Name: finding principal axes in area inertia
Uploaded: 2025-05-22T09:06:03.000Z
Duration: 1 min 14 s
Channel: JoVE

The moments and product of inertia of an area about inclined axes can be calculated using the coordinate transformation relations.
A set of axes corresponding to the maximum and minimum moments of inertia is called the principal axes.
Differentiating either moment of inertia about the inclined axis with respect to the inclination angle and equating the result to zero gives the orientation of the principal axes.
The roots of the resulting equation define two angles ninety degrees apart, specifying the orientation angle of the major and minor principal axes.
The moments of inertia about the inclined axes are rewritten by substituting the sine and cosine terms. The expression is further simplified to obtain the principal moments of inertia.
At these angles, the product of inertia is zero.
If an area possesses an axis of symmetry, then axis of symmetry is the principal axis. In contrast, the principal axes need not be the symmetric axis.
If the origin coincides with the centroid, the principal axes denote the principal centroidal axes.

principal moments of area

Principal Moments of Area

In mechanics, the product of inertia and moments of inertia of area help to calculate the stability and performance of various structures and components. The coordinate transformation relations are used to calculate the moments and products of inertia for an area about the inclined axes. Further, the moments and products of inertia with respect to the principal axes can be determined using the moments and products of inertia about the inclined axes.
The principal moment of inertia axes are the set of axes corresponding to the maximum and minimum values of the moments of inertia. The orientation of these axes is determined by differentiating the moment of inertia about the inclined axis with respect to the inclination angle and equating the result to zero.
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The roots of the above equation define two angles 90° apart from each other. These angles specify the orientation angle with respect to the principal axes.
Substituting these orientation angles into the product of inertia expression gives a zero value. The product of inertia with respect to the principal axis is always zero. Further, the moments of inertia about the inclined axes can be rewritten by incorporating the sine and cosine terms. The expression is further simplified to obtain the principal moments of inertia. The principal moments of inertia can be either the maximum or the minimum, depending on the sign of the expression.
If an area has an axis of symmetry passing through the origin, this is a principal axis of the area about the origin. However, a principal axis does not necessarily have to be an axis of symmetry. When the origin coincides with the centroid, the principal axes denote the principal centroidal axes.

In mechanics, the product of inertia and moments of inertia of area help to calculate the stability and performance of various structures and components. ...

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Mechanical Engineering

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Moment of Inertia

KEY TERMS AND CONCEPTS

finding principal axes in area inertia

Moments of inertia of area and the product of inertia are used in mechanics to describe how an area responds about different axes. They help with the stability and performance of structures and components. To study an inclined axis, coordinate transformation relations are used to calculate both the moment of inertia and the product of inertia for the area.

The principal axes are the special axes that give the maximum and minimum values of the moment of inertia. Their orientation is found by differentiating the moment of inertia for the inclined axis with respect to the angle of inclination and setting that result equal to zero. The equation gives two roots that are 90° apart. These angles show how the principal axes are oriented.

When those orientation angles are placed into the product of inertia expression, the result becomes zero. This means the product of inertia with respect to a principal axis is always zero. The moment of inertia about the inclined axes can also be written with sine and cosine terms. After simplification, this gives the principal moments of inertia, which may be the maximum or the minimum depending on the sign of the expression.

If an area has an axis of symmetry through the origin, that axis is a principal axis for the area about the origin. A principal axis does not have to be an axis of symmetry, though. When the origin is at the centroid, the principal axes are called principal centroidal axes.

Finding Principal Axes in Area Inertia

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key terms and concepts

Learn how inclined axes, product of inertia, and coordinate transforms lead to principal axes and principal moments of inertia in mechanics.

Finding Principal Axes in Area Inertia | JoVE K12

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Finding Principal Axes in Area Inertia

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