Moment Integrals

Moment Integral
This first moment integral sums up the  moments across some area where both the magnitude of the force and the moment arm can change with position.

A moment integral is a general concept that can be applied in a number of ways. As the name implies the moment integral uses calculus to sum the moments of a force distributed over some area (where the force and the moment arm can change with position) or to sum the moments that would be needed to cause a uniform angular acceleration (where the mass and the distance from the rotational axis change with position). Some of the applications of moment integrals include:

  1. Finding the equivalent point load for a distributed force.
  2. Finding the centroid of a 2D or 3D shape.
  3. Finding the center of mass for a 2D or 3D shape.
  4. Finding the rectangular or polar "moment of inertia" for a shape or volume in order to determine the objects relationship between moments and angular accelerations.

When looking at moment integrals, there are number of different types of moment integrals. These types include:

Any combination of these different types is possible (for example a First, Rectangular, 2D, Area Moment of Inertia or a Second, Polar, 3D, Mass Moment of Inertia). However, only some combinations will have practical applications and will be discussed in detail on future pages.

2D vs. 3D Moment Integrals

Technically we can take the moment integral in any number of dimensions, but for practical purposes we will almost always be dealing with either 2D areas or 3D volumes. The number of dimensions will affect the complexity of the calculations (with 3D Moment integrals being more involved), and the nature of the problem (whether it is a 2D or 3D system) will determine the type of moment integral needed. Often this is not listed in the type of moment integral, requiring you to assume the type based on the context of the problem (meaning it may be written as a Second, Polar, Mass Moment of Inertia rather than a Second, Polar, 3D, Mass Moment of Inertia).

Area/Volume vs. Mass Moments Integrals

The next distinction in moments of inertia is between area/volume moment integrals and mass moment integrals. Area or volume moment integrals (used for 2D areas and 3D volumes respectively) are used to find centroids and find equivalent point loads for distributed forces. These are integrated over some area or volume, and assume each little bit of area or volume is equally weighted. Mass moment integrals, used for center of mass or angular acceleration calculations, are also usually integrated over the entire area or volume but take into account possible changes in density, recognizing that not all bits of the shape should be equally weighted. In cases of a uniform density part, the mass moment integral is simply the area/volume moment integral times the density of the part. Since it is usually easier to calculate the area/volume moment integrals, we will sometimes use the area/volume moment integrals to find the mass moment integrals for parts made of a single uniform material.

First vs. Second Moments Integrals

The first moment integral uses the distance from the center axis or point to calculate the moment. First moment integrals are useful in calculating the equivalent point load, the centroid, and the center of mass of an object, because on each of these we need to sum up the moments, which is equal to the force (F) times the moment arm (d).

The First Moment Integral
The first moment integral is useful in finding the total moment exerted by a force distributed over an area.

Second moments of inertia use the distance squared, and are useful in relating moments to angular accelerations.

Second Moment Intergral
The second moment integral is useful in determining the relationship between the moment and angular acceleration of a rigid body.

The figure above shows how the moment and angular acceleration of a mass on the end of a stick are related by the mass times the distance squared. For a rigid bodies we imagine them as an infinite number of point masses, each on the end of a massless stick leading back to the axis of rotation. The sum of all of these mass times distance squared terms relates the overall moment exerted on the body to the overall angular acceleration of the body.

Rectangular vs. Polar Moments Integrals

Finally we will talk about rectangular moments integrals versus polar moments integrals. This is a difference in how we define the distance in our moment integral. Let's start with the distinction in 2D. If our distance is measured from some axis (for example the x-axis, or the y-axis) then it is a rectangular moment integral. If on the other hand the distance is measured from some point (such as the origin) then it is a polar moment integral.

2D rectangular versus polar moment intergrals
In 2D, if we measure the distance from some axis (similar to what x and y do here) then we have a rectangular moment integral. If we measure the distance from some point (such as r does) then we have a polar moment integral.

In three dimensional problems, the definitions change slightly. For rectangular moment integrals the distance will be measured from some plane (such as the xy plane, xz plane, or yz plane). This will correspond to the x, y or z coordinates of a point. For a polar moment integrals the distance will be measured from some axis (such as the the x, y, or z axis).

3D rectangular versus polar moment integrals
In 3D, if we measure from some plane (like x and y do) then we have a rectangular moment integral. If we measure the distance from some axis (such as r does) then we have a polar moment integral.