Shafic S. Oueini

A Nonlinear Vibration Absorber for Flexible Structures

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.

DYNAMICS OF A CUBIC NONLINEAR VIBRATION 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.

A Theoretical and Experimental Implementation of a Control Method Based on Saturation

A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber, which is built in electronic circuitry, takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one. When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode.A working model of using saturation to suppress the vibrations of a rigid beam connected to a DC motor has been built. An electronic oscillator is built, and its frequency is set at one-half the frequency of the beam. The output from a sensor on the beam is multiplied by the output from the electronic oscillator and a suitable gain, and the result is used as the forcing term for the oscillator. At the same time, the output from the oscillator is squared and multiplied by a suitable gain, and that result is used as the input to the motor. The oscillator/actuator and the beam act as the two modes of a two-degree-of-freedom quadratically coupled system with a 2:1 autoparametric resonance. When the beam is excited by a harmonic force, its motion quickly becomes saturated, and most of the energy imparted to the beam by the harmonic force is transferred to the electronic circuit and from there to the actuator. Thus, the harmonic force is made to work against itself. As a result, the motion of the beam always remains small.

Analysis and Application of a Nonlinear Vibration Absorber

The authors investigate theoretically and experimentally the performance of a recently developed quadratic vibration absorber that is based on the saturation phenomenon. They consider the problem of con trolling the vibrations of a single-degree-of-freedom plant, develop the equations governing the response of the closed-loop system, and obtain an approximate solution. They investigate the strategy by studying its steady-state characteristics and comparing its performance with that of a linear tuned absorber. Then, they implement both techniques experimentally using a digital signal processing device. The authors develop a software algorithm that allows for automatic real-time tracking of the system response. They use both tech niques to control high-amplitude responses of a cantilever beam fitted with piezoceramic actuators. When each absorbers frequency is tuned properly, they show that both schemes possess similar suppression band widths. In addition, they demonstrate that the power requirement of the quadratic absorber can be reduced by judicially modifying its control signal.

Response of Two Quadratically-Coupled Oscillators to a Principal Parametric Excitation

The authors study the dynamics of two oscillators coupled with quadratic nonlinearities in the case of two-to-one internal resonance when the higher mode is subjected to a principal parametric excitation. They use the method of multiple scales to obtain an approximate solution to the equations of motion and investigate theoretically its stability. Then, they verify the analysis experimentally. The authors use a cantilever steel beam and an analog second-order circuit to represent the two oscillators. The interaction between the two systems is achieved by fitting the beam with piezoceramic actuators and a strain gage and coupling the beam with the circuit through electronically generated quadratic nonlinearities. They subject the first mode of the beam to a principal parametric excitation and tune the frequency of the circuit to approximately one-half the frequency of the first mode of the beam. The theoretical and experimental results indicate that the system exhibits complicated responses, such as jumps, the saturation phenomenon, types I and II intermittency, as well as periodic, periodically, and chaotically modulated motions.

Multimode Control of Flexible Structures Using Saturation

We propose a new strategy for controlling the response of distributed-p arameter systems subjected to multifrequenc y resonant excitations. The technique exploits the saturation phenomenon exhibited by multidegree-of-freedom systems coupled with quadratic nonlinearities and possessing two-to-one autoparametric resonances. The strategy consists of introducing a series of second-order circuits and coupling them with the plant through an actuator and a quadratic feedback control law. Once the plant is forced near its resonances, the responses of the excited modes saturate and the oscillatory energy is channeled into the circuits. We consider the problem of suppressing the oscillations of a flexible cantilever beam where the actuation is provided by piezoceramic patches, and the feedback signal is generated by a strain gauge attached to the beam. We present theoretical and experimental results of the application of the control strategy. First, the equations of motion are developed and analyzed through perturbation techniques. Second, the strategy is tested by regulating the response of a cantilever beam that is subjected to two simultaneous resonant excitations.

Saturation control of a dc motor

A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber takes advantage of the saturation phenomenon that occurs in autoparametric multi-DOF systems coupled with quadratic nonlinearities. The strategy is based on introducing a supplementary second-order controller that is coupled to the plant via an actuator and a nonlinear feedback control law. Once the plant is forced near resonance, its response becomes quickly saturated and the remaining oscillatory energy is channeled to the controller. We demonstrate theoretically and experimentally the application of the strategy to regulate the oscillations of a dc motor. We analyze the equations of motion and we assemble a circuit that emulates the controller equation and conduct analog simulations to investigate the technique. Finally, we perform experiments by using the circuit to suppress the vibrations of a rigid beam connected to a dc motor. Our initial investigations demonstrate that saturation control can be implemented very successfully to suppress the vibrations of resonantly excited systems. (Author)

Regulation of a Two-Degree-of-Freedom Structure Using Internal Resonance

In this paper, we present a first attempt at using an energy based control technique to regulate the oscillations of a flexible joint, flexible arm device, through computer simulation. This technique takes advantage of the Internal Resonance (IR) phenomenon. The plant is governed by two coupled linear differential equations. The control scheme is implemented by introducing two software based controllers which are coupled dynamically with the plant through a nonlinear feedback control law. At Internal Resonance, the nonlinear coupling generates an energy link between the plant and the controllers. Thus, energy is transferred from the plant to the controllers where two active damping mechanisms subsequently dissipate it. Here the response of the structure is regulated with a single input torque applied to one plant coordinate. The theoretical analysis is based on the two-variable expansion perturbation method. Thereafter, the analytical findings are verified numerically. Simulation results indicate that the IR control strategy is able to effectively quench the oscillations of the plant.

Terfenol-D nonlinear vibration absorber

An active non-linear vibration absorber for flexible structures is developed. The absorber exploits the inherent quadratic nonlinearity of the actuator material Terfenol-D to produce a two-to-one autoparametric resonance between the forced vibrations of a structure and a second-order analog controller circuit. Nonlinear resonance of this type exhibits the well-known saturation phenomenon. When the structure is forced near resonance, its response saturates to a small value. This type of control has been demonstrated by previous researchers using linear actuators where nonlinearities were introduced via the analog circuit. In contrast, we use the natural nonlinearity of the Terfenol-D material to achieve the same results. We develop the theory and present experimental results for the control of the first and second modes of a cantilever beam. We also consider the application of the strategy experimentally when the forcing is due to a rotating imbalance. In this case, the excitation source is nonideal. Our results indicate that the saturation based control technique implemented with a Terfenol-D actuator constitutes an effective nonlinear vibration absorber.

Control of a single-degree-of-freedom system under principal parametric excitation

We consider the problem of suppressing the vibrations of a structure that is subjected to a principal parametric excitation. The vibration amplitudes resulting from such resonance cannot be fully controlled by conventional techniques, such as the addition of linear damping through velocity feedback or by the implementation of conventional mass absorbers. However, it has been shown that the growth of the response is limited by nonlinearities. In this work, we capitalize on this fact and devise a simple nonlinear feedback law to suppress the vibrations of a cantilever beam when subjected to a parametric resonance. We model the dynamics of the first mode of the beam with a second-order nonlinear ordinary-different equation. The model accounts for viscous damping, air drag, and inertia and geometric nonlinearities. We propose a control law based on cubic position and velocity feedback. We use the method of multiple scales to derive two first-order ordinary- differential equations that govern the time variation of the amplitude and phase of the response. We conduct a stability study and analyze the effect of the control gains on the response of the system. The results show that cubic velocity feedback leads to effective vibration suppression and bifurcation control.