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.     

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.     

A review of development and implementation of an active nonlinear vibration absorber

Summary We present an account of an implementation of an active nonlinear vibration absorber that we have developed. The control technique exploits the saturation phenomenon that is known to occur in quadratically-coupled multi-degree-of-freedom systems subjected to primary excitation and possessing a two-to-one internal resonance. The technique is based on introducing an absorber and coupling it with the structure through a sensor and an actuator, where the feedback and control signals are quadratic. First, we consider the case of controlling the vibrations of a single-degree-of-freedom system. We develop the equations governing the response of the closed-loop system and use the method of multiple scales to obtain an approximate solution. We investigate the performance of the control strategy by studying its steady-state and transient characteristics. Additionally, we compare the performance of the quadratic absorber with that of a linear absorber. Then, we present theoretical and experimental results that demonstrate the versatility of the technique. We design an electronic circuit to emulate the absorber and use a variety of sensors and actuators to implement the active control strategy. First, we use a motor and a potentiometer to control the vibration of a rigid beam. We develop a plant model that includes Coulomb friction and demonstrate that the closed-loop system exhibits the saturation phenomenon. Second, we extend the strategy to multi-degree-of-freedom systems. We use PZT ceramics and strain gages to suppress vibrations of flexible steel beams when subjected to single- and simultaneous two-mode excitations. Third, we employ Terfenol-D, a nonlinear actuator, and accelerometers to control the vibrations of flexible beams. In all instances, the technique is successful in reducing the response amplitude of the structures. Received 3 May 1999; accepted for publication 3 June 1999