Multiple Degree-of-Freedom Systems
This chapter analyzes the differential equations governing the behavior of linear discrete multi degree-of-freedom systems with viscous damping. The free response of undamped systems is analyzed first, leading to the eigenvalue problem, the resonance frequencies and the mode shapes, and the orthogonality relationships. Next, this chapter analyzes the modal decomposition of the stationary response with the assumption of modal damping. The modal expansion of the dynamic flexibility matrix is established and its modal truncation is examined. The structures with rigid body modes have peculiar features which are discussed carefully. This chapter gives a special attention to the collocated systems in which the actuator and the sensor operate on the same degree of freedom; for this special type of system, the anti-resonances are defined, and the essential property of interlacing of the resonance and the anti-resonance frequencies is established; the relationship between the anti-resonance frequencies and the resonance frequencies of the constrained system are established. Finally, the eigenvalue problem of a building made of n identical floors is expressed as a difference equation and solved analytically. The chapter ends with a set of problems.