Analysis of objectives:
Towards a conceptual unification of exising approaches to structural analysis,
including aspects of application in engineering and mathematical theory aspects
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Stability of structures
Stability is considered as a property of an equilibrium (or equilibrium series), and, as a consequence, of a trajectory and dynamic evolution of a configuration. Obviously, stilness persisting over time, i.e., a condition where the equilibrium whose stability is under investigation is achieved among non-inertial forces, is part of the definition of trajectory and configuration evolution.
Stability is a property of motion (or, more in general, of the evolution) of systems. This implies that a configuration can be defined for a system as a function of the time parameter, including as essential elements:
the "positions" of its components
the speed of such components.
The system in question is inteded as ruled by physics laws and the evolution to be studied as determined by initial conditions and (possibly) external actions, known functions of the configuration and time.
The analysis of stability necessitates also the intorduction of a measurement of the "distance" between two different configurations. In such a context, two different (increasingly stringent) definitions of stability can be given, which are referred to here as "Second Liapunoff's method or direct method". Usually, only continuous-time systems are addressed, modeled through differential equations. From a physics point of view, the evolution of a system is the satisfaction of equilibrium constraints in a universe where the "principle of equilibrium" can be assumed as one of the foundations of the study of behaviours. Stability, therefore, is a property to be studied for systems in equilibrium.
In many cases,the motions whose stability is investigated are reduced to stilnnes (or relative stilness): it should be stressed that studing stability makes sense only when stilness is the a solution of the equilibrium equations (taking into account the necessary initial conditions and the potential external actions). Stability concerns deviations between a trajectory and its perturbation, or, alternatively, between a trajectory and another trajectory obtained by from slightly deviating initial conditions, when such deviation tends to zero. Since each achieved condition along a trajectory constitutes an initial condition for the remaining of the same trajectory, the deviation can take place at any point in time, and the judgmement of stability obviously affects the trajectory from that point onwards. The two compared trajectories can be annotated with times. Stability concerns the evolution of the distance in space and speed between isochronous points. If such distance decreases with time, there is strong stability; if it does not increase, there is weak stability; if it increases, there is non stability. Clearly, stability is analysed dynamically, and therefore is a dynamic property. Indeed, even if one of the trajectories is stilnnes, the other, which is necessarily a perturbation, can only be obtained through dynamic analysis. However, the concept of "static stability" is also introduced, which does not concern a trajectory as a whole, but the signs of the differences between forces, which are generated due to the deviations mentioned for dynamic stability. Static stability does not ensure dynamic stability. It is a necessary but not sufficient condition for dynamic stability. In structures in particular, even if stilness conditions whose stability is to be investigated need to be determined through non linear analyses, the dynamics which follow stilness perturbation are examined only for minor displacements, so that linearised analyses moving from the intial configuration are sufficient.