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Chapter 8

Multi Degree-of-Freedom

Vibration: Introductory
Topics
"And now we no longer operate in isolation

since we are aware ofothers."

Many engineering systems, such as turbotnachines, bridges, and aircraft, are modeled with hundreds if not thousands of degrees-of-freedom. Each degree-of-freedom has a frequency of vibration, much in the same way as a single degree-of-freedom oscillator has a natural frequency. However, sys tems with multiple degrees-of-freedom are more complicated due to the

physical coupling that exists between pairs of degrees-of-freedom.
In this chapter we introduce a new concept - called the mode of vi bration - which exists in systems of two or more degrees-of-freedom.

A

vibrating linear system, while moving in very complicated patterns, can be modeled as n uncoupled vibrating oscillators if special coordinates -

called principal coordinates - are used. We will demonstrate that the sin gle degree-of-freedom oscillator is the basis for solving many linear multi

degree-of-freedom (MDOF) vibrating systems.

8.1

Example Problems and Motivation

Most engineering systems can be modeled as multi degree-of-freedom sys tems. Three interesting applications motivate the study of multi degree-of503

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CHAPTER 8.

MDOF VIBRATION

7V7\7\7\7\7\7\

Figure 8.1: A periodic structure: an antenna dish.

freedom discrete models. Additional applications are introduced throughout the chapter.

8.1.1

Periodic Structures

Periodic structures have geometry and material properties that are repet itive with a certain pattern. Such structures are very important in appli

cations such as the circularly periodic antenna dish of Figure 8.1, the truss

lattice of a space station, or the reticulated pattern of stiffencrs on the in side shell of an aircraft fuselage. These are well known applications. A less

obvious example is a dynamic model of DNA,1 a biological polymer that plays an essential