It has been established by now that the problem of the absolute stability originates from a paper of B.V. Bulgakov published in 1942. The engineering application generating the problem was the so called gyroscopic pendulum, with various applications in stabilization of ships, aircrafts or armored cars. Even if nowadays the technical implementation of this device might be completely different, t...

We derive a Hamiltonian version of the PT-symmetric discrete nonlinear Schrödinger equation that describes synchronized dynamics of coupled pendula driven by a periodic movement of their common strings. In the limit of weak coupling between the pendula, we classify the existence and spectral stability of breathers (time-periodic solutions localized in the lattice) supported near one pair of cou...

Recently, first limits on putative Lorentz invariance violation coefficients in the pure gravity sector were determined by the reanalysis of short-range gravity experiments. Such experiments search for new physics at sidereal frequencies. They are not, however, designed to optimize the signal strength of a Lorentz invariance violation force; in fact the Lorentz violating signal is suppressed in...

I. PROBLEM FORMULATION The Furuta pendulum consists of an arm rotating in the horizontal plane and a pendulum attached to the end of the arm moving freely in the vertical plane, see Fig. 1. The equations of motion of the Furuta pendulum, derived from the standard Euler-Lagrange dynamics, are1 (α + β sin θ)φ̈ + γ cos θθ̈ + 2βθ̇φ̇ sin θ cos θ − γθ̇ sin θ = τφ, γ cos θφ̈ + βθ̈ − βφ̇ sin θ cos θ − δ sin θ ...

This paper investigates virtual holonomic constraints for Euler-Lagrange systems with n degreesof-freedom and n− 1 controls. In our framework, a virtual holonomic constraint is a relation specifying n − 1 configuration variables in terms of a single angular configuration variable. The enforcement by feedback of such a constraint induces a desired repetitive behavior in the system. We give condi...

A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to t...

The motion of a spherical pendulum is characterized by the fact that all trajectories are relative periodic orbits with respect to its circle group of symmetry (invariance by rotations around the vertical axis). When the rotational symmetry is broken by some mechanical effect, more complicated, possibly chaotic behavior is expected. When, in particular, the symmetry reduces to the dihedral grou...

A controller is developed to balance a two-dimensional walking robot. The equations of motion for the robot, modelled as a double pendulum with pivots at the hip and contact foot, are derived, linearized, and arranged in statespace form. A Linear Quadratic Regulator is developed for the linearized model; its performance simulated using both the linearized and the complete, nonlinear dynamics. T...

The purpose of this paper is to present the design and program implementation of a general procedure which yields the mathematical model for a classical or rotary inverted pendulum system with an arbitrary number of pendulum links. Lagrange equations of the second kind with an integrated Rayleigh dissipation function are employed in model design, and the energetic balance relations, derived for...

We study the Astrojax Pendulum and the N-body problem on the sphere in the light of Lagrangian reduction theory, variational integrators, and pattern evocation.