Response and damping of a rectangular cantilever plate with vibrating beam dampers

In this work the use of beams as auxiliary mass dampers for cantilever plates is considered. Because the cantilever plate problem, which is of strong industrial interest, does not lend itself to a Lévy-type solution, the procedure developed by Ritz is used. Structural damping is incorporated into the main and auxiliary systems by treating them as having a complex elastic modulus. With appropriate selection of the parameters, the fundamental resonance of the plate is split into two new ones with considerably suppressed responses. In order to verify the analysis, an experimental investigation was carried out and the results obtained were compared with the theory developed.     

TRANSVERSE VIBRATION OF ELASTIC-VISCOELASTIC-ELASTIC SANDWICH BEAMS: COMPRESSION-EXPERIMENTAL AND ANALYTICAL STUDY

Experimental and analytical results are presented from an investigation into the compressional vibration of an elastic-viscoelastic-elastic three-layer sandwich beam. Most analytical models make the fundamental assumption that shear deformation in the viscoelastic core yields the largest damping and compressional deformation is negligible. Experimental results from a cantilever beam with a constrained layer viscoelastic damping treatment driven with a sinusoidal input are given which show compressional deformation over a relatively wide driving frequency range. A new analytical model for compressional damping is presented and compared with experimental results, with the Mead and Markus shear damping model, and with the Douglas and Yang compressional damping model. These results indicate that the proposed compressional model is a better predictor of resonance frequencies for the cantilever beams tested and that all models show deficiencies in predicting damping     

Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid

The dry and wet dynamic characteristics of a vertical and a horizontal cantilever square plate [1] immersed in fluid are discussed from the viewpoint of a linear hydroelasticity theory [2–5]. The surface piercing vertical plate is partially immersed in the fluid and the influence of submerged plate length on the resonance frequencies investigated. For the horizontal plate the influence of submerged depth below the free surface on the resonance frequencies is examined. Incorporated into the theoretical model is a free surface boundary condition allowing wave disturbances to be present. The interaction existing between the vibrating cantilever plate and the free surface is clearly exhibited in the calculated curves describing the generalized hydrodynamic coefficients. A limited comparison between predictions and experimental data [1] is also included.     

Dynamic characterization of high damping viscoelastic materials from vibration test data

The numerical analysis and design of structural systems involving viscoelastic damping materials require knowledge of material properties and proper mathematical models. A new inverse method for the dynamic characterization of high damping and strong frequency-dependent viscoelastic materials from vibration test data measured by forced vibration tests with resonance is presented. Classical material parameter extraction methods are reviewed; their accuracy for characterizing high damping materials is discussed; and the bases of the new analysis method are detailed. The proposed inverse method minimizes the residue between the experimental and theoretical dynamic response at certain discrete frequencies selected by the user in order to identify the parameters of the material constitutive model. Thus, the material properties are identified in the whole bandwidth under study and not just at resonances. Moreover, the use of control frequencies makes the method insensitive to experimental noise and the efficiency is notably enhanced. Therefore, the number of tests required is drastically reduced and the overall process is carried out faster and more accurately. The effectiveness of the proposed method is demonstrated with the characterization of a CLD (constrained layer damping) cantilever beam. First, the elastic properties of the constraining layers are identified from the dynamic response of a metallic cantilever beam. Then, the viscoelastic properties of the core, represented by a four-parameter fractional derivative model, are identified from the dynamic response of a CLD cantilever beam.     

Ultrasonic radiation in dynamic force microscopy

In dynamic force microscopy the cantilever of an atomic force microscope is vibrated at ultrasonic frequencies in the range of several 10 kHz up to several MHz while scanning a sample surface. The amplitude and phase of the cantilever vibration as well as the shift of the cantilever resonance frequencies provide information about local sample surface properties. In several operation modes of dynamic force microscopy, for example force modulation microscopy, tapping mode or atomic force acoustic microscopy, the sensor tip is in contact with the sample at least during a fraction of its vibration cycle. The periodic indentation of the tip with the sample surface generates ultrasonic waves. In this paper, the ultrasonic radiation of a vibrating cantilever into a sample and its contribution to the damping of the cantilever vibration are calculated. The theoretical results are compared to experiments.     

Comments on nonlinear dynamics of a non-ideal Duffing-Rayleigh oscillator: Numerical and analytical approaches

An analytical and numerical investigation into the dynamic interaction between a cantilever beam with nonlinear damping and stiffness behavior, modeled by the Duffing-Rayleigh equation, and a non-ideal motor that is connected to the end of the beam, is presented. Non-stationary and steady-state responses in the resonance region as well as the passage through resonance behavior when the frequency of the excitation is varied are analyzed. The influences of nonlinear stiffness, nonlinear damping and the extent of the unbalance in the motor are examined. It is found that in this situation so-called Sommerfeld effects may be observed; the increase required by a source operating near the resonance results in a small change in the frequency, but there is a large increase in the amplitude of the resultant vibration and the jump phenomenon occurs.     

MODAL CHARACTERISTIC OF A ROTATING RECTANGULAR CANTILEVER PLATE

Modal characteristics of a rotating cantilever plate are investigated in the present work. A dynamic modelling method for rectangular plates undergoing prescribed overall motion is employed to derive the equations of motion. The general equations are particularized for the modal analysis of a rotating cantilever plate and dimensionless parameters are identified through dimensional analysis. The effects of the dimensionless parameters on the modal characteristics of the rotating plate are investigated. Incidentally, eigenvalue loci veering and crossing phenomena along with the corresponding modeshape variations are exhibited and discussed.     

Parametric resonance of plates in a sheet metal coating process

The vibration of sheet metal during zinc coating processes can lead to non-uniform coating thickness and overall poor product quality. This vibration develops from two principal mechanisms, namely, the run-out associated with the supporting rollers and/or bearings, and the time-varying tension associated with the manufacturing process. This study focuses on the second of these mechanisms (time-varying tension) that becomes significant under conditions leading to parametric resonance. The parametric resonance of the sheet metal is captured in a proposed model of a plate subject to time-varying and non-uniform edge tension. The model accounts for these effects as well as the non-linear stretching of the plate mid-plane as a result of transverse plate vibration. The linear vibration characteristics of the plate are studied first and are then used in deriving a single mode approximation of the non-linear, parametrically excited plate model. A perturbation solution of this model reveals the major parameters that influence parametric resonance in this application. Theoretical results for plate vibration are compared to experimental measurements of sheet metal vibration in a production facility. This comparison demonstrates that the model accurately captures the physical mechanisms responsible for sheet metal vibration and therefore, the physical parameters (such as damping) have the greatest impact on this vibration.     

Geometrically nonlinear dynamic response of stiffened plates with moving boundary conditions

An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The first approximation solution of the non-resonance is obtained by means of the method of multiple scales. The primary parametric resonance and primary resonance of the stiffened plate are studied by using the same method. The accuracy of the method is validated by comparing the results with those of finite element analysis via ANSYS. Numerical examples for different stiffened plates are presented to discuss the steady response of the non-resonance and the amplitude-frequency relationship of the primary parametric resonance and primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude-frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.     

Electromagnetic oscillations in a rectangular waveguide with a transversely magnetized ferrite plate

The complex propagation constant in a rectangular waveguide with a transversely magnetized ferrite plate has been found by a numerical calculation. The magnetic properties of the ferrite medium are described by the standard permeability tensor for a polycrystalline ferrite, having real and imaginary parts. The solutions are given as curves on the complex propagation-constant plane. The parameter is the relative magnetizing field, which varies over a wide range including a ferromagnetic resonance. From the family of curves obtained for various relative magnetizations and damping parameters, one can quantitatively evaluate the relation between the types of oscillations. This relation depends on the position of the ferrite plate in the waveguide and on the dielectric constant of the ferrite as well as on the magnetization and damping constant.Translated from Izvestiya VUZ. Fizika, No. 4, pp. 74–81, April, 1970.     

A novel multimode negative inductance negative resistance shunted electromagnetic damping and its application on a cantilever plate

Based on the characteristic of the electromagnetic damping, a novel multimode vibration control treatment, negative inductance negative resistance electromagnetic shunt damping (NINR-EMSD), is proposed and employed to control multimode vibration of a cantilever plate. The negative inductance of the shunt impedance can cancel the inherent inductance of the electromagnet, and the impedance of circuit consisted of the coils and shunt will be a pure resistance and the shunt current will be frequency independent when the inductance of the shunt and the electromagnet are equal with each other. The negative resistance cancels the resistance of electromagnet, and as a result the resistance and the current of the closed circuit will change, which make it feasible to control the multimode vibration of the system. Electromechanical coupling coefficient is obtained based on the equivalent current method. The governing equation of the plate with the electromagnetic shunt damping (EMSD) is established according to Hamilton's principle. Multimode vibration control of the system is simulated and experiments are carried out to verify the performance of the NINR-EMSD. The numerical predictions and experimental results agree well with each other and show that: the negative inductance negative resistance (NINR) of the shunt impedance can increase the damping of the structure notably; the decrease of the resistance of shunt impedance have a significant contribution to the improvement of vibration control performance; the first four modes vibration of the plate can be suppressed simultaneously with the NINR-EMSD.     

Resonance-mode effect on piezoelectric microcantilever performance in air,with a focus on the torsional modes

A high quality factor is preferred for a microresonator sensor to improve the sensitivity and resolution. In this paper we systematically investigate the performance of the microcantilever in different resonance modes, which are the first three flexural modes, the first lateral mode, and the first and the second torsional modes. An aluminum nitride-based piezoelectric cantilever is fabricated and tested under controlled pressure from an ultra-high vacuum to a normal atmosphere, using a custom-built vacuum chamber. From the experiment results, it can be seen that the torsional modes exhibit better quality factors than those of the flexural and lateral ones. Finally, an analytical model for the air damping characteristics of the torsional mode cantilever is derived and verified by comparing with experimental results.     

Interplay between nonlinearity,scan speed,damping, and electronics in frequency modulation atomic-force microscopy

Numerical simulations of the frequency modulation atomic force microscope, including the whole dynamical regulation by the electronics, show that the cantilever dynamics is conditionally stable and that there is a direct link between the frequency shift and the conservative tip-sample interaction. However, a soft coupling between the electronics and the nonlinearity of the interaction may significantly affect the damping. A resonance between the scan speed and the response time of the system can provide a simple explanation for the spatial shift and contrast inversion between topographical and damping images, and for the extreme sensitivity of the damping to a tip change.     

Simultaneous imaging of tunneling current and damping energy by noncontact-AFM in ultra-high vacuum

We have simultaneously observed the images of tunneling current and damping energy with the nc-AFM (noncontact atomic force microscopy) image of Si(111)7ǻ. When inverted contrast is observed in the constant frequency shift nc-AFM image, the current image is not inverted, and vice versa. On the other hand, the damping images show a contrast similar to that of the nc-AFM images; the damping decreases at a narrow separation between the tip and the sample. This possibly indicates that the damping decreases as the attractive interaction increases under a constant oscillation amplitude of the cantilever. To keep the oscillation amplitude constant under an attractive interaction between a tip and a sample, the total energy of the cantilever oscillation is reduced and the oscillation velocity of the cantilever decreases. An explanation is proposed that the change in energy dissipation occurs in the cantilever oscillation, depending on the oscillation velocity of the cantilever, and the value is estimated from a simple model.     

Flexural wave suppression by an acoustic metamaterial plate

A new acoustic metamaterial plate is presented for the purpose of suppressing flexural wave propagation. The metamaterial unit cell is made of a plate with a lateral local resonance (LLR) substructure which consists of a four-link mechanism, two lateral resonators and a vertical spring. The substructure presents negative Young’s modulus property in certain frequency range. We show theoretically and numerically that two large low-frequency band gaps are obtained with different formation mechanisms. The first band gap is due to the elastic connection with the foundation while the second is induced by the lateral resonances. Besides, four-link mechanisms can transform the flexural wave into the longitudinal vibration which stimulates the lateral resonators to vibrate and to generate inertial forces for absorbing the energy and thus preventing the wave propagation. Frequency response function shows that damping from the vertical spring has little influence on the band gaps, although the damping can smooth the variation of frequency response (see the dotted line in Figs. 10 and 11). Increasing the damping of the lateral resonators may broaden the second band gap but deactivate its effect. This study provides guidance for flexibly tailoring the band characteristics of the metamaterial plate in noise and vibration controls.     

Damping in micro-scale generalized thermoelastic circular plate resonators

The out-of-plane vibrations of a generalized thermoelastic circular plate are studied under different environmental temperature, plate dimensions and boundary conditions. The analytical expressions for thermoelastic damping of vibration and phase velocity of circumferential surface wave modes are obtained. It is noticed that the damping of vibrations and phase velocities of circumferential surface wave modes significantly depend on thermal relaxation time in addition to thermoelastic coupling in circular plates under resonance conditions. The surface conditions also impose significant effects on the vibrations of such resonators. The expressions for displacement and temperature fields in the plate resonator are also derived and obtained. Some numerical results have also been presented for illustration purpose in case of silicon material plate.     

MODAL ANALYSIS OF ROTATING COMPOSITE CANTILEVER PLATES

A modelling method for the modal analysis of a rotating composite cantilever plate is presented in this paper. A set of linear ordinary differential equations of motion for the plate is derived by using the assumed mode method. Two in-plane stretch variables are employed and approximated to derive the equations of motion. The equations of motion include the coupling terms between the in-plane and the lateral motions as well as the motion-induced stiffness variation terms. Dimensionless parameters are identified and the explicit mass and the stiffness matrices for the modal analysis are obtained with the dimensionless parameters. The effects of the dimensionless angular velocity and the fiber orientation angles of rotating composite cantilever plates on their modal characteristics are investigated. Natural frequency loci veering and crossing along with associated mode shape variations are observed.     

Stability of cantilever plates subjected to biaxial subtangential loading

The stability of cantilever plates which, mathematically, comprises a non-self-adjoint problem is investigated. It is assumed that the plate is acted upon by a subtangential biaxial edge load embodying the dead loading and the follower type loading as its limiting states. The scheme of modal expansions, containing the constrained rigid modes, together with Galerkin's method is employed and the stability of the plate in terms of subtangency and load parameters is analysed. As an example the kinetic stability analysis of a square cantilever plate is carried out in detail.     

Analyses of vibration responses on nanoscale processing in a liquid using tapping-mode atomic force microscopy

An analytical solution of the vibration responses of biological specimens using atomic force microscopy (AFM), which often requires operation in a liquid, is developed. In this study, the modal superposition method is employed to analyze the vibration responses of AFM cantilevers in tapping mode (TM) operated in a liquid and in air. The hydrodynamic force exerted by the fluid on AFM cantilevers is approximated by additional mass and hydrodynamic damping. The tip–sample interaction forces were transformed into axial, distributed transversal, and bending loading, and then applied to the end region of the AFM through the tip holder. The effects of transverse stress and bending stress were adopted to solve the dynamic model. With this model, a number of simulations were carried out to investigate the relationship between the transient responses of the cantilever in a liquid and the parameters considered in nanoscale processing. The simulations show that the vibration of AFM cantilevers in a liquid has dramatically different dynamic characteristics from these of that in air. The liquid reduces the magnitude of the transversal response and reduces the cantilever resonances. Moreover, the magnitudes of response become larger with increasing intermolecular distances and smaller with decreasing tip length. The cantilever vibration amplitudes significantly depend on the damping constant and the mass proportionality constant.     

AERODYNAMIC DAMPING OF RANDOMLY EXCITED PLATES IN STATIONARY AND MOVING AIR

An experimental study of the aerodynamic damping of oscillating plates has been undertaken. Plates of various shapes were placed into an air flow normal to the plate and excited to oscillate parallel to the flow direction by electromagnetic forces of equal amplitudes and random frequencies. The aerodynamic damping of oscillating plates, evaluated in terms of a quality Q-factor from a frequency response resonance curve, was found to vary linearly with the absolute pressure in stationary surrounding air and with the air flow velocity in moving air. The flow velocity was also found to affect the aerodynamic damping more than the absolute pressure. A simple empirical model has been developed to predict the variation of the aerodynamic damping with the flow velocity.