Today I want to talk about the product of inertia.
The product of inertia is kind of like a moment of inertia that we've just done,
I_x or I_y,
except here instead of having x-squared or y-squared,
the product of inertia is defined as the interval of x y dA.
The difference--
one of the big differences between this and the moments of inertia
is that a moment of inertia is always positive.
You've got x-squared in your integral,
and you've got even in the parallel axis theorem
d-squared times A.
So nothing that's ever there is going to be 0 or negative.
But this one, because you've got x y and they're not squared,
these can be 0, greater than 0, less than 0. Either way.
In general, what is a moment of inertia?
You're talking about in a moment of inertia, you have a beam that's bending like this.
You have an axis of rotation. It's a cross-sectional property.
So we say that it is easier to bend the ruler this way
than it is to bend it crosswise
because more of the material is further away from this axis than it is away from this axis.
So when we talk about bending like this, basic beam theory says,
as you bend this, any cross-section that you would take out of the middle,
any chunk right here cut out of the middle here, if it starts flat ends up flat.
But sometimes we have nonsymmetric cross-sections.
So if you have an asymmetric cross-section where you have something like an L-beam,
when you bend this -- think about how this would work.
The flat part, we'll want to have it be easier to bend it up this axis,
but the back part would find that much harder.
So you have competing problems.
You've got one that, one part that wants to bend easily
and the other part that wants to not bend easily.
If you take an L out of the middle of this beam,
even if started flat, it might not end up flat.
So that's what you're talking about when you're talking about products of inertia.
It only comes into play when you're talking about beam bending with asymmetric cross-sections.
Any time you have a cross-section that is symmetric about either the x or y-axis,
you can bend it about the x-axis or the y-axis, and cross-sectional planes remain planar.
So how do you find what the product of inertia is?
You can integrate it just like before
except this is usually done with a double integral instead of a stripe.
So if you're going to integrate it, it's a straight integral.
Composite bodies are a little bit more common.
And you'll see that. That's what we're going to do.
The parallel axis down here says I_xy is I_xy plus A dx dy
where those are the two distances from the axes.
So how does this actually work with composite bodies?
This is the same thing I just had up there
where I_xy -- this part -- is the product of inertia of any individual one of your composite bodies
about its own centroidal axis.
Now that's the part that's generally 0.
Or you could look it up in a chart if you have some other basic shape.
dx and dy are the distances
from the x- and y-axes to each individual part centroid.
A is the area.
And then you put this in the chart
where you've got I_xy, dx, dy, A, and then the sum of them.
All you've got to do is add up this last column.
So if we look at an example,
this is the kind of problem you're going to end up with.
This is the asymmetric beam we just looked at.
If I want to find I_xy about the centroidal axes of these things,
the first thing you have to do is find the centroid of that thing.
So this is the chart we had before: we have x_bar, y_bar, A, x_bar A, and y_bar A.
These are with respect to some axis that you started with.
So I've put my axis at the lower left hand corner and said x and y are,
will be measured from this lower left hand corner for each of my pieces.
A is just the area of each piece.
And that gives me the centroid location with respect to that lower left hand corner.
And then like we've done before, what I want to do is take and move my axes to this point.
Now when I go to fill out the chart for I_xy,
these are zero because I had it broken up into two rectangles.
Each of them are symmetric about their own centroidal x or y axes.
dx is the distance between this x axis and each of the ... well,
well, dx is going to be the distance in the y ...
this distance, the x distance.
Between that point and this line or this point and that line.
dy is the other two.
So you've got, for example, 1.1667 minus 1 gives you this dx.
Notice that this is a negative number. That point is left and down.
So dx and dy are both negative.
dy is positive for one and negative for one like this. There's your A.
This is what I'm going to do to add up that last column.
So a couple of things to remember.
If you have a piece, an individual part,
that is symmetric about its own cross-section, these are 0,
and then these are measured between the centroid of each shape and the new centroidal axis.
So that's how you find the product of inertia most commonly.