of direct series expansions, singular perturbation methods [28], the method of multiple scales and the harmonic balance method [3], the method of averaging [29], etc. The complexification-averaging (CX-A) technique [30] has received much recent attention due to its capacity to provide slow-flow models even for strongly nonlinear transient dynamical interactions; e.g., resonance capture phenomena in coupled oscillators with essential nonlinearities [31]. Focusing on the CX-A method, we demonstrate the extraction of the slow-flow dynamics of a general n-degree-of-freedom (DOF), nonlinear dynamical system of the form 2 ( , ), { } , T T T n t t X f X X x x (1) where x is an n -response vector and f is an n -vector function. Assume that the dynamics possesses N distinct components at frequencies 1 2 , ,..., N , so the response of each DOF of the system can be expressed as a summation of N independent components, (1) ( ) ( ) ( ) ( ) ( ) ( ), 1, , m N k k k k x t x t x t x t k n (2) where ( ) ( ) m kx t indicates the response of the k -th coordinate of (1), associated with the basic frequency m with the ordering 1 2 ... N . We assume at this point that all the basic frequencies in the response are well separated. It turns out that even strongly nonlinear dynamical processes can be analyzed by CX-A, first introduced by Manevitch [30] (for an extensive discussion of this technique and numerous applications refer to Vakakis et al. [31]). In particular, for each component in (2) we assign a new complex variable defined by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) m j t m m m m m k k k k t x t j x t t e (3) where a slow/fast partition of the dynamics in terms of the ‘slow’ (complex) amplitude ( ) ( ) m k t and the ‘fast’ oscillation m j t e is assumed. Substituting (2) and (3) into (1) and performing multi-phase averaging [32] for each of the fast frequencies we obtain the slow flow of (1): 1 2 ( , , , ) N k k n F (4) where (1) (2) ( ) { , , , } , 1, , N T k k k k k n . The number of fast frequencies ( N) determines the dimensionality of the slow flow (4). As an example we consider a weakly damped linear oscillator (LO) coupled to a light attachment by means of essential stiffness nonlinearity of the third degree. In previous work the nonlinear oscillator was termed a ‘nonlinear energy sink’ (NES) due to its capacity to passively absorb and dissipate energy from the LO over broad frequency ranges [31]: 2 3 3 0 1 2 2 ( ) ( ) 0, ( ) ( ) 0 y y y y v C y v v v y C v y (5) where y and v are the displacements of the LO and NES, respectively; 0 is the linearized natural frequency of the LO; the mass ratio of the NES to the LO; C the essentially nonlinear stiffness coefficient; and 1,2 the damping coefficients. We assign 0 1, 1, 0.05 C , and 1,2 0.03 and the initial conditions (0) (0) 0 y v and (0) 0.059443193, (0) 0.014995493 y v . Then, a 1:3 transient resonance capture (TRC [33]) takes place during which the NES engages in transient resonance with the LO, and strong energy exchanges between the two oscillators occur [34]. Figure 1 depicts the responses of the two oscillators in time and frequency (wavelet transform spectra). There exist two dominant fast frequencies in the dynamics, at 2 0 (high-frequency - HF) and at 1 0 /3 (low-frequency - LF), respectively. Given that there are only two fast frequencies in the transient responses we express the responses as (1) (2) (1) (2) 1 2 ( ) ( ) ( ) (), () () ( ) ( ) x t y t y t y t x t v t v t v t , and the slow/fast partitions as, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) m m j t j t m m m m m m m m m m t y t j y t t e t v t j v t t e (6) where 1,2 m . When substituted in (5) and averaged with respect to 1 and 2, the complexification (6) leads to the slow-flow equations 1 1 1 2 2 2 1 2 ( , ), ( , ) F F (7) where (1) (2) 1 1 1 { , } T and (1) (2) 2 2 2 { , } T . The details of the functions 1F and 2F can be found in Kerschen et al. [34]. In the left of Fig. 2 we present the slow flow approximation of the LO and NES responses, demonstrating that the slow flow accurately approximates the strongly nonlinear transient response. 515
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