Bifurcation study of a dynamic model of a landing gear mechanism

This paper presents a new modelling approach for the analysis of landing gear mecha- nisms. By replacing the mechanism's rotational joints with equivalent high-sti ness elas- tic joints, numerical continuation methods can be applied directly to dynamic models of landing gear mechanisms. The e ects of using elastic joints are considered through two applications | an overcentre mechanism, and a nose landing gear mechanism. In both cases, selecting a su cient sti ness for the elastic joint is shown to provide accurate con- tiuation results. The advantages of this new modelling approach are then demonstrated by considering the unlocking of a nose landing gear with a single uplock/downlock mechanism, when subjected to di erent orientations and magnitudes of gravitational loading. The un- locking process is shown to be qualitatively insensitive to changes in both load angle and load magnitude, ratifying the robustness of a previously-proposed control methodology for unlocking a nose landing gear with a single uplock/downlock mechanism

A bifurcation study of a dynamic model of a nose landing gear mechanism subjected to external disturbances

This paper presents a new modelling approach for the analysis of landing gear mechanisms. By replacing the mechanism's rotational joints with equivalent high-stiffness elastic joints, numerical continuation methods can be applied directly to dynamic models of landing gear mechanisms. The effects of using elastic joints are considered through two applications --| an overcentre mechanism, and a nose landing gear mechanism. In both cases, selecting a suffcient stiffness for the elastic joint is shown to provide accurate contiuation results. The advantages of this new modelling approach are then demonstrated by considering the unlocking of a nose landing gear with a single uplock/downlock mechanism, when subjected to different orientations and magnitudes of gravitational loading. The unlocking process is shown to be qualitatively insensitive to changes in both load angle and load magnitude, ratifying the robustness of a previously- proposed control methodology for unlocking a nose landing gear with a single uplock/downlock mechanism

Numerical investigation of aircraft high-speed runway exit using generalized optimal control

To aim at reducing aircraft turnaround time and improving airport operation efficiency, this paper considers the optimization of aircraft ground manoeuvres such as a high-speed runway exit. The aircraft on the ground is a highly nonlinear dynamical system described by a fully parameterized mathematical model. The full aircraft model used in this paper has been further developed to include combined slip tire model. An iterative simulation-based optimization algorithm known as Generalized Optimal Control is employed to investigate the optimal solution for the control input such as nose-gear steering, main-gear brakes and engine thrust. To achieve different control objectives, the cost function is defined accordingly and then minimized by GOC. The optimization results of GOC will help to explore the safety boundary of ground handling and guide the design of a real-time controller

An investigation of a high-speed ground manoeuvre under optimal control

This paper studies the behaviour of a nonlinear aircraft model under optimal control for aircraft ground manoeuvres, specifically for high-speed runway exits. The aircraft’s behavior on the ground is captured by a fully parameterized 6-DOF nonlinear model. A pre-defined cost function is minimized using a Generalized Optimal Control (GOC) algorithm, in order to obtain an optimal control sequence for a particular manoeuvre-cost function combination. In this paper, three scenarios are investigated for a 45-degree high-speed runway exit: the first control sequence minimizes the distance between the aircraft’s CG and the runway centreline; the second maximizes the distance travelled by the aircraft during the 20 seconds of simulation time; the third minimizes tire wear. For each scenario, the GOC algorithm provides the best possible control inputs: such results provide a benchmark against which the effectiveness of future real-time controllers may be judged

Numerical continuation analysis of a three-dimensional aircraft main landing gear mechanism

A method of investigating quasi-static landing gear mechanisms is presented and applied to a three-dimensional aircraft main landing gear mechanism model. The model has 19 static equilibrium equations and 20 equations describing the geometric constraints in the mechanism. In the spirit of bifurcation analysis, solutions to these 39 steady-state equations are found and tracked, or continued, numerically in parameters of interest. A design case-study is performed on the land-ing gear actuator position to demonstrate the potential relevance of the method for industrial applications. The trade-off between maximal efficiency and peak actuator force reduction when positioning the actuator is investigated. It is shown that the problem formulation is very flexible and allows actuator force, length and efficiency information to be obtained from a single numerical continuation computation with minimal data post-processing. The study suggests that numerical continuation analysis has potential for investigating even more complex landing gear mechanisms, such as those with more than one sidestay

A bifurcation analysis of an open loop internal combustion engine

The process of engine mapping in the automotive industry identifies steady-state engine responses by running an engine at a given operating point (speed and load) until its output has settled. While the time simulating this process with a computational model for one set of parameters is relatively short, the cumulative time to map all possible combinations becomes computationally inefficient. This work presents an alternative method for mapping out the steady-state response of an engine in simulation by applying bifurcation theory. The bifurcation approach used in this work allows the engine's steady-state response to be traced through the model's state-parameter space under the simultaneous variation of one or more model parameters. To demonstrate this approach, a bifurcation analysis of a simplified nonlinear engine model is presented. Using "throttle position" and "desired load torque signal", the engine's dynamic response is classified into distinct regions bounded by bifurcation points. These bifurcations are shown to correspond to key physical properties of the open-loop system: fold bifurcations correspond to the minimum throttle angle required for a steady-state engine response; Hopf bifurcations bound a region where self-sustaining oscillations occur. The techniques used in this case study demonstrate the efficiency a bifurcation approach has at highlighting different regions of dynamic behavior in the engine's state-parameter space. Such an approach could speed up the mapping process and enhance the automotive engineer's understanding of an engine's underlying dynamic behavior. The information obtained from the bifurcation analysis could also be used to inform the design of future engine control strategies

Numerical continuation applied to landing gear mechanism analysis

A method of investigating quasi-static mechanisms is presented and applied to an overcenter mechanism and to a nose landing gear mechanism. The method uses static equilibrium equations along with equations describing the geometric constraints in the mechanism. In the spirit of bifurcation analysis, solutions to these steady-state equations are then continued numerically in parameters of interest. Results obtained from the bifurcation method agree with the equivalent results obtained from two overcenter mechanism dynamic models (one state-space and one multibody dynamic model), while a considerable computation time reduction is demonstrated with the overcenter mechanism. The analysis performed with the nose landing gear model demonstrates the flexibility of the continuation approach, allowing conventional model states to be used as continuation parameters without a need to reformulate the equations within the model. This flexibility, coupled with the computation time reductions, suggests that the bifurcation approach has potential for analyzing complex landing gear mechanisms

Numerical continuation analysis of a dual-sidestay main landing gear mechanism

A model of a three-dimensional dual-sidestay landing gear mechanism is presented and employed in an investigation of the sensitivity of the downlocking mechanism to attachment point deflections. A motivation for this study is the desire to understand the underlying nonlinear behavior, which may prevent a dual-sidestay landing gear from downlocking under certain conditions. The model formulates the mechanism as a set of steady-state constraint equations. Solutions to these equations are then continued numerically in state and parameter space, providing all state parameter dependencies within the model from a single computation. The capability of this analysis approach is demonstrated with an investigation into the effects of the aft sidestay angle on retraction actuator loads. It was found that the retraction loads are not significantly affected by the sidestay plane angle, but the landing gear’s ability to be retracted fully is impeded at certain sidestay plane angles. This result is attributed to the landing gear’s geometry, as the locklinks are placed under tension and cause the mechanism to lock. Sidestay flexibilities and attachment point deflections are then introduced to enable the downlock loads to be investigated. The investigation into the dual sidestay’s downlock sensitivity to attachment point deflections yields an underlying double-hysteresis loop, which is highly sensitive to these deflections. Attachment point deflections of a few millimeters were found to prevent the locklinks from automatically downlocking under their own weight, hence requiring some external force to downlock the landing gear. Sidestay stiffness was also found to influence the downlock loads, although not to the extent of attachment point deflection

The influence of contact distribution shaping on the dynamic response of a wiper blade

The primary function of windscreen wipers is to remove water and debris from the windscreen, ensuring the driver has a clear view of the road ahead. Predicting wiper performance at the design stage is therefore important to ensure their safe operation. There is hence a need to develop physics-based models of wiper performance that can be used as evaluative tools early in the design stage. This paper presents an analysis of the impact of changing screen curvature on the contact force distribution of a wiper blade and the subsequent effects on the transient dynamics. The contact distributions for three distinct screen curvatures and three loading points are calculated via FEA (finite element analysis) and subsequently analysed via multiple connected mass spring dampers to model the wiper blade lip transient dynamics. By analysing time and frequency domain data for several calculated contact distributions it is found that decreasing the screen curvature reduces the contact force at the centre of the blade, however, increases the amplitude of vibrations and range of frequencies observed. Additionally, it is found that moving the loading point towards the tip of the blade reduces the amplitude of vibrations, a result analogous to that of increasing the screen curvature. Based upon the understanding gained through this work the influence of design criteria on wiper blades can now be assessed, and several suggestions made as to how to reduce windscreen wiper noise

Numerical continuation applied to internal combustion engine models

This paper proposes tools from bifurcation theory, specifically numerical continuation, as a complementary method for efficiently mapping the state-parameter space of an internal combustion engine model. Numerical continuation allows a steady-state engine response to be traced directly through the state-parameter space, under the simultaneous variation of one or more model parameters. By applying this approach to two nonlinear engine models (a physics-based model and a data-driven model), this work determines how input parameters ‘throttle position’ and ‘desired load torque’ affect the engine’s dynamics. Performing a bifurcation analysis allows the model’s parameter space to be divided into regions of different qualitative types of the dynamic behaviour, with the identified bifurcations shown to correspond to key physical properties of the system in the physics-based model: minimum throttle angles required for steady-state operation of the engine are indicated by fold bifurcations; regions containing self-sustaining oscillations are bounded by supercritical Hopf bifurcations. The bifurcation analysis of a data-driven engine model shows how numerical continuation could be used to evaluate the efficacy of data-driven models