Determinação da freqüência natural de eixos utilizando o software excel

This work aims to determine the first natural frequency of rotation shaft by using a basic software, Excel, and to compare it to the values obtained in laboratory. When an axle is submitted to a rotation, depending on the rotational frequency used, the axle can enter a state of resonance, in which the amplitude of vibration becomes rather high. The frequencies in which the resonance is observed depends on several parameters of the axle, including the number of concentrated masses associated to the axle. Thus, to obtain a computer program of easy use and access, which can preview the frequency of resonance of an axle in rotation with ‘n’ numbers of concentrated masses it has been studied how the frequency varies with each of these parameters. The computer program and the analyses have been made using the Rayleigh Method, which allowed the transformation of a continuous system to discrete through the theory of finite elements, which has proved that, the bigger the number of divisions of the shaft taken into consideration in the calculus of the natural frequency, the more this value gets close to the real value. The results obtained have been considered satisfactory once these have gotten close to the theoretical results expected

On the optimum parameters of a device for harvesting energy from running and walking

This paper investigates the feasibility of using an energy harvesting device tuned such that its natural frequency coincides with higher harmonics of the input to capture energy from walking or running human motion more efficiently. The paper starts by reviewing the concept of a linear resonant generator for a tonal frequency input and then derives an expression for the power harvested for an input with several harmonics. The amount of power harvested is estimated numerically using measured data from human subjects. Assuming that the input is periodic, the signal is reconstructed using a Fourier series before being used in the simulation. It is found that although the power output depends on the input frequency, the choice of tuning the natural frequency of the device to coincide with a particular higher harmonic is restricted by the amount of damping that is needed to maximize the amount of power harvested, as well as to comply with the size limit of the device. It is also found that it is not feasible to tune the device to match the first few harmonics when the size of the device is small, because a large amount of damping is required to limit the motion of the mass.

Robust broadband vibration control of a flexible structure using an electrical dynamic absorber

This paper presents a simple but practical feedback control method to suppress the vibration of a flexible structure in the frequency range between 10 Hz and 1 kHz. A dynamic vibration absorber is designed for this, which has a natural frequency of 100 Hz and a normalized bandwidth (twice the damping ratio) of 9.9. The absorber is realized electrically by feeding back the structural acceleration at one position on the host structure to a collocated piezoceramic patch actuator via an analog controller consisting of a second-order lowpass filter. This absorber is equivalent to a single degree-of-freedom mechanical oscillator consisting of a serially connected mass-spring-damper system. A first-order lowpass filter is additionally used to improve stability at very high frequencies. Experiments were conducted on a free-free beam embedded with a piezoceramic patch actuator and an accelerometer at its center. It is demonstrated that the single absorber can simultaneously suppress multiple vibration modes within the control bandwidth. It is further shown that the control system is robust to slight changes in the plant. The method described can be applied to many other practical structures, after retuning the absorber parameters for the structure under control.

Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness

Engineers often face the challenge of reducing the level of vibrations experienced by a given payload or those transmitted to the support structure to which a vibrating source is attached. In order to increase the range over which vibrations are isolated, soft mounts are often used in practice. The drawback of this approach is the static displacement may be too large for reasons of available space for example. Ideally, a vibration isolator should have a high-static stiffness, to withstand static loads without too large a displacement, and at the same time, a low dynamic stiffness so that the natural frequency of the system is as low as possible which will result in an increased isolation region. These two effects are mutually exclusive in linear isolators but can be overcome if properly configured nonlinear isolators are used. This paper is concerned with the characterisation of such a nonlinear isolator comprising three springs, two of which are configured to reduce the dynamic stiffness of the isolator. The dynamic behaviour of the isolator supporting a lumped mass is investigated using force and displacement transmissibility, which are derived by modelling the dynamic system as a single-degree-of-freedom system. This results in the system dynamics being approximately described by the Duffing equation. For a linear isolator, the dynamics of the system are the same regardless if the source of the excitation is a harmonic force acting on the payload (force transmissibility) or a harmonic motion of the base (displacement transmissibility) on which the payload is mounted. In this paper these two expressions are compared for the nonlinear isolator and it is shown that they differ. A particular feature of the displacement transmissibility is that the response is unbounded at the nonlinear resonance frequency unless the damping in the isolator is greater than some threshold value, which is not the case for force transmissibility. An explanation for this is offered in the paper. (C) 2011 Elsevier Ltd. All rights reserved.

On saturation control of a non-ideal vibrating portal frame foundation type shear-building

In this paper, we examine the nonlinear control method based on the saturation phenomenon and of systems coupled with quadratic nonlinear ties applied to a shear-building portal plane frame foundation that supports an unbalanced direct cut-rent with limited power supply (non-ideal system). We analyze the equations of motion by using the method of averaging and numerical simulation. The interaction of the non-ideal structure with the saturation controller may lead to the occurrence of interesting phenomena during the forward passage through the several resonance states of the systems. Special attention is focused on passage through resonance when the non-ideal excitation frequency is near the portal frame natural frequency and when the non-ideal system frequency is approximately twice the controller frequency (two-to-one internal resonance).

Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: Part II: Nonideal energy source

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On non-ideal simple portal frame structural model: Experimental results under a non-ideal excitation

We present measurements of the non-linear oscillations of a portal frame foundation for a non-ideal motor. We consider a three-time redundant structure with two columns, clamped in their bases and a horizontal beam. An electrical unbalanced motor is mounted at mid span of the beam. Two non-linear phenomena are studied: a) mode saturation and energy transfer between modes; b) interaction between high amplitude motions of the structure and the rotation regime of a real limited power motor. The dynamic characteristics of the structure were chosen to have one-to-two internal resonance between the anti-symmetrical mode (sway motions) and the first symmetrical mode natural frequencies. As the excitation frequency reaches near resonance conditions with the 2nd natural frequency, the amplitude of this mode grows up to a certain level and then it saturates. The surplus energy pumped into the system is transferred to the sway mode, which experiences a sudden increase in its amplitude. Energy is transformed from low amplitude high frequency motion into high amplitude low frequency motion. Such a transformation is potentially dangerous.We consider the fact that real motors, such as the one used in this study, have limited power output. In this case, this energy source is said to be non-ideal, in contrast to the ideal source whose amplitude and frequency are independent of the motion of the structure. Our experimental research detected the Sommerfeld Effect: as the motor accelerates to reach near resonant conditions, a considerable part of its output energy is consumed to generate large amplitude motions of the structure and not to increase its own angular speed. For certain parameters of the system, the motor can get stuck at resonance not having enough power to reach higher rotation regimes. If some more power is available, jump phenomena may occur from near resonance to considerably higher motor speed regimes, no stable motions being possible between these two.

A contribution for nonlinear structural dynamics characterization of cantilever beams

Successful experiments in nonlinear vibrations have been carried out with cantilever beams under harmonic base excitation. A flexible slender cantilever has been chosen as a convenient structure to exhibit modal interactions, subharmonic, superharmonic and chaotic motions, and others interesting nonlinear phenomena. The tools employed to analyze the dynamics of the beam generally include frequency- and force-response curves. To produce force-response curves, one keeps the excitation frequency constant and slowly varies the excitation amplitude, on the other hand, to produce frequency-response curves, one keeps the excitation amplitude fixed and slowly varies the excitation frequency. However, keeping the excitation amplitude constant while varying the excitation frequency is a difficult task with an open-loop measurement system. In this paper, it is proposed a closed-loop monitor vibration system available with the electromagnetic shaker in order to keep the harmonic base excitation amplitude constant. This experimental setup constitutes a significant improvement to produce frequency-response curves and the advantages of this setup are evaluated in a case study. The beam is excited with a periodic base motion transverse to the axis of the beam near the third natural frequency. Modal interactions and two-period quasi-periodic motion are observed involving the first and the third modes. Frequency-response curves, phase space and Poincaré map are used to characterize the dynamics of the beam.

On the undamped free vibration of a mass interacting with a Hertzian contact stiffness

Three methods are used to determine the natural frequency of undamped free vibration of a mass interacting with a Hertzian contact stiffness. The exact value is determined using the first integral of motion. The harmonic balance method is used on a transformed equation for an approximate solution, and the multiple scales method is used on an approximate equation. The maximum initial displacement avoiding contact loss is also determined, and the corresponding exact natural frequency is also obtained analytically. The methods are evaluated by studying the free vibration of an elastic sphere on a flat rigid surface. © 2011 Elsevier Ltd © 2011 Elsevier Ltd. All rights reserved.

On the dynamic behaviour of a nonlinear system weakly connected to a linear single degree-of-freedom system

This paper discusses the dynamic behaviour of a nonlinear two degree-of-freedom system consisting of a harmonically excited linear oscillator weakly connected to a nonlinear attachment that behaves as a hardening Duffing oscillator. A system which behaves in this way could be a shaker (linear system) driving a nonlinear isolator. The mass of the nonlinear system is taken to be much less than that in the linear system and thus the nonlinear system has little effect on the dynamics of the linear system. Of particular interest is the situation when the linear natural frequency of the nonlinear system is less than the natural frequency of the linear system such that the frequency response curve of the nonlinear system bends to higher frequencies and thus interacts with the resonance frequency of the linear system. It is shown that for some values of the system parameters a complicated frequency response curve for the nonlinear system can occur; closed detached curves can appear as a part of the overall amplitude-frequency response. The reason why these detached curves appear is presented and approximate analytical expressions for the jump-up and jump-down frequencies of the system under investigation are given.

Passive vibration isolation using axially loaded curved beams

A nonlinear spring element of a vibration isolator should ideally possess high static and low dynamic stiffness. A buckled beam may be a good candidate to fulfil this requirement provided its internal resonance frequencies are high enough to achieve a wide frequency range of isolation. If a straight beam is used, there is a singularity in the force-displacement characteristic. To smooth this characteristic and eliminate the singularity at the buckling point, beams with initial constant curvature along their length are investigated here as an alternative to the buckled straight beam. Their force displacement characteristics are compared with different initial curvature and with a straight buckled beam. The minimum achievable dynamic stiffness with its corresponding static stiffness is compared for different initial curvatures. A case study is considered where the beams are optimized to isolate a one kilogram mass and to achieve a natural frequency of 1 Hz, considering small amplitudes of vibration. Resonance frequencies of the optimized beams for different curvature are presented. It is shown that an order of magnitude reduction in stiffness compared with a linear spring is achievable, while the internal resonance frequencies of the curved beam are high enough to achieve an acceptable frequency range of isolation.

Um estudo da dinâmica fracamente não-linear de um sistema nanomecânico

Pós-graduação em Matemática - IBILCE

An investigation into the simultaneous use of a resonator as an energy harvester and a vibration absorber

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

Análise dinâmica de mecanismo articulado de suspensão com não-linearidade na rigidez devido à geometria e à excitação por desbalanceamento rotativo na condição não ideal

Pós-graduação em Engenharia Mecânica - FEB

On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface

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