Nonaxisymmetric Vibrations Research Articles

Voltage-controlled non-axisymmetric vibrations of soft electro-active tubes with strain-stiffening effect

Material properties of soft electro-active (SEA) structures are significantly sensitive to external electro-mechanical biasing fields (such as pre-stretch and electric stimuli), which generate remarkable knock-on effects on their dynamic characteristics. In this work, we analyze the electrostatically tunable non-axisymmetric vibrations of an incompressible SEA cylindrical tube under the combination of a radially applied electric voltage and an axial pre-stretch. Following the theory of nonlinear electro-elasticity and the associated linearized theory for superimposed perturbations, we derive the nonlinear static response of the SEA tube to the inhomogeneous biasing fields for the Gent ideal dielectric model. Using the State Space Method, we efficiently obtain the frequency equations for voltage-controlled small-amplitude three-dimensional non-axisymmetric vibrations, covering a wide range of behaviors, from the purely radial breathing mode to torsional modes, axisymmetric longitudinal modes, and prismatic diffuse modes. We also perform an exhaustive numerical analysis to validate the proposed approach compared with the conventional displacement method, as well as to elucidate the influences of the applied voltage, axial pre-stretch, and strain-stiffening effect on the nonlinear static response and vibration behaviors of the SEA tube. The present study clearly indicates that manipulating electro-mechanical biasing fields is a feasible way to tune the small-amplitude vibration characteristics of an SEA tube. The results should benefit experimental work on, and design of, voltage-controlled resonant devices made of SEA tubes.

Theoretical model for investigating three-dimensional effect in integrity test of open-end pipe piles

An innovative theoretical model of non-axisymmetric vibration of pipe pile (NVPP) is proposed in this paper to investigate the three-dimensional effect in the pile integrity test (PIT) for open-end pipe piles. A new dynamic equilibrium equation of the pipe pile is established based on the stress simplification assumption. It overcomes the inaccuracies of existing three-dimensional models in predicting the longitudinal wave velocity of pile material. The surrounding soil and soil plug conform to the classical plane strain assumption, while the pile end soil is simulated by the fictitious soil annular pile developed from the traditional fictitious soil pile. Theoretical solutions for the velocity response of each point on the pipe pile top are deduced and validated against existing solutions and FEM results. The NVPP model is then applied to analyze the characteristics of testing signals of PITs for thick-wall and thin-wall pipe piles. Problems concerned in actual PITs for pipe piles are settled and practical recommendations for acquiring high-quality testing signals with minimal interference are provided.

Numerical and test study on vertical vibration characteristics of pile group in slope soil topography

Topography effects on the vertical vibration responses of pile group are revealed though numerical analysis and model tests. First, a series of model tests with different topography of ground and bedrock are conducted. The results indicate that displacement amplitude of the pile head in sloping ground topography is larger than in horizontal ground. Differential displacement at various positions of the pile cap is observed in non-horizontal topography. Afterwards, a numerical algorithm is employed to further explore the essential response characteristics in group piles of different topography configurations, which has been verified by the test results. The lengths of the exposed and frictional segment, together with the thickness of the subsoil layer, are the dominant factors which cause non-axisymmetric vibration at the pile cap.

Analysis of thermoelastic damping limited quality factor and critical dimensions of circular plate resonators based on axisymmetric and non-axisymmetric vibrations

Thermoelastic damping effects are very important intrinsic losses in microelectromechanical system/nanoelectromechanical system based sensors and filters, which limit the maximum achievable quality factor. Thermoelasticity arises due to coupling between the temperature field and elastic field of the material and its interaction within the material structure. The impacts of axisymmetric and non-axisymmetric vibrations, plate dimensions, material parameters, boundary conditions, mode switching, and temperature on thermoelastic damping limited quality factors (QTED) and critical thickness (hc) were analyzed, and the conditions for an enhanced quality factor were optimized in this work. The analytical models of circular plate resonators have been developed in terms of material performance indices for axisymmetric and non-axisymmetric vibrations. QTED and hc were analyzed based on two boundary conditions: simply supported and clamped–clamped. In order to obtain maximum QTED, micro-circular plates with diamond as the structural material operating at a lower temperature and with non-axisymmetric vibrations are proposed in this paper.

Axisymmetric and Non-Axisymmetric Vibration of Thin Growing Viscoelastic Disc

Axisymmetric and non-axisymmetric in-plane vibrations of thin radially growing disc are considered in the frame of the Kelvin–Voigt model of linear viscoelasticity. The main focus of the research is on asymptotic behavior of solutions to the model. Mixed boundary value problems are formulated and converted into standard form by means of time dependent coordinate transformation. The boundary value problems obtained are transformed into infinite systems of ordinary differential equations using the Galerkin-Kantorovich method. It is shown that both axisymmetric and non-axisymmetric problems can be considered in terms of the same mathematical model. The simplified single mode model of the growing disc vibration is formulated and analysed. In the frame of this model the exact solutions are obtained in some interesting special cases. In the general case, solutions are derived using the Wentzel–Kramers–Brillouin (WKB) method. Depending on the exponent of radial growth, the vibration amplitude of the disc demonstrates qualitatively different behavior: if the exponent is less than 0.5 the vibration amplitude decays, if it is more than 0.5 the amplitude increases without bound no matter what size of damping coefficient is assumed. If the exponent equals 0.5 the vibration behavior depends on damping, namely there exist the critical damping coefficient and if the damping coefficient of a particular mode is less than this coefficient, the vibration amplitude grows without bound. Otherwise, if the damping coefficient of the mode is larger than the critical damping coefficient the vibration amplitude decays to zero. In the case where damping coefficient equals to the critical damping one, the amplitude of vibration remains constant. All of the qualitative results obtained for the single mode model are numerically tested and verified using a truncated multimodal system.

Free vibration analysis of discrete-continuous functionally graded circular plate via the Neumann series method

• The Neumann series method is used for free vibration analysis of functionally graded circular plate with discrete elements. • The equation of motion is presented for the first time for the plate with a very complex system of mechanical elements. • The effects of singularities on eigenfrequencies of the plate with sliding support and elastic constraints are studied. • The characteristic equations can be numerically solved in a simple way to obtain the full spectrum of the eigenfrequencies. The Neumann series method has been used for the first time to solve the boundary value problem of free axisymmetric and nonaxisymmetric vibrations of continuous and discrete-continuous functionally graded circular plate on the basis of the classical plate theory. The equation of motion and the general solution for a functionally graded circular plate with a very complex system of a discrete elements attached, such as concentric ring masses, elastic supports, rotational springs, and damping elements are presented for the first time. The particular continuous solutions to the defined differential equations are obtained as the Neumann power series rapidly, absolutely, and uniformly convergent to the exact eigenfrequencies for any physically justified values of the plate's parameters on the basis of the properties of the obtained closed-form kernels of the Volterra integral equations. The multiparametric nonlinear characteristic equations for plate with classical and nonclassical boundary conditions are defined and numerically solved to obtain the full spectrum of eigenfrequencies in a simple way. The effects of the position and stiffness of ring supports and of singularities as the radii of supports shrink to the center of the plate on the dimensionless eigenfrequencies of homogeneous and functionally graded circular plate with sliding support and elastic constraints are comprehensively studied and presented for the first time. The accuracy of the proposed low-computational-cost method is demonstrated by comparison of the numerical results with those available in the literature.

Stress State of Discretely Stiffened Ellipsoidal Shells Under a Nonstationary Normal Load

The problem of the forced non-axisymmetric vibration of stiffened ellipsoidal shells under nonstationary load is formulated. A numerical algorithm for solving the problem is developed, and the results obtained are analyzed.

Free vibration of biopolymer spherical shells of high structural heterogeneity

A refined elastic shell model is used to study the effect of high structural heterogeneity on natural frequencies and vibration modes of biopolymer spherical shells. With this model, the structural heterogeneity of a biopolymer spherical shell is characterized by an effective bending thickness (which can be quite different from the average thickness) and the transverse shear modulus (which can be much lower than the in-plane shear modulus). Our results show that actual natural frequencies of axisymmetric spheroidal modes of a biopolymer spherical shell can be much lower than those predicted by the classical homogeneous shell model based on the average thickness, although natural frequencies of axisymmetric torsional modes are close to those predicted by the classical model. For example, with physically realistic parameters for virus capsid STMV, the natural frequencies of spheroidal modes predicted by the present model are about 30-50% lower than those predicted by the classical model, in better agreement with known simulation results. In addition, in the low frequency range of several viral capsids, the number of independent non-axisymmetric vibration modes predicted by the present model is considerably larger than that predicted by the classical homogeneous shell model, in qualitative agreement with known atomistic simulations. These results suggest that the refined shell model could offer a relatively simple model to simulate mechanical behavior of biopolymer spherical shells of high structural heterogeneity.

Dynamics of a Three-Layer Elliptic Cylindrical Shell Reinforced with Discrete Rings

The equations of non-axisymmetric vibrations of three-layer elliptical cylindrical shells with reinforcement rings under non-stationary loading are derived. The elements of this elastic structure are described using the refined Timoshenko-type model of shells and rods. The numerical method of solving the dynamic equations is based on the integro-interpolation method of constructing the finite-difference schemes for equations with discontinuous coefficients. The problem of the dynamic behavior of a three-layer elliptical cylindrical shell with discrete ribs under distributed non-stationary loading is solved. The numerical results are given in the form of plots and analyzed.

Free vibration analysis of elastically supported functionally graded annular plates via quasi-Green's function method

Analysis and numerical results for the free axisymmetric and non-axisymmetric vibrations of functionally graded annular plates elastically supported on the ring support have been presented on the basis of classical plate theory. The quasi-Green's function has been employed to solve to boundary value problem of free vibration of functionally graded annular plate with free outer edge and clamped, simply supported and free inner edges. Additional properties of the quasi-Green's function were presented and discussed for the functionally graded annular plate. The influence of volume fraction index, core radius, selected boundary conditions, position and stiffness of ring supports on natural frequencies of an annular plate has been comprehensively studied. Singularities as the core and support radii shrink to zero are calculated for different values of volume fraction index and stiffness of the support. The quasi-Green's function method allows us to obtain universal non-linear multiparametric characteristic equations functional dependent on radius of core, volume fraction index, position and stiffness of any amount of elastic supports or other discrete elements without necessity considering new boundary value problem or find scaling factors. Additionally, the continuity conditions between the supports and plate can be omitted in the obtained solutions of boundary value problem. The presented investigation has not been reported previously. The exact frequencies of vibration presented in non-dimensional form can serve as benchmark values for researchers and engineers to validate their numerical methods applied in many fields of engineering such us mechanical, aeronautical or production engineering.

Non-axisymmetric response of piles in low-strain integrity testing

Low-strain testing of piles is routinely performed by striking the pile head with a small hand-held hammer, usually aiming for the pile centre, and measuring the vertical velocity response by means of a receiver placed also on the pile head. In practice, however, it is impossible for the operator to perfectly align the point of impact with the pile centre, as existing solutions for the interpretation of the velocity measurements assume. In the general case where the impact load is eccentric, the wave field is no longer axisymmetric with respect to the pile axis and waves resulting from the hammer impact can propagate along the vertical, radial and circumferential directions. This is taken into consideration in the analytical solution presented in this note, which provides the non-axisymmetric dynamic response of a pile to an impact load applied at an arbitrary position on the pile head. Arithmetic examples are used to depict the three-dimensional non-axisymmetric vibration characteristics of the soil–pile system, with emphasis on the effect of the eccentricity of the impact load on the optimal position of the receiver, the identification of which is essential for obtaining meaningful results from low-strain integrity tests.

Nonaxisymmetric Vibrations of a Shell of Revolution Partially Filled with Liquid

We propose an algorithm for finding the frequencies and modes of natural vibrations of the shells of revolution partially filled with liquid. The problem of perturbed motion of a liquid is solved under the assumption that its free surface remains flat and perpendicular to the axis of the shell. The solution is based on the use of the method of decomposition of the domain of integration of the equations of the theory of shells in combination with the variational method and the approximate construction of the inverse operator for the hydrodynamic part of the problem. We construct a generalized functional with respect to displacements of the shell for which the conditions of conjugation of the solutions in subdomains are included in the natural boundary conditions. The obtained numerical results are compared with the available exact solutions of the problem under consideration with regard for the wave motions of liquid in a shell in the form of circular cylinder.

Effect of Boundary Conditions on the Natural Frequencies and Vibration Modes of Piezoelectric Plates with Radially Cut Electrodes

The general solution to the problem of the nonaxisymmetric electromechanical vibrations of a piezoceramic ring plate is found. The effect of boundary conditions (clamped edge–free edge, free edge–clamped edge, free edge–free edge) on the natural frequency spectra (for the first circumferential harmonics) of plates with radially cut electrode coating is numerically analyzed

Natural Modes of Vibration of Piezoelectric Circular Plates with Radially Cut Electrodes

The general solution to the problem of the nonaxisymmetric electromechanical vibrations of a piezoceramic ring plate is obtained. The spectra of natural frequencies and modes for the first circumferential harmonics are numerically determined and analyzed for plates with radially cut electrode coating and the following boundary conditions: clamped edge–free edge, free edge–clamped edge, free edge–free edge

Non-Axisymmetric Vibrations of Stepped Cylindrical Shells Containing Cracks

Based on the Donnell’s approximations of the thin shell theory, this paper presents solutions for the problem of free non-axisymmetric vibration of stepped circular cylindrical shells with cracks. The shell under consideration is sub-divided into multiple segments separated by the locations of thickness variations. It is assumed that at thejth step there exists a circumferential surface crack with uniform depthcj. The influence of circular cracks with constant depth on the vibration of the shell is prescribed with the aid of a matrix of local flexibility. The latter is related to the coefficient of the stress intensity known in the linear fracture mechanics. Numerical results are obtained for cylindrical shells of stepped thickness containing cracks at re-entrant corners of steps. Shells with various combinations of boundary conditions can be analyzed by the proposed method. Furthermore, the influences of the shell thicknesses, locations of step-wise variations of the thickness and other parameters on the natural frequencies are examined. The results can be used for the approximate evaluation of dynamic parameters of cylindrical shells with cracks and flaws.

Nonaxisymmetric Vibrations of Conical Shells of Variable Thickness Under a Nonstationary Load

The problem of the nonaxisymmetric vibrations of an inhomogeneous conical shell of variable thickness under a nonstationary load is formulated and solved. An algorithm for solving this problem is presented. The system of differential equations is solved using Timoshenko-type shell theory. The dynamic behavior of a conical panel of variable thickness under a nonstationary load is analyzed as an example

Energy aspects of wave propagation in the infinite cylindrical shell fully submerged into the liquid

The joint non-axisymmetric vibrations of an ideal acoustic liquid and an infinite thin empty cylindrical shell of Kirchhoff type are investigated. The first mode corresponds to this important type of vibrations. Vibrational modes associated with deformation of the cross-section of the shell are also considered. The problem of free vibrations of the shell submerged into the liquid space is studied in the rigorous mathematical statement. The exact analytical solution of this problem is analyzed. The source of vibration and acoustic fields in the system shell-liquid is the wave propagating through the shell from infinity. The high and low frequency asymptotics of the dispersion curves are analyzed. The propagating waves and energy flux in the system shell-liquid are determined. The case of negative group velocity (at positive phase velocity) of the waves and the sign of the energy flux components in the shell are discussed. The energy flux and its component are considered in the vicinity of quasitraverse points of dispersive curves. Different types of these points and their coordinates are asymptotically studied. The influence of relative velocities of the waves in the shell and fluid on the behavior of the entire system is explored. Comparison of various vibration modes is performed from the viewpoint of energy fluxes.

Forced Vibrations of Ellipsoidal Shells Reinforced with Transverse Ribs Under a Nonstationary Distributed Load

The problem of the forced nonaxisymmetric vibrations of reinforced ellipsoidal shells under nonstationary loading is considered. A numerical algorithm for solving the problem is developed. Numerical results are obtained and analyzed

Influence of boundary conditions on the natural frequencies of nonaxisymmetric electroelastic vibrations of piezoceramic plates

The general solution to the problem of the nonaxisymmetric electromechanical vibrations of a piezoceramic ring plate is obtained. For plates with radially cut electrode coating and different boundary conditions (clamped edge–free edge, free edge–clamped edge, free edge–free edge), the natural frequency spectra are determined numerically and analyzed for the first circumferential harmonics

Two classes of vibrations of a short circular cylinder

This paper gives a solution of the stationary dynamic problem of elasticity which describes two classes of natural nonaxisymmetric vibrations of a finite circular cylinder. In the particular case of axial symmetry, the resulting solution describes two well-known classes of axisymmetric vibrations: vibrations of the first class become longitudinal-transverse vibrations and vibrations of the second class become torsional vibrations. The existence of two classes of nonaxisymmetric vibrations is due to the boundary conditions at the ends. It is shown that as the length (height) of the cylinder increases, the effect of the boundary conditions at the ends on the frequency spectrum reduces, and the vibration frequencies of the two classes become similar and then identical.