Mass and Area Moments of Inertia
ENGN 330—Winter 2014
Moments of Inertia
The figures and tables below summarize mass and area moments of inertia. Be careful not to
confuse the two!
Mass Moment of Inertia
The mass moment of inertia describes the rotational inertia of a body subject to torque. This is
the J in Newton’s Law:
X
τi = J θ̈
i
Figure 1 summarizes some of the most commonly encountered geometries1 . Note that they use
I instead of J in the image, which is a common convention throughout many physics courses.
However, do NOT confuse the mass moment of inertia with the area moment of inertia below!
Figure 1: Mass moment of inertia for common geometries.
1
Image: http://theconstructor.org/structural-engg/moment-of-inertia/2825/
1
1
Equivalent Springs of Beams and Such
Support columns, beams, etc. act as springs—they supply a restoring force. The springiness
(stiffness) depends on the geometry of the object, how the beams are supported, etc. Figures 2 and
3 below summarize effective spring constants for commonly encountered situations2 . Note that all
of these formulas contain a material property, E or G (Young’s modulus). The geometrical part
of the equation is hidden in the area moment of inertia I (see next section). The constant out
in front is specified by the geometry.
Figure 2: Equivalent spring constants for common scenarios.
2
Image credit: Mechanical Vibrations, SS. Rao, 5th ed., Pearson Ed. Publishing
2
Figure 3: More equivalent spring constants for common scenarios.
3
Area Moment of Inertia
Why are these important? Because so much of vibrations is figuring out what the spring constant
of a system is, and of course the geometry of the system defines the stiffness of the “spring” which
relies on it’s geometry. Here are two commonly encountered scenarios3 .
1. Rectangular Beam: I =
1
3
12 bh
Figure 4: Rectangular cross section of height h and width b
2. Cylindrical beam: I =
π
4
r24 − r14
Figure 5: Hollow cylindrical cross-section of inner radius r1 and outer radius r2 . Note: If the
cylinder is solid, then r1 = 0.
3
Image credit: http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia
4