Dynamic Systems

This chapter discusses dynamics of single and multiple degrees of freedom undamped and viscously damped systems. It considers single-degree-of-freedom viscously damped systems. The chapter discusses dynamics of multiple-degree-of-freedom undamped systems. It introduces concepts of natural frequencies (eigenvalues) and mode-shapes (eigenvectors). The chapter investigates proportionally damped systems. It considers general non-proportionally damped systems. Equation of motion of a viscously damped system can be obtained from the Lagrange's equation and using the Rayleigh's dissipation function. Caughey and O'Kelly have derived the condition which the system matrices must satisfy so that viscously damped linear systems possess classical normal modes. Dynamic response of proportionally damped systems can be obtained in a similar way to that of undamped systems. Modes of proportionally damped systems preserve the simplicity of the real normal modes as in the undamped case. Dynamic analysis of general viscously damped systems requires the calculation of complex natural frequencies and complex modes.