‫So what really happens if the body is not mass symmetric to see it, let's ride it out, we can write
‫down the moment and then the body frame, angular acceleration relationship like this.
‫And now we're just going to write out this entire system.
‫If you do that, then this is what you will get.
‫You can see that now you have a much more complicated system of equations.
‫If you have a moment about the body frame, x axis, body frame, y axis and body frame Z axis, if
‫you have these three moments here, there are not zero, then this entire system of equations becomes
‫much more complicated.
‫If you're a product of inertia or not zero, then you can see that part of your moment about the body
‫frame x axis that you apply about the body frame x axis.
‫It not only contributes to angular acceleration about the X axis, but this Mexican moment also contributes
‫to angular acceleration about the body frame, Y and Z axis.
‫And the same thing here this moment, m, y, y.
‫Not only it contributes to this thing that it does anyway when all the products of inertia are zeros.
‫But in addition to that, it also contributes to this p dot and our dot and the same thing here.
‫Normally it contributes to this one, but it also contributes to this one and this one.
‫So when all your products of inertia are non zeros and you have moments about all the three axis, then
‫you have a situation where each moment contributes to angular accelerations about all the three axis.
‫That obviously complicates the calculations.
‫That is why engineers try to design things with mass symmetry.
‫Then products of inertia become zeros and your inertia tensor becomes like this.
‫This is much easier to deal with.
‫And now, of course, that is what we will have.
‫We will assume mass symmetry about all the axis.
‫So you have a wave here.
‫This is your body frame x axis, this is your body frame y axis and this is your body frame Z axis.
‫And we assume mass symmetry about all these axis.
‫So we will only deal with this matrix here.
‫And by the way, one final remark about this topic is that if you achieve a situation where products
‫of inertia are zero kilogram times meter squared, in other words, when you achieve this matrix and
‫you only have to deal with these mass moments of inertia, then the axis about which these mass moments
‫of inertia are calculated.
‫These axes are called principle axis.
‫So in the next video, we will start deriving fundamental dynamics equations.
‫So see you there and thank you very much.