Vibrations of an elastic strip with a rough boundary

A plane problem of steady-state forced vibrations of an elastic strip whose lower boundary contains a rough segment is considered. Using Green’s functions for a strip, the problem is reduced to a system of integral equations with integrals over the rough boundary, which is solved by the boundary-element method. The inverse problem of determining the shape of the rough boundary segment from the data on the displacement field of a certain part of the upper boundary is formulated. By the linearization procedure, the inverse problem is reduced to a Fredholm integral equation of the first kind with a smooth kernel, which is solved by Tikhonov’s regularization method.     

FORCED VIBRATIONS OF TRIPLY COUPLED, PERIODICALLY AND ELASTICALLY SUPPORTED, FINITE, OPEN-SECTION CHANNELS

An exact analytical method is presented for the analysis of forced vibrations of uniform, open-section, single- and multi-bay periodic channels. The centre of gravity and the shear centre of the channel cross-sections do not coincide, and hence the flexural vibrations in two mutually perpendicular directions and the torsional vibrations are all coupled. The ends of the channels and the periodic intermediate supports are modelled with springs having finite flexural and torsional stiffnesses. Single-point force excitation has been considered throughout the study, although the developed method is also capable of dealing with multi-point excitation. The channels are assumed to be of Euler-Bernoulli type beams. The study also takes the effects of cross-sectional warping into consideration. The developed method is suitable for structurally damped analysis and in addition to yielding forced vibration characteristics; it also straightforwardly reveals the free vibration properties like the mode shapes.     

Vibration control by recursive time-delayed acceleration feedback

This paper presents a theoretical basis of time-delayed acceleration feedback control of linear and nonlinear vibrations of mechanical oscillators. The control signal is synthesized by an infinite, weighted sum of the acceleration of the vibrating system measured at equal time intervals in the past. The proposed method is shown to have controlled linear resonant vibrations, low-frequency non-resonant vibrations, primary and 1/3 subharmonic resonances of a forced Duffing oscillator. The concept of an equivalent damping and natural frequency of the system is also introduced. It is shown that a large amount of damping can be produced by appropriately selecting the control parameters. For some combinations of the control parameters, the effective damping factor of the system is shown to be inversely related to the time-delay in the small delay limit. Selection of the optimum control parameters for controlling the forced and free vibrations is discussed. It is shown that forced vibration is best controlled by unity recursive gain and smaller values of the time-delay parameter. However, the transient response can be optimally controlled by suitably selecting the time delay depending upon the gain. The delay values for the optimal forced response may be different from that required for the optimum transient response. When both are important, a suboptimal choice of the delay parameters with unity recursive gain is recommended.     

FORCED AND FREE VIBRATIONS OF RECTANGULAR SANDWICH PLATES WITH PARAMETRIC STIFFNESS MODULATION

An active control of the resonant vibrations of a rectangular sandwich plate performed by the parametric stiffness modulation is analyzed. The controlled vibrations are those of the dominantly flexural type excited by the transverse force acting at the first resonant frequency of dominantly flexural vibrations. The stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. The method of direct partition of motions is used that predicts an existence of the modal interaction between these two modes of vibrations due to the parametric stiffness modulation. It is shown that such a parametric control can provide a significant shift of the first eigenfrequency of a controlled plate (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled plate. Heavy fluid loading conditions are accounted for as well as the energy dissipation in the material of a plate. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases. Dynamics of an elementary two-degree-of-freedom model mechanical system is considered to illustrate the mechanism of modal interaction, which is involved in the suggested way of an active control of vibrations of sandwich plates.     

Influence of deformations before instability on the parametric instability of conical shells under periodic pressure

On the basis of the dynamic version of linear Donnell type equations and with deformations before instability taken into account, the dynamic instability of clamped, truncated conical shells under periodic pressure is analyzed. The principal instability regions are determined by combining Bolotin's method and a finite difference procedure. Calculations are carried out for two kinds of conical shell. The effect of bending deformations before instability is found to change the width of the principal instability regions in the vicinities of twice the natural frequencies of asymmetric vibration. Other principal instability regions are detected in the neighborhoods of the resonances of symmetrically forced vibrations.     

Untersuchungen am Kubischen Bariumtitanat Nach der Methode des elektrostriktiven Resonators

The temperature dependence of the constant of elasticity could be measured in the cubic phase on [100] rods in the absence of a d.c. field using the electrostriction resonator method. The temperature range reached from the transition point up to 300°C. A steep decrease in mechanical quality could be observed several degrees above the transition point.The influence of the dielectric non-linearity on the observed electrostriction resonance is discussed. The piezoelectrical resonances found additionally are explained as flexural vibrations excited by a tetragonal boundary layer.     

Secondary critical speed of flexible rotors with inertia slots

Rotors of two-pole generators have longitudinal slots for the electric windings and thus have dual flexural rigidity. Second order (or twice per revolution) forced vibrations are excited by the weight of the rotor and the problem of secondary critical speed, at half of the normal critical speed, arises. To overcome this difficulty transverse saw cuts or inertia slots are made in the pole faces in order to restore equality of the flexural rigidity of the rotor. In this paper, the critical speeds of rotors with inertia slots are calculated by using the transfer matrix method. The flexural rigidity of the element used in the transfer matrix method is determined by a three-dimensional finite element method. The secondary critical speeds of asymmetric rotors with inertia slots were measured experimentally thus demonstrating the validity of the present analysis.     

The acoustic field excited by flexural vibrations of an elastic plate with a circular inclusion

The acoustic field excited by flexural vibrations of a thin elastic plate and the perturbations of this field caused by a homogeneous circular inclusion with other elastic properties are considered. Because the density of air widely differs from that of a metal, this problem can be solved with fair accuracy in two steps: first, by considering the vibrations of the plate in a free space, and, then, by calculating the acoustic field excited by the field of plate’s vertical deflections. The main results of this work are the asymptotic expressions for the far acoustic field excited by each of the Fourier components F _m (r)cosmφ of the flexural wave scattered by the inclusion.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Flexural vibration insulation with an elastic layer for a plate under nonuniform vibratory action

A method is proposed for calculating the vibration insulation of an elastic layer positioned between a finite-size plate with free edges and a vibrationally active base. The elastic layer consists of a discrete set of springs. In the case of an out-of-phase vibration of the base surface, the mean dynamic action on the plate decreases, which leads to an increase in vibration insulation. By contrast, the resonances of flexural plate vibrations reduce the vibration insulation at resonance frequencies. For different cases, the system parameters that correspond to a positive effect of vibration insulation are estimated. The simplest mass-spring system possessing a single degree of freedom is discussed.     

Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem

The harmonic and periodic forced vibrations of rotating rings are derived and investigated. The modal expansion technique yields the forced solution, which is characterized by four generalized co-ordinates associated with each n (circumferential wave number). The inextensional assumption is presumed, when flexural vibration is the only important component, to reduce the order of the system. The closed form solutions to the harmonic load cases, once concentrated, once distributed, are demonstrated and interpreted. The approach is then extended to periodic loads, where Fourier sine and cosine series is applied. Examples depict the numerical responses to all the cases being derived. The solutions of a stationary ring subjected to traveling loads are also solved for comparison. Their difference is investigated and interpreted from various viewpoints.     

Effects of mass layer imperfect bonding on the electrical impedance of a quartz crystal microbalance

We study electrically forced vibrations of a crystal plate of AT-cut quartz carrying a thin mass layer operating as a quartz crystal microbalance for mass sensing. The mass layer is imperfectly bonded to the crystal plate with their interface described by the so-called interface-slip model which allows a discontinuity of the tangential interface displacement. Mindlin’s equations for piezoelectric plates are used. An analytical solution is obtained. The electrical impedance is calculated. The effects of an elastic interface and a viscoelastic interface are examined.     

The influence of proper mechanical vibrations on some properties of TGS tandel

In the present paper the existence is proved of mechanical vibrations in TGS tandels in the frequency range from 5 kc/s to 1000 kc/s. A number of resonances was found in the given range of frequencies. Mainly plane and flexural vibrations occur. A study is made of their influence on the course of the frequency dependence of the complex effective permittivity, dielectric non-linearities of the tandel, and the thermoelectric force measured by a thermocouple on its surface. From the study of the frequency dependences of these parameters at various temperatures the conclusion is drawn that the probable cause of the origin of mechanical vibrations in a tandel is the piezoeffect.At the 2^nd International Conference on Piezoelectricity in Liberec on Sept. 1^st, 1965 the main results of this paper were presented.In conclusion the authors would like to express their deep gratitude to Ing. J. Janta of the Institute of Radio Engineering and Electronics, Czechoslovak Academy of Sciences, to Associate Professor Dr. J. Tichý of the Technical University in Liberec and to Dr. J. Mastner of the Institute of Radio Engineering and Electronics for valuable discussions and suggestions, and to Associate Professor Dr. O. Taraba of the Czech Technical University also for facilitating the ultrasonic experiments and his help in arranging them in his laboratory. We are also indebted to all our colleagues for their friendly help.     

The frequency spectrum of nt-cut quartz plates

An approximate, one-dimensional theory derived and applied by Lee to the solution of coupled odd harmonics of extensional vibration, even orders of flexural vibrations and a width-shear vibration is used for solution of coupled odd orders of flexural vibrations, even harmonics of extensional vibrations and width-shear vibration. Closed-form solution suitable for calculation of the frequency spectrum of mentioned type of vibrations are obtained and the frequency spectrum of NT-cut quartz rectangular plate with simple orientation is calculated. The obtained theoretical results are compared with measurements.     

Study on the prestressed sandwich piezoelectric ceramic ultrasonic transducer of torsional-flexural composite vibrational mode

Based on the classical torsional and flexural vibrational theory of a slender rod, the prestressed sandwich torsional-flexural composite mode piezoelectric ceramic ultrasonic transducer is studied. This type of transducer consists of the slender metal rods and the longitudinally and tangentially polarized piezoelectric ceramic rings. The resonance frequency equations for the torsional and flexural vibrations in the transducers are derived. The simultaneous resonance of the torsional and flexural vibrations in the transducer is acquired by correcting the length of the metal slender rods resulting from the piezoelectric ceramic elements. The experimental results show that the measured resonance frequencies of the transducers are in good agreement with the computed ones, and the measured resonance frequencies of the torsional and the flexural vibrations in the composite transducers are also in good agreement with each other.     

Study on the Langevin piezoelectric ceramic ultrasonic transducer of longitudinal-flexural composite vibrational mode

In this paper, the Langevin longitudinal-flexural composite mode piezoelectric ultrasonic transducer is studied. This type of transducers consists of slender metal rods and longitudinally polarized piezoelectric ceramic rings. The resonance frequency equations for the longitudinal and flexural vibrations in the transducer are derived. By correcting the length of the metal slender rods, the simultaneous resonance of the longitudinal and flexural vibrations in the transducer is acquired. The experimental results show that the measured resonance frequencies of the transducers are in good agreement with the computed ones, and the measured resonance frequencies of the longitudinal and the flexural vibrations in the composite transducers are also in good agreement with each other.     

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

In this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.     

Transient vibrations of a beam/mass system fixed to a rotating body

Transient flexural vibrations of a beam/mass system fixed to a rotating body are investigated. The rotating body is driven so as to have a velocity profile of trapezoidal shape. The governing ordinary differential equations of the beam mass system are derived by use of the extended Galerkin method, and the transient response is obtained by the Laplace transformation. Then the effects of the flexibility of the beam and the rotational period of the rotating body upon the flexural vibrations are investigated.     

Nonlinear free vibrations of beams in space due to internal resonance

The geometrically nonlinear free vibrations of beams with rectangular cross section are investigated using a p-version finite element method. The beams may vibrate in space, hence they may experience longitudinal, torsional and non-planar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The geometrical nonlinearity is taken into account by considering Green’s nonlinear strain tensor. Isotropic and elastic beams are investigated and generalised Hooke’s law is used. The equation of motion is derived by the principle of virtual work. Mostly clamped–clamped beams are investigated, although other boundary conditions are considered for validation purposes. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. One constant term, odd and even harmonics are assumed in the Fourier series and convergence with the number of harmonics is analysed. The variation of the amplitude of vibration with the frequency of vibration is determined and presented in the form of backbone curves. Coupling between modes is investigated, internal resonances are found and the ensuing multimodal oscillations are described. Some of the couplings discovered lead from planar oscillations to oscillations in the three dimensional space.