TABLE OF CONTENTS You're probably all familiar with orthogonality as the term is applied to spatial vectors. It means that the vectors are oriented at right angles to each other. Mathematically, when two vectors are orthogonal, their dot product is zero. Additionally, one way of calculating the dot product of two column vectors A and B is by evaluating A T B. These notions are very close to those that we'll develop in this section. The only twist is that instead of saying certain vectors are orthogonal just to each other, we'll see that the orthogonality conditions include the [ M] and [ K] matrices. Thus orthogonality is very tightly tied to the particular system we're examining.
We'll sneak up on these results in a very straightforward way, namely, by making a simple examination of our equations and then drawing some conclusions from them. For clarity, we'll derive our results with a 2 DOF example. Keep in mind though that the approach is applicable to a problem with any number of degrees of freedom, as we'll indicate after finishing with the 2 DOF example.
To begin, we'll look at our free vibration problem
We know now that the solutions to this problem define our system eigenvectors ( X 1 and X 2) and natural frequencies ( ? 1 and ? 2). Thus, writing (4.8.1) in terms of these known solutions gives us
and
Rearranging these equations gives us
