An approach for implementing an active nonlinear vibration absorber for flexible structures is presented. The technique exploits the saturation phenomenon exhibited by multidegree-of-freedom systems with quadratic nonlinearities possessing two-to-one autoparametric resonances. The strategy consists of introducing second-order controllers and coupling each of them with the plant through a sensor and an actuator, where both the feedback and control signals are quadratic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical and experimental results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. Then, the strategy is tested by assembling the controllers in electronic components and suppressing the vibrations of the first and second modes of two beams.

A Nonlinear Vibration Absorber for Flexible Structures

This letter reports a piezoelectric cantilever-pendulum design for multi-directional energy harvesting. A pendulum is attached to the tip of a piezoelectric cantilever-type energy harvester. This design aims at taking advantage of the nonlinear coupling between the pendulum motion in 3-dimensional space and the beam bending vibration at resonances. Experimental studies indicate that, under properly chosen parameters, 1:2 internal resonance can be induced, which enables the multi-directional energy harvesting with a single cantilever. The advantages of the design with respect to traditional piezoelectric cantilever are examined.

Multi-directional energy harvesting by piezoelectric cantilever-pendulum with internal resonance

Dynamics of autoparametric vibration absorbers using multiple pendulums

In this paper we investigate the dynamics of tuned mass absorber with additional viscous damper and inerter attached to the pendulum. The devices are used to damp out oscillations of non-linear Duffing oscillator. Analysis of how these devices influence the dynamics of tuned mass absorber and its damping properties is shown. We calculate the detailed bifurcation diagrams and show how by changing the parameters of damper and inerter one can eliminate dangerous dynamic instabilities from the systems. Finally, in the last section we present an optimization of TMA׳s parameters in order to achieve best efficiency in damping of the main body vibrations.

The application of inerter in tuned mass absorber

We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.

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Dynamics of a Cubic Nonlinear Vibration Absorber

The autoparametric system consisting of a pendulum attached to a primary spring-mass is known to exhibit 1:2 internal resonance, and amplitude-modulated chaos under harmonic forcing conditions. First-order averaging studies and an analysis of the amplitude dynamics predicts that the response curves of the system exhibit saturation. The period-doubling route to chaos is observed following a Hopf bifurcation to limit cycles. However, to answer questions about the range of the small parameter e (a function of the forcing amplitude) for which the solutions are valid, and about the persistence of some unstable dynamical behavior, like saturation, higher-order non-linear effects need to be taken into account. Second-order averaging of the system is undertaken to address these issues. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of e appearing in the averaged equations. For larger e, second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response. For the response between the two Hopf bifurcation points from the coupled-mode solution branch, the period-doubling as well as the Silnikov mechanism for chaos are observed. The predictions of the averaged equations are verified qualitatively for the original equations.

Resonant dynamics of an autoparametric system : A study using higher-order averaging

Two degree-of-freedom systems with weak quadratic nonlinearities are studied under weak external and parametric excitations respectively. All six possible cases, that arise in the presence of 1∶2 internal resonance, are investigated. The method of averaging is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order asymptotic approximation to the response. An analytical technique, based on Melnikov's method is used to predict the parameter range for which chaotic dynamics exists in the undamped averaged system. Numerical studies show that such chaotic responses are quite common in these quadratic systems, and they seem to persist even in the presence of damping.

Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities

Second Order Averaging Study of an Autoparametric System

The dynamics of two degree-of-freedom dynamical systems with 1:2 subharmonic internal resonance and weak quadratic nonlinearities is analyzed in the neighbourhood of bifurcation points, when the excitation frequency varies slowly through the region of primary resonance. The slowly evolving averaged equations are numerically studied for motions initiated in the vicinity of the stationary responses. An analytical technique, based on the Dynamic Bifurcation Theory is then developed to explain the numerical observations for slow frequency sweep through the bifurcations, and the effects of system parameters on the dynamics near a Pitchfork Point are determined.

Nonstationary Responses in Two Degree-of-Freedom Nonlinear Systems with 1-to-2 Internal Resonance

The dynamics of two-degrees-of-freedom dynamical systems with weak quadratic nonlinearities is analyzed in the neighborhood of bifurcation points when the excitation frequency varies slowly through the region of primary resonance. The two modes of vibration are in 1: 2 subharmonic internal resonance. The slowly evolving averaged equations are numerically studied for motions initiated in the vicinity of stationary responses, and observations are made about the nature of responses of the system near the transition from single-mode to coupled-mode solutions (pitchfork points), and near jump and Hopf bifurcations in the coupled-mode solutions. An analytical technique based on the dynamic bifurcation theory is developed to explain the numerical observations for passage through the bifurcations. A numerical study is carried out to determine the effects of system parameters on the dynamics near the pitchfork bifurcation points and results are compared with analytical and numerical descriptions of dynamics.

Bappaditya Banerjee

Papers

Resonant dynamics of an autoparametric system : A study using higher-order averaging

Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities

Second Order Averaging Study of an Autoparametric System

Nonstationary Responses in Two Degree-of-Freedom Nonlinear Systems with 1-to-2 Internal Resonance

Non-Stationary Responses in Externally Excited Two-Degrees-of-Freedom Nonlinear Systems with 1: 2 Internal Resonance