Research Papers

A Semi-Analytical Method for Calculation of Strongly Nonlinear Normal Modes of Mechanical Systems

[+] Author and Article Information
S. Mahmoudkhani

Faculty of New Technologies and Engineering,
Aerospace Engineering Department,
Shahid Beheshti University,
GC, Velenjak Square,
Tehran 1983969411, Iran
e-mail: s_mahmoudkhani@sbu.ac.ir

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 7, 2017; final manuscript received January 20, 2018; published online February 26, 2018. Assoc. Editor: Anindya Chatterjee.

J. Comput. Nonlinear Dynam 13(4), 041005 (Feb 26, 2018) (12 pages) Paper No: CND-17-1490; doi: 10.1115/1.4039192 History: Received November 07, 2017; Revised January 20, 2018

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

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Grahic Jump Location
Fig. 7

More complete backbone curves corresponding to the first NNM of Eq. (21), obtained by Auto software (thicker solid and dashed lines), and HAM (dotted black lines); Dashed lines: unstable solutions (u1(0) is equivalent to the maximum value of u1)

Grahic Jump Location
Fig. 4

Comparison of the results obtained by numerical and asymptotic method with those obtained by HAM (a), (b): with ℏ=−0.5,H2=c2 cos(3τ) (c),(d): ℏ=−0.5,H2=a2 cos(τ)+c2 cos(3τ)) for (a), (c) backbone curves of the first NNM (b), (d) cross sections of the manifolds at u˙1(0)=0

Grahic Jump Location
Fig. 5

A 2DOF system with quadratic and cubic nonlinearities

Grahic Jump Location
Fig. 6

Backbone curves obtained by the HAM (with ℏ=−1), and numerical, transport, and asymptotic methods for the first NNM of Eq. (21)

Grahic Jump Location
Fig. 1

Backbone curves obtained by NRS (with ℏ=−0.5), numerical, and asymptotic methods for: (a) the first NNM and (b) the second NNM

Grahic Jump Location
Fig. 2

The ℏ-curves of (a), (c) ωNL (b), (d) u2(0), obtained by the NRS scheme for (a), (b) u1(0)=1 and (c), (d) u1(0)=1.4

Grahic Jump Location
Fig. 3

Time histories of the 2DOF, obtained by the NRS and the numerical method for (a) u1(0)=1 and h = −0.8 and (b) u1(0)=1.4 and h = −0.3

Grahic Jump Location
Fig. 11

Backbone curves obtained by the HAM (with ℏ=−0.8) and the high-order HBM

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Fig. 12

Modal functions obtained by the HAM and numerical method, corresponding to variation of (a) W12c (b) W12s, with W11c

Grahic Jump Location
Fig. 8

Backbone curves obtained by HAM with Hi=0 for all DOFs NRS, HAM with Hi=bi cos(2τ)  for i>1, and the numerical method

Grahic Jump Location
Fig. 9

Invariant manifold of the first standing wave NNM of the cylindrical shells representing the variation of (a) W12c and (b) W01, with W11c and W˙11c (V11c)

Grahic Jump Location
Fig. 10

Nonlinear mode shapes at the midpoint cross section of the cylinder at different instants of time (W11c(0)/h=2)



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