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Research Papers

Analytical and Numerical Investigations of Stable Periodic Solutions of the Impacting Oscillator With a Moving Base and Two Fenders

[+] Author and Article Information
Barbara Blazejczyk-Okolewska

Division of Dynamics,
Lodz University of Technology,
Stefanowskiego 1/15,
Lodz 90-924, Poland
e-mail: okolbar@p.lodz.pl

Krzysztof Czolczynski

Division of Dynamics,
Lodz University of Technology,
Stefanowskiego 1/15,
Lodz 90-924, Poland

Andrzej Okolewski

Institute of Mathematics,
Lodz University of Technology,
Wolczanska 215,
Lodz 90-924, Poland

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 17, 2016; final manuscript received March 26, 2017; published online September 7, 2017. Assoc. Editor: Przemyslaw Perlikowski.

J. Comput. Nonlinear Dynam 12(6), 061008 (Sep 07, 2017) (11 pages) Paper No: CND-16-1567; doi: 10.1115/1.4036548 History: Received November 17, 2016; Revised March 26, 2017

A vibrating system with impacts, which can be applied to model the cantilever beam with a mass at its end and two-sided impacts against a harmonically moving frame, is investigated. The objective of this study is to determine in which regions of parameters characterizing system, the motion of the oscillator is periodic and stable. An analytical method to obtain stable periodic solutions to the equations of motion on the basis of Peterka's approach is presented. The results of analytical investigations have been compared to the results of numerical simulations. The ranges of stable periodic solutions determined analytically and numerically with bifurcation diagrams of spectra of Lyapunov exponents show a very good conformity. The locations of stable periodic solution regions of the system with a movable frame and two-sided impacts differ substantially from the locations of stable periodic solution regions for the system: (i) with a movable frame and one-sided impacts and (ii) with an immovable frame and two-sided impacts.

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Figures

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Fig. 1

Impacting oscillator with damping and kinematic external excitation: (a) dimensional form and (b) dimensionless form

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Fig. 2

Regions in which periodic solutions with impacts exist for kr = 0.6 and γ = 0.05: (a) n = 1 and (b) n = 3

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Fig. 3

Regions of existence of periodic solutions with impacts, which do not cross the base for kr = 0.6 and γ = 0.05: (a) n = 1 and (b) n = 3

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Fig. 4

Undisturbed (u) and disturbed (d) trajectory of the impacting oscillator with two fenders and a moving base; example for n = 1

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Fig. 5

Regions of existence of stable periodic solutions with impacts for kr = 0.6 and γ = 0.05: (a) n = 1 and (b) n = 3

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Fig. 6

Regions of existence of stable periodic solutions with impacts for kr = 0.6: (a) γ = 0, (b) γ = 0.05, (c) γ = 0.1, and (d) γ = 0.05 (zoom of Fig. 6(b))

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Fig. 7

Time series for kr = 0.6, γ = 0.05, and d = 1.6: (a) symmetric periodic motion for η = 0.82 (z = 2/3 and pL = pU = 1), (b) symmetric periodic motion for η = 0.99 (z = 2/1 and pL = pU = 1), (c) asymmetric periodic motion for η = 0.7865 (z = 1/2, pL = 1, and pU = 0), (d) asymmetric periodic motion for η = 0.7865 (z = 1/2, pL = 0, and pU = 1), (e) asymmetric periodic motion for η = 0.93 (z = 1/1, pL = 1, and pU = 0), and (f) asymmetric periodic motion for η = 0.93 (z = 1/1, pL = 0, and pU = 1)

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Fig. 8

Bifurcation diagrams of relative displacement w (a) and the corresponding spectra of Lyapunov exponents λ1, λ2 (b) for the impacting oscillator with a moving base for the system (1b); kr = 0.6, γ = 0.05, and d = 1.6 (gray—decreasing η and black—increasing η)

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Fig. 9

Basins of attraction at the frequency: η = 0.7865 (a), η = 0.785 (b), and η = 0.93 (c) for kr = 0.6, γ = 0.05, and d = 1.6

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Fig. 10

Regions of existence of stable periodic solutions with impacts for γ = 0.05: (a) kr = 0.8, (b) kr = 0.9, (c) kr = 0.95, and (d) kr = 0.99

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Fig. 11

Time diagrams (a)–(d) of the motion with impacts and (e) basins of attraction for kr = 0.99, γ = 0.05, η = 10, and d = 50

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Fig. 12

Regions of existence of stable periodic solutions with impacts for kr = 0.99: (a) γ = 0.0 and (b) γ = 0.1

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Fig. 13

Bifurcation diagrams of relative displacement w (a) and the corresponding spectra of Lyapunov exponents λ1, λ2 (b) for the impacting oscillator with a moving base for the system (1b); kr = 0.99, γ = 0.05, and d = 0.4 (gray—decreasing η and black—increasing η)

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