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Numerical continuation applied to automotive powertrains

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posted on 03.05.2022, 09:27 by Shaun SmithShaun Smith

In automotive systems, various elements across the powertrain contain several highly nonlinear components that cause dynamic challenges in the area of noise, vibration and harshness. Traditional approaches to study these problems may involve creating models, either by deriving equations of motion or using software packages, and analysing them in the frequency domain or time domain via simulation. This thesis demonstrates the use of bifurcation analysis in conjunction with numerical continuation as a complementary form of analysis. By first obtaining a single equilibrium, additional neighbouring equilibria can be numerically traced out as a parameter is varied across its operating region. While numerically tracing the equilibria, the stability of the system is also detected and the exact points where any changes in long term behaviour occur are identified as bifurcation points. These bifurcation points can be linked to physical properties and provide boundaries on regions that provide qualitatively similar behaviour. The bifurcations can be also traced for a number of additional parameters and how sensitive the bifurcations are to model alterations can be detected.

To demonstrate the applicability and usefulness of the approach, the methodology is applied to three separate areas across the powertrain: the engine; the braking system; and the rear axle assembly. In the engine study, the methods are applied to both a physics-based and data-driven internal combustion engine model. The analysis is shown to efficiently determine equilibria in the state-parameter space, without the need for exhaustive simulations, presenting the approach as complementary to traditional engine mapping techniques. Furthermore, the bifurcations in the system are linked to key physical properties such as peak torques and minimum throttle angles which are then traced as additional model parameters are varied to determine their sensitivity to mechanical or airflow alterations.

In the area of braking systems, a numerical bifurcation analysis and sensitivity study is conducted to further investigate the phenomena of brake creep groan, the undesirable vibration that occurs in the brake pad and disc when brakes are applied at low speeds. In the literature, the brake system is one of the few areas where an initial bifurcation analysis has already been conducted. This thesis presents novel knowledge in the field by conducting a detailed sensitivity study of the model where it is determined creep groan is highly sensitive to friction parameters. The existing definition of creep groan in the parameter space is updated due to the discovery and tracing of additional Hopf, torus and period doubling bifurcations in the model. The identification of several previously undiscovered period doubling bifurcations may be a route to chaotic behaviour in the system.

Finally the methods are used to study oscillations in the rear axle assembly. One of the main problems in this area is axle tramp, which is a self-sustaining oscillation in the rear axle and wheels with motion in both the vertical and longitudinal plane. In prior literature, the exact nature and cause of axle tramp has been difficult to concisely define. This section develops a 6DOF rear beam axle model which is studied with bifurcation methodology. The main findings show two self-sustaining limit cycles in the system: one at low speeds whose frequency equals the longitudinal eigenfrequency; and one for a wider speed range that is the same frequency as the torsional eigenfrequency. Additional properties such as fold bifurcation clusters at low speeds highlight the potentially dangerous nature of axle tramp and corroborates with findings in the literature. A sensitivity study of the system highlights torsional and longitudinal stiffness as key parameters when looking to mitigate axle tramp. Later the methods are applied on a low order IRS model. A further contribution is made with the implementation and study of a BEV torsional system which is shown to alter the bifurcation behaviour, potentially due the frequency content of the system significantly changing and potentially increasing the risk of modal coupling.


Engineering and Physical Sciences Research Council (ESPRC) EP/L014998/1



  • Aeronautical, Automotive, Chemical and Materials Engineering


  • Aeronautical and Automotive Engineering


Loughborough University

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© Shaun Smith

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A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy of Loughborough University.




James Knowles ; Byron Mason

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