Dynamic equilibrium (engineering mechanics)

In technical mechanics, dynamic equilibrium is the equilibrium between external force and inertial force .

For a body with mass , Newton's second law reads : ${\displaystyle m}$

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

Here, the external force and the acceleration in the inertial system . Having the basic equation of mechanics on the form ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {a}}}$

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

the negative product of mass and acceleration is formally understood as a force, which is called the force of inertia or, more precisely , the force of inertia of D'Alembert . You get: ${\displaystyle {\vec {F}}_{T}}$

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

The dynamic problem is thus reduced to a static problem of the equilibrium of forces. The sum of external force and inertial force is therefore always zero. The d'Alembert inertial force is the result of the acceleration and not its cause.

The advantage of this procedure is that the description takes place uniformly in an inertial system and no further reference systems have to be introduced. For many applications in technical mechanics, a fixed reference system with sufficient accuracy is an inertial system.

application

Motorbike in stationary cornering

In practice, one can take advantage of the fact that inertial force and external force often act at different points. An example is the calculation of the lean angle of a motorcycle when cornering in a stationary position . The weight force in the center of gravity and the tire forces in the wheel contact point act as external forces on the motorcycle. The tire forces are the lateral force radial to the center of the curve and the wheel load vertical (both not shown).

The amount of centripetal force or centrifugal force is calculated from the path speed and the radius of curvature of the path : ${\displaystyle v}$ ${\displaystyle R}$

${\displaystyle \left|F_{\text{Zf}}\right|=m\;{\frac {v^{2}}{R}}}$

If you choose the wheel contact point as the reference point for the moment equilibrium, the resulting force from centrifugal force and weight must go through the wheel contact point if the motorcycle is not to tip over. The tire forces that exert the centripetal force do not need to be taken into account in the moment equilibrium, since they do not have a lever arm with regard to the reference point (wheel contact point) and therefore do not generate any moment. For the inclined position results ${\displaystyle F_{R}}$ ${\displaystyle F_{\text{Zf}}}$ ${\displaystyle F_{G}}$ ${\displaystyle \alpha }$

${\displaystyle \tan {\alpha }={\frac {F_{\text{Zf}}}{F_{G}}}={\frac {v^{2}}{R\;g}}={\frac {a_{y}}{g}}}$

with the acceleration due to gravity , and the radial acceleration . ${\displaystyle g}$ ${\displaystyle a_{y}}$

Individual evidence

1. ^ Alfred Böge, Wolfgang Böge, Klaus-Dieter Arnd a. a .: Handbook of Mechanical Engineering: Fundamentals and Applications of Mechanical Engineering, Hardcover - 22nd edition . Springer Verlag, 2014, ISBN 978-3-658-06597-3 ( limited preview in the Google book search).
2. ↑ Dietmar Gross, Werner Hauger, Jarg Schrader, Wolfgang A. Wall: Technical Mechanics . 10th edition. tape 3 kinetics. Gabler Wissenschaftsverlage, 2008, p. 191 ( limited preview in Google Book Search): "We are writing now and take the negative product of the mass and acceleration formally as a force to which we [...] D'Alembert inertial force call: . This force is not a force in the Newtonian sense, since there is no counterforce to it (it violates the axiom actio = reactio!); we therefore call it a pseudo-force. " ${\displaystyle F-ma=0}$ ${\displaystyle m}$ ${\displaystyle a}$ ${\displaystyle F_{T}}$ ${\displaystyle F_{T}=-ma}$
3. ^ Gerhard Knappstein: Kinematik und Kinetik . Harri Deutsch Verlag, 2004, ISBN 3-8171-1738-8 , pp. 68 ff . ( limited preview in Google Book search).
4. ↑ Cornelius Lanczos: The Variational Principles of Mechanics . Courier Dover Publications, New York 1986, ISBN 0-486-65067-7 , pp. 88-110 . ( limited preview in Google Book Search): "We now define a vector I by the equation I = -m A. This vector I can be considered as a force created by the motion. We call it the "force of inertia". With this concept the equation of Newton can be formulated as follows: F + I = 0. "