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= Three Domain Stability of non-conservative system = The Three Domain Stability is a method for calculation the three domains static, vibration and flutter of non-conservative systems, which was proposed by Ingerle in 2018 in his book. The classical stability theories for a dynamic behaviour of Lyapunov relate to conservative systems with only two stability domains stable and unstable. Since non-conservative systems without non-trivial states of equilibrium have often three stability domains, the classical stability theories can not be applied for theses systems. Infinitesimal small disturbances assumed – the three domains static, vibration and flutter can also described static stability, dynamic stability and instability.

Historical Background
Since the beginning of the 20th century the notion that non-conservative elastic systems without non-trivial states of equilibrium do not have static stability limits predominates in the scientific discourse of the topic. In 1969 Ingerle demonstrated for the first time that such systems have static stability limits. However, the topic continued to be discussed controversially for the remainder of the 20th century. With the presentation of a novel static stability criterion by Ingerle in 2013 the existence of static stability for such systems was excepted. The knowledge of the static stability limits provided the basis for the development of the new theory for the post-critical behaviour of non-conservative systems (Three Domain Stability).

Background and Assumptions
For a better understanding of the following, the deformation behaviour of conservative and non-conservative systems is compared. The double hinged column with the conventional load P at the tip and the Ziegler`s column are shown as examples. Figure 1 depicts the configuration of Ziegler's column.



The conservative system (a) reacts with sudden deformations when p increases beyond $$p_{st} = 0.38$$. Due to damping, non-trivial static equilibrium positions corresponding to the loads $$p$$ are established (two domain stability stable and unstable).

The deformations corresponding to the loads $$p > p_{st}$$ can be calculated with Euler:

$$c\varphi_{2} = Pl \sin(\varphi_1+\varphi_2);\; c\varphi_1 = Pl[\sin\varphi_1+\sin(\varphi_1+\varphi_2)];\; p=\varphi_2/\sin(\varphi_1+\varphi_2) $$

Calculation: $$\varphi_1=30^{\circ};\; \varphi_2=17^{\circ};\; a/l=1.23;\; p=0.41;$$

A small increase in the load from $$p_{st}=0.38$$ to $$p=0.41$$ leads to a large deformation (sudden jump in the disturbance to $$a/l = 1.23 (\alpha\approx90^{\circ})$$).

The non-conservative system (b) has three domains of stability: static, vibration and flutter. The vibration domain in particular differs from the previously known deformation property. The transition from the static domain to the vibration domain can be calculated using the Extra Energy Stability criterion ($$p_{st}=2.0$$). In contrast to the conservative system (a), vibrations occur in the non-conservative system (b) for $$p>p_{st}$$, which depend on the size of the initial disturbance and increase only slightly with increasing $$p$$. The angle $$\alpha$$ is almost zero here. The transition from vibration to flutter ($$p_{flutter}$$) takes place in a similar way to the transition from stable to unstable in the conservative system. However, large vibrations corresponding to $$p>p_{flutter}$$ occur. The calculation of vibration and flutter is based on the exact equations of the motion. The stability value $$p_{flutter}$$ is influenced by the mass distribution, damping and disturbance. Mass and damping are only effective with motion.

Static stability, dynamic stability, instability
This new designation for the three domains of static, vibration and flutter is only permitted if infinitely small disturbances are causing the deformation. The dynamic stability do not differ optically from a static state. The only difference is that on the one hand it is a static and on the other hand a dynamic state.

Example of Beck`s Column
For a better understanding of the Three Domain Stability the behaviour of Beck’s column is shown. Beck’s column – loaded with a tangential load $$P$$ – consist of rod with stiffness $$EI$$ and uniformly distributed mass $$m_0$$(see left configuration in Fig. 2).

For loads $$p$$ in the static domain ($$p < p_{st}$$) the system is in a static stabilty, in the vibration domain ($$p_{st} < p\leq p_{flutter}$$) it is dynamically stable (assuming infinitesimally small disturbances) and in the domain pflutter ($$p > p_{flutter}$$) it is in an instability.

The only difference between static stability and dynamic stability lies in the static and moving state, which cannot be visually recognized. The almost equally large domains of static and dynamic stability are remarkable.

Importance of non-conservative systems
The non-conservative systems with the follower forces are realistic and important for fluid-interacting systems. The physical processes have a large important in the nature. Follower forces occur whenever fluids, like air or water, interact with elastic bodies. At the beginning, the amplitudes of motion might be small (vibration) however, a small increase of the interacting forces might be the start to flutter with very large and destructive amplitudes. A suspension bridge, for example, is in a state of rest when the wind is low (static stability), when the wind speed increases it begins to vibrate (dynamic stability) and in a strong storm it can collapse (instability).

Supplementary Reading
In his book 2018 on non-conservative systems Ingerle expands on the ideas proposed in his article 2013.