Bifurcation analyses on the chatter vibrations of a turning process with state-dependent delay

Bifurcation analyses are performed using the methods of multiple scales and harmonic balance in order to investigate the chatter vibration characteristics of a nonlinear turning system with state-dependent delay. In this study, the tool of the turning system is modeled as a two degrees of freedom oscillator and both the nonlinear stiffness and the nonlinear cutting force are considered. Prior to performing the bifurcation analyses, the nonlinear cutting force is appropriately expanded using a Taylor series, and an eigenanalysis is performed on the linearized system to obtain the linear stability boundaries. The bifurcation analyses are then performed and examined to explain the effect of state-dependent delay on the small- and large-amplitude behavior of the turning system. Analytic results derived from this study are validated through direct comparison with numerical results.     

Laminated piezoelectric plate for active underwater acoustic control

The pressure reflected from a bi-laminated piezoelectric plate has been determined using the Thomson-Haskell matrix method. The plate is composed of a piezoelectric layer with grounded vacuum and an elastic layer in contact with the fluid. An incident plane wave in the fluid medium strikes the plate at different angles. The required electric potential across the piezoelectric layer to cancel the reflection from the fluid/elastic boundary has been determined for the piezoelectric material PZT-5 at various thicknes parameters and incident frequencies. Project supported by the National Natural Science Foundation of China (No. 10172039).     

Free vibrations of a piezoelectric body

We present a systematic analysis of the eigenvalue problem associated with free vibrations of a finite piezoelectric body. The analysis is based on an abstract formulation of the three-dimensional theory of piezoelectricity. A series of fundamental properties of free vibrations of a piezoelectric body are proved concisely. The problem of free vibrations of a piezoelectric plate governed by the two-dimensional plate equations due to Mindlin is treated in a similar manner.Address after September 1, 1994: Department of Mechanical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA.Address after August 16, 1994: Clifton Garvin Professor, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA.     

Optimal placement of piezoelectric active bars in vibration control by topological optimization

A continuous variable optimization method and a topological optimization method are proposed for the vibration control of piezoelectric truss structures by means of the optimal placements of active bars. In this optimization model, a zero-one discrete variable is defined in order to solve the optimal placement of piezoelectric active bars. At the same time, the feedback gains are also optimized as continuous design variables. A two-phase procedure is proposed to solve the optimization problem. The sequential linear programming algorithm is used to solve optimization problem and the sensitivity analysis is carried out for objective and constraint functions to make linear approximations. On the basis of the Newmark time integration of structural transient dynamic responses, a new sensitivity analysis method is developed in this paper for the vibration control problem of piezoelectric truss structures with respect to various kinds of design variables. Numerical examples are given in the paper to demonstrate the effectiveness of the methods.     

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.     

The active control of vibrations of composite beams by parametric stiffness modulation

The paper addresses an active control of the resonant vibrations of composite beams performed by a parametric stiffness modulation. A sandwich beam composition with the continuous core is considered. The stiffness modulation is introduced by some fairly small changes in an orientation of elements of the microstructure of a core ply. The controlled vibrations are those of the dominantly flexural type excited by a transverse force acting at a low resonant frequency, whereas the stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. This difference in time scales of the controlled vibrations and the input signal facilitates a use of the method of direct partition of motion that predicts an existence of the modal interaction between the low-frequency and the high-frequency motions due to so-called vibrational forces. It is shown that such a parametric control can provide a significant favourable shift of the first eigenfrequency of a controlled beam (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled beam. Heavy fluid loading conditions are accounted for as well as material losses in a structure. Then instead of analysis of eigenfrequencies, a problem of forced vibrations is posed and the forced frequency–amplitude response is analysed. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases.     

Non-linear analysis and quench control of chatter in plunge grinding

This paper aims at mitigating regenerative chatter in plunge grinding. To begin with, a dynamic model is proposed to investigate grinding dynamics, where eigenvalue and bifurcation analyses are adopted, respectively, for prediction of grinding stability and chatter. Generally, it is found that most grinding chatter is incurred by subcritical Hopf bifurcation. Compared with supercritical instability, the subcritical generates coexistence of stable and unstable grinding in the stable region and increases chatter amplitude in the chatter region. To avoid these adverse effects of the subcritical instability, bifurcation control is employed, where the cubic non-linearity of the relative velocity between grinding wheel and workpiece is used as feedback. With the increase of feedback gain, the subcritical instability is transformed to be supercritical not only locally but also globally. Finally, the conditionally stable region is completely removed and the chatter amplitude is decreased. After that, to further reduce the chatter amplitude, quench control is used as well. More specifically, an external sinusoid excitation is applied on the wheel to quench the existing grinding chatter, replacing the large-amplitude chatter by a small-amplitude forced vibration. Through the method of multiple scales, the condition for quenching the chatter is obtained.     

Forced axisymmetric vibrations and self-heating of thermoviscoelastic cylindrical shells with piezoelectric actuators

The coupled problem of the forced axisymmetric vibrations and self-heating of electrothermoviscoelastic cylindrical shells with piezoceramic actuators under monoharmonic electromechanical loading is solved. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The coupled nonlinear problem of electrothermoelasticity is solved by using a time-marching method with discrete orthogonalization at each time step (to integrate the equations of elasticity) and an explicit finite-difference method (to solve the heat-conduction equations). An analysis is made of the effect of the boundary conditions at the shell ends, the dimensions of the piezoactuator, and the self-heating temperature on the actuator voltage and the effectiveness of active damping of the forced vibrations of the shell under uniform transverse monoharmonic pressure     

Vibration analysis and active control of nearly periodic two-span beams with piezoelectric actuator/sensor pairs

The piezoelectric materials are used to investigate the active vibration control of ordered/disordered periodic two-span beams. The equation of motion of each sub-beam with piezoelectric patches is established based on Hamilton's principle with an assumed mode method. The velocity feedback control algorithm is used to design the controller. The free and forced vibration behaviors of the two-span beams with the piezoelectric actuators and sensors are analyzed. The vibration properties of the disordered two-span beams caused by misplacing the middle support are also researched. In addition, the effects of the length disorder degree on the vibration performances of the disordered beams are investigated. From the numerical results, it can be concluded that the disorder in the length of the periodic two-span beams will cause vibration localizations of the free and forced vibrations of the structure, and the vibration localization phenomenon will be more and more obvious when the length difference between the two sub-beams increases. Moreover, when the velocity feedback control is used, both the forced and the free vibrations will be suppressed. Meanwhile, the vibration behaviors of the two-span beam are tuned.     

Dynamic modelling and active vibration control of a submerged rectangular plate equipped with piezoelectric sensors and actuators

The active vibration control of a rectangular plate either partially or fully submerged in a fluid was investigated. Piezoelectric sensors and actuators were bonded to the plate, and the assumed mode method was used to derive a dynamic model for the submerged plate. The properties of the piezoelectric actuators and sensors, as well as their coupling to the structure, were used to derive the corresponding equations of their behaviour. The fluid effect was modelled according to the added virtual mass obtained by solving the Laplace equation. The natural vibration characteristics of the plate both in air and in water were obtained theoretically and were found to be consistent with the experimental results, and the changes in the natural frequencies resulting from submersion in fluid can be accurately predicted. A multi-input, multi-output positive position feedback controller was designed by taking the natural vibration characteristics into account and was then implemented by using a digital controller. The experimental results show that piezoelectric sensors and actuators along with the control algorithm can effectively suppress the vibration of a rectangular plate both in air and submerged in a fluid.     

Bifurcation analysis of milling process with tool wear and process damping: regenerative chatter with primary resonance

In this paper, the occurrence of various types of bifurcation including symmetry breaking, period-doubling (flip) and secondary Hopf (Neimark) bifurcations in milling process with tool-wear and process damping effects are investigated. An extended dynamic model of the milling process with tool flank wear, process damping and nonlinearities in regenerative chatter terms is presented. Closed form expressions for the nonlinear cutting forces are derived through their Fourier series components. Non-autonomous parametrically excited equations of the system with time delay terms are developed. The multiple-scale approach is used to construct analytical approximate solutions under primary resonance. Periodic, quasi-periodic and chaotic behavior of the limit cycles is predicted in the presence of regenerative chatter. Detuning parameter (deviation of the tooth passing frequency from the chatter frequency), damping ratio (affected by process damping) and tool-wear width are the bifurcation parameters. Multiple period-doubling and Hopf bifurcations occur when the detuning parameter is varied. As the damping ratio changes, symmetry breaking bifurcation is observed whereas the variation of tool wear width causes both symmetry breaking and Hopf bifurcations. Also, under special damping specifications, chaotic behavior is seen following the Hopf bifurcation.     

Forced vibrations and vibroheating of shallow viscoelastic laminated shells with the piezoelectric effect

The vibrations and dissipative heating of a hinged shallow shell made of viscoelastic piezoelectric material and subject to harmonic electric loading are considered. The basic relations are obtained by using the Kirchhoff-Love mechanical hypotheses supplemented with the respective hypotheses for electric quantities. Analytical solutions of both electromechanical and thermal problems are derived for the case where the temperature is constant along the shell thickness. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 78–87, June, 2000.     

Application of linear and nonlinear vibration absorbers in micro-milling process in order to suppress regenerative chatter

In this paper, a generalized higher-order variable-coefficient nonlinear Schrödinger equation is studied, which describes the propagation of subpicosecond or femtosecond pulses in an inhomogeneous optical fiber. We derive a set of the integrable constraints on the variable coefficients. Under those constraints, via the symbolic computation and modified Hirota method, bilinear equations, one-, two-,three-soliton solutions and dromion-like structures are obtained. Properties and interactions for the solitons are studied: (a) effects on the solitons resulting from the wave number k, third-order dispersion \(\delta _1(z)\), group velocity dispersion \(\alpha (z)\), gain/loss \(\varGamma _2(z)\) and group-velocity-related \(\gamma (z)\) are discussed analytically and graphically where z is the normalized propagation distance along the fiber; (b) bound state with different values of \(\alpha (z)\), \(\delta _1(z)\), \(\gamma (z)\) and \(\varGamma _2(z)\) are presented where some periodic or quasiperiodic formulae are derived. Interactions between the two solitons and between the bound states and a single soliton are, respectively, discussed; and (c) single, double and triple dromion-like structures with different values of \(\alpha (z)\), \(\delta _1(z)\), \(\gamma (z)\) are also presented, distortions of which are found to be determined by those variable coefficients.     

A model of bending vibrations of piezoelectric bimorphs with split electrodes and its applications

We study stationary vibrations and static bending of a bimorph plate consisting of two piezoceramic layers with an infinitely thin split electrode between them. We propose a model taking into account the square root singularity of the electric field structure on the interface between the split electrode regions. For the plane problem, we obtain the equation of motion and formulate the boundary conditions and the transmission conditions on the interface between the split electrode regions. For the piezoceramic PZT-4, we calculate the resonance and antiresonance frequencies and study the dependence of the dynamic electromechanical coupling coefficient on the dimensions of the internal electrode. We show that the use of a plate with a split electrode permits increasing the efficiency of vibration excitation compared with the case of a solid internal electrode. In the case of static bending of the plate-strip, we determine the dimensions of the internal electrode ensuring a significant increase in the deflection at the center of the bimorph.     

Transverse vibrations and vibrational heating of a rectangular bimorphous plate of dissipative piezoelectric material

The problem of induced resonance vibrations and dissipation heating in a rectangular bimorphous plate made of a dissipative piezoelectric material under a harmonic potential difference is tackled. The edges of the plate are considered to be hinged and ideally thermally insulated. The dissipation properties of the material are taken into account on the basis of the concept of complex characteristics, which are assumed to be temperature-independent. An exact solution is found for the problem. The critical value of the load parameter is determined when the maximum temperature reaches the Curie point. A finite-element method has been developed for investigating the dynamic behavior and temperature of vibrational heating that bimorphous plates made of a viscoelastic material undergo under a harmonic load. The results obtained for the electromechanical vibrations of plates by finite-element calculations and by an analytical solution are compared. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 85–93, September, 1999.