Abstract
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.
je sais bien ce que je fuis, et non pas ce que je cherche. Montaigne, Essais (1572–1588)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This assumption means that the poles of the system are complex conjugate pairs localized close to the imaginary axis.
- 2.
\(Q_i=1/2\xi _i\) is often called the quality factor of mode \(i\).
- 3.
It is indeed well known in system theory that the open-loop poles are independent of the actuator and sensor configuration while the open-loop zeros do depend on it.
- 4.
Reading the chapter on continuous systems is recommended before reading this section.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Preumont, A. (2013). Multiple Degree-of-Freedom Systems. In: Twelve Lectures on Structural Dynamics. Solid Mechanics and Its Applications, vol 198. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6383-8_2
Download citation
DOI: https://doi.org/10.1007/978-94-007-6383-8_2
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-6382-1
Online ISBN: 978-94-007-6383-8
eBook Packages: EngineeringEngineering (R0)