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In this paper, we explore the stability of an inverted pendulum under a generalized parametric excitation described by a superposition of $N$ cosines with different amplitudes and frequencies, based on a simple stability condition that does not require any use of Lyapunov exponent, for example. Our analysis is separated in 3 different cases: $N=1$, $N=2$, and $N$ very large. Our results were obtained via numerical simulations by fourth-order Runge Kutta integration of the non-linear equations. We also calculate the effective potential also for $N>2$. We show then that numerical integrations recover a wider region of stability that are not captured by the (approximated) analytical method. We also analyze stochastic stabilization here: firstly, we look the effects of external noise in the stability diagram by enlarging the variance, and secondly, when $N$ is large, we rescale the amplitude by showing that the diagrams for time survival of the inverted pendulum resembles the exact case for $N=1$. Finally, we find numerically the optimal number of cosines corresponding to the maximal survival probability of the pendulum.
The change of the plane of oscillation of a Foucault pendulum is calculated without using equations of motion, the Gauss-Bonnet theorem, parallel transport, or assumptions that are difficult to explain.
This paper presents a general formulation of equations of motion of a pendulum with n point mass by use of two different methods. The first one is obtained by using Lagrange Mechanics and mathematical induction(inspection), and the second one is deri
The topic of this paper is to use an intuitive model-based approach to design a networked controller for a recent benchmark scenario. The benchmark problem is to remotely control a two-wheeled inverted pendulum robot via W-LAN communication. The robo
The analytical solution of the three--dimensional linear pendulum in a rotating frame of reference is obtained, including Coriolis and centrifugal accelerations, and expressed in terms of initial conditions. This result offers the possibility of trea
We set a shortcut-to-adiabaticity strategy to design the trolley motion in a double-pendulum bridge crane. The trajectories found guarantee payload transport without residual excitation regardless of the initial conditions within the small oscillatio