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Research Papers

Dynamic Analysis of an Automotive Belt-Drive System With a Noncircular Sprocket by a Modified Incremental Harmonic Balance Method

[+] Author and Article Information
X. F. Wang

Department of Mechanical Engineering,
University of Maryland,
Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250

W. D. Zhu

Professor
Fellow ASME
Department of Mechanical Engineering,
University of Maryland,
Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 18, 2015; final manuscript received July 1, 2016; published online October 28, 2016. Editor: I. Y. (Steve) Shen.

J. Vib. Acoust 139(1), 011009 (Oct 28, 2016) (13 pages) Paper No: VIB-15-1328; doi: 10.1115/1.4034250 History: Received August 18, 2015; Revised July 01, 2016

A dynamic model of an automotive belt-drive system with a noncircular sprocket instead of a round sprocket is developed in this work to study the effect of reducing the angular variation of camshafts. There are two submodels in the belt-drive system, which are an engine model and a belt-drive model, and they are decoupled to simplify the analysis. When the belt-drive system operates at a steady-state, it is described by a nonlinear model with forced excitation, which can be approximated by a linear model with combined parametric and forced excitations. Steady-state responses of the engine and belt-drive models are calculated by a modified incremental harmonic balance method that incorporates fast Fourier transform and Broyden's method, which is efficient and accurate to obtain a periodic response of a multi-degree-of-freedom system. Steady-state responses of the angular variation of camshafts with different values of sprocket parameters are compared to investigate their optimal values to reduce the angular variation of camshafts. The optimal eccentricity and installation angle are larger and smaller than those from the kinematic model, respectively, which is consistent with published experimental results. This study first shows from a dynamic point of view why use of a noncircular sprocket can reduce the angular variation of camshafts in the operating speed of an engine. Simulation of a speed-up procedure for different sprocket parameters shows results that are consistent with steady-state responses. The belt-drive model developed in this work can be used to select sprocket parameters to minimize the angular variation of camshafts and numerically evaluate the dynamic performance of a belt-drive system with a given design of sprocket parameters.

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Figures

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Fig. 1

Schematic of a single-cylinder engine

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Fig. 2

Schematic of timing of valves

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Fig. 3

Schematic of the belt drive

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Fig. 4

Illustration of the tangential point of SPK on the belt

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Fig. 5

Crankshaft speed when its mean value is at 900 rpm

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Fig. 6

Angular variations of camshafts with a round sprocket and the noncircular sprocket whose sprocket parameters are from the kinematic model, with the latter calculated from the nonlinear and linearized belt-drive models

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Fig. 7

Angular variations of camshafts with different values of Δ and θ0=−45  deg

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Fig. 8

Angular variations of camshafts with different values of θ0 and Δ=0.0008 and 0.0009

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Fig. 9

Angular variations of camshafts with the round sprocket and two noncircular sprockets; sprocket parameters of one noncircular sprocket are from the kinematic model (dotted line) and those of the other noncircular sprocket are optimal sprocket parameters for the steady-state case (dashed line)

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