec {\omega }}}_{i}+{\vec {\omega }}_{i}\times I_{i}\,{\vec {\omega }}_{i}-{\vec {M}}_{i}^{e}\right]\delta {\vec {\varphi }}_{i}^{\,T}\right)=0}.}$

With

${\displaystyle I_{i}}$ Inertial tensor of the body i

${\displaystyle {\dot {\vec {\omega }}}_{i}}$ Angular acceleration of the body i

${\displaystyle {\vec {\omega }}_{i}}$ Angular velocity of the body i

${\displaystyle {\vec {M}}_{i}^{e}}$ impressed moment on the body i

${\displaystyle \delta {\vec {\varphi }}_{i}}$ virtual rotation of the body i.

With N bodies and k bonds there are degrees of freedom. ${\displaystyle f=6\,N-k}$

The virtual shifts or rotations are obtained from the partial derivatives of the translational or rotational position coordinates according to the generalized coordinates:

${\displaystyle \delta {{\vec {r}}_{i}}=\sum _{j=1}^{f}{\frac {\partial {\vec {r}}_{i}}{\partial q_{j}}}\,\delta q_{j}\,}$

${\displaystyle \delta {{\vec {\varphi }}_{i}}=\sum _{j=1}^{f}{\frac {\partial {\vec {\varphi }}_{i}}{\partial q_{j}}}\,\delta q_{j}}$

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

${\displaystyle {\ddot {\vec {r}}}_{i}=\sum _{j=1}^{f}{\frac {\partial {\vec {r}}_{i}}{\partial q_{j}}}\,{\ddot {q}}_{j}+{\vec {a}}_{i}^{\,*}}$ and

${\displaystyle {\dot {\vec {\omega }}}_{i}=\sum _{j=1}^{f}{\frac {\partial {\vec {\varphi }}_{i}}{\partial q_{j}}}\,{\ddot {q}}_{j}+{\vec {\alpha }}_{i}^{\,*}}$.

This allows the system of differential equations of the second order to be represented in matrix form.

${\displaystyle \mathbf {M} \,{\ddot {\vec {q}}}={\vec {F}}^{*}+{\vec {M}}^{*}}$

There are:

${\displaystyle \mathbf {M} }$ the f × f mass matrix

${\displaystyle {\vec {F}}^{*}}$ the vector of generalized forces

${\displaystyle {\vec {M}}^{*}}$ the vector of generalized moments

The elements of the mass matrix are calculated as follows:

${\displaystyle M_{m,n}=\sum _{i=1}^{N}\left(m_{i}\,{\frac {\partial {\vec {r}}_{i}^{\,T}}{\partial q_{m}}}\cdot {\frac {\partial {\vec {r}}_{i}}{\partial q_{n}}}+{\frac {\partial {\vec {\varphi }}_{i}^{\,T}}{\partial q_{m}}}\cdot I_{i}\cdot {\frac {\partial {\vec {\varphi }}_{i}}{\partial q_{n}}}\right)}$

The following results for the components of generalized forces or moments:

${\displaystyle F_{m}^{*}=\sum _{i=1}^{N}\left({\frac {\partial {\vec {r}}_{i}^{\,T}}{\partial q_{m}}}\left[{\vec {F}}_{i}{^{e}}-m_{i}\,{\vec {a}}_{i}^{\,*}\right]\right)}$

${\displaystyle M_{m}^{*}=\sum _{i=1}^{N}\left({\frac {\partial {\vec {\varphi }}_{i}^{\,T}}{\partial q_{m}}}\left[{\vec {M}}_{i}{^{e}}-I_{i}\,{\vec {\alpha }}_{i}^{\,*}-{\vec {\omega }}_{i}\times I_{i}\,{\vec {\omega }}_{i}\right]\right)}$

The calculation of the mass matrix as well as the generalized forces and moments can be carried out numerically in the computer. The differential equation system can also be solved numerically with common programs. The treatment of large multi-body systems with kinematic bonds is only possible in this way.

Example of a thread pendulum

Thread pendulum: is the deflection from the equilibrium position and generalized coordinate
${\displaystyle \varphi }$

In the case of a flat thread pendulum with the mass , the angle at which the thread is deflected from the rest position is chosen as the degree of freedom. The constant thread length represents a holonomic constraint. Position, speed and acceleration of the mass can therefore be expressed as a function of this angle: ${\displaystyle m}$ ${\displaystyle \varphi }$ ${\displaystyle l}$

${\displaystyle {\vec {r}}={\begin{bmatrix}l\sin {\varphi }\\-l\cos {\varphi }\end{bmatrix}}}$

${\displaystyle {\dot {\vec {r}}}={\frac {\partial {\vec {r}}}{\partial \varphi }}\,{\dot {\varphi }}={\begin{bmatrix}l\cos {\varphi }\\l\sin {\varphi }\end{bmatrix}}{\dot {\varphi }}}$

${\displaystyle {\ddot {\vec {r}}}={\begin{bmatrix}l\cos {\varphi }\\l\sin {\varphi }\end{bmatrix}}{\ddot {\varphi }}+{\begin{bmatrix}-l\sin {\varphi }\\l\cos {\varphi }\end{bmatrix}}{\dot {\varphi }}^{2}}$

The virtual shift results from:

${\displaystyle \delta {\vec {r}}={\frac {\partial {\vec {r}}}{\partial \varphi }}\,\delta \varphi ={\begin{bmatrix}l\cos {\varphi }\\l\sin {\varphi }\end{bmatrix}}\delta \varphi }$

The force of weight acts as the applied force:

${\displaystyle {\vec {G}}={\begin{bmatrix}0\\-m\,g\end{bmatrix}}}$

The equation of motion results from the condition that the virtual work of the constraining forces vanishes.

${\displaystyle \left(m{\ddot {\vec {r}}}-{\vec {G}}\right)\cdot \delta {\vec {r}}=0\Rightarrow \left(m{\ddot {\vec {r}}}-{\vec {G}}\right)\cdot {\frac {\partial {\vec {r}}}{\partial \varphi }}=0.}$

By evaluating the scalar products one finally obtains:

${\displaystyle m\,l^{2}\,{\ddot {\varphi }}=-m\,g\,l\,\sin \varphi }$

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

${\displaystyle {\ddot {\varphi }}=-{\frac {g}{l}}\,\sin \varphi }$

receives.

The procedure appears to be very cumbersome in this simple example. However, since only scalar products have to be evaluated, this can be automated in large systems and carried out numerically in the computer. This makes it much easier to set up equations of motion.

literature

• Herbert Goldstein , Charles P. Poole, John L. Safko: Classical Mechanics. VCH.
• Friedhelm Kuypers: Classic mechanics. VCH, 5th edition 1997, ISBN 3-527-29269-1 .
• Georg Hamel : Theoretical Mechanics. Springer 1967.
• Werner Schiehlen: Technical Dynamics. Teubner study books, Stuttgart, 1986.
• Craig Fraser : D'Alembert's Principle: The Original Formulation and Application in Jean D'Alembert's Traité de Dynamique (1743) , Part 1,2, Centaurus, Volume 28, 1985, pp. 31-61, 145-159

Remarks

1. ↑ Infinitesimal displacements are called virtual if they are compatible with the constraints. They should also take place immediately (or instantaneously , at a fixed time).

Individual evidence

1. ↑ ^{a b} Jürgen Dankert, Helga Dankert: Technical mechanics: statics, strength theory, kinematics / kinetics . 7th edition. Springer Vieweg, 2013, ISBN 978-3-8348-2235-2 ( limited preview in Google book search).
2. ↑ Klaus-Peter Schnelle: Simulation models for the driving dynamics of passenger cars taking into account the non-linear chassis kinematics. VDI-Verlag, Düsseldorf 1990, ISBN 3-18-144612-2 . (Progress reports VDI No. 146), p. 73
3. ↑ Script TU Berlin, PDF 120 kB ( Memento from March 5, 2016 in the Internet Archive )
4. ↑ Hans J. Paus: Physics in experiments and examples. Hanser 2007, p. 34.
5. ↑ Istvan Szabo: History of Mechanical Principles. Springer-Verlag, 1987, p. 40.
6. ↑ Cornelius Lanczos: The Variational Principles of Mechanics . Courier Dover Publications, New York 1986, ISBN 0-486-65067-7 , pp. 88–110 ( limited preview in the Google book search): "The addition of the force of inertia I to the acting force F changes the problem of motion to problem of equilibrium."
7. ↑ István Szabó: History of Mechanical Principles and Their Main Applications . Springer DE, 1987, ISBN 978-3-7643-1735-5 , p. 39– (accessed February 8, 2013).
8. ↑ Kurt Magnus, HH Müller-Slany: Fundamentals of technical mechanics . 7th edition. Vieweg + Teubner, 2005, ISBN 3-8351-0007-6 , pp. 258 ( limited preview in Google Book search).
9. ↑ Hamel Theoretical Mechanics , Springer 1967, p. 220