Page 199 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

186 Properties of Vibrating Systems Chap. 6

If A, ^ the foregoing equation requires that
(6.6-6)

It is also evident from Eq. (6.6-2) or Eq. (6.6-3) that as a consequence of Eq.
(6.6- 6),
4>]K4>^ = 0 i ^ j (6.6-7)
Equations (6.6-6) and (6.6-7) define the orthogonal character of the normal modes.

Finally, if / = j\ (A, —A^) == 0 and Eq. (6.6-5) is satisfied for any finite value

of the products given by Eq. (6.6-5) or (6.6-6). We therefore have
= M,,
(6.6-8)
ct>jKcf>, =
The quantities M- and K- are called the generalized mass and the generalized
stiffness, respectively. We will have many occasions to refer to the generalized mass
and generalized stiffness later.
Example 6.6-1
Consider the problem of initiating the free vibration of a system with a specified
arbitrary displacement. As previously stated, free vibrations are the superposition of
normal modes, which is referred to as the Expansion Theorem. We now wish to
determine how much of each mode will be present in the free vibration.
Solution: We will express first the arbitrary displacement at time zero by the equation;
3f(0) = c^(f) I + C2</)2 + C^(j)^ -f • • • + • • •
where (/>, are the normal modes and are the coefficients indicating how much of
each mode is present. Premultiplying the above equation by (¡)JM and taking note of
the orthogonal property of (/>,, we obtain,
= 0 + 0 + 0 + • •• + 0 + • • •

The coefficient c, of any mode is then found as
</>fMX(0)
C: = -----7---------

Orthonormal modes. If each of the normal modes is divided by the
square root of the generalized mass M-, it is evident from the first equation of
Eqs. (6.6-8) that the right side of the foregoing equation will be unity. The new
normal mode is then called the weighted normal mode or the orthonormal mode
and designated as It is also evident from Eq. (6.6-1) that the right side of the
second equation of Eq. (6.6-8) becomes equal to the eigenvalue A,. Thus, in place
of Eqs. (6.6-8), the orthogonality in terms of the orthonormal modes becomes
^JM^, = 1
(6.6-9)
= À,