MOMENTS OF INERTIA

The moments of inertia are inertia properties that result from moment contributions about the axes of the reference frame. In terms of the inertia matrix, the moments of inertia are denoted by I_xx, I_yy, and I_zz.

The moment of inertia of a rigid body is the rotational analog to mass in linear motion. This can be seen in Newton's second law for rotation, which shows that the angular acceleration of a rigid body is proportional to the applied moment, and the constant of proportionality is the moment of inertia.

Consider a single particle undergoing rotational motion. The kinetic energy of the particle is given by

Consider now a system of particles rotating at the same angular rate. The kinetic energy of the system is

where n is the number of particles in the system. A rigid body is a system like the one described above; all particles in a rigid body rotate at the same angular velocity about the mass center during rotation. Compare Eq. 2↑ with the equivalent expression for translation, namely:

Equation 3↑ holds for a rigid body undergoing translation only. Notice that the equivalent “mass” term in the rotational kinetic energy equation is m_ir²_i instead of just m_i in translation. Thus, the quantity mr² may be considered the moment of inertia of a particle undergoing pure rotational motion. Extending Eq. 2↑ to a continuous rigid body with distributed mass per unit volume ρ, the moment of inertia for any rigid body is given by

For a rigid body that is assumed to have a uniform distribution of mass, the moments of inertia in Cartesian coordinates with the axes origin located at the mass center C can be calculated through the following equations:

These integrals can also be evaluated using cylindrical or spherical coordinates depending on what is convenient to solve the integral based on the geometry of the body. In many engineering problems, inertia is represented as a matrix giving the moments of inertia and also the products of inertia for the body.