A note about the appearance of non-hyperbolic solutions in a mechanical pendulum system

The investigation of the behavior of a nonlinear system consists in the analysis of different stages of its motion, where the complexity varies with the proximity of a resonance region. Near this region the stability domain of the system undergoes sudden changes due basically to competition and interaction between periodic and saddle solutions inside the phase portrait, leading to the occurrence of the most different phenomena. Depending of the domain of the chosen control parameter, these events can reveal interesting geometric features of the system so that the phase portrait is not capable to express all them, since the projection of these solutions on the two-dimensional surface can hide some aspects of these events. In this work we will investigate the numerical solutions of a particular pendulum system close to a secondary resonance region, where we vary the control parameter in a restrict domain in order to draw a preliminary identification about what happens with this system. This domain includes the appearance of non-hyperbolic solutions where the basin of attraction in the center of the phase portrait diminishes considerably, almost disappearing, and afterwards its size increases with the direction of motion inverted. This phenomenon delimits a boundary between low and high frequency of the external excitation.

Non-linear dynamics of a tower orchard sprayer based on an inverted pendulum model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A demonstration of dry and viscous damping of an oscillating pendulum

Damped oscillatory motion is one of the most widely studied movements in physics courses. Despite this fact, dry damped oscillatory motion is not commonly discussed in physics textbooks. In this work, we discuss the dry and viscous dampec pendulum, in a teaching experiment that can easily be performed by physics or engineering students.

A structure excited by a DC motor of limited power supply and controlled by a pendulum like device

In this paper, a mathematical model is derived via Lagrange's Equation for a shear building structure that acts as a foundation of a non-ideal direct current electric motor, controlled by a mass loose inside a circular carving. Non-ideal sources of vibrations of structures are those whose characteristics are coupled to the motion of the structure, not being a function of time only as in the ideal case. Thus, in this case, an additional equation of motion is written, related to the motor rotation, coupled to the equation describing the horizontal motion of the shear building. This kind of problem can lead to the so-called Sommerfeld effect: steady state frequencies of the motor will usually increase as more power (voltage) is given to it in a step-by-step fashion. When a resonance condition with the structure is reached, the better part of this energy is consumed to generate large amplitude vibrations of the foundation without sensible change of the motor frequency as before. If additional increase steps in voltage are made, one may reach a situation where the rotor will jump to higher rotation regimes, no steady states being stable in between. As a device of passive control of both large amplitude vibrations and the Sommerfeld effect, a scheme is proposed using a point mass free to bounce back and forth inside a circular carving in the suspended mass of the structure. Numerical simulations of the model are also presented Copyright © 2007 by ASME.

A non-ideal portal frame energy harvester controlled using a pendulum

A model of energy harvester based on a simple portal frame structure is presented. The system is considered to be non-ideal system (NIS) due to interaction with the energy source, a DC motor with limited power supply and the system structure. The nonlinearities present in the piezoelectric material are considered in the piezoelectric coupling mathematical model. The system is a bi-stable Duffing oscillator presenting a chaotic behavior. Analyzing the average power variation, and bifurcation diagrams, the value of the control variable that optimizes power or average value that stabilizes the chaotic system in the periodic orbit is determined. The control sensitivity is determined to parametric errors in the damping and stiffness parameters of the portal frame. The proposed passive control technique uses a simple pendulum to tuned to the vibration of the structure to improve the energy harvesting. The results show that with the implementation of the control strategy it is possible to eliminate the need for active or semi active control, usually more complex. The control also provides a way to regulate the energy captured to a desired operating frequency. © 2013 EDP Sciences and Springer.

Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker

In this paper the dynamical interactions of a double pendulum arm and an electromechanical shaker is investigated. The double pendulum is a three degree of freedom system coupled to an RLC circuit based nonlinear shaker through a magnetic field, and the capacitor voltage is a nonlinear function of the instantaneous electric charge. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space and this behaviour is characterized by power spectral density and Lyapunov exponents. The bifurcation diagram is constructed to explore the qualitative behaviour of the system. This kind of electromechanical system is frequently found in robotic systems, and in order to suppress the chaotic motion, the State-Dependent Riccati Equation (SDRE) control and the Nonlinear Saturation control (NSC) techniques are analyzed. The robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties.

Anelastic spectroscopy in SmBa2Cu3O7

The possibility of variable stoichiometry and the high mobility of oxygen in the CuOx planes of SmBa2Cu3O7 give rise to a rich phase diagram. Measurements of the elastic energy loss and modulus (anelastic spectroscopy) as a function of temperature can distinguish among the different atomic jumps, which occur in the various phases or at different local ordering. In this paper, it is reported anelastic relaxation measurements in SmBa2Cu3O7, above room temperature, using a torsion pendulum operating in frequencies around 40 Hz. The mobility of oxygen atoms in the CuOx planes in the various phases has been discussed and the thermally activated peak of elastic energy dissipation observed around 500 K was interpreted in that framework. (C) 2004 Elsevier B.V. All rights reserved.

Measurement of oxygen atom diffusion in Nb and Ta by anelastic spectroscopy

Impurity interstitial atoms present in metals with BCC structure can diffuse in the metallic matrix by jumps to energetically equivalent crystallographic sites. Anelastic spectroscopy (internal friction) is based on the measurement of mechanical loss or internal friction as a function of temperature. Due to its selective and nondestructive nature, anelastic spectroscopy is well suited for the study of diffusion of interstitial elements in metals. Internal friction measurements were made using the torsion pendulum technique with oscillation frequency of a few Hz, temperature interval from 300 to 700 K, heating rate of about 1 K/min, and vacuum better than 10-5 mbar. The polycrystalline Nb and Ta samples used were supplied by Aldrich Inc. The results obtained showed thermally activated relaxation structures due to stress-induced ordering of oxygen atoms around the Nb (or Ta) atoms of the metallic matrix. The results were interpreted by three methods and led to activation enthalpy values for the diffusion of oxygen in Nb and Ta of 1.15 eV and 1.10 eV, respectively.

Determination of the oxygen and nitrogen interstitial diffusion coeficient in niobium by mechanical spectroscopy

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Study of interstitial oxygen mobility in RuSr2GdCu2O8 by internal friction

The discovery of the spatial uniform coexistence of superconductivity and ferromagnetism in ruthenocuprates, RuSr2GdCu2O8 (Ru-1212), has spurred an extraordinary development in the study of the competition between magnetism and superconductivity. However, several points of their preparation process and characterization that determine their superconductive behaviour are still obscure. The improvement of sample preparation conditions involves some thermal treatments in inert atmosphere. Anelastic spectroscopy measurements were made using an inverted torsion pendulum, operating with an oscillation frequency of 38 Hz, temperature in the 90 and 310 K range, heating rate of 1 K/min, and vacuum better than 10(-3) Pa. The results show anelastic relaxation peaks at 210 K related to the presence of interstitial oxygen atoms. The peaks decrease significantly with the oxygen loss caused by the heat treatments in vacuum, appearing again after the annealing of the sample in an oxygen atmosphere. These observed peaks are clearly related to the additional oxygen atoms, with activation energy 0.13 and 0.36 eV, and can be explained in terms by diffusional jumps of interstitial oxygen in the RuO2 planes. (C) 2009 Elsevier B.V. All rights reserved.

Anelastic spectroscopy in a Ti alloy used as biomaterial

Ti and its alloys have been used thoroughly in the production of prostheses and dental implants due to their properties, such as high corrosion resistance, low elasticity modulus and high mechanical strength/density relation. Among the Ti-based alloys, the Ti-35Nb-7Zr-5Ta (TNZT) is one that presents the smallest elasticity modulus, making it an excellent alternative to be used as a biomaterial. In this paper, mechanical spectroscopy measurements were made in TNZT alloys containing several quantities of oxygen and nitrogen in solid solution. Mechanical spectroscopy measurements were made by using a torsion pendulum, operating at an oscillation frequency in the interval 4-30 Hz, temperature in the range 100-700 K, heating rate of about 1 K/min and vacuum lower than 10(-5) Torr. Complex relaxation structures and a reduction in the elasticity modulus were observed for the heat-treated and doped samples. The observed peaks were associated with the interactions of interstitial atoms and the alloy elements. (C) 2009 Elsevier B.V. All rights reserved.

Anelastic relaxation due to hydrogen in Ti-35Nb-7Zr-5Ta alloy

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Nitrogen diffusion in the Nb-1.0wt%Zr measured by mechanical spectroscopy

The scientific and technological development in the area of new materials contributed to several applications of niobium and its alloys in nuclear power plants as well as in aerospace, aeronautics, automobile and naval industries. This paper presents the interstitial diffusion coefficients of nitrogen in solid solution in the Nb-1.0wt%Zr alloy using internal friction measurements obtained by mechanical spectroscopy, which uses a torsion pendulum operating at an oscillation frequency between 1.0 Hz and 10.0 Hz. The temperature range varies from 300K to 700K, at a heating rate of 1 K/min and vacuum better than 2 x 10(-6) Torr. The results showed an increase of the interstitial diffusion coefficient of nitrogen that was correlated with configurational considerations for the octahedral interstitials.

Diffusion of Interstitial Solutes in Nb-46wt% Ti Alloys Measured by Mechanical Spectroscopy

The mechanical properties of metals with a body-centered cubic (bcc) structure, such as Nb, Ta, V, and their alloys, are modified with the introduction of interstitial impurities, such as O, N, C, or H. These metals can dissolve great amounts of O and N, for example, to form solid solutions. The interstitial solute atoms (ISA) in metals with a bcc structure occupy octahedral sites and cause local distortion with tetragonal symmetry. So ISA in these metals forms an elastic dipole that can align along one of the three cubic axis of the crystal. In the present paper, the torsion pendulum technique was employed for the investigation of various interactions among the metallic matrix and different interstitial solutes in the Nb-46wt%Ti alloy. From the relaxation spectra, we obtained the diffusion coefficients, pre-exponential factors, and activation energies for nitrogen in the Nb-46wt%Ti alloy.