Axisymmetric Vibration Modes Research Articles

Frequency characteristics of an ultrasonic varifocal liquid crystal lens.

Compound lens systems with mechanical actuators are used to focus objects at near to far distances. The focal length of ultrasound varifocal liquid crystal (LC) lenses can be controlled by modulating the refractive index spatial distribution of the medium through the acoustic radiation force, resulting in thin and fast-response varifocal lenses. The frequency characteristics of such a lens are evaluated in this paper, and several axisymmetric resonant vibration modes over 20kHz are observed. The effective lens aperture decreased with the wavelength of the resonant flexural vibration generated on the lens, meaning that this parameter can be controlled with the driving frequency.

Characterization of drying-induced changes in moduli and internal stresses in a constrained gel using laser vibrometry.

Hydrogels, water-saturated polymer networks find widespread use in soft robotics, biomedical, pharmaceutical and food industries. Both solid and water constituents of hydrogels are sensitive to external stimuli such as temperature, humidity, osmolarity, and light. For instance, common hydrogels swell or shrink in the presence of chemical potential gradient between the sample and surrounding environment. Corresponding changes in internal water content lead to significant changes in mechanical properties of hydrogels. Besides, internal stresses build up if the gel samples are constrained during swelling or dehydration. In the present research, we utilize modal analyses technique on drying hydrogels to identify dehydration-induced changes in elastic moduli and internal stresses. In particular, natural frequencies and damping ratios of the first two axisymmetric transverse vibration modes are measured on clamped gelatin disks using non-contact laser vibrometry at various water loss states. Experimental modal frequencies are then compared to the predictions of a pre-stressed thick plate model. The evolutions of elastic moduli and internal stresses for water losses up to 80% are identified. The broadband loss capacity of gelatin is also determined from the measured modal damping ratios. Highly transient mechanical response observed on the gelatin disks further demonstrates the need for non-contact and rapid mechanical characterization of hydrogels. As illustrated in this work, vibration and wave-based techniques are promising candidates to fulfill that need.

Axisymmetric bending vibration analysis of bidirectional functionally graded rotating micro-disk under thermo-mechanical loading

The free vibration behavior of a bidirectional functionally graded rotating micro-disk that is subjected to uniform transverse pressure and high-temperature thermal loading has been studied. The micro-disk is functionally graded along the radial and thickness directions. The problem is mathematically formulated using two distinct but interrelated steps within the framework of Kirchhoff plate theory and modified couple stress theory. The first step determines the time-invariant deformed configuration of the micro-disk under centrifugal, pressure, and thermal loading using minimum potential energy principle. The second step determines the free vibration behavior of the micro-disk in the neighborhood of the deformed configuration using Hamilton’s principle. The solutions of the governing equations for both these steps are obtained using the Ritz method. The mathematical model is successfully validated with various reduced problems. The numerical results for the first four axisymmetric bending vibration modes are presented to investigate the effects of wide range of parameters such as rotational speed, applied pressure, thermal loading, size-dependent thickness, volume fraction indices, and radius ratio. The mode shapes of vibration are illustrated through surface and contour plots.

Hollow cylindrical linear stator vibrator using a traveling wave of longitudinal axisymmetric vibration mode

Ultrasonic motors (USMs) are expected to be used in special environments: high magnetic field environments and space environments, which require lightweight and multiple degrees of freedom. However, when used as linear ultrasonic motors (LUSMs), a linear guide and a preload mechanism are required, complicating the structure. In the present paper, a hollow cylindrical linear stator without an extra linear guide has been considered. The stator consists of a metal pipe and two piezoelectric (PZT) tubes installed at both ends of the metal pipe. Their connected parts are tapered for the first longitudinal axisymmetric vibration mode of the cylinder, namely L(0,1) mode excitation, and the metal pipe is subjected to radial strain. The vibration of the stator is assumed to be one-dimensional and is modeled by an electromechanical equivalent circuit. The principle that the traveling wave is formed on the metal pipe by dual-PZT-tube phase difference excitation was clarified. Finite element analysis and some measurements were conducted to confirm that the theory was consistent. The analyses and measurements were in good agreement. Therefore, the operating principle was confirmed. The results of the transport experiment showed that the average speed of the 8.5-g slider was 7.9 mm/s.

Free vibration behavior of rotating bidirectional functionally-graded micro-disk for flexural and torsional modes in thermal environment

A mathematical model for investigating the asymmetric as well as the axisymmetric free vibration behavior of a rotating annular micro-disk is presented for the first time. The disk is assumed to be functionally-graded (FG) along the radial and thickness directions, and is considered to be operating in high-temperature environment. An energy based approach involving minimum potential energy principle and Hamilton's principle is used to derive the governing equations of motion considering Kirchhoff plate theory. The size-effect is addressed employing modified couple stress theory. A novel tangent stiffness based formulation is employed to derive the governing equations of vibratory motion in the neighborhood of the centrifugally and thermally deformed disk configuration. The governing equations are solved following Ritz method. The model captures both the axisymmetric and asymmetric flexural vibration modes, as well as the torsional mode. The model is successfully validated with the available results for some reduced problems. Numerical results are presented in tabular and graphical form for various material and geometric parameters, and some illustrative mode shape plots are presented showing the mode-switching phenomenon. The work presents a generalized model which can be reduced to theoretically model a wide variety of practical problems.

Axisymmetric vibrations of a circular Chladni plate in air and fully submerged in water

In this study, experimental and numerical analyses were performed to determine the effects of water loading on the axisymmetric modes of vibration of a circular plate. The plate was harmonically excited at its centre through an extension bar and its outer edge was left free. The Chladni technique, which involves exciting the plate at a resonance and waiting for sand grains sprinkled on the plate to collect along the nodal circles, was used to identify and visualize the modes both in air and fully submerged in water. Surprisingly, inverse Chladni patterns were observed in water as particles were drawn towards the zero transversal velocity radii by the induced flow. A coupled acoustic–structural finite element model was built to simulate the same modes, which had been preliminarily validated against theoretical results of a completely free edged plate. A good agreement between experimental and numerical natural frequencies and mode shapes was found. The frequency reduction ratio due to the added mass effect was around 64 %. Moreover, measurable differences due to fluid–structure coupling were observed in the radii of the nodal circles between corresponding dry and wet modes.

High-Frequency Measurement of Ultrasound Using Flexural Ultrasonic Transducers

Flexural ultrasonic transducers are a widely available type of ultrasonic sensor used for flow measurement, proximity, and industrial metrology applications. The flexural ultrasonic transducer is commonly operated in one of the axisymmetric modes of vibration in the low-kilohertz range, under 50 kHz, but there is an increasing demand for higher frequency operation, towards 300 kHz. At present, there are no reports of the measurement of high-frequency vibrations using flexural ultrasonic transducers. This research reports on the measurement of high-frequency vibration in flexural ultrasonic transducers, utilizing electrical impedance and phase measurement, laser Doppler vibrometry, and response spectrum analysis through the adoption of two flexural ultrasonic transducers in a transmit–receive configuration. The outcomes of this research demonstrate the ability of flexural ultrasonic transducers to measure high-frequency ultrasound in air, vital for industrial metrology.

Nonlinear vibration of transversely-loaded spinning membranes

The paper studies the transverse nonlinear vibration of an isotropic and homogeneous annular membrane spinning at constant angular velocity, under the action of a uniform transverse load. The equilibrium configuration of the membrane, clamped along the inner edge and free along the outer edge, and the natural frequencies of vibration of the membrane are calculated. A Galerkin procedure is used to determine a reduced order model describing the weakly nonlinear vibration of the membrane, and it is shown that near-resonance vibration can be modeled as a single degree of freedom Helmholtz-Duffing oscillator. A detailed study at vibration frequencies close to the fundamental, axisymmetric vibration mode shows a transition from softening to hardening behavior, and jump phenomena or hysteretic behavior depending on the angular velocity, the transverse load, the amplitude of dynamic excitation and the damping ratio. The results are in agreement with dynamic implicit finite element simulations in Abaqus/Standard and direct experimental measurements.

Vibration characteristics of a piezoelectric disk laminated with an elastic disk

This paper theoretically and experimentally deals with the vibration characteristics of a piezoelectric disk polarized in the thickness direction and laminated with an elastic disk. Axisymmetric vibration modes include radial and axial motions. Theoretically, in this study, the differential equations of piezoelectric motions were derived in terms of radial and axial displacements and electric potential. The differential equations of elastic motions were expressed in terms of radial and axial displacements. Solving the governing equations and boundary conditions for a coupled structure produced characteristic equations that provided natural frequencies and mode shapes. Experimentally, the natural frequencies were measured using an impedance analyzer and the radial in-plane motions of the fundamental mode were measured using an in-plane laser interferometer. The results of the theoretical analysis were compared with those of a finiteelement analysis and experiments; moreover, the theoretical analysis was verified on the basis of this comparison. It appeared that the natural frequencies of the radial modes of the piezoelectric disk were not affected by the elastic disk; however, those of the thickness modes were reduced by the elastic disk owing to the added mass effect.

Vibration characteristics of an ultrasonic transducer of two piezoelectric discs

This paper considers the influence of the different thickness of the piezoelectric discs used in assembly of an ultrasonic sandwich transducer. The transducer consists of two piezoelectric discs with different thickness between 0 and 2.0mm and with same diameter 28mm. Its vibration characteristics of the radial and axial motions were investigated theoretically and experimentally in axisymmetric vibration modes. Theoretically, the differential equations of piezoelectric motions were solved to produce characteristic equations that provided natural frequencies and mode shapes. The range of the fundamental frequency of radial in-plane mode is 80–360kHz and that of the axial out-of-plane mode is 600–1200kHz. Experimentally, the natural frequencies and the radial in-plane motion were measured using an impedance analyzer and an in-plane laser interferometer, respectively. The results of the theoretical analysis were compared with those of a finite-element analysis and experiments, and the theoretical analysis was verified on the basis of this comparison. It was concluded that the natural frequencies of the radial modes of the transducer were not affected by the stack and thickness of piezoelectric discs; however, those of the thickness modes were affected by the stack and thickness of the piezoelectric discs.

Vibration characteristics of a piezoelectric transducer laminated with piezoelectric disks

This paper considers a piezoelectric transducer polarized in the thickness direction and laminated with a piezoelectric disk. Its vibration characteristics were investigated theoretically and experimentally in the scope of the radial and axial motions in axisymmetric vibration modes. Theoretically, the differential equations of piezoelectric motions were derived in terms of radial and axial displacements and electric potential, and they were solved to produce characteristic equations that provided natural frequencies and mode shapes. Experimentally, the natural frequencies were measured using an impedance analyzer and the radial in-plane motions of the fundamental mode were measured using an in-plane laser interferometer. The results of the theoretical analysis were compared with those of a finite-element analysis and experiments, and the theoretical analysis was verified on the basis of this comparison. It appeared that the natural frequencies of the radial modes of the piezoelectric transducer were not affected by the added piezoelectric disk; however, those of the thickness modes were reduced by the piezoelectric disk owing to the added mass effect.

An exact frequency analysis of annular plates with small core having elastically restrained outer edge and sliding inner edge

The present study deals with an exact analysis of free transverse vibrations of annular plates having small core and sliding inner edge and the outer edge being elastically restrained based on classical plate theory. This study focuses mainly on the influence of variations in the elastic restraint parameters on the fundamental frequencies of plate vibration. The natural frequencies for the first six modes of annular plate vibrations are computed for different materials and varying values of the radius parameter and these natural frequencies may correspond to either axisymmetric and/or non-axisymmetric modes of plate vibration. The extensive data of values of fundamental frequency parameter presented in this paper is believed to be of use in the design of acoustic underwater transducers, ocean and naval structures, compressor and pump elements, offshore platforms. These results may serve as bench mark values for researchers to validate their results obtained using approximate numerical methods.

Two Dimensional Functionally Graded Material Finite Thick Hollow Cylinder Axisymmetric Vibration Mode Shapes Analysis Based on Exact Elasticity Theory

Abstract A thick hollow cylinder with finite length made of two- dimensional functionally graded material (2D-FGM) is considered and its natural modes are determined, based on great importance of mode shapes information in order to understand vibration behaviour of structures. Three dimensional theory of elasticity implemented for problem formulation, since mode shapes of a thick cylinder are three dimensional even with axisymmetric conditions. The axisymmetric conditions are assumed for the 2D-FGM cylinder. The material properties of the cylinder are varied in the radial and axial directions, with power law functions. Effects of volume fraction distribution on the different types of symmetric mode shapes configuration and vibration behaviour of a simply supported cylinder are analyzed. Three dimensional equations of motion are used and the eigen value problem is developed, based on direct variation method.

Experiment and simulation validated analytical equivalent circuit model for piezoelectric micromachined ultrasonic transducers.

An analytical Mason equivalent circuit is derived for a circular, clamped plate piezoelectric micromachined ultrasonic transducer (pMUT) design in 31 mode, considering an arbitrary electrode configuration at any axisymmetric vibration mode. The explicit definition of lumped parameters based entirely on geometry, material properties, and defined constants enables straightforward and wide-ranging model implementation for future pMUT design and optimization. Beyond pMUTs, the acoustic impedance model is developed for universal application to any clamped, circular plate system, and operating regimes including relevant simplifications are identified via the wave number-radius product ka. For the single-electrode fundamental vibration mode case, sol-gel Pb(Zr0.52)Ti0.48O3 (PZT) pMUT cells are microfabricated with varying electrode size to confirm the derived circuit model with electrical impedance measurements. For the first time, experimental and finite element simulation results are successfully applied to validate extensive electrical, mechanical, and acoustic analytical modeling of a pMUT cell for wide-ranging applications including medical ultrasound, nondestructive testing, and range finding.

Analytic solution for N-electrode actuated piezoelectric disk with application to piezoelectric micromachined ultrasonic transducers.

In this work, the deflection equation of a piezoelectrically-driven micromachined ultrasonic transducer (PMUT) is analytically determined using a Green's function approach. With the Green's function solution technique, the deflection of a circular plate with an arbitrary circular/ring electrode geometry is explicitly solved for axisymmetric vibration modes. For a PMUT with one center electrode covering ≈60% of the plate radius, the Green's function solution compares well with existing piece-wise and energy-based solutions with errors of less than 1%. The Green's function solution is also simpler than them requiring no numerical integration, and applies to any number of axisymmetric electrode geometries. Experimentally measured static deflection data collected from a fabricated piezoelectric micro ultrasonic transducer (PMUT) is further used to validate the Green's function model analysis. The center deflection and deflection profile data agree well with the Green's function solution over a range of applied bias voltages (5 to 21 V) with the average error between the experimental and Green's function data less than 9%.

A simple and unified displacement field for three-dimensional vibration analysis of prestressed circular plates

In this study a simple displacement field is proposed for finite element vibration analysis of circular plates. The displacement field is a combination of a conventional field, which is very popular for three-dimensional vibration analysis of circular plates or cylinders, and another field, which is most general for axisymmetric vibration. The investigation here is intended, through mathematical formulation and examples, to verify the existence of coupled axisymmetric vibration modes. These modes cannot be found by conventional displacement field, but can easily be revealed by the present model. Validity and accuracy of the proposed displacement field to show all the prominent vibration modes of circular plates or cylinder-like structures is also demonstrated by comparing results using the present method with those of a three-dimensional finite element analysis and experiments.

Modification of fundamental vibration modes of circular plates with free edges

For an isotropic circular plate with constant thickness and free edges, its fundamental vibration mode takes the form of a twisting mode with two nodal diameters, i.e. n=2. In certain applications, it may be necessary to have an axisymmetric shape for the fundamental vibration mode (i.e. mode shape with no nodal diameter, n=0). In this paper, we show that such an axisymmetric vibration mode can be realized by increasing the bending rigidity of the outer rim of the circular plate by using a larger thickness or by using a material with a larger Young's modulus or both. We also determine the critical flexural rigidity of the outer rim that will trigger this vibration mode change from n=2 type to n=0 type for a given rim width. The ability to alter the mode shape of circular plates with free edges has useful applications in sensing and actuating devices and large pontoon-type floating circular structures.

Vibrations of Complete Spherical Shells With Imperfections

Numerous theoretical investigations on the natural frequencies for complete spherical shells have been reported over the past four decades. However, attempts at correlating the theoretical results with either experimental or simulated results (both for axisymmetric and nonaxisymmetric modes of vibration) are almost completely lacking. In this paper, natural frequencies and mode shapes obtained from axisymmetric and nonaxisymmetric theories of vibration of complete spherical shells and from finite element computer simulations of the vibrations, with and without geometrical imperfections, are presented. Modal tests reported elsewhere on commercially available, thin spherical marine floats (with imperfections) are then utilized as a basis for comparison of frequencies to both the theoretical and numerical results. Because of the imperfections present, “splitting” of frequencies of nonaxisymmetric modes is anticipated. The presence of this frequency splitting phenomenon is demonstrated. In addition, results of a “whole field” measurement on one of the imperfect shells using dynamic holography are presented.

Modal Frequencies of Circular Sandwich Panels

Axisymmetric and non-axisymmetric modes of vibration of circular sandwich panels are investigated by applying first-order shear theory (Mindlin-Reissner Plate). Solutions are obtained for boundary conditions with free, simply supported, and clamped edges. Numerical finite element method (FEM) simulation using ABAQUS is applied to validate the analytical modeling. Both analytical and numerical calculations are compared with experimental measurements of the first few modes of vibration; the frequency differences are less than 4%.

Modal analysis of a unimorph piezoelectrical transducer

Piezoelectric transducers are widely applied to MEMS devices as actuators or sensors due to their simple structure and easy control. A micromachined piezoelectric microjet is introduced and modal analysis of its piezoelectric actuator has been thoroughly presented. Both numerical analysis and experimental measurement are implemented in this study. In numerical analysis, an approximate method is established to evaluate the first vibration mode and finite-element methods are used to simulate higher order vibration modes of the transducer. An experimental system was also set up to measure the axisymmetric vibration modes of the transducer. Compared with experimental results, good correlation was found between the numerical and experimental results, which proved the validation and reliability of the models. Through these models, further optimal design of the transducer can be carried out.