Abstract
A normal mode of a vibrating system is a mode that is orthogonal to all other normal modes of that system. The orthogonality is in a weighted sense. For an undamped discretized linear mechanical system, the orthogonality is defined with respect to stiffness and mass such that normal modes are mutually stiffness and mass orthogonal. Another commonly used definition of an oscillating normal mode is that it is a pattern of motion in which all parts of the system vibrate harmonically with the same frequency and therefore with fixed relative phase relations between parts. The normality of a mode is thus not in a statistical sense. For lightly damped linear systems, a normal observation, i.e. one very common observation in the statistical sense, is that the phase relation between the motion of different parts of the system deviates very little from zero or pi. However, this normally occurring behavior should not lead us to think that that always has to be the case. Here it is shown by example that the normal modes of an undamped system may have arbitrary phase relations. One such mode of vibration may then possess the property of moving nodal lines, which is often attributed to non-proportionally damped or non-selfadjoint systems. The proper normalization of such modes is discussed and their relation to the well-known modal mass and MAC concepts and also to state-space based normalizations that are usually being used for complex-valued eigenmodes.
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Notes
- 1.
This is a rare case in practice, however mathematically interesting. The simplest such case is probably the critically damped single-degree-of-freedom mass-stiffness-damper system.
- 2.
Ref. [6] advocates the use of modal Foss damping and modal Foss stiffness for the a n and b n respectively and gives an abstraction of systems with decoupled complex-valued states. One reason for this naming is that they carry the units of damping and stiffness and the other reason for it is a tribute to K.A. Foss who was one of the first, if not the first, that used the form given by Eq. (7), see [7].
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Abrahamsson, T.J.S., Allemang, R.J. (2016). What Is Normal About Normal Modes?. In: Mains, M. (eds) Topics in Modal Analysis & Testing, Volume 10. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-30249-2_7
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DOI: https://doi.org/10.1007/978-3-319-30249-2_7
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