Page 33 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

30 Chapter 1 Fundamentals oF Vibration
The various classifications of vibration—namely, free and forced vibration, undamped and
damped vibration, linear and nonlinear vibration, and deterministic and random vibration—
are indicated. The various steps involved in vibration analysis of an engineering system are
outlined, and essential definitions and concepts of vibration are introduced.
The concept of harmonic motion and its representation using vectors and complex
numbers is described. The basic definitions and terminology related to harmonic motion,
such as cycle, amplitude, period, frequency, phase angle, and natural frequency, are given.
Finally, the harmonic analysis, dealing with the representation of any periodic function in
terms of harmonic functions, using Fourier series, is outlined. The concepts of frequency
spectrum, time- and frequency-domain representations of periodic functions, half-range
expansions, and numerical computation of Fourier coefficients are discussed in detail.

Learning Objectives
After completing this chapter, the reader should be able to do the following:

• Describe briefly the history of vibration
• Indicate the importance of study of vibration
• Give various classifications of vibration
• State the steps involved in vibration analysis
• Compute the values of spring constants, masses, and damping constants
• Define harmonic motion and different possible representations of harmonic motion
• Add and subtract harmonic motions
• Conduct Fourier series expansion of given periodic functions
• Determine Fourier coefficients numerically using the MATLAB program

1.1 preliminary remarks
The subject of vibration is introduced here in a relatively simple manner. The chapter
begins with a brief history of vibration and continues with an examination of its impor-
tance. The various steps involved in vibration analysis of an engineering system are out-
lined, and essential definitions and concepts of vibration are introduced. We learn here that
all mechanical and structural systems can be modeled as mass-spring-damper systems. In
some systems, such as a motor vehicle, the mass, spring, and damper can be identified as
separate components (mass in the form of the body, spring in the form of suspension, and
damper in the form of shock absorbers). In some cases, the mass, spring, and damper do
not appear as separate components; they are inherent and integral to the system. For exam-
ple, in an airplane wing, the mass of the wing is distributed throughout the wing. Also,
due to its elasticity, the wing undergoes noticeable deformation during flight so that it can
be modeled as a spring. In addition, the deflection of the wing introduces damping due to
relative motion between components such as joints, connections, and support as well as
internal friction due to microstructural defects in the material. The chapter describes the
modeling of spring, mass, and damping elements, their characteristics, and the combina-
tion of several springs, masses, or damping elements appearing in a system. There follows