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Research Papers

# A New Incremental Harmonic Balance Method With Two Time Scales for Quasi-Periodic Motions of an Axially Moving Beam With Internal Resonance Under Single-Tone External Excitation

[+] Author and Article Information
J. L. Huang

Department of Applied Mechanics and Engineering,
Sun Yat-sen University,
Guangzhou, 510275, China
e-mail: huangjl@mail.sysu.edu.cn

W. D. Zhu

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.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 June 18, 2016; final manuscript received October 19, 2016; published online February 15, 2017. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 139(2), 021010 (Feb 15, 2017) (15 pages) Paper No: VIB-16-1307; doi: 10.1115/1.4035135 History: Received June 18, 2016; Revised October 19, 2016

## Abstract

Quasi-periodic motion is an oscillation of a dynamic system characterized by $m$ frequencies that are incommensurable with one another. In this work, a new incremental harmonic balance (IHB) method with only two time scales, where one is one of the $m$ frequencies, referred to as a fundamental frequency, and the other is an interval distance of two adjacent frequencies, is proposed for quasi-periodic motions of an axially moving beam with three-to-one internal resonance under single-tone external excitation. It is found that the interval frequency of every two adjacent frequencies, located around the fundamental frequency or one of its integer multiples, is fixed due to nonlinear coupling among resonant vibration modes. Consequently, only two time scales are used in the IHB method to obtain all incommensurable frequencies of quasi-periodic motions of the axially moving beam. The present IHB method can accurately trace from periodic responses of the beam to its quasi-periodic motions. For periodic responses of the axially moving beam, the single fundamental frequency is used in the IHB method to obtain solutions. For quasi-periodic motions of the beam, the present IHB method with two time scales is used, along with an amplitude increment approach that includes a large number of harmonics, to determine all the frequency components. All the frequency components and their corresponding amplitudes, obtained from the present IHB method, are in excellent agreement with those from numerical integration using the fourth-order Runge–Kutta method.

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## Figures

Fig. 1

Schematic of an axially moving beam

Fig. 2

Example spectrum of quasi-periodic motion described by Eq. (24) with nc0=4, ns0=3, and nf=9

Fig. 3

Scenarios depicting Floquet multipliers moving outside the unit circle in the complex plane for different local bifurcations: (a) Hopf bifurcation, and (b) saddle-node bifurcation

Fig. 4

Frequency response curves of periodic solutions when ω̃20≈3ω̃10 and the excitation frequency Ω is close to ω̃10 with f1=0.0055 and μ11=μ22=0.04: (a) Ω/ω̃10∼A110, and (b) Ω/ω̃10∼A220. Solid lines represent stable periodic solutions obtained from the IHB method; dashed and dash-dotted lines represent unstable periodic solutions with a real Floquet multiplier moving outside the unit circle along the +1 direction and a complex conjugate pair of Floquet multipliers moving outside the unit circle, respectively; and denotes numerical integration (NI), denotes a saddle-node bifurcation point, and denotes a Hopf bifurcation point.

Fig. 5

Fourier spectra of the periodic motion of (a) q1 and (b) q2 with Ω/ω̃10=1.23779 from numerical integration (NI) and the IHB method

Fig. 6

Time histories of the quasi-periodic motion with Ω/ω̃10=1.23334 from numerical integration (NI) and the present IHB method with nc0=5 and nf=9: ((a) and (b)) time histories of q1 and q2, respectively; enlarged views of two zones highlighted in (a) and (b) are shown in (c) and (d), respectively

Fig. 7

Phase plane portraits of the quasi-periodic motion and the unstable periodic solution of (a) q1 and (b) q2 with Ω/ω̃10=1.23334 from numerical integration (NI), the IHB method, and the present IHB method with nc0=5 and nf=9

Fig. 8

Fourier spectra of the quasi-periodic motion of (a) q1 and (b) q2 with Ω/ω̃10=1.23334 from numerical integration (NI) and the present IHB method with nc0=5 and nf=9

Fig. 9

Poincaré sections of the quasi-periodic motion of (a) q1 and (b) q2 with Ω/ω̃10=1.23334 from numerical integration (NI) and the present IHB method with different nc0 and nf

Fig. 10

Frequency response curves of periodic motions with f2=0.0055 and μ11=μ22=0.04 when ω̃20≈3ω̃10 and the excitation frequency Ω is near ω̃20 : (a) Ω/ω̃20∼A110, and (b) Ω/ω̃20∼A220. The legends are the same as those in Fig. 4.

Fig. 11

Fourier spectra of the periodic motion of (a) q1 and (b) q2 with Ω/ω̃20=1.17967 from numerical integration (NI) and the IHB method

Fig. 12

Time histories of the quasi-periodic motion with Ω/ω̃20=1.16936 from numerical integration (NI) and the present IHB method with nc0=5 and nf=9: ((a) and (b)) time histories of q1 and q2, respectively; enlarged views of two zones highlighted in (a) and (b) are shown in (c) and (d), respectively

Fig. 13

Phase plane portraits of the quasi-periodic motion of (a) q1 and (b) q2 with Ω/ω̃20=1.16936 from numerical integration (NI) and the present IHB method with nc0=5 and nf=9

Fig. 14

Fourier spectra of the quasi-periodic motion of (a) q1 and (b) q2 with Ω/ω̃20=1.16936 from numerical integration and the present IHB method with nc0=5 and nf=9

Fig. 15

Poincaré sections of the quasi-periodic motion of (a) q1 and (b) q2 with Ω/ω̃20=1.16936 from numerical integration (NI) and the present IHB method with nc0=5 and nf=9

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