Theory of the shimmy phenomenon

The shimmy phenomenon is the appearance of angular self-excited vibrations of the carriage wheels. Such self-excited vibrations provide a serious safety hazard for motion, which explains the great interest of scientists in this phenomenon [1–6]. This problem is most serious for the aircraft fore wheels.     

Minimal models for squealing of railway block brakes

Block brakes have been used to brake railway vehicles for approximately 200 years. During the last 40 years, disk brakes have replaced the block brakes at passenger cars but block brakes are still used for all freight wagons. One problem of block brakes is that they show an enormous tendency to squeal. Although the block brake is a very common and old technical component, there exists almost no scientific work on its noise behavior regarding squeal. On the other hand, a lot of work has been done on the problem of disk brake squeal especially concerning the modeling of the excitation mechanism. The goal of this paper is to investigate whether and how models from disk brake squeal can be modified to model block brake squeal. The starting point of these investigations is a minimal model for an automotive disk brake introduced by von Wagner et al. (J Sound Vibration, 51(1–2):223–237, 2007) that is adapted to the block brake problem. It can be shown that such a simple converted model does not show any instability at all. A deeper analysis suggests that the reasons for squeal in block brakes could originate from in-plane vibrations of the brake disk or specific geometrical properties of the railway wheel. The self-excited vibrations explaining the squeal occur at relatively low rotational speeds far below the first critical rotor speed which has rarely been observed in rotor dynamics.     

Fast parametrically excited van der Pol oscillator with time delay state feedback

We investigate the effect of a fast vertical parametric excitation on self-excited vibrations in a delayed van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic in the vicinity of the trivial equilibrium. Then, we apply the multiple scales method on this slow dynamic to derive a second-order slow flow system describing the modulation of slow dynamic. In particular we analyze the slow flow to obtain the effect of a fast excitation on the regions in parameter space where self-excited vibrations can be eliminated. We have shown that in the case where the time delay and the feedback gains are imposed, fast vertical parametric excitation can be an alternative to suppress undesirable self-excited vibrations in a delayed van der Pol oscillator.     

Delay effects in shimmy dynamics of wheels with stretched string-like tyres

The dynamics of wheel shimmy is studied when the self-excited vibrations are related to the elasticity of the tyre. The tyre is described by a classical stretched string model, so the tyre-ground contact patch is approximated by a contact line. The lateral deformation of this line is given via a nonholonomic constraint, namely, the contact points stick to the ground, i.e., they have zero velocities. The mathematical form of this constraint is a partial differential equation (PDE) with boundary conditions provided by the relaxation of deformation outside the contact region. This PDE is coupled to an integro-differential equation (IDE), which governs the lateral motion of the wheel. Although the conventional stationary creep force idea is not used here, the coupled PDE-IDE system can still be handled analytically. It can be rewritten as a delay differential equation (DDE) by assuming travelling wave solutions for the deformation of the contact line. This DDE expresses the intrinsic memory effect of the elastic tyre. The linear stability charts and the corresponding numerical simulations of the nonlinear system reveal periodic and quasi-periodic self-excited oscillations that are also confirmed by simple laboratory experiments. The observed quasi-periodic vibrations cannot be explained in single degree-of-freedom wheel models subject to a creep force.     

CFD modelling of the cross-flow through normal triangular tube arrays with one tube undergoing forced vibrations or fluidelastic instability

A CFD methodology involving structure motion and dynamic re-meshing has been optimized and applied to simulate the unsteady flow through normal triangular cylinder arrays with one single tube undergoing either forced oscillations or self-excited oscillations due to damping-controlled fluidelastic instability. The procedure is based on 2D URANS computations with a commercial CFD code, complemented with user defined functions to incorporate the motion of the vibrating tube. The simulation procedure was applied to several configurations with experimental data available in the literature in order to contrast predictions at different calculation levels. This included static conditions (pressure distribution), forced vibrations (lift delay relative to tube motion) and self-excited vibrations (critical velocity for fluidelastic instability). Besides, the simulation methodology was used to analyze the propagation of perturbations along the cross-flow and, finally, to explore the effect on the critical velocity of the Reynolds number, the pitch-to-diameter ratio and the degrees of freedom of the vibrating cylinder.     

Vibrations of a fan jet in running a supersonic jet into a notched obstacle

Some kinds of vibrations of a fan jet are described. A physical model of one kind of self-excited vibrations of the fan jet is formulated. The formulas to calculate the frequencies of self-excited vibrations are given. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 118–124, July–August, 1999.     

Numerical study of the noninertial systems: application to train coupler systems

Car coupler forces have a significant effect on the longitudinal train dynamics and stability. Because the coupler inertia is relatively small in comparison with the car inertia; the high stiffness associated with the coupler components can lead to high frequencies that adversely impact the computational efficiency of train models. The objective of this investigation is to study the effect of the coupler inertia on the train dynamics and on the computational efficiency as measured by the simulation time. To this end, two different models are developed for the car couplers; one model, called the inertial coupler model, includes the effect of the coupler inertia, while in the other model, called the noninertial model, the effect of the coupler inertia is neglected. Both inertial and noninertial coupler models used in this investigation are assumed to have the same coupler kinematic degrees of freedom that capture geometric nonlinearities and allow for the relative translation of the draft gears and end of car cushioning (EOC) devices as well as the relative rotation of the coupler shank. In both models, the coupler kinematic equations are expressed in terms of the car body and coupler coordinates. Both the inertial and noninertial models used in this study lead to a system of differential and algebraic equations that are solved simultaneously in order to determine the coordinates of the cars and couplers. In the case of the inertial model, the coupler kinematics is described using the absolute Cartesian coordinates, and the algebraic equations describe the kinematic constraints imposed on the motion of the system. In this case of the inertial model, the constraint equations are satisfied at the position, velocity, and acceleration levels. In the case of the noninertial model, the equations of motion are developed using the relative joint coordinates, thereby eliminating systematically the algebraic equations that represent the kinematic constraints. A quasistatic force analysis is used to determine a set of coupler nonlinear force algebraic equations for a given car configuration. These nonlinear force algebraic equations are solved iteratively to determine the coupler noninertial coordinates which enter into the formulation of the equations of motion of the train cars. The results obtained in this study showed that the neglect of the coupler inertia eliminates high frequency oscillations that can negatively impact the computational efficiency. The effect of these high frequencies that are attributed to the coupler inertia on the simulation time is examined using frequency and eigenvalue analyses. While the neglect of the coupler inertia leads, as demonstrated in this investigation, to a much more efficient model, the results obtained using the inertial and noninertial coupler models show good agreement, demonstrating that the coupler inertia can be neglected without having an adverse effect on the accuracy of the solution.     

Nonlinear dynamic responses of high-speed railway vehicles under combined self-excitation and forced excitation considering the influence of unsteady aerodynamic loads

The dynamic characteristics of a railway vehicle system under unsteady aerodynamic loads are examined in this study. A dynamic analysis model of the railway vehicle considering the influences of aerodynamic loads was established. The model not only considers the forced excitation effect of unsteady aerodynamic loads but also accounts for the effect of unsteady aerodynamic loads on the change of the wheel–rail contact normal forces as well as changes of the wheelset creep coefficients and creep forces/moments. Therefore, this model also considers the influences of unsteady aerodynamic loads on the self-excited vibration characteristics of the vehicle system. The time-history curves, phase trajectory diagrams, Poincaré sections, and Lyapunov exponents of the vehicle system running on a smooth straight track under unsteady aerodynamic loads were determined. The results show that when the critical speed is exceeded, the vehicle system usually performs quasi-periodic motion under unsteady aerodynamic loads, which is significantly different from the periodic motion under steady aerodynamic loads. In different cases, the amplitude and phase of motion are significantly different. The amplitude of the motions can be increased by more than 159%, and the difference of phase can be up to 173°. (The phase is almost reversed.) The dynamic responses of the vehicle system under unsteady aerodynamic loads contain abundant frequency components, including the frequency of the self-excited vibration, the frequency of the forced excitation, and combinations of their integer multiples. The vibration forms corresponding to the main harmonic components under unsteady and steady aerodynamic loads were compared, and the self-excited vibration component of the vehicle system under unsteady aerodynamic loads was identified. The variations in the critical speed with various parameter combinations were computed. The variation range of the critical velocity can reach 73%.

     

Self-excited stick–slip oscillations of drill bitsOscillations stick–slip auto-entretenues d'un outil de forage

This Note studies the self-excited stick–slip oscillations of a rotary drilling system with a drag bit, using a discrete model which takes into consideration the axial and torsional vibration modes of the bit. Coupling between these two vibration modes takes place through a bit-rock interaction law which accounts for both frictional contact and cutting processes at the bit-rock interface. The cutting process introduces a delay in the equations of motion which is ultimately responsible for the existence of self-excited vibrations, exhibiting stick–slip oscillations under certain conditions. To cite this article: T. Richard et al., C. R. Mecanique 332 (2004).     

Andronov–Hopf bifurcations in wave models of torsional vibrations of drill strings

A wave model of torsional vibrations of rotating drill strings is set up. The ranges of rotational speed in which self-excited vibrations occur are found. Andronov–Hopf bifurcations occur at the limits of these ranges. The conditions for the occurrence and development of self-excited oscillatory processes are established     

Improved lateral-dynamics-intended railway vehicle model involving nonlinear wheel/rail interaction and car body flexibility

To study the vehicle hunting behavior and its coupling with car body vibrations,a simplified lateral-dynamics-intended railway vehicle model is developed.A two-...     

索道吊架失效原因的研究

特种设备索道吊装的失效分析是力学与多学科结合应用的典型事例。本通过有限元模建,结合材料断口分析,指出了某吊架失效原因,并指出了吊架还存在的隐患,其结果得到吊架实际使用情况证实。     

Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction

Nonlinear Dynamics - In this paper, the stick-slip vibrations of an archetypal self-excited smooth and discontinuous (SD) oscillator are investigated. The mathematical model of the self-excited SD...     

Identification of effective properties of the railway substructure in the low-frequency range using a heavy oscillating unit on the track

As the demand for predictions of train-induced vibrations is increasing, it is essential that adequate parameters of the railway structure are given as input in the predictions. Gathering this information can be quite time-consuming and costly, especially when predictions are required for the low-frequency emission. This article presents a procedure for deriving the effective properties of the foundation under the sleepers of a railway track from measurements taken with a heavy oscillating unit on the track. The unit consists of two masses inside a modified freight car that exert a dynamic force in the range 3–30 Hz on one of the two axles. The ratio of force applied on the axle over the resulting response measured with an accelerometer is studied. The foundation of the sleepers is modelled using a frequency-dependent complex-valued dynamic stiffness.     

秦沈客运专线高速列车构架蛇行波标准差的试验研究

国内外大多采用轨道横向不平顺作为列车—桥梁(轨道)时变系统横向振动的激振源。实际上，引起此系统横向振动的因素很多，诸如轨道横向不平顺、车轮踏面锥度、轮轨缺陷、车轮与钢轨的制造误差、车辆质量及其载重的偏心等。仅仅考虑轨道横向不平顺，会忽略其他很多因素的影响。相反，机车车辆构架蛇行波反映了引起此系统横向振动所有因素的影响，同时还反映了轮轨实际接触状态。研究证明，构架蛇行波标准差是输入此系统横向振动的能量，因此，有关它的研究十分重要。本文根据秦沈客运专线高速列车(中华之星)构架蛇行波现场测试结果，采用工程概率数值分析方法，对高速列车构架蛇行波标准差进行了统计分析，得出具有99％概率水平的高速列车构架蛇行波标准差与车速的关系曲线，为高速列车—桥梁(轨道)时变系统横向振动随机分析激振源的确定提供了基础资料；同时还列出了具有代表性的高速列车构架蛇行波实测波形图。     

Comments on “Vibration suppression for high-speed railway bridges using tuned mass dampers” [J.F. Wang,C.C. Lin,B.L. Chen, 2003. Int. J. Solids Struct. 40(2) 465–491]

In this comment, the discusser makes some remarks on the paper “Vibration suppression for high-speed railway bridges using tuned mass dampers” by Wang, J.F., Lin, C.C., Chen, B.L., published in the International Journal of Solids and Structures, 2003, Vol. 40, No. 2, pp. 465–491. First, the formulation of H(t, t_k) on p. 470 is questionable. Second, for a moving suspension mass model, the interaction force between moving mass and bridge is incorrectly given. Third, for a moving mass model for the train and without the installation of PTMD (passive tuned mass damper), the equation of motion of the bridge is incorrect. Lastly, for the train load model, which consists of one-half of a train car, one bogie, two wheel sets, spring and dashpot between bogie and half of a train car, and spring and dashpot between bogie and each wheel set, the authors did not put forward the formulation of interaction force between wheel set and bridge, but the discusser does.     

Self-vibration-induced wear of a friction pair

We study rotational self-excited vibrations of two frictionally coupled disks and consider the case of self-excited vibrations arising in the presence of dry friction, where the disk relative velocity periodically changes sign passing through the point of discontinuity of the friction law. We show that if there are resonances in the system, then undulatory wear arises either on the driving disk or on the driven disks. We find the existence and stability conditions for the corresponding modes.     

Substructure method in high-speed monorail dynamic problems

The study of actions of high-speed moving loads on bridges and elevated tracks remains a topical problem for transport. In the present study, we propose a new method for moving load analysis of elevated tracks (monorail structures or bridges), which permits studying the interaction between two strained objects consisting of rod systems and rigid bodies with viscoelastic links; one of these objects is the moving load (monorail rolling stock), and the other is the carrying structure (monorail elevated track or bridge). The methods for moving load analysis of structures were developed in numerous papers [1–15]. At the first stage, when solving the problem about a beam under the action of the simplest moving load such as a moving weight, two fundamental methods can be used; the same methods are realized for other structures and loads. The first method is based on the use of a generalized coordinate in the expansion of the deflection in the natural shapes of the beam, and the problem is reduced to solving a system of ordinary differential equations with variable coefficients [1–3]. In the second method, after the “beam-weight” system is decomposed, just as in the problem with the weight impact on the beam [4], solving the problem is reduced to solving an integral equation for the dynamic weight reaction [6, 7]. In [1–3], an increase in the number of retained forms leads to an increase in the order of the system of equations; in [6, 7], difficulties arise when solving the integral equations related to the conditional stability of the step procedures. The method proposed in [9, 14] for beams and rod systems combines the above approaches and eliminates their drawbacks, because it permits retaining any necessary number of shapes in the deflection expansion and has a resolving system of equations with an unconditionally stable integration scheme and with a minimum number of unknowns, just as in the method of integral equations [6, 7]. This method is further developed for combined schemes modeling a strained elastic compound moving structure and a monorail elevated track. The problems of development of methods for dynamic analysis of monorails are very topical, especially because of increasing speeds of the rolling stock motion. These structures are studied in [16–18].In the present paper, the above problem is solved by using the method for the moving load analysis and a step procedure of integration with respect to time, which were proposed in [9, 19], respectively. Further, these components are used to enlarge the possibilities of the substructure method in problems of dynamics. In the approach proposed for moving load analysis of structures, for a substructure (having the shape of a boundary element or a superelement) we choose an object moving at a constant speed (a monorail rolling stock); in this case, we use rod boundary elements of large length, which are gathered in a system modeling these objects. In particular, sets of such elements form a model of a monorail rolling stock, namely, carriage hulls, wheeled carts, elements of the wheel spring suspension, models of continuous beams of monorail ways and piers with foundations admitting emergency subsidence and unilateral links. These specialized rigid finite elements with linear and nonlinear links, included into the set of earlier proposed finite elements [14, 19], permit studying unsteady vibrations in the “monorail train-elevated track” (MTET) system taking into account various irregularities on the beam-rail, the pier emergency subsidence, and their elastic support by the basement. In this case, a high degree of the structure spatial digitization is obtained by using rods with distributed parameters in the analysis. The displacements are approximated by linear functions and trigonometric Fourier series, which, as was already noted, permits increasing the number of degrees of freedom of the system under study simultaneously preserving the order of the resolving system of equations.This approach permits studying the stress-strain state in the MTET system and determining accelerations at the desired points of the rolling stock. The proposed numerical procedure permits uniquely solving linear and nonlinear differential equations describing the operation of the model, which replaces the system by a monorail rolling stock consisting of several specialized mutually connected cars and a system of continuous beams on elastic inertial supports.This approach (based on the use of a moving substructure, which is also modeled by a system of boundary rod elements) permits maximally reducing the number of unknowns in the resolving system of equations at each step of its solution [11]. The authors of the preceding investigations of this problem, when studying the simultaneous vibrations of bridges and moving loads, considered only the case in which the rolling stock was represented by sufficiently complicated systems of rigid bodies connected by viscoelastic links [3–18] and the rolling stock motion was described by systems of ordinary differential equations. A specific characteristic of the proposed method is that it is convenient to derive the equations of motion of both the rolling stock and the bridge structure. The method [9, 14] permits obtaining the equations of interaction between the structures as two separate finite-element structures. Hence the researcher need not traditionally write out the system of equations of motion, for example, for the rolling stock (of cars) with finitely many degrees of freedom [3–18].We note several papers where simultaneous vibrations of an elastic moving load and an elastic carrying structure are considered in a rather narrow region and have a specific character. For example, the motion of an elastic rod along an elastic infinite rod on an elastic foundation is studied in [20], and the body of a car moving along a beam is considered as a rod with ten concentrated masses in [21].     

Nonlinear stability analysis of a disk brake model

It has become commonly accepted by scientists and engineers that brake squeal is generated by friction-induced self-excited vibrations of the brake system. The noise-free configuration of the brake system loses stability through a flutter-type instability and the system starts oscillating in a limit cycle. Usually, the stability analysis of disk brake models, both analytical as well as finite element based, investigates the linearized models, i.e. the eigenvalues of the linearized equations of motion. However, there are experimentally observed effects not covered by these analyses, even though the full nonlinear models include these effects in principle. The present paper describes the nonlinear stability analysis of a realistic disk brake model with 12 degrees of freedom. Using center manifold theory and artificially increasing the degree of degeneracy of the occurring bifurcation, an analytical expression for the turning points in the bifurcation diagram of the subcritical Hopf bifurcations is calculated. The parameter combination corresponding to the turning points is considered as the practical stability boundary of the system. Basic phenomena known from the operating experience of brake systems tending to squeal problems can be explained on the basis of the practical stability boundary.     

On plane self-excited vibrations of a cantilever suspended wheel

Torsional vibrations of a wheel about its leg axis arising in the carriage rectilinear motion were dubbed the shimmy phenomenon. Because of insufficient understanding of dry friction laws in the case of point contact, the causes of the shimmy phenomenon were explained by specific features of tyre deformation [1–3].