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Research Papers

Analyzing Bilinear Systems Using a New Hybrid Symbolic-Numeric Computational Method

[+] Author and Article Information
Meng-Hsuan Tien

Department of Mechanical and Aerospace
Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: tien.36@osu.edu

Kiran D'Souza

Department of Mechanical and Aerospace
Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: dsouza.60@osu.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 November 17, 2018; final manuscript received January 7, 2019; published online February 4, 2019. Editor: I. Y. (Steve) Shen.

J. Vib. Acoust 141(3), 031008 (Feb 04, 2019) (9 pages) Paper No: VIB-18-1506; doi: 10.1115/1.4042520 History: Received November 17, 2018; Revised January 07, 2019

In this paper, an efficient and accurate computational method for determining responses of high-dimensional bilinear systems is developed. Predicting the dynamics of bilinear systems is computationally challenging since the piecewise-linear nonlinearity induced by contact eliminates the use of efficient linear analysis techniques. The new method, which is referred to as the hybrid symbolic-numeric computational (HSNC) method, is based on the idea that the entire nonlinear response of a bilinear system can be constructed by combining linear responses in each time interval where the system behaves linearly. The linear response in each time interval can be symbolically expressed in terms of the initial conditions. The transition time where the system switches from one linear state to the other and the displacement and velocity at the instant of transition are solved using a numerical scheme. The entire nonlinear response can then be obtained by joining each piece of the linear response together at the transition time points. The HSNC method is based on using linear features to obtain large computational savings. Both the transient and steady-state response of bilinear systems can be computed using the HSNC method. Thus, nonlinear characteristics, such as subharmonic motion, bifurcation, chaos, and multistability, can be efficiently analyzed using the HSNC method. The HSNC method is demonstrated on a single degree-of-freedom (DOF) system and a cracked cantilever beam model, and the nonlinear characteristics of these systems are examined.

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Figures

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Fig. 1

A n-DOF system with a gap or prestress. Prestress exists when the gap size g <0.

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Fig. 2

(a) Schematic plot and construction of the nonlinear response and (b) the relationship between the two coordinate systems

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Fig. 3

The overall computational process of the HSNC method

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Fig. 4

Bilinear single DOF system

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Fig. 5

Comparison of transient responses computed using HSNC (–) and NI (⋅⋅⋅): (a) NI with the solver tolerance of 10−7, (b) NI with the solver tolerance of 10−10, and (c) NI with the solver tolerance of 10−13

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Fig. 6

(a) Basins of attraction computed using the HSNC method for period-4 motion (•), period-5 motion (□), and period-6 motion (empty area in the figure). (b)–(f) The initial states that have disagreement between HSNC and NI with tolerance values 10−6, 10−7, 10−8, 10−10, and 10−13, respectively. (g) The percentage difference between the basins of attraction obtained using HSNC and NI with different tolerance values. (h) The CPU time for HSNC and NI to obtain the basins of attraction.

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Fig. 7

(a) Chaotic motion computed using HSNC (–) and NI (⋅⋅⋅) and (b) enlarged plot of the area marked by the rectangle

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Fig. 8

(a) Schematic plot of the cracked cantilever beam, (b) discrete element model of the cantilever beam, and (c) coordinate system and the stiffness reduction at crack

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Fig. 9

(a) Response of the cracked beam computed using HSNC (–) and NI (∘) and (b) enlarged plot of the area marked by the rectangle

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Fig. 10

CPU time for HSNC (black) and NI (white)

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Fig. 11

Steady-state amplitude versus crack depth ratio for the cases where (a) Lc = 0.04 m, (b) Lc = 2.30 m, and (c) Lc = 5.13m. The subharmonic motions observed include period-1 motion (– –), period-2 motion (–), period-3 motion (⋅⋅⋅), period-4 motion (–⋅), and period-5 motion (+). The time histories obtained using HSNC and NI are compared in the insets for each type of motion.

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