Axial audio-frequency stiffness of a bush mounting—the waveguide solution

An axial, dynamic stiffness model of an arbitrary wide and long rubber bush mounting is developed within the audible-frequency range, where influences of audible frequencies, material properties, bush mounting length and radius, are investigated. The problems of simultaneously satisfying the locally non-mixed boundary conditions at the radial and end surfaces are solved by adopting a waveguide approach, using the dispersion relation for axially symmetric waves in thick-walled infinite plates, while satisfying the radial boundary conditions by mode matching. The rubber is assumed nearly incompressible, displaying dilatation elasticity and deviatoric viscoelasticity based on a fractional derivative, standard linear solid embodying a Mittag–Leffler relaxation kernel, the main advantage being the minimum parameter number required to successfully model wide-frequency band material properties. The stiffness is found to depend strongly on frequency, displaying acoustical resonance phenomena; such as stiffness peaks and troughs. The presented model agrees fully with a simplified, long-bush model while diverging from it for increased diameter-to-length ratios. To a great extent, the increased influences of higher order modes and dispersion explain the discrepancies reported for the approximate approach.     

Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system

This paper presents a force model for the inter-shaft bearing with a local defect on the surface of the outer race or the inner race, and the nonlinear dynamic characteristics of a dual-rotor system affected by the local defect are investigated. A simplified dual-rotor system is presented with the consideration of the inter-shaft bearing's nonlinearities such as the Hertzian contact force and the radial clearance. The local defect is considered as a regular dent with a constant depth, thus the radial clearance will increase when rolling elements go through the range of the local defect. The motion equations of the system with eight degrees of freedom are formulated by using the Lagrange's equation. The nonlinear vibration responses of the dual-rotor system affected by the local defect are obtained using numerical method. The results show that there exist four abnormal resonances on the amplitude frequency curves of the system due to the effect of the local defect, apart from the couple of primary resonances excited by the unbalance excitations of the two rotors. With the aid of the characteristic defect frequency analysis, it is revealed that one pair of the abnormal resonances are excited by the characteristic defect frequency, and the other pair of the abnormal resonances are excited by the combination frequency. Furthermore, a comprehensive parametric analysis is carried out to give an insight into the nonlinear resonant response characteristics affected by parameters such as the depth and the span of the defect, the rotation speed ratio, the unbalances of two rotors, the stiffness and the damping of the linear elastic spring, and the radial clearance, the stiffness and the roller number of the inter-shaft bearing. The results show that the vibration amplitudes for the abnormal resonances are mainly determined by the depth and the span of the defect, while the resonant frequencies for the abnormal resonances are mainly influenced by the rotation speed ratio and the roller number of the inter-shaft bearing. However, the rotors’ unbalances mainly affect the corresponding primary resonance rather than the abnormal resonances. The obtained results will contribute to a better understanding of the nonlinear resonant response characteristics of dual-rotor systems with a local defect on the inter-shaft bearing, which are helpful for the fault diagnostics of the inter-shaft bearing in a dual-rotor system.     

Shaking table tests of a plate-column structure with semi-active tuned mass damper

This paper presents selected results of experimental tests carried out on the three-storey plate-column structure, which is a physical model of a building. On the top floor the semi-active tuned mass damper (STMD) is installed. That device is a kind of harmonic oscillator with a variable stiffness characteristic. The tuning of STMD is possible in the frequency range from 0 to 5 Hz, which allows to counteract the resonant vibration of plate-column structure for the three basic frequencies. The model's vibration are kinematically excited by using the earthquake simulator. It is assumed that during an experimental tests the plate-column structure's vibration are only excited by horizontal component of base motion (dominant influence on building's vibration). The aim of the experimental analysis presented in this paper is to verify the effectiveness of the semi-active tuned mass damper in vibration's amplitude reduction of the main structure. The tested device is prototype of STMD, which uses a competitive constructional solution of stiffness parameter control. By reason of above, special attention is focused on the testing of vibration eliminator in two aspects: change of stiffness characteristic (the rate of change), and the accuracy of tuning to the resonant frequencies. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

In the present study, a nonlocal continuum model based on the Eringen’s theory is developed for vibration analysis of orthotropic nano-plates with arbitrary variation in thickness. Variational principle and Ritz functions are employed to calculate the size dependent natural frequencies of non-uniform nano-plates on the basis of nonlocal classical plate theory (NCLPT). The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as finite element (FE). Effect of thickness variation on natural frequencies is examined for different nonlocal parameters, mode numbers, geometries and boundary conditions. It is found that thickness variation accompanying small scale effect has a noticeable effect on natural frequencies of non-uniform plates at nano scale. Also a comparison with finite element solution is performed to show the ability of the Ritz functions in fast converging to the exact results. It is anticipated that presented results can be used as a helpful source in vibration design and frequency optimization of non-uniform small scaled plates.     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory

The aim of this paper is to study the free vibration of nanobeams with multiple cracks. The analysis procedure is based on nonlocal elasticity theory. This theory states that stress at a point is a function of strains at all points in the continuum. The nonlocal elasticity theory becomes significant for small length scale in micro and nanostructures. The effects of nonlocality, crack location and crack parameter are investigated on the natural frequencies of the cracked nanobeam. In this study, analytical solutions are given for cracked Euler–Bernoulli nanobeams of different boundary conditions.     

Size dependent analysis of functionally graded microbeams using strain gradient elasticity incorporated with surface energy

In this study, strain gradient theory is used to show the small scale effects on bending, vibration and stability of microscaled functionally graded (FG) beams. For this purpose, Euler–Bernoulli beam model is used and the numerical results are given for different boundary conditions. Analytical solutions are given for static deflection and buckling loads of the microbeams while generalized differential quadrature (GDQ) method is used to calculate their natural frequencies. The results are compared with classical elasticity ones to show the significance of the material length scale parameter (MLSP) effects and the general trend of the scale dependencies. In addition, it is shown the effect of surface energies relating to the strain gradient elasticity is negligible and can be ignored in vibration and buckling analyses. Combination of the well-known experimental setups with the results given in this paper can be used to find the effective MLSP for metal-ceramic FG microbeams. This helps to predict their accurate scale dependent mechanical behaviors by the introduced theoretical framework.     

Modeling of functionally graded circular energy harvesters due to flexoelectricity

Flexoelectric effect can be enhanced in micro/nano scale due to its size-dependency, making it particularly suitable for energy harvesting. In this work, a theoretical model is built to characterize the functionally graded circular flexoelectric energy harvesters based on the Kirchhoff thin plate hypothesis. Using Hamilton's principle, both the force balance equation and current balance equation are obtained. Approximated closed-form solutions of the energy harvesting performances are achieved through the assumed-mode method. Numerical analysis results demonstrate that the clamped circular energy harvesters with the ratio of the electrode radius to the plate radius be 0.64 will generate the maximum electrical output. The volume fraction coefficient has a significant impact on the resonant frequency, electrical output as well as the optimal load resistance. Meanwhile, shrinking the thickness of the circular energy harvester from 10µm to 0.1µm will lead to a remarkable increase of the optimal energy conversion efficiency from 10⁻⁶ to 10⁻². Furthermore, the strain gradient effect is examined to result in a higher resonant frequency while suppress the electrical output particularly if the length scale parameter is relatively large.     

Some features of the natural flexural vibrations of circular rings with initial imperfections

The effect of small initial deviations from an ideal circular shape on the frequencies and modes of the natural oscillations of a thin ring under conditions of plane deformation is investigated. The analysis is based on the equations of motion of the shallow shell theory when the length of the shell tends to infinity. A new approach is proposed to the construction of a finite-dimensional model of a ring in which it is assumed that initial imperfections lead to the interaction of flexural and radial vibrations. A number of assertions are formulated and proved regarding the specific features which accompany the vibrations of an imperfect ring that apparently also have to be taken into account when investigating the dynamic characteristics of thin circular cylindrical shells with initial imperfections.     

Molecular conformation of n-alkanes using terrain/funneling methods

Understanding molecular conformation is a first step in understanding the waxing (or formation of crystals) of petroleum fuels. In this work, we study the molecular conformation of typical fuel oils modeled as pure n-alkanes. A multi-scale global optimization methodology based on terrain methods and funneling algorithms is used to find minimum energy molecular conformations of united atom n-alkane models for diesel, home heating, and residual fuel oils. The terrain method is used to gather average gradient and average Hessian matrix information at the small length scale while funneling is used to generate conformational changes at the large length scale that drive iterates to a global minimum on the potential energy surface. In addition, the funneling method uses a mixture of average and point-wise derivative information to produce a monotonically decreasing sequence of objective function values and to avoid getting trapped at local minima on the potential energy surface. Computational results clearly show that the calculated united atom molecular conformations are comprised of zigzag structure with considerable wrapping at the ends of the molecule and that planar zigzag conformations usually correspond to saddle points. Furthermore, the numerical results clearly demonstrate that our terrain/funneling approach is robust and fast.     

Determination of the resonant frequencies of an anisotropic piezoelectric disk

An equation for the resonant frequencies of a thin anisotropic circular disk is obtained assuming exponential particular solutions. The roots of the equation are determined numerically. The resonant frequencies of two piezoelectric crystals are computed and compared with the resonant frequencies when the piezoelectric effect is not taken into account.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 72–74, 1990.     

Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix

This paper reported the result of an investigation into the effect of magnetic field on wave propagation in carbon nanotubes (CNTs) embedded in elastic matrix. Dynamic equations of CNTs under a longitudinal magnetic field are derived by considering the Lorentz magnetic forces. The results obtained show that wave propagation in CNTs embedded in elastic matrix under longitudinal magnetic field appears in critical frequencies at which the velocity of wave propagation drops dramatically. The velocity of wave propagation in CNTs increases with the increase of longitudinal magnetic field exerted on the CNTs in some frequency regions. The critical/cut-off frequency increases with the increase of matrix stiffness, and the influence of matrix on wave velocity is little in some frequency regions. This investigation may give a useful help in applications of nano-oscillators, micro-wave absorbing and nano-electron technology.     

井下钻柱纵向横向耦合振动模型建立与数值分析

针对井下钻柱运动的复杂性,基于动力学理论,建立了井下钻柱纵向和横向耦合振动的数学模型,并进行数值求解及分析.根据井下钻柱的实际工况,以整个井下钻柱为研究对象,提出了钻柱纵向和横向耦合振动的动力方程,并利用解析法和无量纲法分别求解出其动刚度和动阻尼的表达式,以及钻柱前两阶振动的固有频率.分析结果表明:当井下钻柱振动频率增大时,其动刚度呈幅值衰减的周期性变化,而其动阻尼呈幅值增强的周期性变化;井下钻柱长度和横截面面积越大,其动刚度和动阻尼的幅值越小;井下钻柱的Poisson(泊松)比对其振动的动刚度、动阻尼和前两阶固有频率没有影响;同时,井下钻柱的第二阶固有频率始终大于第一阶固有频率.该文的研究方法和模型为井下钻柱钻具分析和结果优化提供了理论参考和实际意义.     

Correction of stiffness and mass matrices utilizing simulated measured modal data

Measured and analytical data are unlikely to be equal due to measured noise, model inadequacies and structural damage, etc. It is necessary to update the physical parameters of analytical models for proper simulation and design studies. Starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes, this study presents the equations to update the physical parameters of stiffness and mass matrices simultaneously for analytical modelling by minimizing a cost function in the satisfaction of the dynamic constraints of orthogonality requirement and eigenvalue function. The proposed equations are straightforwardly derived by Moore–Penrose inverse matrix without using any multipliers. The cost function is expressed by the sum of the quadratic forms of both the difference between analytical and updated mass, and stiffness matrices. The results are compared with the updated mass matrix to consider the orthogonality requirement only and the updated stiffness matrix to consider the eigenvalue function only, respectively. Also, they are compared with Wei’s method which updates the mass and stiffness matrices simultaneously. The validity of the proposed method is illustrated in an application to correct the mass and stiffness matrices due to section loss of some members in a simple truss structure.     

Modeling of imperfect contact interfaces in sonic composites

In this paper, the tangential and normal discontinuities at the interface between the scatterer and the matrix are independently modeled, and these relative displacements are directly proportional to the corresponding components of tractions at the interface. The generation of full band-gaps is investigated on the basis of the derived imperfect interface conditions for a sonic plate with an array of acoustic piezoceramic hollow spheres scatterers embedded in an epoxy matrix. The scatterers are made from functionally graded materials with radial polarization. Numerical evaluation has shown that imperfect interface may have a significant effect on the increasing of the length of the full band-gaps. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional exact solution of layered two-dimensional quasicrystal simply supported nanoplates with size-dependent effects

A size-dependent plate model is developed to investigate the elastic responses of the multilayered two-dimensional quasicrystal nanoplates based on the nonlocal strain gradient theory for the first time. A nonlocal stress field parameter and a length scale parameter are taken into account in the new model to capture both stiffness-softening and stiffness-hardening size effects. The exact solution for a single-layer two-dimensional quasicrystal simply supported nanoplate is derived by utilizing the pseudo-Stroh formalism in conjunction with the nonlocal strain gradient theory. Afterward, a dual variable and position method is used to deal with the multilayered case. Numerical examples are presented to study the dependence of size-dependent effect on nanoplate length and the influences of scale parameters on the quasicrystal nanoplate subjected to a z-direction mechanical load on its top surface. The proposed model should be useful to verify various nanoplate theories and other numerical methods.     

Noncoaxial vibration of fluid-filled multi-walled carbon nanotubes

This paper is concerned with the free vibration of the fluid-filled multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on simplified Donnell’s cylindrical shell model and potential flow theory, the effect of internal fluid on the coupling vibration of the MWCNTs-fluid system is discussed in detail. The results show that the resonant frequencies are decreased due to the effect of the fluid, and the fluid has only a little influence on the associated amplitude ratio in MWCNTs corresponding to the natural resonant frequency (frequency of the innermost tube), while plays a significant role in the associated amplitude ratios corresponding to the intertube resonant frequency. For the natural resonant frequency, the vibration mode is coaxial. However, for the intertube resonant frequency, the system shows complex noncoaxial vibration, which plays a critical role in electronic and transport properties of carbon nanotubes (CNTs).     

Application of CFRP as a rotor shaft material

Conclusions Theoretical analysis and tests performed on rotors with composite shaft show that there is a sufficiently wide rotation stability region in the rotor parameter space despite comparatively high damping of a polymeric composite with respect to steel. Optimum parameters of the shaft (lay-up, thickness) and bearing (radial stiffness, damping) can be found within this region for each given rotor ensuring a low vibration level at critical frequencies.If rotor system parameters are far enough from the instability threshold, maximum vibration level is observed when rotor passes the first eigenfrequency zone. Further increase of rotation frequency leads to a rotor self-centering, and vibration level does not change passing the second eigenfrequency zone. The rotor response is not sensitive to small changes in rotor system parameters. If rotor system parameters are close to the instability threshold, vibration level at the second eigenfrequency dominates, and a small variation of bearing parameters causes significant changes in the vibration level.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 227–240, March–April, 1995.     

An Asymptotic Dynamic Theory for Cylindrical Shells

By expanding the displacement and stress components together with the axial length scale in terms of a small thin shell parameter, three asymptotic shell theories are obtained which incorporate thickness effects in a systematic way. The expansions are made in the equations of linear three-dimensional elasticity. These theories are used to examine the problem of longitudinal wave propagation in a shell of infinite length.     

Boundary stabilization of non-classical micro-scale beams

In this paper, the problem of boundary stabilization of a vibrating non-classical micro-scale Euler–Bernoulli beam is considered. In non-classical micro-beams, the governing Partial Differential Equation (PDE) of motion is obtained based on the non-classical continuum mechanics which introduces material length scale parameters. In this research, linear boundary control laws are constructed to stabilize the free vibration of non-classical micro-beams which its governing PDE is derived based on the modified strain gradient theory as one of the most inclusive non-classical continuum theories. Well-posedness and asymptotic stabilization of the closed loop system are investigated for both cases of complete and incomplete boundary control inputs. To illustrate the performance of the designed controllers, the closed loop PDE model of the system is simulated via Finite Element Method (FEM). To this end, new strain gradient beam element stiffness and mass matrices are derived in this work.