Research Papers

Multiharmonic Multiple-Point Collocation: A Method for Finding Periodic Orbits of Strongly Nonlinear Oscillators

[+] Author and Article Information
Hamid A. Ardeh

Department of Mechanical Engineering,
University of Wisconsin-Madison,
Madison, WI 53706
e-mail: ansariardeh@wisc.edu

Matthew S. Allen

Department of Engineering Physics,
University of Wisconsin-Madison,
Madison, WI 53706
e-mail: msallen@engr.wisc.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 17, 2013; final manuscript received August 10, 2015; published online November 19, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(4), 041006 (Nov 19, 2015) (11 pages) Paper No: CND-13-1320; doi: 10.1115/1.4031286 History: Received December 17, 2013; Revised August 10, 2015

An iterative method is proposed for finding periodic orbits of strongly nonlinear oscillators. The method combines the strength of analytical approaches, where the candidate solution is assumed in the form of a Fourier series, and the convenience of numerical methods that can be applied to larger systems with strong nonlinearity. The proposed method does not require integration of the vector field over any period of time and examples presented here illustrate that it is faster than traditional collocation algorithms, has a large radius of convergence, and is capable of finding several periodic orbits in each solution.

Copyright © 2016 by ASME
Topics: Algorithms
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Fig. 1

The 10DOF system is a reduced order model of a geometrically nonlinear beam coupled with a linear torsional spring

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Fig. 2

(a) Illustration of two-point collocation condition used in shooting algorithms. (b) Multiple-point collocation condition used in multiple shooting algorithms. (c) Multiple-point collocation condition used in the proposed algorithm. The 2M points are used to create M independent two-point collocation conditions.

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Fig. 3

Schematic representation of a case of homogenous convergence. Left: The candidate periodic solution x¯ changes its shape in a manner that reduces the sum of the norms of the distance vectors di, i.e., the deviation function D. Right: As the deviation function D tends to zero, the candidate solution x¯ converges to periodic orbits of the system.

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Fig. 4

The acceleration and force vectors, i.e., x¨ and f(x), of the candidate solution during the convergence

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Fig. 5

The displacement is shown for the candidate solution x1(k)(t), where it matches the true periodic orbit of the system with x1(36)(t)

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Fig. 6

The path of candidate solution in the state space as it converges to a periodic orbit of the system

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Fig. 7

The path of candidate solution host in the state space as it converges to a periodic orbit of the system using the MMC–CG. (Blue) initial guess, (black) intermediate result, and (red) final solution. For different sets of parameters, the MMC–CG algorithm converged to different solutions on different branches of periodic solutions previously found in Refs. [20,22].

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Fig. 8

A periodic solution of the system (28) found after a homogeneous convergence

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Fig. 9

A case of heterogeneous convergence. The five pairs of collocation points converge on three different periodic solutions. Note that each of the solutions 1–5 was computed by integrating the equations of motion from the collocation points indicated, but starting at time zero, so the solutions do not necessarily intersect the solution host at the collocation point.

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Fig. 10

Phase portraits of the first periodic solutions on the solution hosts in two cases (of 14 and 37, respectively) shown in Table 3. Top: case 2 and bottom: case 3

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Fig. 11

Frequency–energy representation of periodic solutions of the system in Eq. (28) using a shooting-based continuation algorithm (six branches of periodic solutions) and MMC–N. Multiple discrete solutions, found on each solution host, are shown with similar markers.

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Fig. 12

Phase portraits of the periodic solution (T = 0.02225 s) of the system in Eq. (28) marked with a circle on branch 1 in region A

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Fig. 13

Top: phase portraits of the second periodic solution (T = 0.023 s) of the system in Eq. (28) in region A. Bottom: This solution represents a case with multiple internal resonances. The time history of the response reveals cases of 1:5 and 1:18 internal resonances between the first and the fifth and also the first and the eight coordinates, respectively.

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Fig. 14

On the closed disk defined by D2={(x,y)∈ℝ2|x2+y2≤1}, the blue region of the disk highlights the sublevel set and the red region highlights the super-level set. The intersection of the two is the level set (See the online version for color.)



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