An intrinsic multiple-scale harmonic balance method for non-linear vibration and bifurcation problems

A variation of the intrinsic multiple-scale harmonic balance method

“…The shortcoming of the classical harmonic balance method was later eliminated by Atadan and Huseyin to give consistent results [2,3]. More recently, a combination of the intrinsic harmonic balancing technique and the multiple-time-scales (derivative expansion) method was introduced by Huseyin and Lin [4].…”

“…In this section we will introduce a variation of the intrinsic multiple-scale harmonic balance method presented by Huseyin and Lin [4]. To start the development we seek an approximate solution to the Duffing equation given by (5) as which, warping the time via becomes u -u(t;s),…”

“…This was done by replacing the natural frequency of the given equation by a series representation of frequency. In this paper, a combination of the intrinsic multiple-scale harmonic balance method introduced by Huseyin and Lin [4], and the modification of the derivative expansion method given by Veronis [5] are presented. The two methods [4,5] are combined in such a way that the proposed technique has the desirable characteristics of both methods.…”

“…In this paper, a combination of the intrinsic multiple-scale harmonic balance method introduced by Huseyin and Lin [4], and the modification of the derivative expansion method given by Veronis [5] are presented. The two methods [4,5] are combined in such a way that the proposed technique has the desirable characteristics of both methods. Firstly, it is conceptually simple and does not involve secular terms as in the method presented by Veronis [5] (see, e.g., Eq.…”

“…(14) in [5]). The absence of secular terms follows from the intrinsic method of harmonic balancing [2,3] which is the foundation of the intrinsic multiple-scale harmonic balancing technique [4] used here. Secondly, the frequency-parameter relationship obtained is in the form introduced by Veronis [5].…”