Small-amplitude Vibrations Research Articles

Experimental and numerical study on vibrations and static deflection of a thin hyperelastic plate

The hyperelastic behavior of a thin square silicone rubber plate has been investigated analytically, numerically and experimentally; the case of small-amplitude vibrations has been considered, as well as the case of large static deflection under aerostatic pressure. The Mooney-Rivlin hyperelastic model has been chosen to describe the material nonlinear elasticity. The material parameters have been identified by a fitting procedure on the results of a uniaxial traction test. For the analytical model, the equations of motion have been obtained by a unified energy approach, and geometrical nonlinearities are modeled according to the Novozhilov nonlinear shell theory. A numerical model has also been developed by using a commercial Finite-Element code. In the experiments, the silicone rubber plate has been fixed to a heavy metal frame; a certain in-plane pre-load, applied by stretching the plate, has been given in order to ensure a flatness of the surface. An experimental modal analysis has been conducted; results have been used to identify the applied in-plane loads by optimization procedure with two different models: a numerical and an analytical one. The first four experimental and numerical natural modes and frequencies are in good agreement with the experiments after the pre-load identification. The static deflection has been measured experimentally for different pressures. Results have been compared to those obtained by analytical and numerical models. The static deflections are also satisfactorily compared, up to a deflection 50 times larger than the plate thickness, corresponding to a 30 percent strain.

Modeling the spectrum of the 2ν2 and ν4 states of ammonia to experimental accuracy.

The vibrational spectrum of ammonia has received an enormous amount of attention due to its potential prevalence in hot exo-planet atmospheres and persistent challenges in assigning and modeling highly excited and often highly perturbed states. Effective Hamiltonian models face challenges due to strong coupling between the large amplitude inversion and the other small amplitude vibrations. To date, only the ground and ν2 positions could be modeled to experimental accuracy using effective Hamiltonians. Several previous attempts to analyze the 2ν2 and ν4 energy levels failed to model both the microwave and infrared transitions to experimental accuracy. In this work, we performed extensive experimental measurements and data analysis for the 2ν2 and ν4 inversion-rotation and vibrational transitions. We measured 159 new transition frequencies with microwave precision and assigned 1680 new ones from existing Fourier transform spectra recorded in Synchrotron SOLEIL. The newly assigned data significantly expand the range of assigned quantum numbers; combined with all the previously published high-resolution data, the 2ν2 and ν4 states are reproduced to experimental accuracy using a global model described here. Achieving experimental accuracy required inclusion of a number of terms in the effective Hamiltonian that were neglected in previous work. These terms have also been neglected in the analysis of states higher than 2ν2 and ν4 suggesting that the inversion-rotation-vibration spectrum of ammonia may be far more tractable to effective Hamiltonians than previously believed.

Nonlinear dynamic analysis of electrically actuated viscoelastic bistable microbeam system

Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on transition mechanism of nonlinear jumping phenomena and transient chaotic behaviors. This article aims to theoretically investigate the static and dynamic characteristics of electrically actuated viscoelastic bistable microbeam via a low-dimensional model. An improved single-degree-of-freedom model to describe microbeam-based resonators is obtained by using Fractional Kelvin constitutive model, Hamilton’s principle and Galerkin method. Through static bifurcation analysis, three kinds of parameter conditions of the bistable system are obtained, and potential energy function of the Hamiltonian system is theoretically derived. The influence of fractional viscoelasticity on dynamic pull-in phenomena is distinguished from the viewpoint of energy. Then, the method of multiple scales is applied to determine the response and stability of the system for small vibration amplitude and AC voltage. The influence of fractional viscoelasticity on amplitude, frequency and bifurcation behavior is investigated. Results show that compared with the elastic material, nonlinear phenomenon becomes weak, resonance frequency increases and amplitude decreases in the viscoelastic system. Besides, the numerical discretization method of fractional derivative is given to verify theoretical results. To study the influence of fractional viscoelasticity on complicated vibration, Melnikov method is applied to predict the existence of chaos, and numerical simulation is carried out to find the stable regions, chaotic regions and dynamic pull-in regions by using bifurcation diagrams with local maximum method. Rational increase in material modulus ratio parameter and fractional order is effective to reduce the possibility of chaos and dynamic pull-in. This analysis has the potential of developing parameter design in MEMS.

Approaches for Reduced-Order Modeling of Electrically Actuated von-Karman Microplates

This article presents and compares different approaches to develop reduced-order models for the nonlinear von-Karman rectangular microplates actuated by nonlinear electrostatic forces. The reduced-order models aim to investigate the static and dynamic behavior of the plate under small and large actuation forces. A fully clamped microplate is considered. Different types of basis functions are used in conjunction with the Galerkin method to discretize the governing equations. First, we investigate the convergence with the number of modes retained in the model. Then for validation purpose, a comparison of the static results is made with the results calculated by a nonlinear finite element model. The linear eigenvalue problem for the plate under the electrostatic force is solved for a wide range of voltages up to pull-in. Results among the various reduced-order modes are compared and are also validated by comparing to results of the finite-element model. Further, the reduced-order models are employed to capture the forced dynamic response of the microplate under small and large vibration amplitudes. Comparison of the different approaches is made for this case.

Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.

Dynamics of masonry walls connected by a vibrating cable in a historic structure

An original parametric model is presented to describe the modal interactions characterizing the linearized free dynamics of a cable–beam system. The structural system is composed by two vertical cantilever beams, connected by a suspended nonlinear cable. A closed form solution is achieved for the linear eigenproblem governing the undamped small-amplitude vibrations. The eigensolution shows a rich scenario of frequency crossing and veering phenomena between global modes, dominated by the beam dynamics, and local modes, dominated by the cable vibrations. Veering phenomena are accompanied by modal hybridization processes. The role played by different parameters is discussed, with focus on the resonance regions. The discussion allows the identification of an updated set of parameters, which offers a satisfying explanation of some unexpected dynamic interactions, observed in the experimental behavior of the masonry walls of the Basilica of Collemaggio, a historic structure heavily damaged by the 2009 L’Aquila earthquake.

Vortex suppression and nano-vibration reduction of aerostatic bearings by arrayed microhole restrictors

Air vortex flow can induce small amplitude vibration of aerostatic bearings on the order of nanometers, which is known to be harmful to positioning accuracy of aerostatically supported devices in ultra-precision applications. In this paper, in order to suppress the vortex flow and reduce vibration of aerostatic bearings, a novel design of arrayed microhole restrictor (AMR) is proposed, and its effects are numerically and experimentally investigated. By computational fluid dynamics analysis, the transient flow features are studied for aerostatic bearings with AMR and conventional restrictors, and static performances of the bearings are also compared. Vibration strength of the bearing is measured experimentally to validate the effectiveness of AMR. The results show that vortex shedding in the recess is suppressed and the vibration can be effectively reduced by AMR, while load capacity and stiffness of the bearing remain unchanged.

Vibrations and spatial patterns in biomimetic surfaces: using the shark-skin effect to control blood clotting.

We study the effect of small-amplitude fast vibrations and small-amplitude spatial patterns on various systems involving wetting and liquid flow, such as superhydrophobic surfaces, membranes and flow pipes. First, we introduce a mathematical method of averaging the effect of small spatial and temporal patterns and substituting them with an effective force. Such an effective force can change the equilibrium state of a system as well as a phase state, leading to surface texture-induced and vibration-induced phase control. Vibration and patterns can effectively jam holes in vessels with liquid, separate multi-phase flow, change membrane properties, result in propulsion and locomotion and lead to many other multi-scale, nonlinear effects including the shark-skin effect. We discuss the application of such effects to blood flow for novel biomedical 'haemophobic' applications which can prevent blood clotting and thrombosis by controlling the surface pattern at a wall of a vessel (e.g. a catheter or stent).This article is part of the themed issue 'Bioinspired hierarchically structured surfaces for green science'.

Inverse Problem of Lagrangian Mechanics for Classically Damped Linear Multi-Degrees-of-Freedom Systems

Lagrangians for classically damped linear multi-degrees-of-freedom dynamical systems are obtained using simple and elementary methods. Such dynamical systems are very widely used to model and analyze small amplitude vibrations in numerous naturally occurring and engineered systems. An invariant of the motion is also obtained.

On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations

Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.

Isoscalar monopole and dipole excitations of cluster states and giant resonances inC12

The isoscalar monopole(ISM) and dipole(ISD) excitations in $^{12}$C are theoretically investigated with the shifted antisymmetrized molecular dynamics(AMD) plus $3\alpha$-cluster generator coordinate method(GCM). The small amplitude vibration modes are described by coherent one-particle one-hole excitations expressed by small shift of single-nucleon Gaussian wave functions within the AMD framework, whereas the large amplitude cluster modes are incorporated by superposing $3\alpha$-cluster wave functions in the GCM. The coupling of the excitations in the intrinsic frame with the rotation and parity transformation is taken into account microscopically by the angular-momentum and parity projections. The present calculation describes the ISM and ISD excitations in a wide energy region covering cluster modes in the low-energy region and the giant resonances in the high-energy region, though the quantitative description of the high-energy part is not satisfactory. The low-energy ISM and ISD strengths of the cluster modes are enhanced by the radial motion of $\alpha$ clusters, and they split into a couple of states because of the angular motion of $\alpha$ clusters. The low-energy ISM strengths exhaust 26% of the EWSR, which is consistent with the experimental data for the $^{12}$C($0^+_2$;7.65 MeV) and $^{12}$C($0^+_3$;10.3 MeV) measured by $(e,e')$, $(\alpha,\alpha')$, and ($^6$Li,$^6$Li$'$) scatterings. In the calculated low-energy ISD strengths, two $1^-$ states (the $1^-_1$ and $1^-_2$) with the significant strengths are obtained in $E=10-15$ MeV. It is indicated that the ISD excitations can be a good probe to experimentally search for new cluster states such as the $^{12}$C($1^-_2$) obtained in the present calculation.

Quantum corrections to parameters of interatomic distance distributions in molecular dynamics simulations

The problem of computation of interatomic distance distributions is considered in the context of gas electron diffraction method. A new theoretical approach has been developed for calculation of mean square amplitudes, corrections for interatomic distances and asymmetry parameters on the basis of ab initio molecular dynamics simulations with a posteriori quantum corrections. Several approximations have been evolved, and corresponding algorithms have been coded. Their testing has been performed by calculation of parameters for a set of diatomic molecules, ethane and 9,12-I2-1,2-dicarba-closo-dodecaborane. Comparison of the obtained amplitudes and distance corrections with those calculated by conventional methods demonstrates the superiority of the new approach. By contrast, asymmetry parameters remain numerically unstable after introduction quantum corrections. The best of the assessed approximations, termed MDC(5), is recommended for routine application to large molecules with small-amplitude vibrations.

A QTAIM and stress tensor perspective of large-amplitude motions of the tetrasulfur tetranitride S4N4molecular graph

The nature of the bonding interactions including the intramolecular SS interactions in tetrasulfur tetranitride, S4N4 are probed by performing very large amplitude vibrations of all of the 18 normal modes of vibration. The QTAIM and stress tensor point properties are -then investigated and found to be highly dependent on the mode of vibration. A considerable degree of metallicity ξ(rb) is found for the SS and SN bonding interactions. A unique bonding feature is found for a small amplitude vibration of the most anharmonic mode of this investigation, mode 2, where the SS bond critical point (BCP) transforms from a closed-shell SS BCP to a shared-shell SS BCP. We find 17 new unique QTAIM topologies for the molecular graphs corresponding to the 18 modes of vibration along with seven “missing” topologies that are mapped onto a spanning 2-D Quantum topology Phase Diagram (QTPD). In addition, eleven unique topologies existing on 3-D QTPDs are found due to the presence of non-nuclear attractors (NNAs). We use the stress tensor eigenvalues to explain the invariance of the numbers and types of QTAIM critical points. The applicability of both the stiffness S and stress tensor stiffness Sσ are also explored. Two new bond measures are introduced, a polarizability P and the stress tensor polarizability Pσ which are derived from the stiffness S and stress tensor stiffness Sσ, respectively. © 2016 Wiley Periodicals, Inc.

Analysis of a resonance liquid bridge oscillation on board of the International Space Station

We study the singular event which took place when conducting an experiment with a liquid bridge aboard the International Space Station. The liquid bridge vibrated unexpectedly for several tens of seconds with an oscillation amplitude larger than 15% of its radius. At first glance, the analysis of the mass force measured by the accelerometer during the oscillation did not show any significant perturbation. However, our study reveals the existence of two small-amplitude vibrations of the experimental setup with practically the resonance frequency of the first lateral mode. These vibrations occurred a few tens of seconds before the liquid bridge oscillation reached its maximum amplitude, produced a mass force with a magnitude of the order of 10−5g. The numerical integration of the non-linear Navier–Stokes equations reproduces remarkably well the free surface oscillations measured in the experiments. It allows us to reconstruct the three-dimensional liquid bridge motion which took place in the experiment. The present study illustrates the sensitivity of liquid bridges in a microgravity environment, where tiny perturbations may produce significant vibrations which survive over long periods of time.

Modeling of a hydraulic damper with shear thinning fluid for damping mechanism analysis

This paper presents an analytical model for computing the force-displacement loops of the passive hydraulic dampers filled with shear thinning fluid when the damper is subjected to a sinusoidal excitation. The analytical model is developed on the basis of Navier-Stokes equations by considering the rheological behavior of silicone oil. The obtained computational results agree well with those by experimental measurements, and both of them suggests that viscous friction, fluid compressibility, and friction loss are the three major damping mechanisms of a hydraulic damper with shear thinning fluid. The damper in a low-frequency and large-amplitude vibration may have bigger viscous damping coefficient than that when it is in a high-frequency and small-amplitude vibration, which suggests that a hydraulic damper filled with shear thinning fluid can satisfy the requirement of the isolation system better.

Methods for Calculating Partition Functions of Molecules Involving Large Amplitude and/or Anharmonic Motions.

We present a method for calculating partition functions taking into account anharmonic contributions for systems involving both small-amplitude vibrations and hindered rotations. The Wang-Landau scheme is used in the first case, while two alternative schemes are used for hindered rotation based on imaginary time propagation and fitting of the exact energy levels as a function of quantum number. These two schemes are shown to be complementary in their ranges of applicability (in terms of the torsional rotational constant and the relevant potential). Partition functions for four different molecules are calculated and compared to simpler ones obtained using a harmonic model.

Topology optimization: a review for structural designs under vibration problems

This article provides a comprehensive review of structural optimization employing topology methods for structures under vibration problems. Topology optimization allows creative and radical design modifications, compared to shape and size optimization techniques. Various works of structural topology optimization, which are subjected to vibration as the response function of the optimization process, are reviewed. Different types of calculus and numerical methods commonly used for solving structural topological optimization problems are briefly discussed. Moreover, different aspects of topology optimization related to vibration problems are explained. The articles reviewed are largely confined to linear systems that concern small vibration amplitudes. Accordingly, the works related to vibration topological optimization are classified according to the method employed (homogenization, evolutionary structural optimization, solid isotropic material with penalization, or level set). The reviewed works are tabulated according to their methodology, year, and the objective functions and applications of each work. Although the homogenization and evolutionary methods were common in the past, the solid isotropic material with penalization (SIMP) method is the most popular method applied in recent years. The advantages of the level set method show promise for future applications.

Влияние ПАВ на генерацию среднего течения около искривленной поверхности раздела

The present work is devoted to the average flow generation near curved interface with a surfactant adsorbed on the surface layer. The investigation was carried out for a liquid drop embedded in a viscous liquid with a different density. The liquid flows inside and outside the drop are generated by small amplitude and high frequency vibrations. Surfactant exchange between the drop surface and the surrounding liquid is limited by the process of adsorption-desorption. It was assumed that the surfactants are soluble in the surrounding liquid, but not soluble in the liquid drop. Surrounding liquid and the liquid in the drop are considered incompressible. Normal and shear viscous stresses balance at the interface is performed under the condition that the film thickness of the adsorbed surfactant is negligible. The problem is solved under assumption that the shape of the drop in the presence of adsorbed surfactant remains spherical symmetry. The effective boundary conditions for the tangential velocity jump and shear stress jump, describing the above generation have been obtained by matched asymptotic expansions method. The conditions under which the drop surface can be considered as a quasi-solid are determined. It is shown that in the case of the significant effect of surfactants on the surface tension, the dominant mechanism for the generation is the Schlichting mechanisms under vibrations. Received 25.10.2016; accepted 29.11.2016

Mechanistic Modeling of Milling Process Damping Including Velocity and Ploughing Effects

In this study, a practical method for modeling of process damping in milling process is presented. The cutting forces are decomposed into two components, i.e., the shearing and ploughing forces. The tangential and radial shearing forces are modified considering the direction changing of cutting velocity. With the assumption of small amplitude vibration, ploughing forces are simplified as linear forms through introducing equivalent viscous damper. The effect of cutting velocity and the equivalent viscous damper are integrated into milling forces model since they all have contributions to process damping. The equation governing the dynamics of the milling system is constructed to predict the stability lobe. The prediction is verified by previous researchers’ experiments. It is shown that when the proposed process damping model is taken into consideration, the accuracy of stability prediction can be improved significantly at low cutting speeds.

Fluid-structure interaction simulation of three-dimensional flexible hydrofoil in water tunnel

The closely coupled approach combined with the finite volume method (FVM) solver and the finite element method (FEM) solver is used to investigate the fluid-structure interaction (FSI) of a three-dimensional cantilevered hydrofoil in the water tunnel. The FVM solver and the coupled approach are verified and validated by comparing the numerical predictions with the experimental measurements, and good agreement is obtained concerning both the lift on the foil and the tip displacement. In the noncavitating flow, the result indicates that the growth of the initial incidence angle and the Reynolds number improves the deformation of the foil, and the lift on the foil is increased by the twist deformation. The normalized twist angle and displacement along the span of the hydrofoil for different incidence angles and Reynolds numbers are almost uniform. For the cavitation flow, it is shown that the small amplitude vibration of the foil has limited influence on the developing process of the partial cavity, and the quasi two-dimensional cavity shedding does not change the deformation mode of the hydrofoil. However, the frequency spectrum of the lift on the foil contains the frequency which is associated with the first bend frequency of the hydrofoil.