In this chapter, we discuss dynamics of single- and multiple-degrees-of-freedom (SDOF and MDOF) undamped and viscously damped systems. There are many excellent books [MEI 67, MEI 80, PAZ 80, NEW 89, MEI 97, GÉR 97, INM 03, RAO 11], which discuss these topics very well. Here, we give only the details as a necessary background for further developments in the following chapters.
The outline of this chapter is as follows. Dynamic analysis of SDOF undamped systems is discussed in section 2.1. SDOF viscously damped systems are considered in section 2.2. Dynamics of MDOF undamped systems are discussed in section 2.3. The concepts of natural frequencies (eigenvalues) and mode-shapes (eigenvectors) are introduced. Proportionally damped systems are investigated in detail in section 2.4. General non-proportionally damped systems are considered in section 2.5. The idea of Rayleigh quotient for damped systems is proposed in section 2.6. Stationarity properties for systems with proportional damping and non-proportional damping are discussed. Finally, a summary of the topics discussed in this chapter is given in section 2.7.
An undamped SDOF system is possibly the simplest system and is of fundamental interest in structural dynamics. A schematic diagram of the system is shown in Figure 2.1. The equation of motion of the system can be expressed as