Himanshu Roy Pota
This short note explains how the area moment of inertia enters the vibration analysis and the expression for the area moment of inertia is given by .
Figure 1: Flexible Slewing Beam
The dynamics of the flexible slewing beam shown in Figure 1 is described by the classical Bernoulli-Euler equation. The equation needs the area moment of inertia. In the following note the formula for the area moment of inertia is derived. Hopefully the derivation is clear and will enable derivation of the area moment of inertia for other situations as well.
Figure 2: Unbent Beam
The beam in its rest position is shown in Figure 2.
Figure 3: Bent Beam
Due to the torque at the base or any external disturbance the flexible beam gets bent and vibrates. Here we look at a snap shot during the vibration. The bent beam is shown in Figure 3.
Figure 4: Top-view of the Bent Beam
The top-view of the beam is shown in Figure 4.
The curvature is where is the radius of the circle of which
the neutral axis is an arc. Note that .
Looking along the line mn in
Figure 4 it can be seen that in an unbent position the line
mn is at a distance dx away from the centre but after bending, the change
distance as a function of y is given as ; giving,
The stress is then:
Figure 5: Beam Cross-sections
From Figure 5 it can be seen that half the cross-section
of the beam will have compressive stress and the other half will have
expanding stress. This force distribution gives rise to the bending moment:
The term inside the integral in the equation (3) is defined as
the area moment of inertia:
If the beam is vibrating in the x-z plane then the area moment of inertia
of interest is:
In general fo the experimental beams h > > w.
And for the unlikely case when the vibrations in the y-z plane are of
interest then is needed which is: