Creep groan is the undesirable vibration observed in the brake pad and disc as brakes are applied during low-speed driving. The presence of friction leads to nonlinear behavior even in simple models of this phenomenon. This paper uses tools from bifurcation theory to investigate creep groan behavior in a nonlinear 3-degrees-of-freedom mathematical model. Three areas of operational interest are identified, replicating results from previous studies: region 1 contains repelling equilibria and attracting periodic orbits (creep groan); region 2 contains both attracting equilibria and periodic orbits (creep groan and no creep groan, depending on initial conditions); region 3 contains attracting equilibria (no creep groan). The influence of several friction model parameters on these regions is presented, which identify that the transition between static and dynamic friction regimes has a large influence on the existence of creep groan. Additional investigations discover the presence of several bifurcations previously unknown to exist in this model, including Hopf, torus and period-doubling bifurcations. This insight provides valuable novel information about the nature of creep groan and indicates that complex behavior can be discovered and explored in relatively simple models.
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 stateparameter 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 demand" 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 selfsustaining 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.
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.
Axle tramp is a self-sustaining vibration in the driven axle of a vehicle with a beam axle layout, known to occur under heavy braking or acceleration. A 6DOF mathematical model of this phenomenon is used to identify how the key parameters of driveline stiffness, axle mass and fore/aft stiffness change the system’s dynamics. A bifurcation analysis is performed to study this nonlinear system’s dynamics. Four Hopf bifurcations in the underlying equilibria, along with a fold bifurcation in the outermost limit cycle branch, are shown to bound the parameter space where tramp occurs. The severity of tramp was found to be minimised by increasing drivetrain stiffness, reducing axle mass or increasing fore/aft stiffness: the trade-off for minimising tramp severity is that it may be easier to excite tramp when the drivetrain stiffness is increased, and the speed range over which tramp can occur is increased as fore/aft stiffness is increased. A key outcome from this work is that future electrified powertrains may experience more tramp, albeit at a reduced magnitude, than their combustion-powered counterparts.
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