In this paper we consider the onset of shimmy oscillations of an aircraft nose landing gear. To this end we develop and study a mathematical model with torsional and lateral bending modes that are coupled through a wheel-mounted elastic tyre. The geometric effects of a positive rake angle are fully incorporated into the resulting five-dimensional ordinary differential equation model. A bifurcation analysis in terms of the forward velocity and the vertical force on the gear reveals routes to different types of shimmy oscillations. In particular, we find regions of stable torsional and stable lateral shimmy oscillations, as well as transient quasiperiodic shimmy where both modes are excited.

/pdf/interaction-of-torsion-and-lateral-bending-in-aircraft-nose-4rsim6siqk.pdf

Interaction of torsion and lateral bending in aircraft nose landing gear shimmy

The work reported in this paper concentrates on the possibility of suppressing shimmy oscillations of landing gear more effectively using a linear passive suppression device incorporating inerter. The inerter is a one-port mechanical device with the property that the applied force is proportional to the relative acceleration between its terminals. A linear model of a Fokker 100 aircraft main landing gear equipped with a shimmy-suppression device is presented. Time-domain optimizations of the shimmy-suppression device are carried out using cost functions of the maximum amplitude and the settling time of torsional-yaw motion. Applying two types of excitations that trigger the shimmy oscillations, the performance advantages of inerter-based configurations for suppressing shimmy of main landing gear, together with corresponding parameter values, are identified.

/pdf/inerter-based-configurations-for-main-landing-gear-shimmy-2h9tk3sngx.pdf

Inerter-Based Configurations for Main-Landing-Gear Shimmy Suppression

In this article, we provide an overview of a selection of topics of current interest in nonlinear dynamics and vibrations of mechanical systems. Specifically, we cover the traditional topics of structural dynamics, rotating systems, vibration control, vehicle dynamics, and machining, as well as some topics that have emerged more recently, namely micro- and nano-electromechanical systems, piecewise smooth systems, and structural health monitoring and system identification. In the introductory and closing sections, we offer our perspective on the state of affairs in nonlinear dynamics and its utility in the study of mechanical systems.

https://www.jstage.jst.go.jp/article/jsdd/2/3/2_3_611/_pdf

A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008

Nowadays, throughput has become a limiting factor in road transport. An effective means to increase the road throughput is to decrease the intervehicle time gap. A small time gap, however, may lead to string instability, being the amplification of velocity disturbances in upstream direction. String-stable behavior is thus considered an essential requirement for the design of automatic distance control systems, which are needed to allow for safe driving at time gaps well below 1 s. However, the formal notion of string stability is not unambiguous in literature, since both stability interpretations and performance interpretations exist. Therefore, a novel definition for string stability of nonlinear cascaded systems is proposed, using input–output properties. This definition is shown to result in well-known string stability conditions for linear cascaded systems. Employing these conditions, string stability is obtained by a controller that uses wireless intervehicle communication to provide information of the preceding vehicle. The theoretical results are validated by implementation of the controller, known as Cooperative Adaptive Cruise Control, on a platoon of six passenger vehicles. Experiments clearly show that the practical results match the theoretical analysis, thereby indicating the practical feasibility for short-distance vehicle following.

Analysis and design of controllers for cooperative and automated driving

Aircraft landing gear assemblies comprise of various subsystems working in unison to enable functionalities such as taxiing, take-off and landing. As development cycles and prototyping iterations begin to shorten, it is important to develop and improve practical methodologies to meet certain design metrics. This paper presents an efficient methodology that applies high-fidelity multi-disciplinary design optimization techniques to commercial landing gear assemblies, for weight, cost, and structural performance by considering both structural and dynamic behaviours. First, a simplified landing gear assembly model was created to complement with an accurate slave link subassembly, generated based of drawings supplied from the industrial partner, Safran Landing Systems. Second, a Multi-Body Dynamic (MBD) analysis was performed using realistic input motion signals to replicate the dynamic behaviour of the physical system. The third stage involved performing topology optimization with results from the MBD analysis; this can be achieved through the utilization of the Equivalent Static Load Method (ESLM). Lastly, topology results were generated and design interpretation was performed to generate two designs of different approaches. The first design involved trying to closely match the topology results and resulted in a design with an overall weight savings of 67%, peak stress increase of 74%, and no apparent cost savings due to complex features. The second design focused on manufacturability and achieved overall weight saving of 36%, peak stress increase of 6%, and an estimated 60% in cost savings.

/pdf/design-optimization-of-aircraft-landing-gear-assembly-under-3x1txoq3gc.pdf

Design optimization of aircraft landing gear assembly under dynamic loading

The landing gear is an important aircraft system, which has to meet many different design requirements. It is a highly loaded structure, which is designed for minimum weight. Shimmy is a dynamic instability of the landing gear, which is caused by the interaction of the dynamic behaviour of the landing gear structure and tyres. The unstable lateral and yaw vibration of the landing gear can reach considerable amplitudes and may even result in severe damage to the aircraft. Shimmy is easily ignored in the design process, which may be caused by a of lack of knowledge on the shimmy phenomenon, absence of suitable analysis tools or the non-availability of e.g. tyre characteristics. Computer simulations are very important to evaluate the shimmy stability of a landing gear. Experience has shown that it will be very difficult to rigorously prove shimmy stability from experiments, e.g. full-scale flight tests or laboratory tests using a drum. Three fields of research are covered in this thesis: • shimmy fundamentals • modelling of the tyre dynamic behaviour • the development and validation of a detailed landing gear model Analytical expressions for the shimmy stability have been derived for a number of relatively simple systems using the Hurwitz criterion. In particular, an analytical solution has been found for a system where the wheel has a mechanical trail and both the yaw and lateral stiffness of the hinge point are taken into account. The stability boundaries can be represented by two shifted parabolas in the mechanical trail versus yaw stiffness plane; this analytical result is very important to understand the interaction between the different variables. The model may be enhanced by including the gyroscopic behaviour of the rotating wheel and structural damping. The shimmy stability can also be analysed in the frequency domain by considering the landing gear structure and tyre as a feedback system and applying the Nyquist criterion. A design study is performed using a twin wheeled landing gear, having three mechanical degrees of freedom (lateral, roll and yaw). The stability of the baseline configuration can be improved considerably by modifying the length of the mechanical trail, lateral stiffness, yaw stiffness and wheel track. It appears that a small positive mechanical trail is better avoided; this is substantiated by the analytical results. Other methods to improve the stability have been investigated: modification of the cant angle, the introduction of a bob mass, tuned mass, shimmy damper or co-rotating wheels. Furthermore the stability of a bogie landing gear has been evaluated both analytically and using a more complex model; the results indicate that this configuration is far less susceptible to shimmy. Different linear tyre models have been developed for application in a shimmy analysis; in particular the models of Von Schlippe, Smiley, Pacejka (straight tangent and parabolic approximation), Kluiters, Rogers, Keldysh and Moreland are discussed. Expressions for the transfer functions with respect to side and turn slip are derived and equivalence conditions can be established between some of the tyre models. A comparison is made using transfer functions, step response and energy considerations. In addition, the impact of the tyre model on system stability is studied for a number of simple mechanical systems. Some guidelines regarding the values of different tyre parameters are given using measurement data and literature. A detailed model will be required to assess shimmy stability in the design stage or when solving actual shimmy problems. The stiffness of a landing gear is dependent on the shock absorber deflection due to changes in torque link geometry and distance between upper and lower bearing. The flexibility of the back-up structure and wing results in a significant reduction of the lateral stiffness of the landing gear at wheel axle level. Modal testing can be performed to assess eigenfrequencies and mode shapes of the landing gear, but measurements show that the results may be highly amplitude dependent due to free-play and friction. Free-play and friction are also important for the shimmy stability and will have to be included in a detailed model. The shimmy damper may have a non-linear characteristic consisting of a preloaded spring and velocity squared damping force. Various component tests will be required to determine parameters or to validate the characteristics of the model. A detailed simulation model was developed using the MECANO multi-body software package. The flexible slider element proved to be very convenient for modelling the landing gear structure. Full-scale tests on the aircraft may be used to perform a limited validation of the simulation model. During taxi runs an external disturbance is required to provoke a dynamic response of the landing gear. This may be achieved by running over a diagonally positioned plank, introducing an unbalance mass or asymmetrical braking. In a landing event the asymmetrical spin-up of the wheels is the main excitation source. Generally, only limited data will be available when a shimmy event occurs, which makes it difficult to perform a detailed assessment. An interesting exception is a shimmy vibration which occurred on a test aircraft, equipped with an instrumented landing gear. The unstable motion is analysed in detail. This event has also been simulated using the MECANO model, aiming to match the landing conditions as closely as possible. A reasonable agreement can be obtained between simulation model and measurement. Future research may aim at an accurate determination of tyre characteristics and correlation between different tyres. The dynamic tyre model can be extended to describe the non-linear tyre behaviour at large side slip angles more accurately. Also some enhancements of the landing gear and airframe model are possible, in particular the dynamic behaviour of the wing and brakes may be included. Friction may be rather important for an accurate simulation of the landing gear behaviour; in this field both additional experimental data and improved modelling techniques may be required.

/pdf/shimmy-of-aircraft-main-landing-gears-4v4paok387.pdf

Ijm Igo Besselink

Papers

Shimmy of Aircraft Main Landing Gears