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Research Papers

# An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

[+] Author and Article Information
Ashu Sharma

Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: azs0111@auburn.edu

S. C. Sinha

Life Fellow ASME
Alumni Professor Emeritus,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu

Contributed by Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 6, 2017; final manuscript received August 17, 2017; published online November 9, 2017. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 13(2), 021008 (Nov 09, 2017) (18 pages) Paper No: CND-17-1104; doi: 10.1115/1.4037797 History: Received March 06, 2017; Revised August 17, 2017

## Abstract

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

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## Figures

Fig. 1

Stability diagram of the QP Hill equation with  ω1=π and ω2=7.0.  ω min=0.274334. Solid: 2Ta periodic and dashed: Ta periodic.

Fig. 2

Stability diagram of the QP Hill equation with  ω1=π and ω2=7.0.  ω min=0.0619600. Solid: 2Ta periodic and dashed: Ta periodic.

Fig. 3

Stability diagram of the QP Hill equation with  ω1=π and ω2=7.0.  ω min=0.0442270. Red/solid: 2Ta periodic and blue/dashed: Ta periodic.

Fig. 4

Convergence diagram of bifurcation points of the main instability regions of QP Hill equation with  ω1=π and ω2=7.0

Fig. 5

Poincaré maps of approximate and exact solutions constructed at a few typical points in the stability diagram of QP Hill equation with  ω1=π and ω2=7.0. (ω min=0.0442270; see Fig. 3). Left: approximate system (Eq. (19)), right: original QP system, Eq. (18). (i) Point A: aa=0.268080  b=3.5  and  d=0.0, (ii) point B: aa=0.356044  b=3.6  and  d=0.0, (iii) point C: aa=3.976490   b=2.5  and  d=0.0 , (iv) point D: aa=5.556   b=4.5  and  d=0.0, (v) point F: aa=8.730260   b=5.0  and  d=0.0, (vi) point G: aa=10.52374  b=4.0  and  d=0.0, and (vii) point H: aa=11.1  b=1.0  and  d=0.0.

Fig. 6

Poincaré map of an exact solution computed near the exact stability boundary (ae=0.358719  b=3.6  and  d=0.0) of QP system with ω1=π andω2=7.0

Fig. 7

Frequency spectrum of solutions: (i) approximate solution generated at point B (aa=0.356044  b=3.6and  d=0.0 ; see Fig. 3) using Floquét theory with Ta=142.067 and (ii) exact solution generated near the exact boundary (ae=0.358719  b=3.6  and  d=0.0)

Fig. 8

(i) and (ii) Poincaré maps of approximate and exact solutions, respectively, constructed at point C1 in Fig. 2 with ω1=π and ω2=7.0. ω min=0.0619600. (iii) Frequency spectrum of the approximate solution generated at point C1.

Fig. 9

Frequency spectrum of solutions: (i) approximate solution generated at point D (aa=5.556b=4.5and  d=0.0 ; see Fig. 3) using Floquét theory with Ta=142.067 and (ii) exact solution generated near the exact boundary (ae=5.593  b=4.5  and  d=0.0)

Fig. 10

Comparison of instability regions of the QP system (ω1=π and ω2=7.0, ω min=0.0442270) with periodic systems: (i) QP system versus periodic system with ω1=π and (ii) QP system versus periodic system ω2=7.0

Fig. 11

Comparison between the stability boundaries of the main instability regions computed using Floquét theory (Ta=142.067) and Hill’s type approach

Fig. 12

Stability diagram of the damped QP Hill equation with  ω1=π and ω2=7.0.  ω min=0.0442270, and d=0.1.

Fig. 13

Poincaré maps of the approximate and the exact solutions of the damped QP Hill equation with  ω1=π and ω2=7.0. ωmin=0.0442270 (see Fig. 12). (i) Point Â : a=2.467, b=0.315, and d=0.1, (ii) point B̂ : a=2.467, b=0.313, and d=0.1, (iii) point Ĉ : a=4.120, b=3.2, and d=0.1, (iv) point D̂ : a=4.130, b=3.2, and d=0.1, (v) point Ê : a=4.140, b=3.2, and d=0.1, (vi) point F̂ : a=11.230, b=2.2, and d=0.1, (vii) point Ĝ : a=11.250, b=2.2, and d=0.1, and (viii) point Ĥ : a=11.280, b=2.2, and d=0.1.

Fig. 14

Comparison between the stability boundaries of the main instability regions of the damped QP Hill equation ( ω1=π and ω2=7.0) computed using three approaches: maximal Lyapunov exponent, proposed method using Floquét theory (Ta=142.067) and Hill’s type approach

Fig. 15

Effect of b2 on the main instability region corresponding to ω1=π, i.e., R1 in the damped QP Hill equation with  ω1=π,  ω2=7.0, and ω min=0.0442270

Fig. 16

Convergence diagram of bifurcation points of the main instability regions of QP Hill equation with  ω1=1 and ω2=(1+5)/2

Fig. 17

Stability diagram of the QP Hill equation with  ω1=1,  ω2=(1+5)/2,  ω min=0.0557281. Red: 2Ta periodic and blue: Ta periodic.

Fig. 18

Comparison between the stability boundaries computed using Floquét theory ( ω min=0.0557281) and those computed by Broer and Simó [31]

Fig. 19

Stability diagram of the damped QP Hill equation with  ω1=1 and ω2=(1+5)/2. ( ω min=0.0557281 and d=0.1).

Fig. 20

Convergence diagram of bifurcation points of the main instability regions of QP Hill equation with  ω1=1,  ω2=3,  and ω3=11 (see Fig. 21)

Fig. 21

Stability diagram of the QP Hill equation with  ω1=1,  ω2=3,  ω3=11, ω min=0.0216711. Red: 2Ta periodic and blue: Ta periodic.

Fig. 22

Stability diagram of the QP Hill equation with ω1=1  and  ω2=(1+5)/2 using symbolic FTM (ω min=0.0557281, mc=15, and p=20). Solid: 2Ta periodic and dashed: Ta periodic.

Fig. 23

Stability diagram of the QP Hill equation with  ω1=1 and ω2=(1+5)/2 using symbolic FTM (ω min=0.0557281, mc=20, and p=30). Solid: 2Ta periodic and dashed: Ta periodic.

Fig. 24

Stability diagram of the QP Hill equation with ω1=1.0, ω2=3, and ω3=11 using symbolic FTM (ω min=0.0216711, si=40, mc=25, and p=35). Red: 2Ta periodic and blue: Ta periodic.

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