Euler force

In the classical mechanics which is Euler force (designated by Leonhard Euler ) acting on a body Scheinkraft , in a rotating reference system occurs when the axis of rotation or the rotational speed of change over time. The name was introduced by Cornelius Lanczos in his book The Variational Principles of Mechanics in 1949 , pointing out at the same time that there was no common name for this inertial force at the time.

General

The Euler force is given by:

{\ displaystyle {\ vec {F}} _ {\ mathrm {Euler}} = - m \ cdot {\ frac {d {\ vec {\ omega}}} {dt}} \ times {\ vec {r}} = -m \ cdot {\ vec {\ alpha}} \ times {\ vec {r}}}

With

the angular velocity of the reference system, ${\ displaystyle {\ vec {\ omega}}}$
the angular acceleration of the reference system, ${\ displaystyle {\ vec {\ alpha}}}$
the position vector of the point in the reference system. ${\ displaystyle {\ vec {r}}}$

The Euler acceleration (also azimuthal acceleration or transverse acceleration ) is caused by the angular acceleration of the reference system. In technical mechanics, it gives the dependent part of the guide acceleration that would be experienced by a point that is permanently connected to the reference system. It is caused by a non-uniformity of the rotational movement of the reference system . ${\ displaystyle {\ vec {a}} _ {\ mathrm {Euler}}}$ ${\ displaystyle {\ dot {\ vec {\ omega}}}}$

{\ displaystyle {\ vec {a}} _ {\ mathrm {Euler, TM}} = {\ vec {\ alpha}} \ times {\ vec {r}}}

The Euler force defined above is the associated inertial resistance. The term Euler acceleration is rarely used in physics.

Examples

Moving carousel

A person sitting on a horse in a children's carousel feels the Euler's force when starting off. It is the inertial force that pulls the person backwards from the horse when starting (and pushing them forward when stopping). The direction of the Euler force lies here in the plane of rotation perpendicular to the centrifugal force . In this example with a fixed direction of the axis of rotation, the Euler force is nothing else than the inertial force that a body opposes to any acceleration of its movement ( is the mass of the body and the acceleration of its orbital velocity). ${\ displaystyle -m {\ vec {a}}}$ ${\ displaystyle m}$ ${\ displaystyle {\ vec {a}}}$

If the person does not hold on to the jerky start, they will not feel any inertia in their own frame of reference, but will slide backwards off the horse. Seen from the outside, her position remains unchanged and the horse under her drives away. From a standpoint in the accelerated rotating reference system, however, the person appears to be accelerated backwards, which is interpreted as a consequence of the Euler force effective in the accelerated rotating reference system . ${\ displaystyle {\ vec {F}} _ {\ mathrm {Euler}}}$

Tilting the axis of rotation

If the axis of rotation inclines increasingly to one side while the speed of rotation of a carousel remains the same, the person traveling with it will always experience the Euler force most strongly when their position vector is perpendicular to the plane in which the inclination takes place. In addition to the curvature due to the rotary movement, your trajectory shows a maximum curvature away from the current trajectory plane at these points. In contrast, the Euler force is zero when the trajectory of the person goes through the plane in which the axis of rotation inclines. Then the increasing inclination of the axis corresponds only to a uniform speed perpendicular to the plane of the path.

supporting documents

↑ Jerrold E. Marsden, Tudor S. Ratiu: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems . Springer, 1999, ISBN 038798643X , p. 251.
↑ Lanczos: The variational principles of mechanics. University of Toronto Press 1949, p. 103: “This third apparent force has no universally accepted name. The author likes to call it the 'Euler force' in view of the outstanding investigations of Euler in this subject. "
↑ Ralf Greve: Continuum Mechanics . Gabler Wissenschaftsverlage, 2003, ISBN 978-3-540-00760-9 , p. 36 (accessed on May 11, 2012).
^ David Morin: Introduction to Classical Mechanics. With problems and solutions . Cambridge University Press, 2008, ISBN 0521876222 , p. 469.
^ Grant R. Fowles, George L. Cassiday: Analytical Mechanics . 6th edition. Harcourt College Publishers, 1999, p. 178 .
^ Richard H. Battin: An Introduction to the Mathematics and Methods of Astrodynamics . American Institute of Aeronautics and Astronautics, Reston, VA 1999, ISBN 1563473429 , p. 102.
↑ Note: Note that the analog Coriolis acceleration is defined in physics and technical mechanics with opposite signs.

[1] Jerrold E. Marsden, Tudor S. Ratiu: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems . Springer, 1999, ISBN 038798643X , p. 251.

[2] Lanczos: The variational principles of mechanics. University of Toronto Press 1949, p. 103: “This third apparent force has no universally accepted name. The author likes to call it the 'Euler force' in view of the outstanding investigations of Euler in this subject. "

[Greve2003-3] Ralf Greve: Continuum Mechanics . Gabler Wissenschaftsverlage, 2003, ISBN 978-3-540-00760-9 , p. 36 (accessed on May 11, 2012).

[Morin-4] David Morin: Introduction to Classical Mechanics. With problems and solutions . Cambridge University Press, 2008, ISBN 0521876222 , p. 469.

[Fowles-5] Grant R. Fowles, George L. Cassiday: Analytical Mechanics . 6th edition. Harcourt College Publishers, 1999, p. 178 .

[Battin-6] Richard H. Battin: An Introduction to the Mathematics and Methods of Astrodynamics . American Institute of Aeronautics and Astronautics, Reston, VA 1999, ISBN 1563473429 , p. 102.

[7] Note: Note that the analog Coriolis acceleration is defined in physics and technical mechanics with opposite signs.