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Analytical Method for Stroboscopically Sampling General Periodic Functions With Arbitrary Frequency Sweep Rates

[+] Author and Article Information
Dane Sequeira

Dynamical Systems Laboratory,
Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: dane.sequeira@duke.edu

Xue-She Wang

Dynamical Systems Laboratory,
Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: xueshe.wang@duke.edu

Brian Mann

Dynamical Systems Laboratory,
Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: brian.mann@duke.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 12, 2018; final manuscript received April 18, 2018; published online May 10, 2018. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 140(6), 064701 (May 10, 2018) (6 pages) Paper No: VIB-18-1018; doi: 10.1115/1.4040046 History: Received January 12, 2018; Revised April 18, 2018

Parameter sweeps are commonly used to explore the behavior of dynamical systems. This paper derives exact solutions for the instances in time to stroboscopically sample the response of a dynamical system subject to varying input excitations. This work will enable more accurate bifurcation diagrams and Poincaré sections in parameter regimes where numerical approaches may lead to incorrect behavior characterization. The simplest case of a linear frequency sweep is first considered before generalizing the results to include more complex functions with nonlinear sweep rates and arbitrary phase shifts.

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Grahic Jump Location
Fig. 1

A system exhibiting period doubling to chaos as the forcing frequency is varied where x* represents the sampled x values: (a) uses analytical solutions for stroboscopic sampling points, while (b) allows for 0.5% random error for each sampling point

Grahic Jump Location
Fig. 2

Results showing a computational method for approximating stroboscopic sampling points using Δt = 0.02 (s) to illustrate error progression for increasing frequency: (a) and (b) were conducted using β = 0.2, while (c) and (d) use β = 0.02 (rad/s2)

Grahic Jump Location
Fig. 3

Plots showing the error from calculating the stroboscopic sampling points in a linear frequency sweep using (a) zero crossing, (b) first-order Newton–Raphson, (c) second-order Newton–Raphson, and (d) the proposed analytical method

Grahic Jump Location
Fig. 4

Comparison showing the accumulation of error using four different methods outlined as a function of step size, where β=1×10−4 (rad/s2) and the final frequency ωf = 40 (rad/s)

Grahic Jump Location
Fig. 5

Plots showing the analytical method applied to more general scenarios including a (a) phase shift, (b) quadratic sweep rate, and (c) arbitrary periodic function. All plots were constructed using a value of β = 2π.



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