Page 474 - Dynamics of Mechanical Systems

0593_C13_fm Page 455 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 455

FIGURE 13.7.4 FIGURE 13.7.5
Particle movement at lowest frequency ω 1 . Particle movement at intermediate frequency ω 2 .

FIGURE 13.7.6
Particle movement at highest
frequency ω 3 .

13.8 Analysis and Discussion of Three-Particle Movement:
Modes of Vibration

In reviewing the foregoing analysis of the three-particle system we see a striking similarity
to the analysis for the eigenvalue problem of Sections 7.7, 7.8, and 7.9. Indeed, a comparison
of Eqs. (13.7.13), (13.7.14) and (13.7.15) with Eq. (7.7.10), shows that they are in essence
the same problem. This means that analyses similar to those of Sections 7.7, 7.8, and 7.9
(such as determination of eigenvalues, eigenvectors, orthogonality, etc.) could also be
conducted for the three-particle vibration problem. In our relatively brief introduction to
vibrations, however, it is not our intention to develop such detail. Instead, we plan to
simply introduce the concept of vibration modes (analogous to eigenvectors).
To this end, consider again the governing differential equations for the spring-supported
particles (see Eqs. (13.7.1), (13.7.2), and (13.7.3)):

mx ˙˙ + 2 kx − kx = 0 (13.8.1)
1 1 2

mx ˙˙ − kx + 2 kx − kx = 0 (13.8.2)
2 1 2 3

mx ˙˙ − kx + 2 kx = 0 (13.8.3)
3 2 3