Active damping of the vibrations of a plate using sensor data: effect of boundary conditions

A new approach is followed to study the effect of mixed mechanical boundary conditions on the effectiveness of active damping of the forced resonant vibrations of thermoviscoelastic orthotropic plates. The problem is solved by the Bubnov–Galerkin method. Formulas for the voltage that should be applied to the actuator to damp the first vibration mode are derived. It is shown that the mechanical boundary conditions, the dissipative properties of the material, and the dimenstions of the sensors and actuators have a strong effect on the effectiveness of active damping of the vibrations of plates     

Forced axisymmetric vibrations and self-heating of a circular viscoelastic plate,controlled with piezoelectric sensors and actuators

The forced monoharmonic vibrations and self-heating of a circular thermoviscoelastic plate with piezoelectric sensor and actuator are studied. The viscoelastic behavior of the passive (without piezoeffect) and piezoactive materials is described using the concept of complex moduli. The problems of electroelasticity and heat conduction are solved numerically, assuming that the mechanical load is unknown. The effect of self-heating on the active damping of the vibrations of the plate is analyzed     

Basic equations of the theory of thermoviscoelastic plates with distributed sensors and actuators

The basic equations of the theory of thermoviscoelastic thin-walled plates with piezoelectric sensors and actuators under monoharmonic mechanical and electric loading are derived using the Kirchhoff–Love hypotheses. The thermomechanical behavior of passive and piezoactive materials is described using the concept of complex characteristics. Methods of solving nonlinear problems of active damping of thermomechanical vibrations of plates with sensors and actuators are considered. The effect of dissipative heating on the damping of axisymmetric vibrations of a thermoviscoelastic solid circular plate is analyzed as an example     

Resonant vibrations of a clamped thermoviscoelastic rectangular plate with sensors and actuators

The paper discusses the active damping of the resonant flexural vibrations of a clamped thermoviscoelastic rectangular plate with distributed piezoelectric sensors and actuators. The thermoviscoelastic behavior of the passive and active materials is described using the concept of complex characteristics. The interaction of the mechanical and thermal fields is taken into account. The Bubnov–Galerkin method is used. The effect of self-heating, the dimensions of the piezoelectric inclusions, and the feedback factor on the effectiveness of active damping of the resonance vibrations of the plate is studied     

Resonant vibrations of a hinged thermoviscoelastic rectangular plate with sensors and actuators

The paper discusses the active damping of the resonant flexural vibrations of a hinged thermoviscoelastic rectangular plate with distributed piezoelectric sensors and actuators. The thermoviscoelastic behavior of the passive and active materials is described using the concept of complex characteristics. The interaction of mechanical and thermal fields is taken into account. The Bubnov–Galerkin method is used. The effect of dissipative heating, the dimensions of the piezoelectric inclusions, and the feedback factor on the effectiveness of active damping of resonance vibrations of the plate is studied     

Control of axisymmetric resonant vibrations and self-heating of shells of revolution with piezoelectric sensors and actuators

The coupled problem of forced vibrations and self-heating of thermoviscoelectroelastic shells of revolution with piezoceramic sensor and actuator under monoharmonic 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 thermoelectroelasticity is solved by time-marching integration, using discrete orhogonalization to integrate the equations of elasticity and explicit finite differencing to solve the heat conduction equation. The effect of the dimensions of the sensor and actuator and self-heating on the sensor voltage and on the active damping of forced vibrations of a circular plate under uniform monoharmonic transverse pressure is studied     

Basic equations for thermoviscoelastic plates with distributed actuators under monoharmonic loading

The basic equations for thin-walled thermoviscoelastic plates with distributed piezoelectric actuators under monoharmonic mechanical and electric loads are derived. The thermomechanical behavior of materials is described using the concept of complex characteristics. Variational methods of solving nonlinear problems of active damping of the bending vibrations of plates are considered. The effect of dissipative heating on the damping of the axisymmetric bending vibrations of a circular plate is examined as an example Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 107–123, February 2009.     

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     

Flexural vibrations and vibrational heating of a ring plate with thin piezoceramic pads under single-frequency electromechanical loading

The paper deals with the coupled problem of flexural vibrations and dissipative heating of a viscoelastic ring plate with piezoceramic actuators under monoharmonic electromechanical loading. The temperature dependence of the complex characteristics of passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoviscoelasticity is solved by an iterative method. At each iteration, orthogonal discretization is used to integrate the equations of elasticity and an explicit finite-difference scheme is used to solve the heat-conduction equation with a nonlinear heat source. The effect of the dissipative heating temperature, boundary conditions, and the thickness and area of the actuator on the active damping of the forced vibrations of the plate under uniform transverse harmonic pressure is examined __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 99–108, February 2008.     

Forced Vibration and Self-Heating of a Thermoviscoelastic Cylindrical Shear Compliant Shell with Piezoelectric Actuators and Sensors*

International Applied Mechanics - The problem of the forced resonant vibrations and dissipative heating of a hinged thermoviscoelastic cylindrical shell with piezoelectric actuators and sensors is...     

Axisymmetric Dynamic Problem of Coupled Thermoviscoplasticity

An axisymmetric dynamic thermoviscoelastic problem is formulated with allowance for the coupling of mechanical and thermal fields. The behavior of the material is described by the Bodner–Partom model. A technique for numerical solution of the problem is developed. The laws governing the stress–strain state and the temperature field of a circular disk under forced flexural vibrations are studied     

Influence of shear strains on damping the vibrations of a rectangular plate with piezoelectric actuators

The problem of active damping of the nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators is solved using Timoshenko’s hypotheses. To this end, two methods are employed: (i) a classical method of balancing the principal vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamical-programming method that reduces the problem to the matrix algebraic Riccati equation. The results obtained by both approaches are compared. The effect of the shear modulus on the amplitude of the damping potential difference and the deflection of the plate is analyzed     

Damping the vibrations of a rectangular plate with piezoelectric actuators

The paper addresses the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators. The problem is solved by two methods: (i) the classical method of balancing the fundamental vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to an algebraic Riccati equation. The results produced by both approaches are presented and compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 78–84, February 2008.     

Thermomechanical response of a viscoelastic rod driven by a sinusoidal displacement

The forced vibrations of a rod of thermoviscoelastic material are studied. The rod is considered to be laterally insulated but not constrained, such that a one-dimensional analysis may be employed. Temperature dependence of the material properties and the resulting thermomechanical coupling effects are included. The vibrations are forced by the imposition of a sinusiodal displacement of known amplitude and frequency at one end of the rod. This problem corresponds to a dissipative material bonded to the surface of a relatively rigid, vibrating structure.Initial transient behavior is not considered. A steady-state response is found by means of a finite difference formulation. Material properties of a Lockheed solid propellant are used.The presence of critical frequencies, characterized by high stresses and temperatures, is found for small amplitudes of vibration. Nonlinearities and instabilities lead to a lack of one-to-one correspondence between stress and displacement boundary conditions. No relationship is found between the critical frequencies of the driven rod and the natural frequencies of a rod with an equivalent temperature profile.     

Modelling the Stationary Vibrations and Dissipative Heating of Thin-Walled Inelastic Elements with Piezoactive Layers

A coupled dynamic problem of thermoelectromechanics for thin-walled multilayer elements is formulated based on a geometrically nonlinear theory and the Kirchhoff–Love hypotheses. In the case of harmonic loading, an approximate formulation is given using the concept of complex moduli to characterize the cyclic properties of the material. The model problem on forced vibrations of sandwich beam, whose core layer is made of a passive physically nonlinear material, and face layers, of a viscoelastic piezoactive material, is considered as an example to demonstrate the possibility of damping the vibrations by applying harmonic voltage to the oppositely polarized layers of the beam. Substantiation is given for a linear control law with a complex coefficient for the electric potential, which provides damping of vibrations in the first symmetric mode at the linear and nonlinear stages of deformation. The stress–strain state and dissipative-heating temperature are studied     

Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers

This paper aims to study the nonlinear-forced vibrations of a viscoelastic cantilever with a piecewise piezoelectric actuator layer on its top surface using the method of Multiple Scales. The governing equation of motion is a second-order nonlinear ordinary differential equation with quadratic and cubic nonlinearities which appear in stiffness, inertia, and damping terms. The nonlinear terms are due to the piezoelectricity, viscoelasticity, and geometry of the system. Forced vibrations of the system are investigated in the cases of primary resonance and non-resonance hard excitation including subharmonic and superharmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including damping coefficient, thickness to width ratio of the beam, length and position of the piezoelectric layer, density of the beam, and the piezoelectric coefficient on the frequency-response curves are discussed for each case. It is shown that in all these cases, the response of the system follows a softening behavior due to the existence of the piezoelectric layer. The piezoelectric layer provides an effective tool for active control of vibration. In addition, the effect of the viscoelasticity of the beam on passive control of amplitude of vibration is illustrated.     

Resonant vibrations of a hinged viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a hinged thermoviscoelastic rectangular plate undergoing resonant flexural vibrations. The thermoviscoelastic behavior of materials is described using the concept of complex characteristics. The coupling of the electromechanical and thermal fields is taken into account. The nonlinear problem is solved by the variational and Bubnov–Galerkin methods. The effect of the dissipative-heating temperature and the dimensions of the sensor on its performance during resonant vibrations of the plate is analyzed.     

Experimental investigation on using the piezoelectric and electromagnetic vibration absorbers in milling

The paper presents an active control system that counteracts the development of chatter vibration. The vibration amplitude depends on the dynamic properties of the machine tool, cutting tool and work-piece. In the paper we analyze the case when the loss of machining stability is caused by the work-piece. The proposed active control system employs electromagnet or piezoelectric actuator to suppress vibration during milling. The active control introduces damping into the system, thereby raising the critical depth of cut and reducing forced vibration amplitude. It enables stable cutting under a much wider range of cutting parameters that for the uncontrolled system. Cutting tests are performed on JAFO FYN-50 machine with mill DIN 845 B-25 K-N HSS to demonstrate an effectiveness of the proposed systems.     

Damping of Forced Vibrations in Systems of Connected Rigid Bodies

The problem of damping forced vibrations in a system of bodies linked by one-degree-of-freedom elastic joints is considered. The vibrations are damped by introducing additional bodies into the system. The controlled motion of these bodies compensates the effect of the external perturbation. The motion of a system of three bodies is studied in detail. To determine the sensitivity of the perturbation compensation method to the damping parameters, a numerical simulation is carried out     

Infinite-Dimensional Control of Nonlinear Beam Vibrations by Piezoelectric Actuator and Sensor Layers

An infinite-dimensional approach for the active vibration control of a multilayered straight composite piezoelectric beam is presented. In order to control the excited beam vibrations, distributed piezoelectric actuator and sensor layers are spatially shaped to achieve a sensor/actuator collocation which fits the control problem. In the sense of von Kármán a nonlinear formulation for the axial strain is used and a nonlinear initial boundary-value problem for the deflection is derived by means of the Hamilton formalism. Three different control strategies are proposed. The first one is an extension of the nonlinear H-design to the infinite-dimensional case. It will be shown that an exact solution of the corresponding Hamilton–Jacobi–Isaacs equation can be found for the beam under investigation and this leads to a control law with optimal damping properties. The second approach is a PD-controller for infinite-dimensional systems and the third strategy makes use of the disturbance compensation idea. Under certain observability assumptions of the free system, the closed loop is asymptotically stable in the sense of Lyapunov. In this way, flexural vibrations which are excited by an axial support motion or by different time varying lateral loadings, can be suppressed in an optimal manner. A numerical example serves both to illustrate the design process and to demonstrate the feasibility of the proposed methods.