Comments on “finite element model for dynamic analysis of Timoshenko beam”

An analysis is presented of the effect of dry friction on vibrations of two self-excited systems. Attention is directed to cases where practical quenching of self-excited vibrations cannot be achieved merely by the action of linear additional damping. It is shown that at correct tuning of the systems discussed, a combination of linear additional damping with dry friction is so effective in limiting their amplitudes that self-excited vibrations may be regarded as practically suppressed.     

A model for the characterization of friction contacts in turbine blades

Stresses produced by the forced vibrations can lead to a significant reduction of the life of turbo engine blades. To predict the vibration amplitudes of this components an accurate dynamic analysis is necessary. The forced response calculation of these dynamic systems is strongly affected by the presence of the contact interfaces (i.e., underplatform dampers, shrouds, root joints). Different contact models are available in literature. These models make use of contact parameters, contact stiffness and friction coefficient to evaluate the damping and stiffness related to the contact interfaces. In this paper a model is proposed to characterize friction contact of non-spherical contact geometries obeying the Coulomb friction law with constant friction coefficient and constant normal load. The hysteresis curves of the oscillating tangential contact forces vs. relative tangential displacements and the dissipated energy at the contact are obtained for different contact geometries. The developed model is suitable to be implemented in numerical solvers for the calculation of the forced response of turbine blades with embedded friction contacts.     

Damping lateral vibrations in rotary machinery using motor speed modulation

This paper outlines a new principle for damping lateral vibrations of rotary systems. According to this principle, no changes in the visco-elastic properties of the system to be damped are required. The method is based on the generation of a harmonic additive to the constant speed of rotation that provides significant damping of lateral vibrations at critical speeds of rotation. This concept is validated analytically using the method of averaging and additionally with the help of direct numerical integration. The solution is shown to represent Fourier series containing Bessel functions. Consequently, proper choice of the parameters of the additional harmonic component ensuring that the Bessel functions have minimum values results from a minimization of the solution itself. Thus, the analytical solution and numerical results prove this concept by showing an essential decrease of the amplitudes of lateral vibrations of the damped system compared with those of the undamped system. The physical explanation of this effect is presented.     

Stability and transient dynamics of a propeller–shaft system as induced by nonlinear friction acting on bearing–shaft contact interface

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.     

A numerical investigation of curve squeal in the case of constant wheel/rail friction

Curve squeal is commonly attributed to self-excited vibrations of the railway wheel, which arise due to a large lateral creepage of the wheel tyre on the top of the rail during curving. The phenomenon involves stick/slip oscillations in the wheel/rail contact and is therefore strongly dependent on the prevailing friction conditions. The mechanism causing the instability is, however, still a subject of controversial discussion. Most authors introduce the negative slope of the friction characteristic as a source of the instability, while others have found that squeal can also occur in the case of constant friction due to the coupling between normal and tangential dynamics. As a contribution to this discussion, a detailed model for high-frequency wheel/rail interaction during curving is presented in this paper and evaluated in the case of constant friction. The interaction model is formulated in the time domain and includes the coupling between normal and tangential directions. Track and wheel are described as linear systems using pre-calculated impulse response functions that are derived from detailed finite element models. The nonlinear, non-steady state contact model is based on an influence function method for the elastic half-space. Real measured wheel and rail profiles are used. Numerical results from the interaction model confirm that stick/slip oscillations occur also in the case of constant friction. The choice of the lateral creepage, the value of the friction coefficient and the lateral contact position on the wheel tread are seen to have a strong influence on the occurrence and amplitude of the stick/slip oscillations. The results from the interaction model are in good qualitative agreement with previously published findings on curve squeal.     

Vibration control by recursive time-delayed acceleration feedback

This paper presents a theoretical basis of time-delayed acceleration feedback control of linear and nonlinear vibrations of mechanical oscillators. The control signal is synthesized by an infinite, weighted sum of the acceleration of the vibrating system measured at equal time intervals in the past. The proposed method is shown to have controlled linear resonant vibrations, low-frequency non-resonant vibrations, primary and 1/3 subharmonic resonances of a forced Duffing oscillator. The concept of an equivalent damping and natural frequency of the system is also introduced. It is shown that a large amount of damping can be produced by appropriately selecting the control parameters. For some combinations of the control parameters, the effective damping factor of the system is shown to be inversely related to the time-delay in the small delay limit. Selection of the optimum control parameters for controlling the forced and free vibrations is discussed. It is shown that forced vibration is best controlled by unity recursive gain and smaller values of the time-delay parameter. However, the transient response can be optimally controlled by suitably selecting the time delay depending upon the gain. The delay values for the optimal forced response may be different from that required for the optimum transient response. When both are important, a suboptimal choice of the delay parameters with unity recursive gain is recommended.     

Quenching of self-excited vibrations: one- and two-frequency vibrations

An analysis is presented of the possibility of initiation of two-frequency self-excited vibrations, and it is shown on this basis, and amply documented by analogue solutions, that the occurrence of such a phenomenon is rather exceptional in the systems being examined. It also is shown that unless the value of the coefficient of additional damping is very close to that lying on the boundary of the stability of the equilibrium position, it will have little effect on reducing the amplitude of self-excited vibrations even in the case of optimum tuning.     

PASSIVE DAMPING AUGMENTATION OF A VIBRATING BEAM USING PSEUDOELASTIC SHAPE MEMORY ALLOY WIRES

This paper examines the effectiveness of pseudoelastic shape memory alloy (SMA) wires for passive damping of flexural vibrations of a clamped-free beam with a tip mass. A finite-element model of the system is developed and validated with experimental results. The SMA behavior is modelled using amplitude-dependent complex modulus. Numerical simulations indicate that the damping introduced by the SMA wires will increase for higher excitation-force amplitudes that produce higher strain levels in the SMA wires. Increasing the wire cross-section area provides more damping at low force-excitation amplitudes, but reduced damping at higher amplitudes. The angle between the beam and the SMA wires is an influential parameter, and a value in the 10-20° range was found to introduce maximum damping. The underlying physical mechanisms are examined in detail. System damping depends only mildly on the SMA wire length, and is unaffected by the tip mass.     

Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

Internal resonance in a plane system of rods

The transverse vibrations of a plane system of rods is considered. The analysis of internal resonance in the system is a primary purpose of the paper. The internal resonance analyzed has an autoparametric nature. The couplings of the elements of the system through internal longitudinal forces, which are transverse forces at the ends of neighbouring rods, are taken into account. The amplitudes of the vibrations in the stationary states of internal resonance are investigated. Non-linear terms appear in the equations of motion. These terms are non-linear damping and non-linear inertia, and have a geometrical nature. The approximate method of calculation gives formulae for the vibration amplitudes of the rods. Plots of the amplitudes against frequency are presented. The stabilizing effect of masses placed at the articulated joints of the system is shown. The influences of the inertia and damping values on the character of the curves is considered. The results obtained are of a qualitative character.     

A note on safety-relevant vibrations induced by brake squeal

Brake squeal is mostly considered as a comfort problem only but there are cases in which self-excited vibrations of the brake system not only cause an audible noise but also result in safety-relevant failures of the system. In particular this can occur if lightweight design rims having very low damping are used. Considering the special conditions of lightweight design rims, a minimal model for safety-relevant self-excited vibrations of brake systems is presented. It is shown that most of the knowledge emanated from investigations of the comfort problem can be used to understand and avoid safety-relevant failures of the brake system.     

Lock-in and quasiperiodicity in hydrodynamically self-excited flames: Experiments and modelling

Hydrodynamically self-excited flames are often assumed to be insensitive to low-amplitude external forcing. To test this assumption, we apply acoustic forcing to a range of jet diffusion flames. These flames have regions of absolute instability at their base and this causes them to oscillate at discrete natural frequencies. We apply the forcing around these frequencies, at varying amplitudes, and measure the response leading up to lock-in. We then model the system as a forced van der Pol oscillator.Our results show that, contrary to some expectations, a hydrodynamically self-excited flame oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, it responds at both frequencies as well as at several nearby frequencies, indicating quasiperiodicity. When forced at high amplitudes, it locks into the forcing. The critical forcing amplitude for lock-in increases both with the strength of the self-excited instability and with the deviation of the forcing frequency from the natural frequency. Qualitatively, these features are accurately predicted by the forced van der Pol oscillator. There are, nevertheless, two features that are not predicted, both concerning the asymmetries of lock-in. When forced below its natural frequency, the flame is more resistant to lock-in, and its oscillations at lock-in are stronger than those of the unforced flame. When forced above its natural frequency, the flame is less resistant to lock-in, and its oscillations at lock-in are weaker than those of the unforced flame. This last finding suggests that, for thermoacoustic systems, lock-in may not be as detrimental as it is thought to be.     

A finite element study on rail corrugation based on saturated creep force-induced self-excited vibration of a wheelset-track system

The present work proposes friction coupling at the wheel-rail interface as the mechanism for formation of rail corrugation. Stability of a wheelset-track system is studied using the finite element complex eigenvalue method. Two models for a wheelset-track system on a tight curved track and on a straight track are established. In these two models, motion of the wheelset is coupled with that of the rail by friction. Creep force at the interface is assumed to become saturated and approximately equal to friction force, which is equal to the normal contact force multiplied by dynamic coefficient of friction. The rail is supported by vertical and lateral springs and dampers at the positions of sleepers. Numerical results show that there is a strong propensity of self-excited vibration of the wheelset-track system when the friction coefficient is larger than 0.21. Some unstable frequencies fall in the range 60-1200 Hz, which correspond to frequencies of rail corrugation. Parameter sensitivity analysis shows that the dynamic coefficient of friction, spring stiffness and damping of the sleeper supports all have important influences on the rail corrugation formation. Bringing the friction coefficient below a certain level can suppress or eliminate rail corrugation.     

Monoharmonic approximation in the vibration analysis of a sandwich beam containing piezoelectric layers under mechanical or electrical loading

The formulation for the coupled electromechanical problem of forced vibration of a simply supported inelastic sandwich beam with piezoelectric layers is developed. An approximate formulation for the problem in terms of the amplitudes of the main electromechanical field variables is produced by applying the monoharmonic (single-frequency) approach along with the concept of complex moduli to characterize the cyclic properties of the material. Accuracy of the developed monoharmonic approach is estimated. It is achieved through the comparison of the results computed for the transient response of the beam using the complete model with those found using the approximate model. Limitations on the approximate monoharmonic method application are specified. The effect of physically nonlinear behaviour of the passive layer on the beam response is investigated. The possibility of damping the forced vibrations of a structure with the help of harmonic voltages applied to the external piezoactive layers is also discussed.     

Soil influence on unbalance response and stability of a simple rotor-foundation system

The equations of motion are set up for a simple rotor (Jeffcott or Laval rotor) on a rigid foundation mass resting on an elastic half space (soil). The unbalance response and the stability limit against self-excited vibrations caused by the internal damping of the rotating shaft are calculated. The numerical results presented as response diagrams and stability graphs show that the damping effect of the soil on the system, due to radiation of energy, may have a very positive influence on the smooth running of the rotors.     

Structural optimization for the avoidance of self-excited vibrations based on analytical models

Self-excited vibrations are a severe problem in many technical applications. In many cases they are caused by friction as for example in disk and drum brakes, clutches, saws and paper calenders. The goal to suppress self-excited vibrations can be reached by active and passive techniques, the latter ones being preferable due to the lower costs. Among design engineers it is known that breaking the symmetries of structures is sometimes helpful to avoid self-excited vibrations. This has been verified from an analytical point of view in a recent paper. The goal of the present paper is to use this analytical insight for a systematic structural optimization of rotors in frictional contact. The first system investigated is a simple discrete model of a rotor in frictional contact. As a continuous example a rotating beam in frictional contact is considered and optimized with respect to its bending stiffness. Finally a brake disk is optimized giving some attention to the feasibility of the modifications for the production process.     

器壁滑动摩擦力对受振颗粒体系中冲击力倍周期分岔过程的影响

颗粒物质由离散的固体颗粒组成, 受到周期性振动时可以表现出复杂的动力学行为. 这些行为往往受众多因素的影响, 如空气阻力和器壁摩擦力等. 针对受振颗粒体系中冲击力的倍周期分岔现象, 通过抽真空或将容器底镂空消除空气阻力, 单独研究器壁滑动摩擦力的影响. 结果表明在仅有器壁摩擦力作用的情况下, 倍周期分岔过程仅受约化振动加速度的控制, 与颗粒的尺寸、颗粒层数及振动频率无关. 将器壁摩擦力处理成一个大小恒定、方向与颗粒和器壁相对速度反向的阻力, 并包含到完全非弹性蹦球模型中, 能够对所观察到的现象给出很好的解释. 通过对倍周期分岔点测量平均值的拟合, 得到器壁滑动摩擦力的大小约为颗粒总重量的10%. 关键词： 颗粒物质 器壁摩擦力 倍周期分岔 冲击力     

Validation of estimated isotropic viscoelastic material properties and vibration response prediction

This paper is concerned with finite element (FE) prediction of forced vibrations using a linear viscoelastic constitutive vibration damping modelling technique. A combined numerical and experimental investigation was performed on two bonded aluminium-PMMA (polymethyl methacrylate) plates with different geometry. Three-dimensional FE models were established using experimentally estimated PMMA material properties (elastic and damping) from previously published procedures. The viscoelastic material damping parameters are here validated from the perspective of accurate estimation of constitutive material properties. Vibration responses were predicted from the FE models and measured on the two composite plate structures at a large number of points. Comparisons between the numerical FE simulations and corresponding measured responses show that the estimated material damping properties used as input to the computations are very accurate and may be treated as independent of the geometry and boundary conditions of the plate structures, i.e., as constitutive damping parameters.     

Pole assignment of friction-induced vibration for stabilisation through state-feedback control

Friction-induced self-excited linear vibration is often governed by a second-order matrix differential equation of motion with an asymmetric stiffness matrix. The asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface. When the friction coefficient becomes high enough, the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues (poles) coalesce (when viscous damping is low).This short paper presents a receptance-based inverse method for assigning complex poles to second-order asymmetric systems through (active) state-feedback control of a combination of active stiffness, active damping and active mass, which is capable of assigning negative real parts to stabilise an unstable system.     

Complex dynamics of an archetypal self-excited SD oscillator driven by moving belt friction

We propose an archetypal self-excited system driven by moving belt friction, which is constructed with the smooth and discontinuous(SD) oscillator proposed by the Cao et al. and the classical moving belt. The moving belt friction is modeled as the Coulomb friction to formulate the mathematical model of the proposed self-excited SD oscillator. The equilibrium states of the unperturbed system are obtained to show the complex equilibrium bifurcations. Phase portraits are depicted to present the hyperbolic structure transition, the multiple stick regions, and the friction-induced asymmetry phenomena. The numerical simulations are carried out to demonstrate the friction-induced vibration of multiple stick-slip phenomena and the stick-slip chaos in the perturbed self-excited system. The results presented here provide an opportunity for us to get insight into the mechanism of the complex friction-induced nonlinear dynamics in mechanical engineering and geography.