3,282 research outputs found

    Stepwise Precession of the Resonant Swinging Spring

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    The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the well-known three-wave equations that also apply, for example, in laser-matter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this system's dynamics, namely, the stepwise precession of its azimuthal angle.Comment: 28 pages, 10 figure

    Study of the apsidal precession of the Physical Symmetrical Pendulum

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    We study the apsidal precession of a Physical Symmetrical Pendulum (Allais' precession) as a generalization of the precession corresponding to the Ideal Spherical Pendulum (Airy's Precession). Based on the Hamilton-Jacobi formalism and using the technics of variation of parameters along with the averaging method, we obtain approximate solutions, in terms of which the motion of both systems admits a simple geometrical description. The method developed in this paper is considerably simpler than the standard one in terms of elliptical functions and the numerical agreement with the exact solutions is excellent. In addition, the present procedure permits to show clearly the origin of the Airy's and Allais' precession, as well as the effect of the spin of the Physical Pendulum on the Allais' precession. Further, the method can be extended to the study of the asymmetrical pendulum in which an exact solution is not possible anymore.Comment: 20 pages, 8 figures, LaTeX2

    Parametrically excited helicopter ground resonance dynamics with high blade asymmetries

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    The present work is aimed at verifying the influence of high asymmetries in the variation of in-plane lead-lag stiffness of one blade on the ground resonance phenomenon in helicopters. The periodical equations of motions are analyzed by using Floquet's Theory (FM) and the boundaries of instabilities predicted. The stability chart obtained as a function of asymmetry parameters and rotor speed reveals a complex evolution of critical zones and the existence of bifurcation points at low rotor speed values. Additionally, it is known that when treated as parametric excitations; periodic terms may cause parametric resonances in dynamic systems, some of which can become unstable. Therefore, the helicopter is later considered as a parametrically excited system and the equations are treated analytically by applying the Method of Multiple Scales (MMS). A stability analysis is used to verify the existence of unstable parametric resonances with first and second-order sets of equations. The results are compared and validated with those obtained by Floquet's Theory. Moreover, an explanation is given for the presence of unstable motion at low rotor speeds due to parametric instabilities of the second order

    Gauge Theory for Finite-Dimensional Dynamical Systems

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    Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This theory has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems with implications to numerical integration of differential equations. We distinguish between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry is, in essence, re-scaling of the independent variable, while descriptive gauge symmetry is a Yang-Mills-like transformation of the velocity vector field, adapted to finite-dimensional systems. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a remarkable connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse engineering and scientific fields, including quantum mechanics, chemistry, rigid-body dynamics and information theory

    Arnold diffusion for a complete family of perturbations with two independent harmonics

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    We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps is also provided.Comment: 23 pages, 14 figure

    Optimal manoeuvres and aeroservoelastic co-design of very flexible wings

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    The single shooting method is applied to the optimal control of very flexible aeroelastic wings and the combined structural and control design (co-design) of geometrically nonlinear beam models in vacuum. As large deflections occur, the dynamical properties of these systems can undergo substantial changes. Efficient actuation strategies require characterising, and possibly exploiting, these phenomena. With this purpose, geometrically-nonlinear models are built using composite beams and an unsteady vortex-lattice aerodynamics description. Optimal control is employed to identify actuations time-histories. Numerical solutions are obtained via single-shooting and sequential quadratic programming upon parametrisation of the control input. The approach is also extended to assess the feasibility of an integrated design strategy for active geometrically-nonlinear structures. Numerical studies are first presented for a very flexible actuated pendulum with large rigid-body motion. The impact of local (B-splines) and global (discrete sines) basis functions is investigated for increasing levels of actuation authority, underlining the importance of the time-frequency resolution of the parametrisation on the convergence properties and outcome quality of the process. Locking between control and structural vibrations around specific design points is found, thus highlighting the limitations of a sequential design approach. Simultaneous designing of control law and structure is seen, instead, to explore more efficiently larger portions of the design space. The lateral manoeuvring of very flexible partially-supported wings is then considered. A flight-dynamics model based on elastified stability derivatives is shown to capture the relevant dynamics either under slow actuation or for stiff wings, and it is hence used as a reference. Embedding the full aeroelastic description into the optimisation framework expands the space of achievable manoeuvres, allowing for quick wing response with low structural vibrations or large lateral forces with minimal lift losses.Open Acces
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