Free vibration analysis of moderately thick asymmetric piezoelectric adaptive cantilever beams using the distributed transfer function approach

A semi-analytical distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with a single surface-bonded piezoelectric patch. The asymmetric piezoelectric adaptive structure is decomposed into three segments; the first and third segments are bare beam parts before and after the patch, while the second segment contains the beam part with attached piezoelectric patch bonded to its upper surface. The theoretical formulation assumes first-order shear deformation kinematics and linear electric potential through the patch thickness with an electrode equipotential physical condition, and uses the extended Hamilton?s principle to derive the equations of motion and electromechanical boundary conditions. The latter, together with the continuity and equilibrium conditions at the segments interfaces, are then transformed into a first-order state space equation that is solved using the DTF approach. The electrodes of the piezoelectric patch are considered either in short-circuit (SC) or open-circuit (OC); this leads to two free-vibration problems to be solved for the corresponding SC and OC frequencies, from which the Electro-Mechanical Coupling Coefficient (EMCC) is post-treated. Four benchmarks from the open literature are simulated in order to validate the proposed approach. Very satisfactory correlations are obtained for all examples with maximum errors less thank 5 percent in all results. For future reference, an additional benchmark is proposed to assess the influence of the patch-to-composite host width ratio on the effective modal EMCC. It was found that the latter is mode-dependent (as expected) and decreases with increasing the former.     

The refined theory of transversely isotropic piezoelectric rectangular beams

The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur’e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.     

An efficient coupled layerwise theory for dynamic analysis of piezoelectric composite beams

An efficient new coupled one-dimensional model is developed for the dynamics of piezoelectric composite beams. The model combines third order zigzag approximation for the displacement with layerwise approximation of the electric field as piecewise linear for sublayers. By enforcing the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces, the displacement field is expressed in terms of three primary displacement variables and potentials. The governing coupled equations of stress and charge equilibrium and boundary conditions are derived from Hamilton's principle. Analytical solutions are obtained, for free vibrations and forced response under harmonic load, for simply supported hybrid beams and the results are compared with the exact three-dimensional solution and uncoupled first order shear deformation theory solution. The present results show significant improvement over the first order solution and agree very well with the exact solution for both thin and thick hybrid beams. The results demonstrate the capability of the developed theory to adequately model open and closed circuit electric boundary conditions to accurately predict their influence on the response.     

Lifting the singular nature of a model for peeling of an adhesive tape

We investigate the dynamics of peeling of an adhesive tape subjected to a constant pull speed. Due to the constraint between the pull force, peel angle and the peel force, the equations of motion derived earlier fall into the category of differential-algebraic equations (DAE) requiring an appropriate algorithm for its numerical solution. By including the kinetic energy arising from the stretched part of the tape in the Lagrangian, we derive equations of motion that support stick-slip jumps as a natural consequence of the inherent dynamics itself, thus circumventing the need to use any special algorithm. In the low mass limit, these equations reproduce solutions obtained using a differential-algebraic algorithm introduced for the earlier singular equations. We find that mass has a strong influence on the dynamics of the model rendering periodic solutions to chaotic and vice versa. Apart from the rich dynamics, the model reproduces several qualitative features of the different waveforms of the peel force function as also the decreasing nature of force drop magnitudes.     

Interaction between two active structural paths for source mass motion control over mid-frequency range

The interaction between two active structural paths is analytically and experimentally studied as part of a resonating source-path-receiver system, where each path consists of a piezoelectric stack actuator in series with an elastomeric (passive) mount. An analytical model of the system is first developed, and then an experiment is constructed to verify the feasibility. Good agreement is found between the model and experiment. A performance index to characterize the active path interaction for source mass motion control up to 1000 Hz is analytically defined; it considers the passive phase interaction (caused by system dynamics) between the active mounts and the resulting system motion. Two passive system parameters (rubber path structural damping and disturbance moment arm) emerge as key design variables that drastically change the performance index, and guidelines are developed for desirable path interactions. Limited experimental validation demonstrates that active source mass motion control is achieved at 400 Hz using piezoelectric stack actuators.     

Modal analysis of a rotating multi-packet blade system

A modeling method for the modal analysis of a multi-packet blade system undergoing rotational motion is presented in this paper. Blades are idealized as tapered cantilever beams that are fixed to a rotating disc. The stiffness coupling effects between blades due to the flexibilities of the disc and the shroud are modeled with discrete springs. Hybrid deformation variables are employed to derive the equations of motion. To obtain more general information, the equations of motion are transformed into a dimensionless form in which dimensionless parameters are identified. The effects of the dimensionless parameters and the number of packets on the modal characteristics of the rotating multi-packet blade system are investigated with numerical examples.     

Dynamic instability of a wheel moving on a discretely supported infinite rail

The parametric instability of a wheel moving on a discretely supported rail is discussed. To achieve this, an analysis method is developed for a quasi-steady-state problem which can represent an exponential growth of oscillation. The temporal Fourier transform of the rail motion is expanded by a Fourier series with respect to the longitudinal coordinate, and then the response of the rail deflection due to a quasi-harmonic moving load is derived. The wheel/track interaction is formulated by the aid of this function and reduced to an infinite system of linear equations for the Fourier coefficients of the contact force. The critical velocities between the stable and unstable states are calculated based on the nontrivial condition of the homogeneous matrix equation. Through these analyses the influences of the modeling of rail and rail support on the unstable speed range are examined. Moreover, not only the first instability zone but also other zones are evaluated.     

Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

Vibratory source identification by using the Finite Element Model of a subdomain of a flexural beam

This paper proposes the identification of vibration excitations by an inverse method. The use of the local operator of a structure, for which the vibration field is obtained by measurements, is investigated. This method has already been developed on analytically known structures, but there are as yet few applications. The goal of this paper is to adapt the same technique by using an operator obtained with the Finite Element Method rather than a discretised differential motion equation. The first difficulty is the measurement of rotational degrees of freedom. A concept analogous to an observation matrix enabling only the measurement of displacements is then proposed. The method of extracting the operator on a part of the structure is also discussed. Two variants are proposed: an operator with free boundary conditions and a truncated operator without boundary conditions. The first permits identifying internal forces at the boundaries of the subdomain, whereas the second ignores them. In order to regularise the inverse method, a Tikhonov approach is used. An important point of this paper is the discussion of the effect of this regularisation which implies ambiguities between reconstructed forces and moments in the results. The possibility of an a priori “force only” assumption is then demonstrated and recommended. After illustrations obtained from numerical simulations on beams, an experimental validation is provided for a beam excited by a shaker where several signals are used, giving several signal to noise ratios. The results are discussed and compared to the force measured directly by a piezoelectric sensor.     

Closed-form solution for free vibration of piezoelectric coupled annular plates using Levinson plate theory

Free vibration analysis of annular moderately thick plates integrated with piezoelectric layers is investigated in this study for different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Levinson plate theory (LPT). The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study the closed-form solution for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. To demonstrate the accuracy of the present solution, comparison studies is first carried out with the available data in the literature and then natural frequencies of the piezoelectric coupled annular plate are presented for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric and boundary conditions. Present analytical model provides design reference for piezoelectric material application, such as sensors, actuators and ultrasonic motors.     

物面单反射镜波前分割三光束全息干涉计量研究

提出了一种用于分析物体三维位移场的全息干涉计量新方法。该方法将一个小平面反射镜贴于被测物体的表面，用三束呈空间分布的发散光波，在干版的三个不同区域或同一部位，记录被测物体的三个独立的双曝光干涉图。这些干涉图被由小平面反射镜运动造成的参考光虚点光源的位移所调制。基于对这种调制的理论分析，导出计算参考光虚点光源和被测物体三维位移的二个线性方程组。     

The equations of motion of initially stressed Timoshenko tubular beams conveying fluid

In this paper the equations of motion of an initially stressed Timoshenko tubular beam subjected to a tensile follower load and conveying fluid are derived by using the appropriate statement of Hamilton's principle. This latter is obtained first for “open” systems, the instantaneous total mass of which does not necessarily remain constant in the course of deformation—“open” denoting that there is momentum transport in and out of the system. The equations of motion are derived separately for a cantilevered system and for one with both extremities of the tube clamped. Yet another derivation for the cantilevered tube is presented with the system considered to be quasi-closed, where all flow-induced effects are incorporated through the virtual work, as if they were “external” forces. All three sets of equations are found to be identical. These equations are then compared with those obtained, more simply, by the Newtonian force-balance approach. Some differences between them are found to exist, the principal of which are associated with the follower or other tensile forces; these are discussed at some length, and the equations of motion obtained here are compared to those obtained by other researchers for Timoshenko beams subjected to follower or tensile forces.     

Dynamics of a camphoric acid boat at the air–water interface

We report experiments on an agarose gel tablet loaded with camphoric acid (c-boat) spontaneously set into motion by surface tension gradients on the water surface. We observe three distinct modes of c-boat motion: harmonic mode where the c-boat speed oscillates sinusoidally in time, a steady mode where the c-boat maintains constant speed, and an intermittent mode where the c-boat maintains near-zero speed between sudden jumps in speed. Whereas all three modes have been separately reported before in different systems, controlled release of Camphoric Acid (CA) from the agarose gel matrix allowed the observation of all the three modes in the same system. These three modes are a result of a competition between the driving (surface tension gradients) and drag forces acting on the c-boat. Moreover we suggest that there exist two time scales corresponding to spreading of CA and boat motion and the mismatch of these two time scales give rise to the three modes in boat motion. We reproduced all the modes of motion by varying the air–water interfacial tension using Sodium Dodecyl Sulfate (SDS).     

Equations of motion for damped longitudinal,torsional and flexural oscillations of piezoelectric bars

The equations of motion for pure (mutually independent) longitudinal, torsional and flexural oscillations of very thin piezoelectric bars are derived systematically and in detail first in integral and then in differential form. During derivation attention is paid to the effect of external and internal damping and the method of exciting the individual oscillations is considered.     

Software package for modeling spin–orbit motion in storage rings

A software package providing a graphical user interface for computer experiments on the motion of charged particle beams in accelerators, as well as analysis of obtained data, is presented. The software package was tested in the framework of the international project on electric dipole moment measurement JEDI (Jülich Electric Dipole moment Investigations). The specific features of particle spin motion imply the requirement to use a cyclic accelerator (storage ring) consisting of electrostatic elements, which makes it possible to preserve horizontal polarization for a long time. Computer experiments study the dynamics of 10⁶–10⁹ particles in a beam during 10⁹ turns in an accelerator (about 10¹²–10¹⁵ integration steps for the equations of motion). For designing an optimal accelerator structure, a large number of computer experiments on polarized beam dynamics are required. The numerical core of the package is COSY Infinity, a program for modeling spin–orbit dynamics.     

Spatial disorder in a smectic A mesophase

Starting with the equations of motion for a stiff chain, a projection operator approach is utilized to develop diffusional equations for the dynamics of the end-to-end distance. The diffusion equation resulting has a spatial-dependent diffusion coefficient calculable from equilibrium properties of the chain, and a frequency-dependent part which requires dynamical information. The analysis is applied, in so far as the spatial dependence of D is determined, for three and four bond chains. A critique of this procedure is provided.     

Effect of fringing fields in storage rings

Electron beams in storage rings of third-generation synchrotron radiation sources feature a low emittance and extended lifetime. The provision of such characteristics requires a detailed study of higher order effects related to magnetic fields and cooperative effects associated with beam density. Fringing fields, being an unavoidable attribute of magnets of any type, may significantly affect the beam dynamics, since they appear in equations of particle motion of first and higher orders. A simple technique for evaluating the effect of fringing fields on the beam dynamics is suggested. Numerical results obtained with this technique for the quadrupoles of the CANDLE storage ring [1] are reported.     

The possibility of amplification and generation of higher harmonics in a microwave device with turbulent electron flow (Natural experiment and phenomenological model)

A generator of turbulent electron beams in an independent mode and in the mode of amplification of an external single-frequency signal is experimentally investigated. The formation of higher harmonic components in the spectrum of output radiation of the generator is shown experimentally. The phenomenological model of the turbulent electron flow is constructed on the basis of a chain of interacting “electron vortices” described by the modified nonlinear Van-der-Pol differential equations. The modes of independent dynamics and the mode of amplification of an external harmonic signal are considered. The results of numerical modeling and experimental investigations are compared. The qualitative correspondence of the behavior of the dependences obtained from the experimental investigation of the generator of turbulent electron beams and the results obtained from numerical modeling is shown.     

Vibration of a two-member open frame

The dynamics of a two-member open frame structure undergoing both in- and out-of-plane motion is examined. The frames are modelled using the Euler-Bernoulli beam theory and are further generalized by permitting an arbitrary angle between the beams and the attachment of a payload at the end of the second beam. The equations of motion are derived using Hamilton's principle and the orthogonality conditions are presented. It is shown that the in- and out-of-plane motions can be decoupled by including the axial deformation components in the assumed displacement fields. The natural frequencies of the system and the contribution of each member into the system potential energy are examined via numerical examples.     

An integral equation analysis of the harmonic response of three-layer beams

The integral equations of harmonic motion have been derived and solved for three-layer sandwich beams with a constrained linear viscoelastic core. The method of solution required first the construction of the Green's vector for a beam in analytical form. Following this, the integral equations were derived and readily approximated by matrix equations which were finally solved numerically. In addition to this analysis, the corresponding eigenvalue problem has been solved so that the modal frequencies and the beam loss factor could be calculated directly. The integral equation analysis offers a fast and efficient alternative to the traditional methods based on the solution of the differential equations of motion. The method has been verified by comparison with experimental results for three-layer cantilevers and simply supported beams.