Shimmy Phenomenon Research Articles

Seven‐degree‐of‐freedom‐based electric wheel sampled‐data active shimmy control method considering unknown sensor measurement error

AbstractThe active shimmy control methods for electric vehicle driven by in‐wheel motors (EV‐DIM) have been proposed in the recent years. However, these methods assume that data obtained from sensors are accurate, despite the fact that sensor measurements are prone to error. This unknown measurement error can make shimmy control difficult. Additionally, current shimmy models are low degree‐of‐freedom, which simplifies control but decreases accuracy. In this paper, we address these issues using a sampled‐data output control method based on a higher seven‐degree‐of‐freedom (7DOF) shimmy model which includes the steering system, suspension, and electric wheel. We first construct a 7DOF electric wheel shimmy model and use Lagrange's theorem to derive the electric wheel shimmy dynamic equations. We then obtain system state equations that account for unknown sensor measurement error based on the 7DOF shimmy model. A sampled‐data observer and controller are designed to attenuate or eliminate the shimmy phenomenon via a domination gain. Finally, we conduct numerical simulations and experiments to verify the effectiveness of our proposed method.

Distribution of the residual tolerable imbalance between correction plans

Shimmy is a self-accident lateral and yaw vibration of a wheel caused by the interaction between the dynamic behavior of the tire and the supporting structure. The shimmy affects road vehicles, motorcycles, and aircraft landing gears. This undesired vibration can increase to a considerable amplitude, producing loss of control and may even result in severe damage to the wheel-suspension system. Suitable tools and models are required to assess the system’s stability properties and understand shimmy mechanisms. This research focuses on the technical solutions to reduce the shimmy phenomenon. The suspension, axle, and bearing part on the axle work together in a united way to protect against the forces transmitted by the wheels’ imbalances. This physical phenomenon suggests balancing the wheel by considering a mechanical system consisting of suspensions, axles, and wheels. In the design phase, the research suggests balancing the axle and wheels, comparing two planes, and influencing coefficient methods. An inertial measurement unit (IMU) sensor can capture mechanical vibrations on the wheel at various vehicle speeds.

Research and Analysis of Coulomb Friction in Landing Gear Shimmy

Aircraft are subject to small disturbances during taxiing, many of which are accidental and difficult to explain. Although nonlinear factors are considered in traditional analysis, Coulomb friction is generally ignored, and the interaction and common effects of nonlinear factors are mostly not discussed. In this paper, the dynamic model of shimmy is established, and the influence of Coulomb friction on shimmy is studied by using bifurcation theory and the structural mechanics analysis method. The results show that the system with Coulomb friction has subcritical Hopf bifurcation and that the system with square damping has supercritical Hopf bifurcation. Although the two do not change the stability of the system, their cooperation can improve the stability of the system. The Coulomb friction torque of the system has a complex piecewise functional relationship with the stability distance, rake angle, and acceleration. Some combinations will lead to very low Coulomb friction and deteriorate the anti-interference ability of the landing gear. This paper provides theoretical basis and support for the rational design of structural parameter collocation, enhancing the antidisturbance ability of the system during constant-speed or variable-speed taxis and explaining the shimmy phenomenon in the process of variable-speed taxis of some aircraft.

Study on Shimmy Phenomenon on Two-wheeled Vehicle Steering System Considering Tire Dynamics

Study on Shimmy Phenomenon on Two-wheeled Vehicle Steering System Considering Tire Dynamics

Nonlinear energy sink to enhance the landing gear shimmy performance

The shimmy phenomenon is a significant concern in aircraft landing gear dynamics. The prediction of the shimmy instability is an essential issue in landing gear design to develop a passive or active suppression method. This work investigates the application of the nonlinear energy sink (NES) concept to mitigate the effects of shimmy in landing gears. The NES concept has been used in recent research on mechanical vibrations. It comprises a passive target energy transfer method that refers to a one-way energy transfer from a primary to a nonlinear subsystem. The landing gear model is based on torsional displacement coupled with the tyre classical elastic string analogy model. The NES device connects to the wheel shaft, and it comprises a mass, a linear damper, and a pure cubic spring. The numerical integration in time was used to assess the shimmy onset speed and the post-shimmy limit cycle oscillations. A parametric analysis of the landing gear nonlinear dynamics without the NES is presented. The design space of possible NES parameters is given, obeying design constraints, and inspected to assess adequate NES designs. The best NES samples are included in the landing gear dynamics to study their influence. Results have shown that the NES can adequately expand the operational speed range for no shimmy and lead to lower LCO amplitudes in the post-shimmy for a reasonable range of speeds. The NES concept’s successful employment for the landing gear dynamics suggests an enormous potential form of passive shimmy control.

Finite‐time active shimmy control based on uncertain disturbance observer for electric vehicle with independent suspension

For attenuating the shimmy phenomenon appeared in an electric vehicle (EV) with independent suspension, this study proposes a finite-time active shimmy stability control method based on an uncertainty estimation observer. Firstly, a four-degree-of-freedoms shimmy model of an EV with independent suspension is constructed. Secondly, in order to deal with the uncertainties in the shimmy model, a finite-time control method via a non-linear uncertain disturbance observer is proposed. The direct Lyapunov function method is used to analyse the global stability of the closed-loop system, and the results show that the system outputs globally converge to zero. Simulation and hardware-in-the-loop simulation test results verify the built shimmy model and show the effectiveness of the designed control method compared with the sliding mode control method.

Multiple Limit Cycles Shimmy of the Dual-Front Axle Steering Heavy Truck Based on Bisectional Road

The shimmy problem causes considerable harm to vehicles and is difficult to solve, especially multiple limit cycle shimmy. Moreover, the dynamic behavior of the multiple limit cycle shimmy of vehicles based on a bisectional road is more complex. Shimmy is practically observed in trucks of cooperative factories during utilization. Thus, we take a heavy truck of a cooperative factory as the prototype and establish a dynamic model of the vehicle-road coupling shimmy system, considering the road adhesion coefficient and dry friction between the suspension and steering system. Based on the dynamic model, the Hopf bifurcation theory is used to qualitatively analyze the existence of the limit cycle for the vehicle shimmy system, and the multiple limit cycle shimmy phenomenon is successfully reproduced using a numerical method. Moreover, the effect of the road adhesion coefficient on the multiple limit cycle shimmy characteristic is studied. Results show that the speed interval and amplitude of the multiple limit cycle shimmy decrease with the road adhesion coefficient; when the coefficient is reduced to a certain extent, the multiple limit cycle shimmy phenomenon is not observed. In addition, the adhesion coefficient of the second axle has a stronger effect on the shimmy characteristic than that of the first axle.

ОБОБЩЕНИЕ КАЧЕСТВЕННО НОВОЙ ТЕОРИИ КАЧЕНИЯ КОЛЕСА ПРИ ОПИСАНИИ ЯВЛЕНИЯ ШИММИ

ОБОБЩЕНИЕ КАЧЕСТВЕННО НОВОЙ ТЕОРИИ КАЧЕНИЯ КОЛЕСА ПРИ ОПИСАНИИ ЯВЛЕНИЯ ШИММИ

A new approximate model of tyre accounting for both deformed state and dry friction forces in the contact spot on the background of the coupled model

A new approximate model of the tire rolling accounting for coupled longitudinal and lateral sliding as well as the spinning and the deformed state resulting in elastic forces is proposed. The main goal of this investigation consists in the construction of simple models with a few of degrees of freedom allowing one to implement them analytically in the engineering practice, primarily for the estimation of the rolling stability and for the prognosis of the instable rolling so-called ”shimmy phenomenon” and to numerical simulation of the transient dynamics of rolling wheels with sliding effects. Such a model could become useful at the earlier stages of the engineering design instead of complex numerical models that are usually resource consuming. The known models of the shimmy phenomenon are usually based on the nonholonomic condition of the steady rolling and consider as the main cause of instability only the tire deformation while the sliding and spinning are assumed to vanish, i. e. the dry friction effects are neglected. Another type of models consists in the ”rigid wheel” assumption, in other words only the dry friction effects are accounted on the background of the coupled dry friction theory whereas the deformed state effects are neglected. Such a theory is based on the complete accounting of the combined kinematics due to simultaneous sliding and spin and shows its efficiency for slightly deformed wheels and the unsteady rolling regimes. Our goal consists in the formulation of the combined model that takes into account as well as the deformed state as the dry friction effects. This model is based on the solution of some model problems for the tire using solid finite element simulation or various shell theories, the computing of a set of specific generalized rigidity factors for the tire model, and on the accounting for the shape of the contact spot and the contact pressure distribution after the numerical simulation of the tire-road contact interaction in quasi-statics. As a result, we obtain a model with only a few degrees of freedom but more general that the Keldysh’s or Klimov’s ones.

Study on Suppression of Shimmy Phenomena of Casters

Study on Suppression of Shimmy Phenomena of Casters

Shimmy model for electric vehicle with independent suspensions

A 5-degrees-of-freedom shimmy model is established to analyse the dynamic responses of an electric vehicle with independent suspensions. Tyre elasticity is considered by means of Pacejka’s magic formula. Under the nonslip assumption for the leading contact point, tyre–road constraint equations are derived. Numerical simulation is conducted with different structural parameters and initial conditions to observe the shimmy phenomenon. Simulation results indicate that Hopf bifurcation occurs at a certain vehicle forward speed. Moreover, suspension structural parameters, such as caster angle, affect wheel shimmy. The linearized model of the system presents the stability boundaries, which agree with the simulation results. The results of this study not only provide a theoretical reference for shimmy attenuation, but also validate the effectiveness of the provided model, which can be used in further dynamic analysis of vehicle shimmy.

Linear Stability Analysis and Dynamic Response of Shimmy Dampers for Main Landing Gears

This paper presents the analysis and study of common shimmy dampers used today for main landing gears with the use of analytical and numerical tools. The shimmy phenomenon is studied by using the tire stretched string theory model and by developing linear approximations of the dynamics of a single tire landing gear. The dynamics of commonly used shimmy dampers are then incorporated into the model. The objectives of this paper are to study already developed shimmy damper designs and to develop tools to design a new innovative and better shimmy damper for main landing gears, those which have nonsteerable wheels. Two shimmy damper designs are studied in this paper, one developed by Boeing and another by UTC Aerospace Systems (UTAS). A linear approximation of the dynamics of these dampers is obtained, omitting the freeplay, saturation, and nonlinear dynamics. Stability plots are then created by changing the system's parameters, such as the velocity, caster length, and the shimmy damper stiffness and damping coefficients. These plots show the comparison of using a UTC two-arm design against the Boeing damper, for which the former spans larger zones of stability but requires higher damping coefficients due to the UTC damper's geometry which is very impractical. In addition, a multibody model is developed in MSC adams (from MSC Software Corporation) to study the dynamic response of these systems and to create a modeling tool that can be used to design a new and improved shimmy damper for main landing gears. The simulation results from the model show the disadvantages of using the UTC two-arm damper, which include an asymmetrical vibration response. Further recommendations are given to design an improved shimmy damper.

Studies of Shimmy Phenomenon by Statistical Approaches

Studies of Shimmy Phenomenon by Statistical Approaches

Controlling and Optimization of Steering Wheel Shimmy of the Vehicle at High-Speed Based on FEA Method

In this paper, finite element method was adopted to solve the Steering wheel shimmy problem. Firstly, finite element model of the whole steering system from the knuckle to the steering wheel is conducted, and be verified by test. After modeling and verification, it is possible for the FE model to identify a natural frequency that contributes reasonably to the shimmy phenomenon in the steering wheel. Secondly the frequency spectrum of acceleration which is obtained based on test is loaded at the knuckle to simulate steering wheel shimmy. Finally, the sequential quadratic programming is performed to optimize steering system structure and improve the isolation performance based on this model. The plate thickness and stiffness of bushing are set as discrete optimization variables, and the Y-direction acceleration of steering wheel at 12 o'clock is set as the objective function. The successful solution of the steering wheel shimmy of a passenger car proves that this method is efficacious.

Application of nonlinear tyre models to analyse shimmy

This paper focuses on the application of different tyre models to analyse the shimmy phenomenon. Tyre models with the Magic Formula and a non-constant relaxation length are introduced. The energy flow method is applied to compare these tyre models. A trailing wheel suspension is used to analyse shimmy stability and to evaluate the differences between tyre models. Linearisation and nonlinear techniques, including bifurcation analysis, are applied to analyse this system. Extending the suspension model with lateral flexibility and structural damping reveals more information on shimmy stability. Although the nonlinear tyre models do not change the stability of equilibria, they determine the magnitude of the oscillation. It is concluded that the non-constant relaxation length should be included in the shimmy analysis for more accurate results at large amplitude.

A new model of shimmy

Shimmy is the phenomenon of intensive angular self-excited vibrations of a vehicle wheel. Such self-excited vibrations seriously threaten the safety ofmotion, which explains scientists’ profound interest in this phenomenon. This problem is most serious for the front wheels of aircraft. Tyre deformation is usually viewed to be the main cause of shimmy. Without casting doubt on this point, we still note that this is not the only cause. The shimmy phenomenon can also be observed in everyday life and for various hand carts, where any reference to tyre elasticity is most often irrelevant if the wheels are rigid.

A new shimmy model

Shimmy is a phenomenon of intense angular self-excited vibrations of the wheels of a carriage. Such self-excited vibrations present a serious threat to traffic safety, which accounts for the great interest of researchers in this phenomenon. The problem is of highest importance for the front wheels of aircraft. Usually, the deformation of pneumatic tires is considered to be the main factor responsible for shimmy. Without challenging this thesis, we nevertheless note that this is not the only factor. The shimmy phenomenon can be observed in everyday life in the case of various carriages that often have nothing to do with pneumatics if the wheels are rigid. Below we will show that the theory of polycomponent dry friction fully explains the shimmy phenomenon for absolutely rigid wheels and, hence, is at least one of the factors responsible for shimmy in the general case. The reason why researchers have not taken dry friction into account when explaining shimmy is that the theory of this kind of friction has not been well developed; at the same time one has failed to explain shimmy in the framework of other existing theories.

Castor Dynamic Investigation

Dynamic behaviour of vehicles having castored wheels can be affected by wheel shimmy phenomenon, that is a self-sustained vibration of the whole castor around the steering axis. This article presents a shimmy investigation conducted on a laboratory castor characterized by low lateral stiffness; numerical models show that this characteristics leads to three different stability conditions depending on the forward velocity: at low speeds the system is unstable and the castor is affected by shimmy oscillations; then there is an intermediate speed range in which the system recovers stability; finally, above a threshold speed value, the system returns to be unstable. The experimental investigation mainly aims to compare the numerical results with the experimental ones and to define the steering damping required to stabilize the castor according to its main features.

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].

Stability of delay integro-differential equations using a spectral element method

This paper describes a spectral element approach for studying the stability of delay integro-differential equations (DIDEs). In contrast to delay differential equations (DDEs) with discrete delays that act point-wise, the delays in DIDEs are distributed over a period of time through an integral term. Although both types of delays lead to an infinite dimensional state-space, the analysis of DDEs with distributed delays is far more involved. Nevertheless, the approach that we describe here is applicable to both autonomous and non-autonomous DIDEs with smooth bounded kernel functions. We also describe the stability analysis of DIDEs with special kernels (gamma-type kernel functions) via converting the DIDE into a higher order DDE with only discrete delays. This case of DIDEs is of practical importance, e.g., in modeling wheel shimmy phenomenon. A set of case studies are then provided to show the effectiveness of the proposed approach.