Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension

Non-linear vibrations of axially moving beam with time-dependent tension are investigated in this paper. The beam material is modelled as three-parameter Zener element. The Galerkin method and the fourth order Runge-Kutta method are used to solve the governing non-linear partial-differential equation. The effects of the transport speed, the tension perturbation amplitude and the internal damping on the dynamic behaviour of the system are numerically investigated. The Poincare maps and bifurcation diagrams are constructed to classify the vibrations. For small values of the transport speed and the amplitude of periodic perturbation the system is asymptotically stable with its response tending to zero. With the increase of parameters one can observe the coexistence of attractors. Regular and chaotic motion occur when the internal damping increases.     

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.     

Nonlinear dynamics of a system of rigid bodies with controlled electromagnetic dampers

A nonlinear mathematical model of a system of n rigid bodies undergoing translational vibrations under inertial loading is constructed. The system includes ball supports as a seismic-isolation mechanism and electromagnetic dampers controlled via an inertial feedback channel. A system of differential dynamic equations in normal form describing accelerative damping is derived. The frequencies of small undamped vibrations are calculated. A method for analyzing the dynamic coefficients of rigid bodies subject to accelerative damping is developed. The double phase–frequency resonance of a two-mass system is studied     

On the damping of nonlinear vibrations in some second-order dynamical systems

This paper continues the study of damping of nonlinear vibrations in second-order dynamical systems for the case considered in [1]. The estimates obtained for energy scattering are applied to a system of two bodies connected by a weightless elastic cable. The system rotates in a plane, and the viscous friction forces in the cable are taken into account. The external drag forces are neglected. Such a problem arises, for example, if the motion of a system of elastically connected spacecraft far from attracting centers is considered [2, 3].     

Equivalent linearizations for practical hysteretic systems

Experimental testing of a friction damped base isolation system has indicated a need for a new model of friction damping and for an appropriate equivalent linearization technique. The model for the damping adopted is a combination of viscous damping, constant Coulomb friction and linear Coulomb friction.This model is incorporated into the equation of motion for a single-degree-of-freedom system and the exact solutions are given for free vibrations and for steady-state vibrations excited by a harmonic force. The exact solution is taken as a basis for an equivalent linearization technique that can be used in conjunction with conventional design spectra for a practical design of such a system.     

The effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers

We investigate analytically and experimentally the effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers (CPVAs), which are used to reduce torsional vibrations in rotating machinery. The analysis is based on perturbation methods applied to the nonlinear equations of motion for a rotor subjected to an engine order applied torque and equipped with a circular path CPVA with viscous and Coulomb damping. The experimental work is based on quantifying parameters for the damping model using free vibration measurements with a viscous and Coulomb damping identification scheme that is enhanced to better handle measurement noise, and running tests for steady-state operation under a range of loading conditions. The level of Coulomb damping is varied by adjusting the friction of the absorber connection bearing. Good agreement is found between the analytical predictions and the experimental data. It is shown that the absorber sticks up to a level of excitation that allows it to release, after which the Coulomb damping acts in the expected manner, resulting in lowered response amplitudes. The results obtained are of general use in assessing absorber performance when dry friction is present in absorber suspensions.     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.     

Effects of damping on the speed of increase and amplitude of limit cycle for an aircraft braking system subjected to mode-coupling instability

A nonlinear model of an aircraft braking system is presented and used to investigate the effects of damping on the stability in Chevillot et al. (Arch Appl Mech 78(12):949–963, 2008). It has been shown that the addition of damping into the equations of motion does not lead systematically to the stabilization of the system. In the case of a mode-coupling instability, there is indeed an optimal ratio between the modal damping coefficients of the two modes in coalescence, that maximize the stable area. But the stable area is not a sufficient criterion. In dynamics, the amplitude of the vibrations and the transient behavior characterized by the speed of increase of the oscillations are best indicators. In this paper, the same nonlinear model of the aircraft braking system is used to compute time-history responses by integration of the full set of the nonlinear dynamic equations. The aim of the study is to evaluate the effects of damping on the nonlinear dynamics of the brake. It is shown that damping may be very efficient to significantly reduce and slow down the increase of the friction-induced vibrations. But, in the same way as for the stability area, there exists a value of the damping ratio that optimizes the effects of damping.     

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.     

Study of the damping characteristics of sectional flight vehicles of different length

The dependence of the damping characteristics of axisymmetric tri-sectional flight vehicles executing plane oscillations about the zero angle of attack on their geometric parameters is investigated on the supersonic range of uniform oncoming air flow Mach numbers. Systematic data are obtained using the least laborious approach, namely, by calculating the steady-state inviscid flow past equivalent bodies determined from the principle of the local similarity of force interaction. Ranges of the vehicle design parameters on which the dependence of the damping moment coefficient in pitch on the length of the conical or cylindrical-conical stabilizer of the vehicle is nonmonotonic are found. Typical viscosity effects on the characteristics under study are estimated using a method based on the assumptions of the linear theory for finite-thickness bodies. It is established that the damping coefficient increases monotonically with decreasing Reynolds number; moreover, even transition to oscillation antidumping regimes is possible when the sign of the damping coefficient changes.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 153–162. Original Russian Text Copyright © 2004 by Antonets and Shmanenkov.     

Stability of motion of two rigid bodies connected by a cable in the atmosphere

The spatial motion of two rigid bodies connected by a weightless inextensible cable in the atmosphere is considered. They are assumed to be bodies of revolution with static and dynamic symmetry. The condition of static stability of the system angular motion with respect to the direction of the incident airflow velocity vector is written out and analyzed. The influence of gyroscopic terms and damping moments on the stability condition is studied. An example of analysis of motion in the atmosphere of two connected bodies that are cones with a spherical tip is given. It is shown that the stable motion in the atmosphere can always be ensured by an appropriate matched choice of the parameters of the entire system on the basis of the obtained stability conditions. A numerical example of estimating the cable tension forces arising as the system descends on a ballistic trajectory in the atmosphere is presented.     

Nonlinear lateral-torsional coupled motion of a rotor contacting a viscoelastically suspended stator

Interaction of a rotor with a stationary part is a kind of serious malfunction that could result in a catastrophic failure if remained undetected. Past analytical and numerical simulation work on rotor?Cstator interactions mainly focus on the vibrations along the lateral directions. The torsional degree of freedom (dof) is usually ignored. The present work is aimed to study the influence of a rotor to stator contact on the lateral-torsional coupled vibrations. A mathematical model consisting of interacting vibratory systems of rotor and stator is presented. The contact is modeled using contact stiffness, damping and Coulomb friction. Equations derived for kinetic, potential and dissipation energies and non-conservative external forces are used in the Langrange??s equations for deriving the motion equations for the rotor?Cstator system. Equations revealed that the lateral-torsional motion coupling exists twofold for the rotor. The unbalance couples lateral-torsional motion of rotor through inertia and damping matrices. Coupling due to the rotor?Cstator friction occurs through a force vector. The nonlinear equations are solved using a Runge?CKutta fourth-order numerical integration scheme using relatively small time step. Results obtained through the proposed model are compared with the identical rotor?Cstator system without torsional dof and differences are identified. Effect of several parameters such as speed, relative inertia, coefficient of friction and contact damping on the bifurcation behavior of the rotor?Cstator motion has been investigated. Vibration motions presented in the forms of spectrum cascade of the coast-up response, and orbit and Poincaré plots of the steady-state response are exhibiting rich dynamic behavior of the system.     

Vibrations of rod-rigid element systems with a local non-linearity

The paper deals with vibrations of systems consisting of non-coaxial rods connected by rigid bodies and of a local non-linearity. The motion of the rods is described by classical wave equation and the solution of the d’Alembert type is applied in the study. This leads to solving ordinary differential equations with a retarded argument. The local non-linearity is described through irrational functions and in a special case it includes the polynomial of the third degree. Detailed considerations are given for a system consisting of three rods and two rigid bodies. In numerical analysis non-linear effects are discussed. The results concerning harmonic vibrations are presented for the local non-linearities having characteristics of a soft type as well as of a hard type.     

柔性机械臂刚柔耦合运动的摄动解法

本文在建立了柔性机械臂无量纲形式摄动方程的基础上,提出了求解在慢变驱动力矩作用下有阻尼简单柔性机械臂刚柔耦合运动的摄动解法。数值模拟结果较好的证实了本方法的有效性。     

Interaction of a Traveling Pressure Perturbation with a Boundary Layer

The interaction between a traveling pressure perturbation and a laminar compressible boundary layer is investigated for a perturbation level higher than that needed to initiate steady-state flow separation. It is found that if the velocity of the pressure perturbation is fairly high the flow may remain unseparated and its direction of motion determines the nature of the perturbation propagation in the boundary layer. It is shown that even in the linear approximation the perturbations are mainly induced by the linear wall layer and not by the critical layer, which always remains nonlinear. It is also found that in the unsteady case shortwave perturbation oscillations are damped with time while the longwave ones grow and that the growth of the perturbations with time amplifies their damping along the streamwise coordinate while damping reduces it.     

A novel aseismic foundation system for multipurpose asymmetric buildings

The vulnerability of civil engineering structures with fundamental frequency, say roughly above 1?Hz, (or buildings having less than ten stories), when exposed to the strong motion phase of an earthquake is considerably reduced by means of base isolation. The low-pass filter for isolating horizontal vibrations is redesigned where the classical elastomeric bearings are substituted by a number of prestressed helical steel springs with pivoted columns along their vertical axes carrying a fraction of the dead weight and guiding the remaining horizontal motion. The base-isolated building in its fundamental mode is considered to be rigid and low-cost tuned liquid column gas dampers (TLCGDs), in optimal arrangement within the plan of the basement of the building, supply the effective damping of the remaining horizontal vibrations. TLCGD-tuning in a first step is performed by a simple transformation of the well-documented optimal parameters of the tuned mass damper (TMD) followed by fine-tuning in state space. The action of the passive damping device is commonly considered to be sufficient. Since the gas-spring effect somewhat counter acts changes in fluid mass, the absorber can be used as a water reservoir. Compatible sliding elements are innovatively designed to resist the motion of the building relative to the ground for sufficiently small disturbances by static friction, thus complete the isolation system. However, during seismic excitation, the frictional contact is released over much of the time to avoid excessive wear.     

Dynamics of vibration isolation system with a quasi-isochronous roller shock absorber

The low-frequency vibrations of a vibration isolation system of rigid bodies (roller shock absorber and carrying body) under external harmonic loading are considered. The working surface of the absorber has the form of a brachistochrone. The equations describing the slip-free motion of the absorber over the hinged roller and the motion of the carrying body are derived. A graphical method for optimizing the parameters of the roller absorber as a component of the vibration isolation system is proposed     

伴随变阻尼作用的干摩擦下的车辆系统非线性动力学分析

对分段线性阻尼和干摩擦共同作用下的车辆悬挂系统进行了非线性动力学分析研究，阐述了判定系统周期运动稳定性的理论方法；利用数值模拟方法分析了具有不同阻尼参数组合的系统对简谐激励的振动响应，并分析了由干摩擦引起的粘-滑振动行为．结果表明：提高摩擦力对抑制响应有利，但车辆系统在低速下运行时会出现复杂的粘-滑振动，轮轨之间产生较大的瞬时刚性冲击；而通过增加轮对与侧架的弹性悬挂可以有效减弱这种瞬时刚性冲击．