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Concept
Radius of gyration
Summary
Radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there.
Mathematically the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application. It is actually the perpendicular distance from point mass to the axis of rotation. One can represent a trajectory of a moving point as a body. Then radius of gyration can be used to characterize the typical distance travelled by this point.
Suppose a body consists of particles each of mass . Let be their perpendicular distances from the axis of rotation. Then, the moment of inertia of the body about the axis of rotation is
If all the masses are the same (), then the moment of inertia is .
Since ( being the total mass of the body),
From the above equations, we have
Radius of gyration is the root mean square distance of particles from axis formula
Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation. It is also known as a measure of the way in which the mass of a rotating rigid body is distributed about its axis of rotation.
In structural engineering, the two-dimensional radius of gyration is used to describe the distribution of cross sectional area in a column around its centroidal axis with the mass of the body. The radius of gyration is given by the following formula:
Where is the second moment of area and is the total cross-sectional area.
The gyration radius is useful in estimating the stiffness of a column. If the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment. For example, a column with an elliptical cross-section will tend to buckle in the direction of the smaller semiaxis.
Official source
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