Nonaxisymmetric Vibrations Research Articles

Nonaxisymmetric electroelastic vibrations of piezoceramic ring plates with radially cut electrodes

The general solution describing the nonaxisymmetric electromechanical vibrations of a piezoceramic ring plate is obtained. The spectra of natural frequencies and modes of plates with radially cut electrode coating with various boundary conditions (both edges simply supported; one edge simply supported and the other clamped; both edges clamped) are numerically determined and analyzed for the first circumferential harmonics. The results are presented in the form of graphs and tables

Solution of the problem of nonaxisymmetric free vibrations of piezoceramic hollow cylinders with axial polarization

We consider the nonaxisymmetric problem of free vibrations of hollow piezoceramic cylinders for some types of boundary conditions at their end faces. The piezoceramics is polarized in the axial direction. The cylinder lateral surfaces are free from external loads and covered with thin electrodes that are shortcircuited. The initial three-dimensional problem is reduced to two-dimensional after separation of variables and representation of the components of the displacement vector and electrostatic potential in the form of standing waves in the circumferential direction. The method of solution of the problem is based on the joint use of the spline-collocation method along the longitudinal coordinate and the method of discrete orthogonalization combined with step-by-step search by the radial coordinate. We also present some results of numerical analysis of the behavior of a cylinder made of PZT 4 piezoceramics over a wide range of changes in the geometrical characteristics of the cylinder. Piezoceramic structural elements of cylindrical shape are widely used in radio electronics, various automatic devices, computing machinery, and measuring tools. A high efficiency of the transformation of electron energy into mechanical and the inverse process, together with comparatively small sizes, enable one to use such materials in nanotechnologies. The solution of dynamic problems for thick-walled elements as three-dimensional problems of the theory of elasticity is connected with significant difficulties, which are due to the complexity of the system of initial partial differential equations as well as by the necessity to satisfy boundary conditions on the surfaces confining the body. These difficulties grow substantially in the cases where the fields under study are coupled and the piezoelectric materials are anisotropic [1, 2, 8, 9]. Therefore, it becomes necessary to develop efficient numerical– analytic approaches for the solution of these classes of problems. During recent years, for the solution of such problems of computational mathematics, mathematical physics, and mechanics, spline functions are widely used. This is explained by the advantages of the apparatus of spline approximations as compared with other methods. Among the main advantages of this apparatus, we should mention

Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials

The nonaxisymmetric problem of natural vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials is solved. The properties of the material change continuously along a radial coordinate according to an exponential law. The lateral surfaces of the cylinder are free of external tractions and short circuited by electrodes. After separation of variables and representation of the components of the displacement vector in the form of standing circumferential waves, the initially three-dimensional problem is reduced to a two-dimensional partial differential equation problem. By using the method of spline-collocations with respect to a longitudinal coordinate, this two-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed in terms of ordinary differential equations. This problem is solved by the stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Results were obtained numerically and subsequently analyzed in this paper.

Dynamic Characteristics Analysis of Stepped Circular Plates

Natural frequencies and mode shapes of axisymmetric and non-axisymmetric vibration of stepped thin circular plates were calculated with self-programmed program. Three-node annular elements were used in the program.Results of axisymmetric vibration obtained by the self-programmed program and those by ANSYS program were compared.It shows that self-programmed program can calculate axisymmetric vibration as well as non-axisymmetric vibration.The accuracy of self-programmed program is higher than ANSYS method.

Dynamic Characteristics Analysis of Stepped Circular Plates

Natural frequencies and mode shapes of axisymmetric and non-axisymmetric vibration of stepped thin circular plates were calculated with self-programmed program. Three-node annular elements were used in the program.Results of axisymmetric vibration obtained by the self-programmed program and those by ANSYS program were compared.It shows that self-programmed program can calculate axisymmetric vibration as well as non-axisymmetric vibration.The accuracy of self-programmed program is higher than ANSYS method.

Free nonaxisymmetric vibrations of radially polarized hollow piezoceramic cylinders of finite length

The three-dimensional problem of free nonaxisymmetric vibrations of hollow piezoceramic cylinders with axial polarization is considered. An efficient numerical analytic method to solve boundary-value problems is proposed. The original three-dimensional problem of electroelasticity is reduced to a two-dimensional problem by representing the displacement components as standing circumferential waves. Spline collocation with respect to the axial coordinate is used to reduce this two-dimensional problem to an eigenvalue boundary-value problem with respect to the radial coordinate. This problem is solved by the stable discrete-orthogonalization and incremental-search methods. Numerical results are presented and the natural frequencies of the cylinders are analyzed in a wide range of their geometric characteristics

Analysis of the nonaxisymmetric vibrations of flexible ellipsoidal shells discretely reinforced with transverse ribs under nonstationary loads

The problem of forced nonaxisymmetric vibrations of reinforced ellipsoidal shells under nonstationary loads is formulated. A numerical algorithm of solving it is developed and the results obtained are analyzed

Nonaxisymmetric vibrations of ellipsoidal shells under nonstationary distributed loads

Problems of forced nonaxisymmetric vibrations of ellipsoidal shells under nonstationary loads are formulated. A numerical algorithm to solve them is developed. The solutions obtained are analyzed

A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates

A finite element is presented to analyze the three-dimensional (3-D) vibration of piezoelectric coupled circular and annular plates. The proposed finite element is a modification of a conventional axisymmetric finite element and is capable of conducting both axisymmetric and nonaxisymmetric vibration analysis of circular and annular laminated plates, with piezoelectric layers therein. The present formulation, a two-dimensional model itself, can investigate 3-D vibration of those plates for a preselected number of nodal diameters, and is therefore more economical than the conventional 3-D finite element analysis, yet still has almost the same accuracy and versatility as the 3-D analysis. In cases such as analysis of stators of traveling wave ultrasonic motors where only vibration modes with particular numbers of nodal diameters are of interest, the proposed approach is very convenient and useful.

Free Vibration of Composite Annular Plates Embedded with SMA Fibers in the Circumferential Direction

Based on Brinson’s one-dimensional thermo-constitutive law of SMA and classical plate theory, linear non-axisymmetric vibration of uniform heated composite annular plate embedded with circumferential SMA fibers is investigated. Natural frequencies of the annular plates with immovably clamped boundary condition, depending on the temperature rise are obtained by using shooting method. The characteristic curves of first four natural frequencies of non-axisymmetric vibration versus temperature rise are plotted. Influences of the volume fraction and the initial strain of SMA on the natural frequencies of plate are analyzed. The numerical results show that, the activation of SMA can enhance the natural frequency both in the inverse martensite transformation temperature range and the thermal buckling temperature.

B114 柔らかいフィンをもつ円管から流出する噴流

As a part of the study on the control of jet diffusion, the jet flow issuing from a circular pipe with fluttering fins was investigated. It has been already obtained from the previous study that the fluttering fins were effective for promotion of the jet diffusion. In this paper, the effects of the fin width on the jet diffusion were addressed. The jet flow was visualized by means of the PIV analysis and the LIF method. Moreover, the fin vibration was observed by the high-speed camera. As a result, it has been found that there exists a suitable fin width for the jet diffusion. Furthermore, it has been recognized that the amplitude of the fin vibration becomes smaller and the frequency higher with the decrease of the fin width. And then, the non-axisymmetric vibration appears and the regularity of vibration period weakens.

Acoustic backscattering from fluid‐filled submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The fluid‐filled shell is insonified by a plane wave with an arbitrary angle of incidence. The external and internal fluid loading impedances were computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far‐field backscattered acoustic pressure spectra are presented as a function of the angle of incidence for several shell thickness‐to‐half‐length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). A comparison of the backscattering from fluid‐filled and empty shells is presented at selected plane wave incident angles. [Work supported by the ONR/ASEE Summer Faculty Research Program and the NAVSEA Newport ILIR Program.]

Observation of the shape of a water drop on an oscillating Teflon plate

This is an experimental investigation of the dynamics of water drops under large-amplitude forcing within a frequency range of 10–700 Hz. Water drops on an oscillating horizontal plate displayed various behaviors, such as bouncing, dividing, ejecting, axisymmetric polygonal vibration, and polygonal but not axisymmetric vibration. These last two were alike in that the capillary waves of a water drop gave rise to various polygons. However, the planar directions of the polygons of the non-axisymmetric vibrations seemed stable, and their frequencies were the same as the induced frequencies of the oscillating plate. These two features make non-axisymmetric polygonal vibration different from the axisymmetric polygonal vibration reported to date, and the phenomenon has not yet been successfully analyzed. Trials of simulating some shapes of non-axisymmetric polygonal vibrations according to Rayleigh’s theory are described in this paper.

Nonaxisymmetric acoustic scattering from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamiltons principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The shell is insonified by a plane wave with arbitrary incident angle. The external (heavy) fluid-loading impedance was computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far-field backscattered acoustic pressure spectra are presented as a function of incident angle for several shell thickness-to-half-length ratios ranging from .005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic scattering is presented at selected frequencies and plane wave incident angles. [Work supported by the ONR/ASEE Summer Faculty Research Program and the NAVSEA Newport ILIR Program.]

Vibration and stability of rotating polar orthotropic sandwich annular plates with a viscoelastic core layer

A sandwich annular plate is built with two face layers of polar orthotropic material and a viscoelastic core layer. The non-axisymmetric vibration and stability problem of the rotating sandwich plate are analyzed by using the finite element method. By employing the discrete layer annular element and Hamilton’s principle, the finite element equations of motion that consider the effects of transverse shear and rotary inertia are derived. The viscoelastic material in the core layer is assumed to be incompressible, and its extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved. The effects of various properties, such as orthotropic properties of face layers, and the viscoelastic core layer stiffness and thickness are discussed. Numerical results show that the larger modulus ratio and higher rotational speed tend to increase the nature frequency of the sandwich annular plate system. In addition, the critical speeds decrease while increasing the thickness of the viscoelastic core layer.

Axisymmetric acoustic scattering from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The shell is insonified by a plane wave incident along the major axis. The external (heavy) fluid loading impedance was computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far-field scattered acoustic pressure spectra are presented for several shell thickness-to-half-length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic scattering is presented at selected frequencies. [Work supported by the ONR/ASEE Summer Faculty Research Program.]

Nonaxisymmetric Electroelastic Vibrations of a Hollow Cylinder with Radial Axis of Physicomechanical Symmetry

An approach is proposed to calculate the natural frequencies and modes of vibrations of cylindrically anisotropic piezoelectric cylinders. The frequency equation is derived, and the frequency spectrum is analyzed

Forced Nonstationary Vibrations of a Sandwich Cylindrical Shell with Cross-Ribbed Core

The equations of nonaxisymmetric vibrations of sandwich cylindrical shells with discrete core under nonstationary loading are presented. The components of the elastic structure are analyzed using a refined Timoshenko theory of shells and rods. The numerical method used to solve the dynamic equations is based on the integro-interpolation method of constructing finite-difference schemes for equations with discontinuous coefficients. The dynamic problem for a sandwich cylindrical shell under distributed nonstationary loading is solved with regard for the discreteness of the core

Axisymmetric acoustic radiation from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamiltons principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. For axisymmetric vibration of a submerged shell, the external (heavy) fluid loading impedance was computed using expansions of prolate spheroidal wavefunctions. The shell was excited by axisymmetric normal surface forces, including a point load at the shell apex and ring load at other locations. Far-field radiated pressure spectra are presented for several shell thickness-to-half-length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic radiation is presented at selected frequencies. [Work supported by the NUWC ILIR Program and the ONR/ASEE Summer Faculty Research Program.]

Vibration actuator with two vibration modes

A vibration actuator includes a vibration element, which simultaneously generates a radial symmetric expansion vibration mode in which it expands and contracts in the radial direction and a non-axisymmetric planar vibration mode in which it bends to and fro in a non-axisymmetric manner within a single plane, and thereby drives a relative movement member, and a base member to which the vibration element is fixed. The vibration element includes at least one superimposed layer structure comprising a pair of electrical energy to mechanical energy conversion elements and an elastic member sandwiched between the pair of electrical energy to mechanical energy conversion elements. The elastic member includes a ring shaped portion to both the sides of which the pair of electrical energy to mechanical energy conversion elements are stuck, and a cylindrical shaped support portion, formed integrally with the ring shaped portion, which fixes the vibration element to the base member. The support portion includes an elastic portion which suppresses the transmission of vibration generated by the vibration element to the base member.