Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions

In this study, nonlinear vibrations of an axially moving multi-supported string have been investigated. The main difference of this study from the others is in that there are non-ideal supports allowing minimal deflections between ideal supports at both ends of the string. Nonlinear equations of the motion and boundary conditions have been obtained using Hamilton’s Principle. Dependence of the equations of motion and boundary conditions on geometry and material of the string have been eliminated by non-dimensionalizing. Method of multiple scales, a perturbation technique, has been employed for solving the equations of motion. Axial velocity has been assumed a harmonically varying function about a constant value. Axially moving string has been investigated in three regions. Vibrations have been examined for three different cases of the velocity variation frequency. Stability has been analyzed and stability boundaries have been established for the principal parametric resonance case. Effects of the non-ideal support conditions on stability boundaries and vibration amplitudes have been investigated.

     

Understanding root cause of stick–slip vibrations in deep drilling with drag bits

In search for the root cause of stick–slip, a mode of torsional vibrations of a drilling assembly, a linear stability analysis of coupled axial–torsional vibrations has been carried out. It has been shown that in a rotary drilling system with axial and torsional degree of freedom two distinct modes of self-excited vibrations are present: axial and torsional. These axial (torsional) modes of vibrations are due to resonance between the cutting forces acting at the bit and the axial (torsional) natural modes of drillstring vibrations. It has been demonstrated that although axial and torsional modes of vibrations do affect each other the underlying mechanisms driving these modes of vibrations are completely different. In particular, the only driving mechanism of the axial vibrations is the regenerative effect, while there are two distinct mechanisms that drive the torsional vibrations: (i) the cutting action of the bit, and (ii) the wearflat/rock interaction. Moreover, in the case of the torsional vibrations the regenerative effect plays only a secondary role. The results of the present study indicate that the axial compliance can play a stabilizing role. In particular, the stabilizing role of the axial compliance increases as the ratio of the torsional to the axial natural frequency of the drillstring vibrations decreases.     

Internal resonances in non-linear vibrations of a laminated circular cylindrical shell

Internal resonances in geometrically non-linear forced vibrations of laminated circular cylindrical shells are investigated by using the Amabili?CReddy higher-order shear deformation theory. A harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retain dissipation. Numerical results are obtained by using the pseudo-arc length continuation method and bifurcation analysis. A one-to-one-to-two internal resonance is identified, giving rise to pitchfork and Neimark?CSacher bifurcations of the non-linear response. A threshold level in the excitation has been observed in order to activate the internal resonance.     

Receptivity of Hypersonic Boundary Layer to Wall Disturbances

Theoretical analysis of hypersonic boundary-layer receptivity to wall disturbances is conducted using a combination of asymptotic and numerical methods. Excitation of the second mode by distributed and local forcing on a flat-plate surface is studied under adiabatic and cooled wall conditions. Analysis addresses receptivity to wall vibrations, periodic suction/blowing, and temperature disturbances. A strong excitation occurs in local regions where forcing is in resonance with normal waves. It is shown that the receptivity function tends to infinity as the resonance point tends to the branch point of the discrete spectrum that is typical for boundary layers on cool surfaces. Asymptotic analysis resolves this singularity and provides the receptivity coefficient in the branch-point vicinity. Numerical results indicate extremely high receptivity to vibrations and suction/blowing in the vicinity of the branch point located near the lower neutral branch of the Mack second mode. Received 5 September 2000 and accepted 7 September 2001     

端部激励相位差下斜拉索的非线性振动

斜拉桥中拉索承受着多种端部激励,可激发大幅空间振动．以斜拉索为对象,探究不同端部激励间相位差对其非线性振动的影响．首先,推导斜拉索无量纲离散控制方程,引入考虑相位的三向端部激励得到一般化模型;然后,针对拉索下端存在的纵桥向、竖向和横桥向激励的两两组合,受大幅或小幅激励,及其在主共振区或主参数共振区几组因素,共计12种工况,采用数值分析法分别研究了各工况下不同激励相位差时的斜拉索稳态响应．研究发现：激励相位差能加剧与激励频率相近的面内、外模态振动;在任意端部激励组合下,激励相位差不仅可使斜拉索非线性振动出现定量变化,还可改变内共振的表现形式．面内、外激励组合下,相位差对拉索响应幅值的影响以π为周期变化,且当相位差趋于π/2 + kπ (k = 0, 1, 2…)时影响最为突出;而面内激励组合下,以2π为变化周期,当相位差为π + 2kπ (k = 0, 1, 2, …)时其对稳态幅值的影响最显著．其原因是：面外激励关于拉索所在的竖直面对称,故其本质上以π为周期;而面内激励无此对称性,仍以2π为周期．因此,有无面外激励参与决定了激励间相位差对斜拉索响应的影响规律．     

Supercritical forced response of coupled motion of a nonlinear transporting beam

Steady-state periodic responses of nonlinear coupled planar motions are investigated for transporting beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates into multiple equilibrium positions in the supercritical regime. The finite-difference schemes are developed to calculate the non-trivial static equilibrium and the steady-state response under simply supported or clamped boundary conditions. The forced vibration is assumed to be spatially uniform and temporally simple harmonic. Based on the long time series, the steady-state transversal amplitudes of nonlinear planar motions are recorded with changing load frequencies. A?resonance exists if the external load frequency approaches the fundamental frequency. The effects of material parameters and vibration amplitude on the resonance responses are investigated. The coupled planar model can be reduced to two nonlinear models on transversal vibrations, an integro-partial?Cdifferential equation and a partial?Cdifferential one. Numerical examples are displayed for the pros and cons between the two transversal models. It is also revealed that the increased axial speed converts the hardening-type behavior into the softening-type one.     

Parametric vibrations of flexible hoses excited by a pulsating fluid flow,Part II: Experimental research

The paper presents the results of experimental studies of vibrations of an elastic hose which are induced by a pulsating fluid flow. It was found that there is a possibility of parametric resonances: principal and combination associated with certain modes of vibrations. The influence of frequency and the amplitude of pulsation, average flow velocity, pressure inside pipe, the length of the hose, and the temperature on the ranges of parametric vibrations were analysed. The character of vibrations in resonance ranges was determined by showing time histories and the results of spectral analyses. A flexible hose applied in high-pressure hydraulic systems was used as an object of research. The values of basic parameters which describe the hose׳s mechanical properties were identified experimentally. The results of the experiments were compared with the results of numerical simulations conducted on the basis of the methodology proposed in Part I of this paper.     

Experimental and theoretical analysis of the transverse vibrations of a beam having bilinear support

Experimental and theoretical investigations have been conducted to study the transverse vibrations of a beam having nonlinear constraint. One end of the beam is fixed while the other is supported on a bilinear spring and carries a concentrated mass. Free-vibration curves, obtained for different values of the spring constants and the end mass, indicate that free periodic vibrations with frequencies which can lie within any one of an infinite number of ranges may occur. Forced harmonic response may exhibit the multiplicity of jump phenomena within the frequency ranges of free vibrations.     

Chaos in the three-well potential system

A new approach based on analysis of the wandering trajectories is applied to investigate an appearance of chaotic vibrations in many-well potential systems. The chaotic behavior regions were found in the both amplitude–frequency of excitation and amplitude–damping coefficient plane. The phase plane of initial conditions has been investigated taking into account different values of an external periodic excitation. It demonstrated remarkable agreement with investigations based on homoclinic and heteroclinic bifurcation criteria for chaos, computations of Lyapunov exponents and fractal basin boundaries. The presented technique is very effective, convenient to use, and can be applied to the investigation of a wide class of problems.     

Local resonance phononic band gaps in modified two-dimensional lattice materials

In this paper,modified two-dimensional periodic lattice materials with local resonance phononic bandgaps are designed and investigated.The design concept isto introduce some auxiliary structures into conventional periodic lattice materials.Elastic wave propagation in this kindof modified two-dimensional lattice materials is studied using a combination of Bloch’s theorem with finite elementmethod.The calculated frequency band structures of illustrative modified square lattice materials reveal the existenceof frequency band gaps in the low frequency region due tothe introduction of the auxiliary structures.The mechanismunderlying the occurrence of these frequency band gaps isthoroughly discussed and natural resonances of the auxiliarystructures are validated to be the origin.The effect of geometric parameters of the auxiliary structures on the width ofthe local resonance phononic band gaps is explored.Finally,a conceptual broadband vibration-insulating structure basedon the modified lattice materials is designed and its capability is demonstrated.The present work is anticipated to beuseful in designing structures which can insulate mechanicalvibrations within desired frequency ranges.     

Single-mode response and control of a hinged–hinged flexible beam

The problem of suppressing the vibrations of a hinged–hinged flexible beam that is subjected to primary and principal parametric excitations is tackled. Different control laws are proposed, and saturation phenomenon is investigated to suppress the vibrations of the system. The dynamics of the beam are modeled with a second-order nonlinear ordinary-differential equation. The method of multiple scales is used to derive two-first ordinary differential equations that govern the time variation of the amplitude and phase of the response. These equations are used to determine the steady-state responses and their stability. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters on the steady-state amplitude and on saturation phenomena at resonance have been reported.     

Determining the resonance frequencies of forced vibrations of turbogenerators with foundations

The model method for determination of dynamic properties of framed foundations on plastic scale models is elaborated. Position shifting of resonance peaks has been found by evaluation of different response curves, measured on models. Theoretical and experimental analysis of forced vibrations of complex damped mechanical system with several degrees of freedom has shown that resonance peaks, measured at different points of the same system, correspond to various frequencies of exciting force. To locate eigen frequencies and dangerous resonances from results of measurements, it is necessary to plot the envelope of the set of curves and to determine the positions of its peaks, or, at least, to find the concentration of resonance maxima of all individual curves.     

Nonlinear dynamic analysis of space cable net structures with one to one internal resonances

The nonlinear dynamic analysis of cable net structures becomes more and more significant for their space applications required high surface accuracy, especially mesh reflector antennas. In this work, the resonant multi-modal dynamics due to 1:1 internal resonances in the finite-amplitude vibrations of cable net structures subjected to harmonic loads are investigated. The nonlinear dynamic equation of space cable net structures is first developed using the extended Hamilton principle, which belongs to the self-excited vibration with quadratic and cubic nonlinearities. Linear modal analysis is then performed to decouple the nonlinear differential equations, and yields a complete set of system quadratic/cubic coefficients. With the aim of parametrically revealing nonlinear behaviors of space cable net structures, the second-order asymptotic analysis under 1:1 internal resonance is accomplished by the method of multiple scales. The nonlinear phenomena of a planar cable net and cable net reflector, such as the bending of response curve, jump phenomena, instability regions, saddle-node bifurcation, are verified by means of numerical analysis.     

Three-to-one internal resonance in multiple stepped beam systems

In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales （a perturbation technique）. The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and .frequency- response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.     

An analysis of nonlinear normal contact microvibrations excited by a harmonic force

This paper presents the results of numerical investigations of nonlinear normal contact microvibrations excited by a harmonic force in a system of two bodies in planar contact that models the slideway connections of machine tools. The aim of the computation analysis was to determine the plots of the forced vibrations as a function of frequency and amplitude and to study the resonances taking place in the system. The studies showed that beside the main resonance, at exciting frequencies lower than the natural frequency (f _w < f _o ), a number of interesting phenomena can occur in the investigated system such as: high amplitude vibration with complex motion, resonance peaks, amplitude jumps, bifurcations, and chaotic vibrations.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

Enhanced damping of a cantilever beam by axial parametric excitation

A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.     

Numerical analysis of nonlinear modes of piecewise linear systems torsional vibrations

The nonlinear modes of essentially nonlinear piecewise-linear finite degrees of freedom systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the systems motions in configuration space. The numerical method for the nonlinear modes of essentially nonlinear piecewise-linear systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous system nonlinear modes. The nonlinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonlinear piecewise-linear system with fifteen degrees of freedom. The NNMs of this system forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the systems with dissipation are close to NNMs of the system without dissipation.     

Combination Internal Resonances in Heated Annular Plates

Nonlinear modal interactions in the forced vibrations of a thermally loaded pre-buckled annular plate with clamped–clamped immovable boundary conditions are investigated. The mechanism responsible for the interaction is a combination internal resonance involving the natural frequencies of the three lowest axisymmetric modes. The in-plane thermal load acting on the plate is assumed to be axisymmetric and the plate is externally excited by a harmonic force. The nonlinear von Kármán plate equations along with the heat conduction equation are combined to model the behavior of the system. An analytical/numerical approach is used to examine the plate vibrations to a harmonic excitation near primary resonance of one of the modes.     

Hydrodynamic Damping of Beam Oscillations near a Surface

Fluid Dynamics - The hydrodynamic action on long thin beams executing bending resonance vibrations in a fluid near a plane rigid surface (surface) is investigated. The model of...