Complex Natural Frequencies Research Articles

Lateral kinematic properties of offshore pipe piles embedded in saturated soil considering soil plug effect

AbstractThis study establishes a theoretical framework for analyzing the lateral oscillation of marine pipe piles. The additional mass model is introduced herein to consider the inertial fluctuation effect of the soil plug. Analytical mathematical methods are used to determine the complex impedance variation of the pile over a range of frequency effects. An investigation is performed to determine how the presence of soil plugs changes the lateral complex stiffness and natural frequency of pipe piles. Additionally, comparisons of the applicability of the plane strain model and continuous medium model have been conducted to enable the easy use of the theoretical model. The main conclusions can be drawn as (1) if the fluctuation inertia effect of the soil plug is not taken into consideration, the dynamic active length and the dynamic stiffness of the pipe pile will be underestimated; (2) for the soft soil, the plane strain model may give rise to substantial calculation errors attributed to them regardless of the vertical continuity of the soil, nevertheless, the calculation error decreases rapidly with the increase of soil shear modulus and vibration frequency.

On the complex mode shapes and natural frequencies of clamped-clamped fluid-conveying pipe

In the present study, the closed-form expression for complex mode shapes of a fluid-conveying pipe using Timoshenko beam theory is developed for the first time. By applying the method of separation of variables, the complex mode shapes and eigenvalues are obtained. To the best knowledge of the authors, there is no study carried out on the complex eigenvalue problem for vibration analysis of the Timoshenko fluid-conveying pipes. Given this oversight, in this study the effects of fluid velocity on the imaginary and real parts of the mode shapes and natural frequencies of a clamped-clamped fluid-conveying pipe are investigated. The results show that for very low fluid velocities, the first and the second mode shapes are similar to that of an equivalent beam. For high fluid velocities, the first mode shape experiences some deviation so that it looks like the second mode shape of a pipe having very low fluid velocity. The results show that fluid flow in the pipeline effectively reduces its bending moment as well as the energy of vibration. In addition, the nodal points corresponding to the second mode are replaced with quasi-node points.

Generalized frequency shift and attenuation in simply-supported micro/nano-beam resonators with thermoelastic dissipation

The low frequency shift, weak frequency attenuation, and large quality factor (Q) are strongly required for micro/nano-resonators with excellent operation performance. Therefore, the accurate predictions of frequency shift, frequency attenuation, and Q based on thermoelastic dissipation (QTED ) are significant when optimally designing micro/nano-resonators. In this work, employing the modified-couple-stress (MCS) and nonlocal-single-phase-lag (NSPL) models, the generalized analytical expressions of the frequency-shift ratio, frequency-attenuation ratio, and QTED are first proposed for the simply-supported micro/nano-beam resonator with two-dimensional (2D) heat conduction. In the modeling procedure, utilizing the Galerkin approach, the function of 2D fluctuation temperature is solved from the governing equation of NSPL thermoelasticity. For simplicity, the function of one-dimensional (1D) fluctuation temperature is presented directly. Various expressions of complex natural frequency are obtained from the governing equation of MCS vibration with thermoelastic coupling, and then the models of the frequency-shift ratio, frequency-attenuation ratio, and QTED are developed based on the relevant expressions of complex natural frequency. It is confirmed that the present developed models with generality can effectively degenerate into the classical models. In the simulation cases, silicon (Si) that is a representative semiconductor material is selected. Results reveal that the MCS size-dependence effect can affect the frequency shift more markedly than the NSPL and heat-conduction-dimension (HCD) effects. Moreover, for the beam with a small thickness and operated in a high-order mode, the NSPL and HCD effects will significantly determine behaviors of the frequency shift, frequency attenuation, and QTED .

Determination of Frequency-Dependent Shear Modulus of Viscoelastic Layer via a Constrained Sandwich Beam.

Viscoelastic material can significantly reduce the vibration energy and radiated noise of a structure, so it is widely used in lightweight sandwich structures. The accurate and efficient determination of the frequency-dependent complex modulus of viscoelastic material is the basis for the correct analysis of the vibro-acoustic behavior of sandwich structures. Based on the behavior of a sandwich beam whose core is a viscoelastic layer, a combined theoretical and experimental study is proposed to characterize the properties of the viscoelastic layer constituting the core. In this method, the viscoelastic layer is bonded between two constraining layers. Then, a genetic algorithm is used to fit the analytical solution of the frequency¬ response function of the free–free constrained beam to the measured result, and then the frequency-dependent complex modulus is estimated for the viscoelastic layer. Moreover, by varying the length of the beams, it is possible to characterize the frequency-dependent complex modulus of the viscoelastic material over a wide frequency range. Finally, the characterized frequency-dependent complex modulus is imported into a finite element model to compute the complex natural frequencies of a sandwich beam, and a comparison of the simulated and measured results displays that the errors in the real parts are within 2.33% and the errors in the imaginary parts are within 3.31%. It is confirmed that the proposed method is feasible, accurate, and reliable. This provides essential technical support for improving the acoustic vibration characteristics of sandwich panels by introducing viscoelastic materials.

Диссипативные свойства трехслойных композитных структур. 2. Метод решения

Object and purpose of research. This paper discusses a three-layered plate made up by two rigid anisotropic layers and soft isotropic medium layer of viscoelastic polymer. Each of the two rigid layers is an anisotropic structure formed by a finite number of arbitrarily oriented orthotropic viscoelastic composite layers. The purpose of this work is to develop a numerical solution method for decaying vibration equations of three-layered unsupported rectangular plates. Materials and methods. The system of algebraic equations is constructed as per Ritz method using Legendre polyno-mials as coordinate functions. The first step is to find real solutions. To find complex natural frequencies of the system, their initial values are taken as real natural frequencies thus found, with subsequent calculation of complex frequencies as per the method of third-order iterations. Main results. This paper discusses the results of convergence analysis for a numerical solution of differential motion equations with respect to an unsupported rectangular three-layered plate with transversally isotropic rigid layers. The material for these rigid lay-ers is unidirectional carbon-reinforced plastic (CRP) with elastic dissipation properties, within the investigated range of frequencies and temperatures, independent on its vibration frequency. For the soft isotropic medium layer of viscoelastic polymer, temperature-frequency curve governing the real part of complex elasticity modulus and mechanical loss coefficient is taken into account. Validation of the mathematical model and the numerical solution method, the comparison of calculated and experimental natural frequencies and mechanical loss coefficients for the two variants of three-layered unsupported plate has demonstrated their good correlation. Conclusion. This paper suggests and validates the numerical solution method for decaying vibration equations of three-layered unsupported rectangular plate made up by two rigid monoclinic layers and soft isotropic medium layer of viscoelastic polymer.

Tension Estimation Method for Cable With Damper Using Natural Frequencies With Uncertain Modal Order

In the maintenance of cable structures, such as cable-stayed bridges, it is necessary to estimate the tension acting on the cables. In current Japanese practice, the cable tension is estimated from the cable’s natural frequency using vibration-based methods. However, in recent years, dampers have been installed onto the cables to suppress aerodynamic vibrations. Because the damper changes the cable’s natural frequencies, the methods used for cables without dampers are not appropriate for cables with dampers. With this background, the authors previously proposed a method (Method 0F) for estimating the tension of a cable with a damper from the natural frequencies and their modal order. Method 0F partially ignores the imaginary part of the complex natural frequencies to simplify the problem. This study proposes a new method (Method 1F) that does not ignore the imaginary part of a complex natural frequency. Method 1F still needs both the natural frequencies and their modal order to be input. If the modal order is not correctly specified, the accuracy deteriorates. Therefore, a new method (Method 2F) that only requires the natural frequencies is also proposed. The validity of the proposed methods was confirmed by numerical simulations and experiments. The numerical and experimental verifications confirmed that the new methods were superior compared with previous methods, and that Method 2F has the highest effectiveness.

Desempeño del método de Prony y del método del método del haz de matrices en la determinación de las frecuencias complejas naturales de resonancia de un sistema lineal

Characterization of a physical system is an important issue to approach some applied physics and engineering problems. The complex natural resonance frequencies of the system which are included in its impulsive response are characteristic of such system and are part of its description. Few works written in English language show a comparison among discrete methods that extract natural complex frequencies from a system impulsive response. Much less common is to find works written in Spanish language about this important research topic. Given this situation, important discrete numeric methods to estimate the complex natural resonance frequencies of a system through its impulsive response are described, tested and compared in different simulation scenarios in this document. According to the obtained results, the matrix pencil method with a SVD filter is the less sensitive method to the noise, while the Prony method and its different versions are the fastest ones. Scenarios that could be more suitable for each method are discussed.

Graphical Method for Examining Complex Natural Frequencies of Dispersive Materials

A graphical method is extended to and applied for visual inspection of solution patterns (region of complex natural frequencies s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) in the complex s-plane for dispersive materials characterized by the Debye and Lorentz models. These patterns are in the form of exponential decay and C-shaped profiles for the Debye and Lorentz models, respectively. Their forms are observed to be affected (e.g., moving left/right) in different manners with a variation of dispersion parameters. This method, thanks to its visual inspection capability, can help in analyzing/synthesizing filters and frequency-selective surfaces.

Flow Induced Vibration of Cantilever Tapered Pipes Transporting Fluid

A cantilevered tapered slender pipe conveying an incompressible, inviscid fluid of one material is not a conserved system. For certain large fluid velocity, the pipe with uniform cross section would go unstable via flutter Hopf bifurcation. In this paper, the flow induced vibration for cantilever tapering pipe transporting a fluid is presented. Euler Bernoulli and Hamilton’s theories are applied to develop the mathematical model which will be solved using well known Galerkan’s procedure. The effect of smooth tapering of the circular cross sectional area, flow velocity and pipe to fluid mass fraction on the complex natural frequencies and stability will be investigated.

Nonclassical vibrations of a monoclinic composite strip

A mathematical model of damped flexural-torsional vibrations of monoclinic composite strip of constant length rectangular cross section is proposed. The model is based on the refined bending theory Timoshenko beams, the theory of generalized Voigt — Lekhnitskii torsion and the elastic-viscoelastic correspondence principle in the linear theory of viscoelasticity. A two-stage method for solving a coupled system of differential equations is developed. First, using the Laplace transform in spatial variable, real natural frequencies and natural forms are found. To determine the complex natural frequencies of the strip in found real values are used as their initial values of natural frequencies, and then the complex frequencies are calculated by the method iterations of the third order. An assessment of the reliability of the mathematical model and method of numerical solution, performed by comparing calculated and experimental values of natural frequencies and loss factors is given. The results of a numerical study of the effect angles of orientation of reinforcing fibers and lengths by the values of natural frequencies and loss factors for free-free and cantilever monoclinic stripes are discussed. It is shown that for the free-free strip the region of mutual transformation eigenmodes of coupled vibration modes arise for quasi-bending and quasi-twisting vibrations of either even or odd tones. In the console strip of the region of mutual transformation of eigenforms of coupled modes vibrations occur for both even and odd tones.

Numerical algorithm for searching for layouts of electroelastic bodies with external electric circuits for obtaining the best damping properties

This paper presents an algorithm which allows finding such layouts of electromechanical systems that provide the best vibration damping whether for one mode or a set of vibration modes within some continuous frequency ranges. The basis for the algorithm is the problem solution about natural vibrations. An elastic structure with a piezoelectric element located on its surfaces, which electrodes are connected to a passive external electric circuit, is treated as an electromechanical system. The piezoelectric elements shunted with an electric circuit are the devices where energy dissipation occurs, thus leads to damping of vibrations. A change in damping properties of such systems can be reached by a proper choice of parameters of the electric circuits and corresponding location of the piezoelectric element, which provides the highest energy withdrawal into the electric circuit. The paper presents the mathematical formulation of the natural vibrations problem for piecewise-homogeneous electroelastic bodies shunted with passive external electric circuits. Within the proposed mathematical statement, the problem solution of natural vibrations for such objects is based on values of complex natural vibration frequencies. Real parts of the complex natural vibration frequencies are the circular frequency of vibrations, and the imaginary parts are the damping indices of vibrations. We proposed techniques aimed at determining the location of the piezoelectric element and selecting parameters of shunting the external electric circuit. These approaches are based on values of the complex natural vibration frequencies obtained as results of solving the natural vibrations problem. The proposed approach is demonstrated using a specimen of a thin-walled shell in the semi-cylinder form. The piezoelectric element has a form of a segment of a ring made of PZT-4 piezoceramics. Electrodes of the piezoelectric element are connected to the series resonant RL -circuit. The formulated problem is solved numerically using the finite element method and ANSYS commercial package.

Analysis of free vibrations of sandwich panel with electrorheological layer based on two models of laminated shells

Based on two models of laminated shells, free low frequency vibrations of a three-layered cylindrical panel with the internal layer fabricated of an electrorheological composite are studied. Both models lead to the same governing equations accounting for shears in layers, but differ in equations for calculating the reduced shear parameter which depends on the electric field strength and the temperature of a composite. In the case of a simple support of all edges, the formula for the complex natural frequency is obtained explicitly. The influence of the electric field strength and the temperature of the electrorheological composite on the lowest natural frequencies and associated vibration decrements is investigated. It was detected that both models give very close results for the heated composite at an electric field strength of more than 1.5 kV/mm. It is also shown that the frequency of natural vibrations of the electrorheological panel is a monotonically increasing function of the electric field strength, while the decrement – strength curve shows the presence of a local maximum corresponding to the best damping of viscoelastic vibrations.

Dynamics of structural - inhomogeneous coaxial-multi-layered systems "cylinder-shells"

A mathematical model and a technique for assessing the efficiency of the dissipative ability of structurally inhomogeneous mechanical systems consisting of multilayer cylinders bonded to a thin viscoelastic shell of finite length have been developed. A detailed analysis of the known works devoted to this problem is given. A model, methodology, and algorithm for studying the natural and forced oscillations of a system to assess the damping ability of structurally inhomogeneous elastic and viscoelastic mechanical systems, taking into account the influence of the geometric and physico-mechanical parameters of the shell and cylinderhave been developed. In solving the problems considered, the method of divided variables, the method of the theory of potential functions, the Mueller method, the Gauss method and the orthogonal sweep method were used. The complex eigenfrequencies, amplitudes of forced oscillations are determined, and the largest dephasing abilities of the considered structurally inhomogeneous systems are estimated. It has been revealed that, the effect of the greatest damping ability in structurally heterogeneous systems is manifested when the real parts of complex natural frequencies come closer due to the interaction of close natural forms with each other.

The Vibration of Simply Supported Non-Uniform Cross Sectional Pipe Conveying Fluid Resting on Viscoelastic Foundation

The induced flexural vibration of slender pipe systems with continuous non uniform cross sectional area containing laminar flowing fluid lying on extended Winkler viscoelastic foundation is considered. The Euler Bernoulli model of the pipe has hinged ends. The inlet flow is considered constant steady that interacts with the wall of the pipe. The mathematical model is developed and its corresponding solution is obtained. The influence of the combination of variation of cross section, foundation stiffness and damping on the critical velocities, complex natural frequencies and stabilization of the system is presented.

ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES

Analytical solution of thermoelastic damping for rectangular Mindlin micro plate with the four edges simply supported is presented for the first time. Based on the Mindlin plate theory and the one-way coupled heat conduct theory, governing differential equations for thermo-elastically coupled free vibration of the micro plate are formulated. Ignoring the in-plane variation of the temperature gradient, analytical solution of temperature field in terms of the kinematic parameters is obtained under the adiabatic boundary conditions at the top and bottom surfaces. Furthermore, the equations of structural vibration including the thermal bending moment are transformed into a fourth-order partial differential equation only in terms of the deflection. By using the mathematical similarity between the eigenvalue problems under the simply supported boundary conditions, analytical solution of the complex natural frequency of the Mindlin plate is expressed in terms of the frequency of isothermal Kirchhoff plate. Then the inverse quality factor which represents the level of the thermoelastic damping is obtained. Finally, the effects of the shear deformation, the material and the geometry parameters on the thermoelastic damping are examined in detail by the numerical results. The numerical results show that thermoelastic damping estimated by Mindlin plate theory is less than that by Kirchhoff plate theory. The difference between the values evaluated by the two plate theories becomes very significant near the critical thickness. Moreover, along with the increase of the side-to-thickness aspect ratio,the maximum of the thermoelastic damping in micro Mindlin plate increase monotonically, however, that of micro Kirchhoff plate keeps constant.

Coupled Vibrations of Viscoelastic Three-Layer Composite Plates. 1. Formulation of the Problem

A mathematical model of damped vibrations of three-layer plates formed by two rigid anisotropic layers and a soft middle isotropic viscoelastic polymer layer is proposed in this paper. The model is based on the Hamilton’s variational principle, the refined theory of first-order plates, the model of complex modules, and the principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. The frequency-temperature dependence of the elastic-dissipative characteristics is considered negligible for rigid layer materials; however, this dependence is taken into account for the viscoelastic polymer of the soft layer. By minimizing the Hamilton functional we reduce the problem of damped vibrations of anisotropic structures to the algebraic problem of complex eigenvalues. The Rietz method using the Legendre polynomials as coordinate functions is applied to form the system of algebraic equations. The real solutions are found. When determining the complex natural frequencies of the plate, real natural frequencies obtained are used as their initial values, and then the complex frequencies are calculated by the third-order iteration method. The results of the study of the convergence of the numerical solution are discussed. The estimation of reliability of the mathematical model and the numerical solution method obtained by comparing the calculated and the experimental values of natural frequencies and loss factors is presented.

Torsional vibrations of a cylindrical shell in a linear viscoelastic medium

In this paper, we consider the natural vibrations of inhomogeneous mechanical systems, i.e., cylindrical bodies located in a deformable viscoelastic medium. The theory and methods for studying the natural vibrations of a cylindrical shell in a viscoelastic medium are constructed. The viscoelastic properties of the medium are taken into account using the hereditary Boltzmann-Walter theory. For the statement of the problem, the general equation of the theory of viscoelasticity in the potentials of displacements in a cylindrical coordinate system is used. An algorithm has been developed to solve the tasks posed on a computer using the Bessel, Hankel, and Mueller and Gauss methods.The considered problems were reduced to finding complex natural frequencies for the system of equations of motion of a cylindrical shell in an infinite viscoelastic medium using radiation conditions. It is shown that the problem has a discrete complex spectrum. The eigen frequencies of oscillations of a low-contrast heterogeneity are found. Revealed that the imaginary part of the eigen frequencies is comparable with the real one, which can lead to aperiodic movements of the systems considered.

A Phonetic Level Analysis of Dysarthria in Kannada Language

Dysarthria is a motor speech disorder and the most common neurodegenerative disease characterized by low volume in precise articulation, poor coordination of respiratory and phonatory subsystems, irregular pauses and so on. The challenging factor in improving the communication with dysarthria is speech intelligibility. The intelligilibility of speech can be determined by measuring the acoustic characteristics and phonetic structures of speech. In our analysis, we have concentrated on bisyllabic words in the Kannada language. The analysis is carried out with respect to basic acoustic parameters and spectrogram to extract fundamental frequency, formant frequency, jitter, shimmer, HNR and Standard Deviation using PRAAT tool. These results are helpful to identify and differentiate the complex natural frequency of the vocal tract functions with respect to normal subjects and dysarthria subjects. This study provides inputs to increase efficacy in the clinical assessments for monitoring and diagnosing the speech disorders for improving communication among human beings.

Performance Analysis of Kannada Phonetics: Vowels, Fricatives and Stop Consonants Using LP Spectrum

Kannada is an alpha syllabary language wherein each individual akshara/alphabet having different phonetic structures can be divided into sequence of numerous consonants and vowels. In our analysis, we concentrate on vowels, fricatives and stop consonants of prerecorded samples of various subjects. The analysis is carried out with respect to basic acoustic features and spectrogram to extract formant frequency and formant analysis to identify the complex natural frequency of the vocal tract functions using linear prediction method.

Dynamic response spectra of fractionally damped viscoelastic beams subjected to moving load

The dynamic response spectra of fractionally damped viscoelastic beams subjected to concentrated moving load are presented. This work quantitatively emphasizes the need for fractional-order damping model in the dynamics of viscoelastic structures, by demonstrating the effect of various orders of fractional derivative damping in beams subjected to concentrated moving loads. The fractional-order Kelvin–Voigt model describes the rheological properties of the viscoelastic material. The Riemann–Liouville fractional derivative of order 0<α ≤ 1 is applied. The complex natural frequencies of viscoelastic beams are studied with respect to α. The forced vibration response is investigated for a wide range of speeds of the moving load, encompassing all practical applications. The results obtained from the fractional-order damping model and the integer-order damping model are compared. With an increase in the order of the fractional derivative, the system damping of the system increases and the dynamic amplification factor (DAF) decreases, especially in the dynamic zone of the sweep parameter. Thus, an integer-order mechanical model over-predicts the damping and under-predicts the dynamic deflections and stresses. The fractional-order of derivative significantly influences the natural frequency, damping coefficient, and DAF. The results are verified with the literature. Communicated by Jie Yang.