Two Rectangles; Second Moments
- Author:
- Dr. Doug Davis, 3D
- Topic:
- Rotation
Background
This is similar to the Two Rectangles Centroid illustration. There are two rectangles defined by the points A,B and C,D. If the rectangles overlap the overlap rectangle is subtracted from the A,B rectangle.
In the following description I will use moment of inertia for area moment of inertia or second moment of area. The moment of inertia is useful in rotational dynamics and for calculating bending and twisting.
Similar to the centroid a table/spreadsheet is a good way to calculate the moment of inertia of a shape made up of simple shapes. The moments of inertia about the centroid of simple shapes is readily available. The moment of inertia is about an axis, the dot dash line represents the axes which are parallel to the x and y axes. Moving point AA will move the axes.
The mass moment of inertia of a thin flat uniform plate about an axis parallel to the axis is related to angular acceleration of the plate about that axis. The moment of inertia is indicated with a capital with a subscript indicating the axis. Mathematically, the moments about axes parallel to the and axes are:
where is the location of the axis parallel to the axis and is the value of each element.
and
where is the location of the axis parallel to the axis and is the value of each element.
Two other moments are the product of inertia, and the moment about an axis parallel to the axis through the point, the polar moment of inertia, . The product of inertia can be used to calculate moments about axes that are rotated. Because of the Pythagorean Theorem the polar moment of inertia is simply
Axis of Rotation
When talking about moment of inertia the axis of rotation should be specified. Below is an illustration of different axes of revolution. You can choose an axes parallel to any of the principle axes. The axis goes through the point which you can move. The three dimensional image then animates the thin plate rotating about the axis.
Note that parts of the shape further from the axis move faster than parts close to the axis. This is where the like terms in the integral originate.
Note that rotation about the axis, , involves terms, rotation about the axis, , involves terms and rotation about the axis, , involves terms.
Description
In the applet below you can move the corners of the rectangles to change there shape. If the rectangles overlap, the intersection is subtracted from the A,B rectangle instead of adding Area 2. The point AA controls the axis location. Axes parallel to the x and y axes are shown as dash-dot lines. The table shows the calculation results for moments about axes parallel to the principle axes and the product of inertia. The calculations are described below.
The orange cross indicates the combined centroid of the two rectangles. Notice how the moments vary as the point AA moves closer to the centroid.
Calculations
From tables of moments of inertia, the moment of inertia of a rectangle for an axis parallel to the -axis through it's centroid is I_{xx}= where is the rectangles height in direction and is the rectangles base width in the direction. The height and base of each rectangle are indicate by and in the applet. Similarly for an axis parallel to the axis the moment of inertia through the centroid is I_{yy}=.
To find the moment of inertia about an axis not through the centroid use the Parallel Axis Theorem which adds the distance from the centroid squared times the area to the moment of inertia. It is always the distance from the centroid. In the table d_x is the distance of the axis parallel to from the centroid and d_{y} is the distance of the centroid from the axis parallel to . In formula form, d_x= and d_{y}= where indicates the axis parallel to and indicates the axis parallel to and the subscript indicates the coordinate.
Application of the Parallel Axis Theorem then gives I_{yy}= and I_{xx}=. The Parallel Axis for Product of Inertia is similar as but since a rectangle is symmetric about the axes through the centroid, . This gives the Product of Inertia for a rectangle as I_{xy}=.
The polar moment of inertia, , through the point AA is the sum of the parallel moments of inertia, .
For composite shapes the moment of inertia through the same axis can be added. An area subtracted to make the composite shape would have a negative moment of inertia about it's centroidal axes. The calculation is the same with an inverted sign. It is an important point the the moments need to be through the same axis before they can be added.