Jon R. Pratt

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.

Perturbation Methods in Nonlinear Dynamics—Applications to Machining Dynamics

The role of perturbation methods and bifurcation theory in predicting the stability and complicated dynamics ofmachining is discussed using a nonlinear single-degree-of-freedom model that accounts for the regenerative effect, linear structural damping, quadratic and cubic nonlinear stiffness of the machine tool, and linear, quadratic, and cubic regenerative terms. Using the width of cut w as a bifurcation parameter, we find, using linear theory, that disturbances decay with time and hence chatter does not occur if w w c . In other words, as w increases past W c , a Hopf bifurcation occurs leading to the birth of a limit cycle. Using the method of multiple scales, we obtained the normal form of the Hopf bifurcation by including the effects of the quadratic and cubic nonlinearities. This normal form indicates that the bifurcation is supercritical; that is, local disturbances decay for w w c . Using a six-term harmonic-balance solution, we generated a bifurcation diagram describing the variation of the amplitude of the fundamental harmonic with the width of cut. Using a combination of Floquet theory and Hills determinant, we ascertained the stability of the periodic solutions. There are two cyclic-fold bifurcations, resulting in large-amplitude periodic solutions, hysteresis, jumps, and subcritical instability. As the width of cut w increases, the periodic solutions undergo a secondary Hopf bifurcation, leading to a two-period quasiperiodic motion (a two-torus). The periodic and quasiperiodic solutions are verified using numerical simulation. As w increases further, the torus doubles. Then, the doubled torus breaks down, resulting in a chaotic motion. The different attractors are identified by using phase portraits, Poincare sections, and power spectra. The results indicate the importance of including the nonlinear stiffness terms.

Chatter control and stability analysis of a cantilever boring bar under regenerative cutting conditions

A theoretical and experimental investigation into the stability of a slender boring bar under regenerative cutting conditions is presented. The bar has been equipped with actuators and sensors for feedback control of its structural dynamics. It is modelled at the tool point by a mass–spring–damper system free to move in two mutually perpendicular directions. Our aim is to demonstrate the effect of simple feedback control on the parameter space of chatter–free machining in a boring process using theory and experiment. We reinforce the notion that the system design for control should provide actuation in two orthogonal directions because the cutting forces couple the principal modes of the tool in a complex fashion. Active control of the tool damping in each of the principal modal directions is implemented and shown in theory and experiment to be quite effective at suppressing chatter. Problems caused by jumps from stable to unstable cutting due to nonlinear regenerative chatter effects are also considered. The case where the cutting forces are described by polynomial functions of the chip thickness is examined. We use a perturbation technique to calculate the nonlinear normal form of the governing equations to determine the post–linear instability (bifurcation) behaviour. The predicted bifurcation corresponds to a subcritical Hopf bifurcation, and hence the predicted transition from stable to unstable cutting is not smooth and may possess hysteresis. This result is in qualitative agreement with experimental observations. An active control technique for changing the form of this bifurcation from subcritical to supercritical is presented for a prototypical, single–degree–of–freedom model.

Design and Modeling for Chatter Control

Boring bars for single-point turning on a lathe are particularly susceptible to chatter and have been the subject of numerous studies. Chatter is, in general, caused by instability. Clearly, the cutting process can be limited to regions of known stable operation. However, this severely constrains the machine-tool operation and causes a decrease in productivity. The more aggressive approach is to attack the stability problem directly through application of vibration control. Here, we demonstrate a new biaxial vibration control system (VPI Smart Tool) for boring bars. We present the experimentally determined modal properties of the VPI Smart Tool and demonstrate how these properties may be used to develop models suitable for chatter stability analysis, simulation, and development of feedback compensation. A phenomenological chatter model that captures much of the rich dynamic character observed during experiments is presented. We introduce the notion that the mean cutting force changes direction as the width of cut increases due to the finite nose radius of the tool. This phenomenon is used to explain the progression from chatter that is dominated by motions normal to the machined surface at small widths of cut to chatter that is dominated by motions tangential to the machined surface at large widths of cut. We show experimental evidence to support our assertion that a biaxial actuation scheme is necessary to combat the tendency of the tool to chatter in both directions. We then present some preliminary theoretical results concerning the persistence of subcritical instability as we expand consideration to high-speed machining.

ACTIVE VIBRATION CONTROL FOR CHATTER SUPPRESSION

The vibration absorber, in its various forms, is a well-known control strategy that reduces the forced vibratory response of structures by introducing a carefully tuned, auxiliary degree of freedom. The method is simple, effective, and widely used. Recently, investigations have focused on the use of socalled active vibration absorbers for the suppression of machine-tool chatter. These absorbers are distinguished from their passive counterparts by the incorporation of various sensors and actuators to facilitate feedback control and thereby enhance the combined system response. In this paper, we consider a device, which we call an electronic vibration absorber; it is so-called because the auxiliary degree of freedom is simply a compensator realized via an electronicanalog circuit. In this regard, the technique is similar to the positive-position-feedback control algorithm and other popular second-order-compensation schemes in the literature. First, we develop the general theory for this device. Then, we consider its application to the problem of machine-tool chatter. The plant we consider in this study is a nonlinear single-degree-offreedom, machine-tool-chatter model that is known to possess a jump-type instability. A linear analysis is performed to determine the stability lobes of the controlled system. This analysis reveals that an electronic vibration absorber can increase the limit width of cut predicted in the modelling by approximately an order of magnitude. Furthermore, analogcomputer simulation of the complete nonlinear system predicts that the jump-type instability is eliminated by application of the absorber. We present a biaxial, electronic-vibration-absorber control system

Chatter Identification and Control for a Boring Process

Previous work on machine-tool dynamics has identified three mechanisms that lead to the dynamic instability known as chatter, namely the regenerative effect, velocity-dependent friction or stick-slip, and modecoupling. It has long been recognized that the underlying physics of these mechanisms is nonlinear, yet traditional treatments of the problem focus on linearized reduced-order approaches.

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.

Experimental stability of a time-delay system

The stability and complicated dynamics of a nonlinear single-DOF system subject to time-delay feedback is explored via analog-computer simulation. The system modeled is taken from machine-tool dynamics but is analogous to many other practical engineering systems. The experimental results validate the analytical and computational methods developed by Nayfeh et al. (1995). Good agreement between the theory and experiment is shown. A cyclic-fold bifurcation that leads to large-amplitude limit cycles is observed as predicted. Also, evidence of torus breakdown route to chaos is presented. The dynamics of the system is analyzed using a variety of techniques, including phase portraits, time traces, Poincare sections, power spectra, pseudo-state space construction, and pointwise dimension calculations. (Author)

Smart structures for chatter control

We discuss a vibration measurement and control system developed to investigate the complex dynamics of boring-bar chatter. A fundamental question is how best to integrate the sensors and actuators for effective control. Is it sufficient to control only motions normal to the machined surface? We consider a smart structure that consists of two actuator/sensor pairs oriented orthogonally to control the motions of a boring tool in directions normal and tangential to the machined surface. Actuation is achieved with Terfenol-D struts that sting the tool near its base. We develop a control strategy by considering a single-degree- of-freedom chatter model for tool motions normal to the machined surface, showing that enhanced structural damping is an effective chatter control. We adapt a second-order feedback compensation scheme from the literature and point out the special design considerations engendered by the use of Terfenol-D actuation. We consider chatter signatures obtained using the system with and without feedback control and show that the system is very effective at chatter suppression. Because we may control each of the dominant structural modes independently, we examine the validity of a single-mode approximation by considering chatter signatures obtained with only tangential control active. We find that so-called mode coupling effects persist; hence, we expand our modeling efforts to include a coupling due to the cutting forces. We see that a variety of chatter modes exist, depending on the operating parameter, and that, to achieve the most robust performance from the controlled system, it is advisable to have control of both directions.