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TECHNICAL BRIEFS

# Higher-Order Pseudoaveraging via Harmonic Balance for Strongly Nonlinear Oscillations

[+] Author and Article Information
K. Nandakumar1

Department of Mechanical Engineering,  Indian Institute of Science, Bangalore 560012, Indianandakumar.k@geind.ge.com

Anindya Chatterjee

Department of Mechanical Engineering,  Indian Institute of Science, Bangalore 560012, Indiaanindya@mecheng.iisc.ernet.in

In the method of multiple scales, this step is called “elimination of secular terms.”

Alternate parametrizations in terms of energy or action are also conceivable in some systems.

See Sec. 2.2 of 9 for a discussion on the convergence of these approximations.

The averaged equation for $dτ∕dt¯$ was also integrated numerically, and then terms like $A¯sin(τ+ϕ¯)$ were computed as functions of $t¯$, for plotting.

The oscillator need not actually be conservative. See Sec. 4.3 of 9.

1

Present address: John F. Welch Technology Center, Bangalore.

J. Vib. Acoust 127(4), 416-419 (Sep 26, 2004) (4 pages) doi:10.1115/1.1924639 History: Received September 14, 2003; Revised September 26, 2004

## Abstract

Some strongly nonlinear conservative oscillators, on slight perturbation, can be studied via averaging of elliptic functions. These and many other oscillations allow harmonic balance-based averaging (HBBA), recently developed as an approximate first-order calculation. Here, we extend HBBA to higher orders. Unlike the usual higher-order averaging for weakly nonlinear oscillations, here both the dynamic variable and time are averaged with respect to an auxiliary variable. Since the harmonic balance approximations introduce technically $O(1)$ errors at each order, the higher-order results are not strictly asymptotic. Nevertheless, as we show with examples, for reasonable values of the small expansion parameter, excellent approximations are obtained.

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## Figures

Figure 1

Exact and averaged solutions of Eq. 21, for ϵ=0.1. Initial conditions are x(0)=3.1411, ẋ(0)=0, and A¯(0)=3, ϕ¯(0)=π∕2.

Figure 2

Time period of limit cycle of Eq. 32: Numerical and Eq. 37

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