TABLE OF CONTENTS This chapter offers a brief introduction to vibration theory. This introduction is necessary because many of the SHM methods to be discussed in later chapters will utilize concepts and formulae from vibration theory.
The chapter starts with the theory of vibration of a single degree of freedom (1-dof) system, the particle vibration. This simple system will be used as a springboard for the analysis of more complicated system later in the chapter. The 1-dof particle vibration will be used to introduce fundamental basic concepts such as the differential equation of motion, harmonic solutions, free vs. forced vibrations, and damped vs. undamped vibrations. Energy methods approach to vibration analysis will also be discussed.
The second part of the chapter covers the vibration of continuous systems. Partial differential equations (PDE) in space and time will govern this type of vibrations. Assuming harmonic behavior in time, the equation of motion is reduced to an ordinary differential equation (ODE) in the space domain. This is a boundary value problem, which yields eigenvalues and eigenmodes, and the associate natural frequencies and mode shapes. The axial vibration of bars, flexural vibration of beams, and torsional vibration of shafts will be considered. In each case, the study of free vibrations is followed by the study of forced vibrations.
The chapter ends with a set of problems and exercises that will assist the student in consolidating the understanding of the basic concepts and in applying the theory to practical situations.
