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Prototype Problems: Bifurcations of Different Kinds Read chapter Read first chapter

2020 | OriginalPaper | Chapter

3. Nonconservative Systems with a Finite Number of Degrees of Freedom

Authors : Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

Published in: Stability of Axially Moving Materials

Publisher: Springer International Publishing

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Abstract

In this chapter we present some results on the stability and bifurcations of the systems with a finite number of degrees of freedom. We consider damping-induced destabilization in nonconservative systems. We start with a general theoretical treatment of the topic. As the model problem, we consider the double pendulum subject to both a follower force and gravitational loading. A special case of interest is treated with the theoretical framework. The chapter finishes with a thorough presentation and analysis of the model problem including the nonlinear dynamics, quasistatic equilibrium paths and their stability, and special cases of interest. In numerical examples, we show equilibrium paths and trajectory density visualizations of the time evolution of the nonlinear system. Sample-based uncertainty quantification is employed to capture both branches of a bifurcation in the same visualization.

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previous chapter Bifurcation Analysis for Polynomial Equations
next chapter Some General Methods
Footnotes
1
This \(\mathbf {u}\) is unrelated to the displacement that was used when deriving the model; we will not need the displacement any more.
 
2
As is seen from (3.6.54)–(3.6.57), setting \(\mu _{\rho }=\mu _{m}=0\) completely eliminates the inertial generalized force, and thus removes all occurrences of second time derivatives from the dynamical equation. It also eliminates the external force in the dead-weight case. In such a system, time evolution is driven by the damping terms in (3.6.54). If there is no damping, then there are no time derivatives anywhere in the dynamical equation, and it becomes simply a set of constraints for admissible combinations of \(q_{1}\) and \(q_{2}\). In the follower-force case, for example, the admissible positions form a continuous path, by the continuity of (3.6.54) and (3.6.55), possibly branching at certain points as \(q_{2}\) increases, due to the sine in (3.6.55).
 
3
Lighthill [12], pp. 113–114 suggested, in the name of being constructive, renaming d’Alembert’s paradox to d’Alembert’s theorem: a mathematical demonstration showing that if one were able to physically construct shapes which, when subjected to an external flow, have the flow fulfill the very stringent conditions of potential flow, then very low (essentially zero) drag could be observed.
 
4
Strictly speaking, at the critical load, Eq. (3.6.105) itself—when considering only the expressions on both sides of the equals sign—admits any value of \(q_{1}\), no matter how large. Thus we may say that according to the linearized model, at the bifurcation point the angular displacement \(q_{1}\) shoots off to infinity. However, to remain rigorous, we must consider the assumptions made when deriving (3.6.105). We have used a Taylor series, up to the linear term, requiring \(\left| q_{1}\right| \ll 1\) for validity.
 
5
Solving at least in principle. What matters is that, given a state vector \((q_{1},q_{2},\dot{q}_{1},\dot{q}_{2})\), we somehow extract the corresponding values for \((\ddot{q}_{1},\ddot{q}_{2})\) from the dynamical equation. It does not matter whether the solution process is an explicit formula, is based on iterating an implicit system of equations, or is based on some entirely different approach.
 
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Metadata
Title
Nonconservative Systems with a Finite Number of Degrees of Freedom
Authors
Nikolay Banichuk
Alexander Barsuk
Juha Jeronen
Tero Tuovinen
Pekka Neittaanmäki
Copyright Year
2020
Book
Stability of Axially Moving Materials

Print ISBN: 978-3-030-23802-5

Electronic ISBN: 978-3-030-23803-2

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-23803-2_3