3.5 Orthogonality of Natural Modes

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Let us write Equation 3.13 explicitly for the two distinct modes i and j: Distinct modes are defined as

those having distinct natural frequencies (i.e., vi – vj).

v2i

Mci 2 Kci ¼ 0 ð3:15Þ

v2j

Mcj 2 Kcj ¼ 0 ð3:16Þ

Premultiply Equation 3.15 by cT

j and Equation 3.16 by cT

i

v2i

cT

j Mci 2 cT

j Kci ¼ 0 ð3:17Þ

v2j

cT

i Mcj 2 cT

i Kcj ¼ 0 ð3:18Þ

Take the transpose of Equation 3.18, which is a scalar:

v2j

cT

j MTci 2 cT

j KTci ¼ 0

This, in view of the symmetry of M and K as expressed in Equation 3.8 and Equation 3.3, becomes

v2j

cT

j Mci 2 cT

j Kci ¼ 0

3-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

By subtracting this result from Equation 3.17,

we get

ðv2i

2 v2j

ÞcT

i Mcj ¼ 0

Now, because vi – vj; it follows that

cT

i Mcj ¼

0 for i – j

Mi for i ¼ j

(

ð3:19Þ

Equation 3.19 is a useful orthogonality condition

for natural modes.

Even though the foregoing condition of

M-orthogonality was proved for distinct (unequal)

natural frequencies it is generally true, even if two or

more modes have repeated (equal) natural frequencies. Indeed, if a particular natural frequency is repeated

r times, there will be r arbitrary elements in the modal vector. As a result ,we are able to determine r

independent mode shapes that are orthogonal with respect to the M matrix. An example is given later in the

problem of Figure 3.6. Note further that any such mode shape vector corresponding to a repeated natural

frequency will also be M-orthogonal to the mode shape vector corresponding to any of the remaining

distinct natural frequencies. Consequently, we conclude that the entire set of n mode shape vectors is

M-orthogonal even in the presence of various combinations of repeated natural frequencies.

3.5.1 Modal Mass and Normalized Modal Vectors

Note that, in Equation 3.19, a parameter Mi has been defined to denote cT

i Mci: This parameter is termed

the generalized mass or modal mass for the ith mode. Since the modal vectors ci are determined for up to

one unknown parameter, it is possible to set the value of Mi arbitrarily. The process of specifying the

unknown scaling parameter in the modal vector, according to some convenient rule, is called the

normalization of modal vectors. The resulting modal vectors are termed normal modes. A particularly

useful method of normalization is to set each modal mass to unity ðMi ¼ 1Þ: The corresponding normal

modes are said to be normalized with respect to the mass matrix, or M-normal. Note that, if ci is normal

with respect to M, then it follows from Equation 3.18 that 2ci is also normal with respect to M.

Specifically,

ð2ciÞTMð2ciÞ ¼ cT

i Mci ¼ 1 ð3:20Þ

It follows that M-normal modal vectors are still arbitrary up to a multiplier of 2 1. A convenient practice

for eliminating this arbitrariness is to make the first element of each normalized modal vector positive.

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