Radius of gyration

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The radius of gyration is the distance from the given axis of rotation at which the point-like mass m of the body has to be attached in order to obtain the moment of inertia J. ${\ displaystyle i}$

{\ displaystyle {\ begin {aligned} J & = m \ cdot i ^ {2} \\\ Leftrightarrow i ^ {2} & = {\ frac {J} {m}} \\\ Leftrightarrow i & = {\ sqrt { \ frac {J} {m}}} \ end {aligned}}}

For a good material utilization z. B. in flywheels a radius of gyration is sought which is large compared to the outer dimension, d. H. is as far out as possible.

In strength theory, there is an analogous relationship between the area A and the moment of area of the second degree I:

{\ displaystyle {\ begin {aligned} I & = A \ cdot i ^ {2} \\\ Leftrightarrow i ^ {2} & = {\ frac {I} {A}} \\\ Leftrightarrow i & = {\ sqrt { \ frac {I} {A}}} \ end {aligned}}}

The radius of gyration is used here as a calculation variable in the verification of buckling loads .

Calculation of the radius of gyration with known geometry

Radius of gyration for a cylinder cross-section

Cylinder: , , ${\ displaystyle {r_ {i}} = 0}$ ${\ displaystyle i =?}$ ${\ displaystyle {r_ {a}} = known}$

{\ displaystyle {\ begin {aligned} A_ {1} & = A_ {2} \\\ Leftrightarrow {i ^ {2} * \ pi} & = {r_ {a}} ^ {2} * \ pi -i ^ {2} * \ pi \\\ Leftrightarrow i & = {\ sqrt {\ frac {{r_ {a}} ^ {2}} {2}}} \\\ end {aligned}}}

Radius of gyration for any cross section and central axis of rotation

Any cross-section: -> The radius of gyration can be calculated from this equation.
${\ displaystyle A_ {1} = A_ {2}}$

Polymer chemistry

In polymer chemistry , the mean square distance between the molecular chains and the center of gravity of the molecule is referred to as the scattering mass radius , sometimes also as the radius of gyration .

literature

Martin Mayr: Technical mechanics: statics - kinematics - kinetics - vibrations - strength theory . Part 3, ISBN 978-3-446-22608-1 , chap. 5 .

Individual evidence

^ Alfred Böge (Ed.): Vieweg manual mechanical engineering: Basics and applications of mechanical engineering . 18th edition. Vieweg, 2007, ISBN 978-3-8348-0110-4 ( limited preview in Google Book Search).
^ Friedrich R. Schwarzl: Polymer mechanics: Structure and mechanical behavior of polymers . Springer, 1990, ISBN 3-540-51965-3 ( limited preview in Google book search).

Web links

Strength of Materials (accessed October 18, 2019)
Area moments of inertia (accessed on October 18, 2019)
Faculty of Physics and Geosciences Basic physics internship (accessed October 18, 2019)
Load-bearing capacity of structural steel bars - non-linear load-bearing behavior, stability, verification methods (accessed on October 18, 2019)

[Boege-1] Alfred Böge (Ed.): Vieweg manual mechanical engineering: Basics and applications of mechanical engineering . 18th edition. Vieweg, 2007, ISBN 978-3-8348-0110-4 ( limited preview in Google Book Search).

[Schwarzl-2] Friedrich R. Schwarzl: Polymer mechanics: Structure and mechanical behavior of polymers . Springer, 1990, ISBN 3-540-51965-3 ( limited preview in Google book search).