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Lagrangian formulation in mechanics
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
In the theory of small oscillations one considers the perturbed behavior near the equilibrium configuration of a mechanical system and inquires if the system has a tendency to return to its original position given a slight disturbance from the position of equilibrium. The system is said to be in stable equilibrium if such a feature holds. The case of a suspended pendulum in which a point mass, tied to an inextensible string of negligible mass and hanging vertically down, furnishes one such example. On the other hand, if a slight disturbance causes a significant deviation of the system from its original position the system is said to be in a state of unstable equilibrium. A rod standing on its one end is an example of this type.
Power System Transient Stability Study by Involving Higher Order Corrections in Improved Quadratic Lyapunov Function for Singular Perturbed Model of Synchronous Generator
Published in IETE Journal of Research, 2022
Sunitha Anup, Ashu Verma, T.S. Bhatti
Note: For an autonomous system where is a vector field in a domain . Let is an equilibrium point, i.e.. The purpose of analysis is studying the stability of . For convenience, the equilibrium point is shifted to origin, that is . In doing so, there is no loss of generality, because coordinate transformation enables the shifting of equilibrium point to origin. For a non-linear system, there are more than one equilibrium point which may be stable and unstable. For a stable equilibrium, the trajectory of the state variable converges to stable equilibrium point and for unstable equilibrium, the trajectory of the state variable moves away from the equilibrium point. For transient stability study, this domain of equilibrium is advisable to relate to the fault location. The acceleration gained by a machine after the occurrence of a three phase fault enables to provide an accurate step for the determination of stability. The controlling unstable equilibrium point is closely related to boundary of system separation and trajectory of critically stable condition.
Modelling and control of a spherical pendulum via a non–minimal state representation
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Ricardo Campa, Israel Soto, Omar Martínez
An equilibrium which is not stable is said to be an unstable equilibrium. Definition 3. An equilibrium point is (uniformly) asymptotically stable if it is (uniformly) stable and there exists a such that implies Moreover, the equilibrium is globally (uniformly) asymptotically stable if it is (uniformly) stable and (19) is satisfied for all .