Stability problems in constrained pendulum systems and time-delayed systems
In this dissertation, we study the boundary of stability of a class of linear mechanical systems as a function of a parameter. We consider two different systems under this class: a constrained double pendulum connected by a rigid rod and a state-feedback-controlled mechanical system with time delay. In the first system, the destabilizing parameter is the distance between the supports of the two pendulums. In the second system, the destabilizing parameter is the time delay. In the constrained double pendulum system, linear perturbation analysis is used to determine the natural frequency of the system. Our analysis reveals a zone of instability in what seemingly is an inherently stable configuration. This paradoxical behavior, which is not mentioned in the literature until now, is explained and a simple experiment confirms the instability predicted by the analysis. The approach is extended to a chain of pendulums consisting of n masses and n+1 links, which is a lumped parameter model for small vibrations of a catenary. Our work confirms the existence of asymmetric stable equilibrium configurations for a symmetric system. The problem of determining the critical distance for instability between two supporting points of a catenary has potential application in the design of novel mechanical switches, sensors, and valves. In the second part of the dissertation, we consider a linear mechanical system where a time delay exists in the linear state feedback control input. We seek a closed-form solution for the problem of determining the critical time delay for instability of the closed-loop system. Such a closed-form solution, which to the best of our knowledge is inexistent in the literature, offers an exact value for the critical time delay whereas a numerical solution is only approximate. We show that in the single-input/multi-output (SIMO) case of the class of systems under consideration, the problem may be reduced by using singular value decomposition to that of finding the roots of a certain polynomial. The obtained closed-form solution accurately predicts the smallest time delay that would render the SIMO system unstable when the control gain matrices have a unit rank. This technique however cannot be extended to the multi-input/multi-output case. Two numerical methods are therefore developed to solve this case. One method involves Newton’s iterations and the other method involves bisection for multiple functions.
Stability, Double Pendulum, Time-Delay, Transcendental Eigenvalue Problem
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