Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

Journal of Sound and Vibration 333 (2014) 2054–2072

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Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification Mehmet Kurt a,n, Melih Eriten b, D. Michael McFarland c, Lawrence A. Bergman c, Alexander F. Vakakis a a

Department of Mechanical Science and Engineering, University of Illinois at Urbana – Champaign, Urbana, IL 61801, USA Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA c Department of Aerospace Engineering, University of Illinois at Urbana – Champaign, Urbana, IL 61801, USA b

a r t i c l e in f o

abstract

Article history: Received 13 June 2012 Received in revised form 8 November 2013 Accepted 8 November 2013 Handling Editor: W. Lacarbonara Available online 23 December 2013

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumedmodes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end. & 2013 Elsevier Ltd. All rights reserved.

Abbreviations: DOF, degree-of-freedom; EMD, empirical mode decomposition; FEP, frequency–energy plot; FT, Fourier transform; WT, wavelet transform; IMF, intrinsic mode function; IMO, intrinsic modal oscillator; NIM, nonlinear interaction model; NSI, nonlinear system identification; POD, Proper Orthogonal Decomposition; NNM, nonlinear normal mode; ROM, reduced-order model; RGM, reduced Guyan model n Corresponding author. E-mail address: [email protected] (M. Kurt). 0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.11.021

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1. Introduction System identification, modal analysis and reduced order modeling of linear dynamical systems have been well-studied (e.g., [1]). However, in analyzing time series obtained from experimental measurements, one is faced with a variety of problems, such as insufficient data sets, nonstationarity and nonlinearity in the dynamics. For instance, strongly nonlinear dynamics due to effects such as local buckling, plastic deformations, clearance and backlash, hysteresis, friction-induced oscillations, and vibro-impact motions cannot be accurately modeled and analyzed by linear techniques such as experimental modal analysis. Because of these issues, Fourier spectral analysis, which assumes linearity and stationarity in the data, can exhibit spurious harmonic components that can lead to a misleading energy–frequency distribution, even if the signal contains pure harmonic components. This highlights the need for developing system identification and reducedorder modeling techniques capable of characterizing broad classes of dynamical systems with nonlinear and nonstationary effects. Although techniques for nonlinear system identification (NSI) and reduced order modeling (ROM) have been presented in the literature, in general their range of applicability is limited to special classes of nonlinear systems, and they are broadly applicable. Reviews of NSI and ROM methods are well-presented in works such as Kerschen et al. [2,3]. We note that typical nonparametric NSI methods such as Proper Orthogonal Decomposition (POD) [4], smooth orthogonal decomposition [5], or methods based on neuro-fuzzy models and wavelet networks [6–8], and the use of splines [9] are only applicable to specific classes of dynamical systems and usually are based on a priori assumptions regarding the nonlinearities to be identified. Recently, a new NSI technique with promise of broad applicability was presented. It employs empirical mode decomposition (EMD) [10] under the assumption that the measured time series can be decomposed in terms of a finite number of harmonic components in the form of ‘fast’ nearly monochromatic oscillations that are modulated by ‘slow’ varying amplitudes [11,12]. Hence, the method is based on slow-fast partition of the measured dynamics and on the correspondence between analytical and empirical (i.e., derived from EMD of the measured time series) slow-flow models [13]. This yields local nonlinear interaction models (NIMs) [14] consisting of sets of intrinsic modal oscillators (IMOs) that can reproduce the measured time series over different time scales and can account for nonlinear modal interactions in the measured dynamics. Construction of the set of IMOs constitutes the local aspect of the NSI methodology since they model specific nonlinear transitions (time series) of the dynamics. The global aspect of the methodology is based on identifying the dynamics in the combined energy–frequency domain, thus determining possible co-existing attractors and bifurcations as the frequency and energy of the dynamics vary. Up to now this NSI methodology has been applied to the study of targeted energy transfers in dynamical systems [14], of aeroelastic instabilities in rigid wing models [15], of elastic continua with essentially nonlinear end attachments [16] or systems undergoing vibro-impacts [17], and in friction identification in jointed structures [18]. Other NSI techniques with the promise of broad applicability include a similar time-domain NSI approach using conjugate-pair decomposition (CPD) along with EMD [19], or frequency-domain subspace identification [20]. At this point, it might be useful to provide some definitions concerning certain of the strongly nonlinear dynamical phenomena that will be addressed in this work. Starting from the definition of internal resonance, this denotes a strongly nonlinear energy transfer phenomenon, whereby two structural modes (even widely spaced in the frequency domain) become coupled by the system nonlinearity and start exchanging energy between them giving rise to nonlinear beat phenomena [21]. In linear systems this type of nonlinear beats can occur only when two modes possess closely spaced frequencies, whereas in nonlinear systems this is not necessary. Due to such internal resonances mode-mixing or nonlinear modal interactions occur, whereby with varying energy a certain structural mode starts interacting with another structural mode in the form of a beat phenomenon, until the former mode is completely ‘transformed’ to the latter. It follows that when nonlinear mode mixing occurs, the dynamical response is composed of two (or more) structural modes coupled by the nonlinearity; depending on the frequencies of the ‘mixed modes’ the overall response can be time-periodic (for rational frequency ratios between modes), quasi-periodic (for irrational frequency ratios), or of other type (e.g., non-periodic) [22]. We note that the conventional wisdom in the theory of nonlinear dynamics is that an internal resonance between modes of a dynamical system can occur only when their corresponding linearized eigenfrequencies are related by rational relations. As we discuss in this work (and as shown previously by Kerschen et al. [23]) this does not need to be the case in strongly nonlinear systems that possess nonlinear normal modes [24] with strong frequency–energy dependencies; in such systems (and also in the system discussed in this work) internal resonances can occur at higher energy ranges where rational relations between the frequencies of nonlinear normal modes can be realized, even if their corresponding linearized eigenfrequencies are not necessarily rationally related. Then, the resulting nonlinear dynamical phenomena depend strongly on the energy level of the dynamics. In this work, we consider nonlinear system identification of a cantilever beam with an essentially (nonlinearizable) nonlinear stiffness at its free end. Our aim is two-fold: first, we aim to show that this system can exhibit an (unexpected) nonlinear beat phenomenon resulting from internal resonances between beam modes that are coupled through the strong nonlinearity; second we will develop a nonlinear system identification methodology capable of identifying the parameters of the system and the participating modes for which nonlinear beats occur. In general, our motivation is to study and identify the strongly nonlinear dynamics that arise in flexible structures with local stiffness nonlinearities. Unexpected nonlinear beat phenomena involving low-frequency modes are observed in this system caused by strongly nonlinear modal interactions induced by the local essential stiffness nonlinearity. We provide an explanation of these nonlinear phenomena and then proceed to extend the NSI methodology reviewed in [11,12] in order to account for this type of complex multi-scale dynamics.

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We will do this by considering the nonlinear normal modes (NNMs) of the system in the frequency–energy domain. NNMs were defined as vibrations-in-unison of dynamical systems starting from the seminal works of Rosenberg [25,26], and were proposed in [23,27] as a way to extend experimental modal analysis to the nonlinear regime. In the next section, we review the basic elements of the nonlinear system identification and reduced order modeling methodology considered in this work. In Section 3 we perform an NSI study of the nonlinear dynamics of the cantilever beam considering the linearized, weakly nonlinear and strongly nonlinear response regimes. The source of the strongly nonlinear beat phenomena in the dynamics of this system is investigated using a reduced order model based on Guyan reduction in Section 4, showing the generation of multi-scale nonlinear modal interactions due to the presence of a local, strong stiffness nonlinearity. Finally, in the last section we extend the NSI technique to account for nonlinear beats in the measured time series, and then apply it to the NSI of the low-frequency nonlinear dynamics of the cantilever beam. 2. Basic elements of the nonlinear system identification (NSI) methodology 2.1. Slow flow dynamics [11–14] _ ¼ f ðX; tÞ; The slow-flow model of the dynamics of an n-degree-of-freedom (DOF) dynamical system (e.g., X 2n _ X ¼ fx; xg A ℝ ; t A ℝ) is constructed by assuming that the dynamics possesses N distinct components at frequencies ω1 ; …; ωN , respectively, and expressing the response of each DOF as a summation of independent components ðtÞ; xk ðtÞ ¼ xkð1Þ ðtÞ þ⋯ þxðNÞ k

k ¼ 1; …; n

(1)

ðtÞ xðmÞ k

denotes the response of the kth coordinate of (1) associated with the basic frequency ωm with the ordering where ω1 4 ⋯ 4 ωN . We will denote these as fast frequencies of the problem. For each component of (1) we assign a new complex variable ðtÞ ¼ x_ ðmÞ ðtÞ þ jωm xðmÞ ðtÞ ¼ φðmÞ ðtÞ ψ ðmÞ k k k k |fflfflffl{zfflfflffl}

ejωm t ; |ffl{zffl}

m ¼ 1; …; N

(2)

SlowFlow FastFlow

ðtÞ and the ‘fast’ with j ¼ ð  1Þ1=2 , where a slow/fast partition of the dynamics in terms of the ‘slow’ (complex) amplitude φðmÞ k oscillation ejωm t is assumed. That such a partition holds is a central assumption in our methodology. Obviously, such a slow/ fast partition is by no means unique or universal; yet at this point we restrict our analysis to measured signals that can be decomposed as in (1) and (2). Substituting (1) and (2) into the equations of motion and performing N averaging operations with respect to each of the fast frequencies yields a set of n complex differential equations that provides the slow flow of the system φ_ k ¼ F k ðφ1 ; …; φn Þ;

φk ¼ fφð1Þ ; …; φðNÞ g A CN ; k k

k ¼ 1; …; n

(3)

The slow-flow (3) captures the slow modulations of the N harmonic components of each coordinate of the response. The number of fast frequencies, N, determines the dimensionality of the slow-flow. 2.2. Empirical mode decomposition (EMD) [13,14] The second element of the proposed methodology is empirical mode decomposition (EMD) combined with the Hilbert Transform, which has been used previously for analyzing nonstationary and nonlinear time series and for nonlinear system identification and damage detection of structures [28–30]. This decomposition identifies the characteristic time scales of a measured oscillatory time series, is adaptive, highly efficient, and especially suitable for nonlinear and nonstationary processes. In particular, EMD decomposes a measured time series xk ðtÞ (following the previous notation, this is the response of the kth coordinate of the considered dynamical system) into a nearly orthogonal basis of oscillatory intrinsic mode functions (IMFs). These are nearly mono-component oscillatory modes embedded in a measured time series xk ðtÞ, each with its own characteristic time scale (or fast frequency ωm ), whose linear superposition reconstructs the measured time series. Lee et al. [15] applied the EMD method for reduced-order modeling of aeroelastic systems, whereas Tsakirtzis et al. [16] applied it to nonlinear system identification of a linear continuum (rod) with a local essential stiffness nonlinearity. The EMD algorithm (denoted as the sifting algorithm) consists of the following steps: (i) Identify all extrema of xk ðtÞ. (ii) Interpolate minima and maxima of xk ðtÞ by spline approximations, forming two envelopes emin ðtÞ and emax ðtÞ, respectively. (iii) Compute the average curve RðtÞ ¼ ðemin ðtÞ þ emax ðtÞÞ=2. (iv) Extract the remainder cðtÞ ¼ xk ðtÞ –RðtÞ. (v) Apply steps (i)–(iv) repetitively until the residual RðtÞ becomes smaller than a prescribed tolerance tol. Once this criterion (through the sifting process) is met, the remainder cðtÞ  ckð1Þ ðtÞ is regarded as the first, or highestfrequency, IMF of the measured time series. By subtracting this IMF from the original time series and applying the algorithm

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ðlÞ iteratively; that is to say, substituting xk ðtÞ  ∑m l ¼ 1 ck ðtÞ instead of xk ðtÞ in (i) above for each step m, we extract additional IMFs, so that the original signal xk ðtÞ is decomposed sequentially from high to low frequency into components as K

xk ðtÞ ¼ ∑ cðlÞ ðtÞ þ Rðlk þ 1Þ ðtÞ; k l¼1

Rðlk þ 1Þ ðtÞ o tol

(4a)

Hence, the original time series can be reconstructed by superposition of the K leading IMFs; however, only a subset of these IMFs is physically meaningful. A comparison of wavelet transforms of the original time series and the computed nearly mono-component IMFs is necessary to identify the dominant (i.e., physically meaningful) IMFs. Given that EMD is applied to a time series obtained from a physical coordinate of our dynamical system in Section 2.1, the decomposition in terms of dominant IMFs finally yields xk ðtÞ ¼ cð1Þ ðtÞ þ cð2Þ ðtÞ þ⋯ þ cðNÞ ðtÞ k k k

(4b)

cðmÞ ðtÞ k

where is the mth dominant nearly mono-component IMF of the response xk ðtÞ associated with the fast frequency ωm . Given that EMD is applied in an ad hoc manner, it has certain deficiencies, related to issues such as uniqueness of the EMD results, and lack of orthogonality between the computed IMFs [12]. These issues can be addressed through the use of masking and mirror-image signals [13,14] that lead to well-decomposed, nearly monochromatic and orthogonal sets of IMFs. The resulting “step-by-step” EMD method is summarized in Fig. 1. In short, instead of applying the EMD method all at once, we start by extracting the first IMF out of the given time signal, xk ðtÞ and treat each reminder as our new time signal. This can be summarized as follows: Rð1Þ ðtÞ ¼ xk ðtÞ  cð1Þ ðtÞ k k ⋮

(5)

 1Þ ðtÞ ¼ Rðm ðtÞ cðmÞ ðtÞ; RðmÞ ðtÞ otol RðmÞ k k k k

a final issue concerns the lack of a theoretical foundation for the derived near-orthogonal basis of the IMFs, so the next element of the proposed NSI methodology needs to address this by providing a link between the theoretical slow flow discussed in Section 2.1 and the numerically derived IMFs discussed herein. 2.3. Correspondence between theoretical and empirical slow flows [14] To formulate the correspondence between slow flows and IMFs derived by EMD we shift the analysis to the complex plane. For the slow flow this was performed through relation (2) for the response xk of the kth coordinate of the dynamical system. To perform a similar complexification of the dominant IMFs in (4b) we employ the Hilbert transform, complexifying the mth IMF cðmÞ ðtÞ of the measured signal xk ðtÞ through the relation k ðtÞ  cðmÞ ðtÞ þ jH½cðmÞ ðtÞ ψ^ ðmÞ k k k

(6)

where H½ U  represents the Hilbert transform. This leads to estimates of the instantaneous amplitude and phase of the mth IMF 2 ðmÞ ðtÞ þ H½cðmÞ ðtÞ2 g1=2 ; A^ k ðtÞ ¼ fcðmÞ k k

ðmÞ tan θ^ k ðtÞ ¼ H½cðmÞ ðtÞ=cðmÞ ðtÞ k k

from which the instantaneous frequency of the IMF is computed as of the complexified IMF (5) ðmÞ ^ ψ^ ðmÞ ðtÞ  A^ k ðtÞejθk k

ðmÞ

ðtÞ

_ ðmÞ ^ ðmÞ ω ðtÞ ¼ θ^ k ðtÞ. k

ðmÞ ðmÞ ^ ¼ A^ k ðtÞej½θk ðtÞ  ωk t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðmÞ

0 Slow0

component

(7)

This leads to the slow-fast representation ðmÞ

ejωk t |fflffl{zfflffl}

0 Fast0

(8)

component

we note that this is in the same form as the complexification (2) introduced in the slow flow construction in Section 2.1. This provides a direct way to relate the IMFs to the slow flow dynamics and physically interpret the dominant IMFs in terms of the slow flow dynamics. In summary, we relate the theoretical slow flow decomposition of Section 2.1 and the extracted IMFs by EMD of the response of the kth coordinate by the two expansions Slow flow: xk ðtÞ ¼ xkð1Þ ðtÞ þ⋯ þxðNÞ ðtÞ; k

ψ ðmÞ ðtÞ ¼ x_ ðmÞ ðtÞ þ jωm xðmÞ ðtÞ  k k k

φðmÞ ðtÞ k |fflfflffl{zfflfflffl} 0 Slow

0

component

Fig. 1. Step-by-step description of the EMD method.

0 Fast0

ejωm t |ffl{zffl} component

(9a)

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EMD: ðtÞ; xk ðtÞ ¼ ckð1Þ ðtÞ þ ⋯ þcðNÞ k

ðmÞ ðmÞ ^ ψ^ ðmÞ ðtÞ  cðmÞ ðtÞ þjH½cðmÞ ðtÞ ¼ A^ k ðtÞej½θk ðtÞ  ωk t k k k |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðmÞ

0 Slow0

ðmÞ

ejωk t |fflffl{zfflffl}

0 Fast0

component

(9b)

component

 ωm for given that (9a) and (9b) represent identical expansions of the same measured time series it holds that ωðmÞ k m ¼ 1; …; N, where N is the number of dominant harmonic components in the slow flow decomposition of the dynamics (i.e., the dimensionality, or the number, of significant frequency–time scales in the dynamics). This implies the correspondence between the theory and measurement [14] xðmÞ ðtÞ - cðmÞ ðtÞ ; |fflfflkffl{zfflfflffl} |fflfflkffl{zfflfflffl} Theory

k ¼ 1; …; n; m ¼ 1; …; N

(10)

Measurement

Noting that the complexification (6) is different than the corresponding complexification (2) for the slow flow analysis in Section 2.1, we introduce the alternative complexification ðmÞ ðmÞ ψ^^ k ðtÞ ¼ c_ ðmÞ ðtÞ þjωm cðmÞ ðtÞ  φ^^ k ðtÞ k k |fflfflffl{zfflfflffl} 0 Slow0

part

ðmÞ

^^ ðmÞ ejωm t ¼ A^ k ðtÞej½θk ðtÞωm t |ffl{zffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 0

0 Fast

0 Slow0

part

part

ejωm t |ffl{zffl}

0 Fast0

(11)

part

^^ ðmÞ ðmÞ ðmÞ ^ ðmÞ ðtÞ ¼ ðjωm Þ  1 ψ^^ k ðtÞ. We conclude that the where ψ^^ k ðtÞ ¼ A^ k ðtÞejθk ðtÞ . In terms of the complexification (8) it holds that ψ^ ðmÞ k correspondence between the theoretical and measured complexifications can be shown to be ðmÞ ðmÞ ^ ðtÞ-ψ^^ k ðtÞ ¼ jωm ψ^ ðmÞ ðtÞ ) φðmÞ ðtÞ-jωm A^ k ðtÞej½θk ðtÞ  ωm t ψ ðmÞ k k k |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðmÞ

(12)

Equivalence of 0 slow0 complex amplitudes

This final relation provides a physics-based theoretical foundation for EMD, whereby the dominant IMFs represent the underlying slow flow of the dynamics and, hence, capture all the important (multi-scale) dynamics. This correspondence between analytical and measured slow flows will be the foundation of the reduced-order modeling of the local dynamics. The standard EMD often cannot separate distinct frequency components, especially when these components are closely spaced to each other (that is, when their ratios are between 0.5 and 2). However, the use of masking signals [31], and heterodyning [32] can greatly enhance EMD's capacity to separate distinct frequency components. 2.4. Reduced order models: intrinsic modal oscillators – IMOs [11,12,14] Based on the previous formulation and results one can construct local reduced-order models that capture the transient dynamics of the dynamical system. To this end, intrinsic modal oscillators (IMOs) were introduced in the form of forced linear oscillators that reproduce a measured time series over different time scales. Since many structural dynamics problems involve slowly-modulated oscillatory responses, a linear, damped oscillator with a forcing function containing all modal interactions and nonlinearity is an obvious candidate for reproducing the mth dominant IMF in the form ðmÞ x€ k ðtÞ þ 2λðmÞ ωm x_ ðmÞ ðtÞ þ ω2m xðmÞ ðtÞ ¼ F ðmÞ ðtÞ; k k k k

k ¼ 1; …; n; m ¼ 1; …; N

(13)

we note that this model is based on the central assumption that the fast frequencies of the measured time series are wellseparated. The time-dependent forcing term can be written in terms of slow/fast partitions of all dominant fast frequencies in the general form F ðmÞ ðtÞ ¼ RefΛð1Þ ðtÞejω1 t þ Λkð2Þ ðtÞejω2 t þ ⋯ þ ΛðmÞ ðtÞejωm t þ ⋯ þ ΛðNÞ ðtÞejωN t g k k k k

(14)

ΛðmÞ k

are slowly varying complex modulations of the corresponding fast oscillations ejωm t . where the forcing amplitudes However, we note that since the IMOs constitute a set of linear oscillators with well-separated natural frequencies, only forcing terms with fast frequencies identical to the natural frequencies of the IMOs can cause significant responses of the IMOs. It follows that the IMO equations simplify to ðmÞ x€ k ðtÞ þ2λðmÞ ωm x_ ðmÞ ðtÞ þω2m xðmÞ ðtÞ ¼ F ðmÞ ðtÞ  RefΛðmÞ ðtÞejωm t g; k k k k k

k ¼ 1; …; n; m ¼ 1; …; N

(15)

using the complexification introduced in (2), we express ðtÞ ¼ xðmÞ k

φðmÞ ðtÞejωm t  φðmÞ ðtÞe  jωm t k k 2jωm

and apply averaging with respect to the fast frequency ωm to compute the slowly varying complex forcing amplitude ΛðmÞ ðtÞ  2½φ_ ðmÞ ðtÞ þ λðmÞ ωm φðmÞ ðtÞ k k k k

(16)

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Finally, we use the equivalence of slowly-varying analytical and experimental slow modulations (12), to express the forcing amplitudes in terms of quantities that can be directly identified by post-processing the measured time series   d ðmÞ 2 ðmÞ ðmÞ j½θðmÞ ðtÞωm t j½θðmÞ ðtÞ  ωm t ðjω ðtÞ  2 A ðtÞe Þ þjλ ω A ðtÞe ΛðmÞ (17) m m k k dt in this relation the instantaneous amplitude and phase can be computed directly from the measured data using the correspondence (12) as AðmÞ ðtÞ ¼ jψ^ ðmÞ ðtÞj and θðmÞ ðtÞ ¼ ∠ψ^ ðmÞ ðtÞ. The damping ratio of the IMO is the only parameter that needs to be evaluated by minimizing the difference between the measured and reproduced time series based on (15). Examples and applications of the outlined NSI methodology were given in [11–18], together with a discussion of an additional component of the method involving global identification of the dynamics in the frequency–energy domain. Starting from the next section we consider the main problem addressed in this work; namely, the study and identification of strongly nonlinear beat phenomena in a linear flexible beam with a strong nonlinearity at its free end, and discuss how the previous NSI methodology can be extended to account for this type of strongly nonlinear dynamics. 3. Beat phenomena caused by a local nonlinear stiffness element 3.1. System configuration We now consider the system depicted in Fig. 2a, composed of a uniform, homogeneous cantilever beam made of steel with length 1.311 m, width 0.0446 m, and thickness 0.008 m. The system includes a strongly nonlinear (nonlinearizable) spring with pure cubic force–displacement law attached at its tip to ground, which provides the local nonlinearity. The system is forced at the 0.4 m location with the impulsive force depicted in Fig. 2b; this force was computed from actual measurements. The transient dynamics of this system is discretized by an assumed-modes approach with 10 modes and the partial differential equations are solved using a Runge–Kutta fourth- and fifth-order method with the ode45 command in MATLAB. The aim of these numerical simulations is two-fold. First, to investigate the strongly nonlinear effects in the global dynamics of this system caused by the local strong nonlinearity. Indeed, due to the nonlinearizable (essential) nature of the local stiffness nonlinearity we expect it to affect beam modes in arbitrary frequency ranges [31]. In particular, we wish to show that the strong nonlinearity can generate nonlinear beat phenomena involving widely spaced frequencies by nonlinearly coupling beam modes. Second, to extend the previous NSI methodology to account for these types of transient, strongly nonlinear energy exchanges between widely spaced modes. In a series of simulations the dynamics of the system of Fig. 2 is studied for two different values of the nonlinear cubic stiffness (specifically, knl ¼ 108 N=m3 , and knl ¼ 1010 N=m3 ), and for the case when the nonlinear spring is detached (i.e., the

Fig. 2. The system under consideration: (a) configuration and (b) applied impulsive force.

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Nonlinear effects

Nonlinear effects

Fig. 3. Acceleration of the tip of the beam: (a) linear case (no nonlinear spring attached), (b) case of weaker nonlinearity, knl ¼ 108 N=m3 , and (c) case of stronger nonlinearity, knl ¼ 1010 N=m3 .

linear beam). The results of the simulations are depicted in Fig. 3. For each simulated case we depict the transient acceleration of the tip of the beam, together with its Morlet wavelet transform and Fourier spectrum. The wavelet transform provides information regarding the temporal evolutions of the dominant frequency components of the measured signals, and, hence, it can be regarded as a ‘dynamic analog’ of the classical Fourier transform that provides averaged information regarding the freuqency content of a signal under the assumption of stationarity and linearity. The wavelet transform (WT) is especially useful for post-processing the nonstationary and nonlinear time series considered in this work. From the examination of the linear (Fig. 3a) and nonlinear (Fig. 3b and c) we can readily deduce the effects of the strong nonlinearity on the dynamics. As evidenced by the WT plots and the Fourier spectra, the local nonlinearity considerably affects the frequency content of the response. Moreover, whereas the weak nonlinear stiffness case mainly affects the lower structural modes, the stronger nonlinear stiffness seems to affect the entire frequency range of the response. 3.2. Empirical mode decomposition In this section, we post-process the previous nonlinear time series of the tip response of the beam by EMD. Although the EMD of the linear response of Fig. 3a is somewhat straightforward, this is not the case for the nonlinear time series of Fig. 3b and c, for

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which the analysis will pose some distinct challenges. First, due to the realization of nonlinear internal resonances between distinct nonlinear normal modes of the beam, we will encounter difficulities in extracting nearly monofrequency IMFs. Second, we need to determine a certain frequency value around which to extract a narrowband IMF, which is not generally trivial for the strongly nonlinear case. We now proceed with the results of the empirical mode decomposition of the acceleration time series of the linear (cf. Fig. 3a – no nonlinear spring attached) and strongly nonlinear (cf. Fig. 3c – case of the nonlinear spring with stiffness constant knl ¼ 1010 N=m3 ) beams discussed in Section 3.1. Although as many as 10 IMFs were extracted for both systems, in Fig. 4, we only present the 1st and 6th IMFs for both linear and strongly nonlinear systems, since those IMFs are good representatives of linear and strongly nonlinear behaviors, respectively. For each IMF we also depict its corresponding wavelet spectrum depicting the temporal variation of its frequency content. We note that, whereas in the linear case the harmonic components of the IMFs are expected to be at fixed frequencies (identical to the damped natural frequencies of the linear beam), this is not so for the strongly nonlinear case, where temporal variation of the harmonic content of the IMFs is expected (especially for the lowest frequency IMFs). From the results of EMD and the comparison between the linear and nonlinear IMFs, we note that the nonlinear effects are more pronounced in the 6th IMF; i.e., the nonlinear effects seem to predominantly affect this beam mode. In addition, in the nonlinear case for the 6th IMF, we observe beating phenomena which dominate the entire IMF response. This indicates a continuous exchange of energy between nonlinearly interacting modes for the entire duration of the dynamical response. This type of nonlinear beating is generated by strongly nonlinear modal interactions induced in the dynamics by the local strong stiffness nonlinearity. In Fig. 5 we show the transient response reconstruction of both linear and nonlinear time series using the superposition of all the IMFs extracted for both systems. As discussed in Section 2, the IMFs do not form an orthogonal basis of functions (although they are nearly orthogonal); however, their superposition reconstructs the measured time series. Therein lies the usefulness of EMD, as it provides a multi-scale decomposition of the oscillatory time series in terms of embedded oscillatory components (the IMFs) at the dominant time scales of the dynamics.

Fig. 4. The 1st and 6th intrinsic mode functions (IMFs) and their wavelet transforms: (a) linear system and (b) strongly nonlinear system.

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Fig. 5. Original reconstructed time series, the latter obtained by summing intrinsic mode functions (IMFs) and the responses of intrinsic modal oscillators, respectively, (IMOs) for (a) the linear system and (b) the strongly nonlinear system; ———— original time series, sum of IMF responses, and – – – sum of IMO responses.

3.3. Intrinsic modal oscillators (IMOs) Based on the EMD results of Section 3.2, we construct sets of intrinsic modal oscillators – IMOs that reproduce the nonlinear transient dynamics (as described in Section 2). In Fig. 5, it is demonstrated that the original time series can be reconstructed by summing the IMFs identified in the previous section, and by summing the responses of the corresponding intrinsic modal oscillators (IMOs – see Section 2.4) that will be computed below. As discussed in Section 2.4, the forcing amplitudes of the IMOs contain essential information about the slow-flow dynamics of the system and the nonlinear interactions that occur between the nonlinear modes of the beam. In linear modal analysis the structural response is composed of nearly decoupled modes, which are the corresponding IMOs in our methodology. Hence, in a digression we consider a linear system with a forcing for the mth IMO (mode) obtained from a physical response linked to the kth degree of freedom given by F ðmÞ δðtÞ. Then the corresponding IMO (mode) has the k simple solution ðtÞ ¼ cðmÞ k

ðmÞ F ðmÞ k e  ζk ωm ωd;m

t

sin ωd;m t

(18)

1=2 is the damped frequency of the mth dominant frequency (natural frequency) and ζ ðmÞ the where ωd;m ¼ ωm ½1 ζ ðmÞ2 k k damping ratio associated with the mth IMO of the kth degree of freedom. Using the equivalence between the analytical and measured slow-flows (see discussion in Section 2), and substituting this solution into (2) yields the corresponding slow complex amplitude φðmÞ ðtÞ which, in turn, provides the forcing amplitude for the mth IMF of the kth degree of freedom k n ðmÞ ðtÞ ¼ 2F ðmÞ e  tðj þ ζk Þωm ð  2ζ ðmÞ þ λðmÞ Þωm cos ωd;m t þ ΛðmÞ k k k k o ζ ðmÞ λðmÞ Þω2m =ωd;m Þ sin ωd;m t (19) ½ ωd;m þð1 þ ζ kðmÞ 2 þjλðmÞ k k k as shown in Eriten et al. [18], the natural logarithm of the forcing amplitude (19) yields an expression of the form ðtÞj  CðλðmÞ ; ωm ; ζ ðmÞ ; F ðmÞ Þ  ζ ðmÞ ωm t ln jΛðmÞ k k k k k λðmÞ k

CðλðmÞ ; ωm ; ζ ðmÞ ; F ðmÞ Þ k k k

(20)

is the damping ratio for the corresponding IMO equation and is a coefficient that is constant where with respect to time. Hence, when the proposed NSI technique is applied to a linear structure and an IMO is forced properly to reproduce the measured modal coordinate response (IMF), the corresponding forcing amplitude jΛðmÞ ðtÞj is expected to k scale linearly with the modal damping in a log-linear plot with a slope value equal to  ζ ðmÞ ω . m k In Table 1, we provide the expected slopes for the modes of the linear beam under consideration (i.e., with nonlinear spring detached), along with the corresponding modal damping factors and natural frequencies (fast frequencies in EMD notation). These estimates are computed by analyzing the 10 IMFs (IMOs), some of which are shown in Fig. 4, which are extracted by EMD of the cantilever tip response. From these results and the previous discussion it should be clear that linearity in an identified IMO is demonstrated by a linear plot of the logarithm of the magnitude of its forcing function. Hence, any nonlinear deviation from linearity in this plot provides a clear indication of nonlinear effects in the measured IMOs. We show this by computing the logarithms ln jΛðmÞ ðtÞj for the IMFs of the linear and nonlinear beams. The corresponding results for the IMFs depicted in Fig. 4 are Tip depicted in Fig. 6, from which we deduce that not all cantilever modes are affected to the same extent by the strongly nonlinear local stiffness attached to the boundary of the beam. Specifically, and referring to the plot of Fig. 6a, we observe

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Table 1 Damping factors, fast frequencies and line slopes for each linear cantilever mode based on the beam tip response. Mode no. (m)

ζ ðmÞ Tip

ωm (rad/s)

ω (rad/s)  ζ ðmÞ Tip m

1 2 3 4 5 6 7 8 9 10

0.0071 0.0011 0.0012 0.0029 0.0068 0.0034 0.0074 0.0018 0.0023 0.0015

23 146 409 798 1319 1979 2758 3657 4719 5818

 0.17  0.16  0.49  2.3  9.0  6.7  20  6.6  11  8.7

Fig. 6. Plots of ln jΛðmÞ ðtÞj for IMOs corresponding to the (a) 1st IMF and (b) 6th IMF of the linear and strongly nonlinear cantilever beams; ———— Linear, Tip strongly nonlinear, and – . – . – predicted linear slope.

that for the first (highest-frequency) IMO the logarithms of the forcing amplitudes are on almost perfect match with the predicted slope lines for both the linear and nonlinear beams, indicating that this highest frequency mode is nearly unaffected by the local stiffness nonlinearity. However, considering the lower frequency IMO depicted in Fig. 6, we note a different trend. Focusing on Fig. 6b we observe a strong nonlinear effect in this IMO. Recalling the result depicted in Fig. 4, we observed that due to the high modal damping and the mode shape effect, the 6th IMF (corresponding to the 5th beam mode) exhibited strongly nonlinear effects at various time instants; that is, new nonlinear modes appeared near the fast frequency of this mode caused by mode mixing due to internal resonances. Similarly, in Fig. 6b, we observe that the logarithm of the forcing amplitude of the 6th IMO of the linear beam lines up well with the predicted slope line until 0.2 s after which noise effects introduce perturbations. For the strongly nonlinear beam, however, the corresponding logarithm of the forcing amplitude exhibits a completely different behavior and deviates significantly from the predicted linear trend. This is caused by the nonlinear effects generated by internal resonances, and shows that this particular mode is strongly affected by the local stiffness nonlinearity of the beam. Thus, by examining the temporal dependence of the magnitude of the forcing function of each IMO, one is able to detect the presence of nonlinear effects in its respective IMO. It follows that by employing the outlined methodology it is possible to identify which of the modes (IMOs) of the cantilever beam with the local stiffness nonlinearity at its end are significantly affected by nonlinear effects. We conclude from the analysis of these IMOs that whereas lower frequency modes are affected by the nonlinear effects, higher frequency modes are less affected by the nonlinearity appearing to be nearly linear. In the following section we construct a reduced-order model of the system of Fig. 2a in order to investigate (and explain) the origin of the nonlinear beat phenomena detected in the IMFs in Fig. 4. Based on this understanding, in a later section we will develop a method for system identification of these strongly nonlinear dynamics.

4. Analysis of nonlinear beat phenomena by reduced order modeling To analyze nonlinear beat phenomena in the system of Fig. 2a it is necessary to consider a reduced-order model of this flexible system. To this end we apply Guyan reduction, a static model reduction method [33]. Ignoring the damping effects

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for the moment, a linear discrete dynamical system with mass matrix ½m and stiffness matrix ½k can be partitioned in terms of driven ðfzb gÞ and driver ðfza gÞ coordinates as " #( )   za kaa kab Fa ½kfzg ¼  fFg (21) ¼ zb kba kbb 0 where we assume that degrees of freedom fza g are excited by the external forces fFa g, and we ignore forces (external and inertial) associated with the degrees of freedom fzb g. This yields the coordinate transformation ( ) " # za I ½kba fza g þ½kbb fzb g ¼ 0 ) fzb g ¼ ½kbb   1 ½kba  fza g ) fzg ¼ fza g  ½Tfza g (22) ¼ zb Tba |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} ½Tba  |fflfflffl{zfflfflffl} ½T

returning to the original linear dynamical system (with inertia, damping and all forcing terms included) we apply the static transformation (22) to reduce the system to a form that involves only the driver coordinates fza g ( ) Fa T T T T € _ ½T ½m½T fz g þ ½T ½c½T fz g þ ½T ½k½T fz g ¼ ½T (23) ¼ fFa g þ ½Tba fFb g Fb |fflfflfflfflfflffl{zfflfflfflfflfflffl} a |fflfflfflfflffl{zfflfflfflfflffl} a |fflfflfflfflfflffl{zfflfflfflfflfflffl} a |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ~ R ½M

½C~ R 

½K~ R 

fF~ R g

where fFb g represents the vector of forces applied to the driven coordinates, and fF~ R g the vector of forces applied to the reduced model. We now consider the nonlinear cantilever system of Fig. 2a. We apply Guyan reduction to the global mass and stiffness matrices assembled through finite element (FE) discretization of the cantilever beam. In the application considered in this section the FE model consists of two planar beam elements leading to a four-DOF full model (two translations and two rotations), which reduces to a two-DOF reduced Guyan model (RGM) involving only the two translations. The mass, damping and stiffness matrices of the RGM are given by       ~ R  ¼ 1:66 0:33 ; ½C~ R  ¼ 0:5329 0:1140 ; ½K ~ R  ¼ 17709  5534 ½M (24) 0:33 0:51 0:1140 0:1675 5534 2214 reproducing the two lowest modes of the cantilever beam with natural frequencies equaling 23 and 146 rad/s (3.7 Hz and 23.3 Hz). Moreover, in correspondence with the original beam we assume proportional damping in the RGM, selecting the damping coefficients to match those of the two lowest modes of the cantilever beam, as given in Table 1. In summary, the RGM of the system of Fig. 2 is given in the form   ~ R fza g ¼ fF~ R g ¼ fFnl g þ F e ðtÞ 0 T ~ R fz€ a g þ ½C~ R fz_ a g þ½K (25) ½M  T 3 T where fza g ¼ x1 x2 , fFnl g ¼ ½ 0  knl x2  represents the force due to the strong nonlinearity connected to the second DOF of the RGM (with coefficient knl ¼ 1010 N=m3 , representing the strongly nonlinear case studied in Section 3), andF e ðtÞthe external shock excitation (given in Fig. 2b) applied to the first DOF. In the previous notation x1 and x2 represent the amplitudes of the two translations that represent the two-DOF of the RGM. Next, we consider the nonlinear normal modes (NNMs) [24] of the RGM (25) by considering the Hamiltonian system resulting when we omit the damping and set the external excitation equal to zero. The nonlinear modes of the reduced system are depicted in a frequency–energy plot (FEP) as discussed in [34]. This plot depicts the periodic orbits of the RGM in the frequency–energy plane. Of special interest will be strongly nonlinear modal interactions generated by internal resonances since, as shown below, they are responsible for nonlinear beat phenomena of the type identified in the transient responses of the full flexible system. In Fig. 7a we depict the FEP of the RGM; this plot was computed using the numerical continuation code developed by Peeters et al. [27]. There are two branches of nonlinear normal modes for this system denoted as backbone branches. These branches are labeled S11 8 since they correspond to in-phase (þ sign) and out-of-phase (  sign) synchronous oscillations of the two masses of the RGM with both degrees of freedom oscillating with identical frequencies; i.e., they satisfy the condition of 1:1 internal resonance between the two degrees of freedom of the RGM. Due to the stiffening effect of the cubic nonlinearity, the frequency of the in-phase NNM converges to a certain asymptotic limit with increasing energy, as the cubic stiffness becomes approximately rigid and the system approaches a beam with pinned right end. The out-of-phase NNM ‘decouples’ from the in-phase mode during this limiting process, becoming a purely nonlinearity-governed NNM. Apart from the backbone branches, of particular interest for our discussion are the subharmonic tongues appearing as bifurcating branches out of the lower frequency in-phase backbone branch S11 þ , generated by 1 : n internal resonances between the in-phase and out-of-phase NNMs of the RGM. In Fig. 7b we depict the subharmonic tongues S31 and S51 corresponding to 3:1 and 5:1 internal resonances, respectively, between the two backbone modes. The reason that we find two of each of these subharmonic tongues (labeled as S311;2 and S511;2 in Fig. 7b) is that the ratio between the frequencies of the in-phase and out-of-phase backbone branches reaches 3 and 5, respectively, at two different energy levels. We mention that, if the ratio of the linearized frequencies of these two NNMs were smaller than 3, it would be impossible to obtain these tongues. The stability of the NNMs on the backbone curves and the subharmonic tongues was studied by means of Floquet theory [27], and the results are depicted in Fig. 7.

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S312

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S512

S311

S511

Fig. 7. FEP of the RGM with end stiffness nonlinearity: (a) backbone branches, (b) detail of the in-phase backbone branch indicating subharmonic tongues due to internal resonances; stable NNMs, and ———— unstable NNMs.

As discussed in [34] on subharmonic tongues there occurs mode conversion from the in-phase to the out-of-phase NNM, since the orbit of each subharmonic tongue corresponding to maximum energy correspond to a point of bifurcation with the out-of-phase NNM (S33  for the branches S311;2 , and S55  for the branches S511;2 ). Hence, on each subharmonic tongue mode mixing occurs and the corresponding periodic orbits are in the form of nonlinear beats [27]. It turns out that it is the excitation of such subharmonic tongues due to nonlinear internal resonance that gives rise to the nonlinear beat phenomena observed in the numerical simulations of Section 3 for the beam with the nonlinear stiffness at its end. To demonstrate the effect of these internal resonances on the transient response we now consider the full RGM (25) and perform numerical simulations of its dynamics for the transient force used in Section 3 and depicted in Fig. 2b. In Fig. 8 we depict the responses of the two degrees of freedom (DOFs) of the RGM, together with their wavelet spectra and Fourier transforms. In Fig. 8b we clearly observe that at t  9 s the response of the second DOF approaches the neighborhood of the S312 subharmonic tongue, as can be deduced by considering the frequency ratios of the two modes of the response in the wavelet transforms around this particular time instant. It is evident from the WTs that the dynamics in the second DOF is richer, due to the fact that the second mass of the RGM is directly attached to the strongly nonlinear end spring. Therefore, in the remaining part of this section, we will focus our analysis on the transient response x2 ðtÞ. In Fig. 9 we depict the WT spectrum of x2 ðtÞ superimposed onto the Hamiltonian FEP of Fig. 7. This WT spectrum was computed from the corresponding spectrum of Fig. 8b by replacing time with the instantaneous energy of the forced and damped system at every time instant. The comparison of the Hamiltonian FEP to the WT spectrum of the damped and forced response of the RGM is purely phenomenological; however, it shows how the underlying Hamiltonian dynamics (periodic orbits and nonlinear internal resonances) affect the response of the same RGM when damping and forcing are added. Indeed, at high energies (that is, in the initial highly energetic phase of the dynamics) the WT spectrum approximately follows the Hamiltonian S11  branch, indicating that initially the dynamics is dominated by 1:1 resonance between the

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Fig. 8. RGM (24): (a) response of the first reduced DOF x1 ðtÞ and (b) response of the second reduced DOF x2 ðtÞ; both in meters (wavelet transform spectra and Fourier transforms are also depicted).

S312

S511

S311

Nonlinear beats

Fig. 9. Wavelet transform spectrum of the second DOF of the RGM superimposed on the Hamiltonian FEP of Fig. 7; nonlinear beats in the neighborhood of subharmonic tongues S312 and S512 are evidenced by the corresponding broadband frequency components that emanate from these tongues.

two DOFs. However, complex modal interactions are found in the vicinity of the subharmonic tongues S312 and S512 ; these lead to complex nonlinear beat phenomena, as evidenced by multiple higher and lower harmonic components emanating from the neighborhood of these tongues. Hence, the study of the dynamics of the RGM reveals that internal resonances due to the strong local stiffness nonlinearity lead to nonlinear beat phenomena. We investigate the two dominant modes separately by carrying out empirical mode decomposition (EMD), through which we separate the in-phase and out-of-phase dynamics. Note that, although a computed IMF is expected to possess a narrow frequency bandwidth (see discussion in Section 2), it makes more sense to separate the damped physical modes as ‘single’ IMFs, rather than partitioning them into smaller narrow-band components. We extract the two IMFs of the response of the second DOF, x2 ðtÞ, corresponding to the out-of-phase and in-phase modes of the RGM, respectively. The realization of complex nonlinear modal interactions in these IMFs is evident. To study these modal interactions in more detail and highlight the resulting nonlinear beat phenomena, in Fig. 10 we depict the regions of nonlinear beats in the 1st and 2nd IMFs, and in the transient response x2 ðtÞ. In the time interval 9–15 s, we clearly observe the realization of strong nonlinear beats in both IMFs. This particular time interval was examined since this region corresponds to the S312 subharmonic tongue, as discussed above. Note that the beats tend to vanish as soon as the transient dynamics leaves the S312 region as

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Fig. 10. Details of (a) the 1st IMF, (b) the 2nd IMF, and (c) the transient response x2 ðtÞ in the region of nonlinear beats.

the total energy of the motion decreases due to damping dissipation. It is clear from the WT spectrum of Fig. 10c that in this time interval there exist strong nonlinear interactions (mode mixing) between the out-of-phase and in-phase modes, which cause the nonlinear beats observed in the corresponding IMFs. The results of this section based on the RGM demonstrated that the source of the nonlinear beats in the transient dynamics of the cantilever system of Fig. 2a is the nonlinear internal resonances realized due to the local strong stiffness nonlinearity, leading to complex modal interactions between cantilever modes. It follows that this type of nonlinear ‘mode coupling’ is rather generic in linear flexible systems with local strong stiffness nonlinearities, so complex nonlinear beat phenomena are expected to be present in the dynamics of this type of structures. We note that these nonlinear beating phenomena signal energy exchanges between different modes, which we attempted to model by using RGMs. This calls for the extension of the nonlinear system identification (NSI) methodology discussed in Section 2 to account for such nonlinear beat phenomena or nonlinear modal interactions. This is performed in the next section. 5. Slow flow modeling and NSI of resonance-induced beating phenomena 5.1. Preliminaries: analysis of beating signals We start the analysis and modeling of beating signals by considering a preliminary example. As discussed in Section 2, since the general ‘nearly monochromatic’ IMO (15) models a nearly monochromatic IMF (i.e., possessing a single dominant frequency), any analysis based on IMOs of the form (14) fails to be valid for a beating signal which contains at least two different closely spaced frequency components. The simplest model that can reproduce beating signals is the following linear damped oscillator with a harmonic forcing y€ þ 2ζωy_ þ ω2 y ¼ cos ½ωð1 þ εÞt  cos Ωt

(26)

where Ω ¼ ωð1 þ εÞ ¼ ω þεω  ωþ ωd . The response of oscillator (26) is expressed as the superposition of a homogeneous and particular solution, yh ðtÞ and yp ðtÞ, respectively 1 yðtÞ ¼ yh ðtÞ þ yp ðtÞ ¼ Ce  ζωt cos ðωd t þ ϕÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ðΩt ψÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 2 ðΩ ω Þ2 þ ð2ζωΩÞ2 yh ðtÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(27)

yp ðtÞ

which can be expressed as yðtÞ ¼ Ce  ζωt cos 2

 Ω  ω  ϕ þ β 

 β d t 2 cos Ω þ2 ωd t  ϕ  þ 2 2 3

1 6 7 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ce  ζωt 5 cos ðΩt βÞ 2 2 ðΩ  ω Þ2 þ ð2ζωΩÞ2

(28)

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where β ¼ tan  1 ð2ζωΩ=Ω2  ω2 Þ. Viewed in the context of the previous sections the response yðtÞ can be regarded as the only IMF of system (26). The amplitude Cand the phase difference ϕ are obtained by imposing the initial conditions of the problem, but this is not critical for our further analysis in this section. At this point, we apply a slow flow analysis by introducing the complexification y_ ðtÞ þ jωyðtÞ ¼ ψðtÞ ¼ φðtÞejωt ;

φðtÞ ¼ jωAðtÞejθðtÞ

(29)

note the different definition of the complex amplitude φðtÞ compared to the corresponding definition in Section 2 for the nearly monochromatic signal. Upon matching (28) and (29), we evaluate the slow envelope and the slow phase of the beating signal 2 3 



Ω  ωd ϕþβ 1 6 7 AðtÞ ¼ Ce  ζωt cos þ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Ce  ζωt 5 t 2 2 ðΩ2  ω2 Þ2 þð2ζωΩÞ2  

 

Ω ωd ϕ þβ 1  1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ Ce  ζωt cos (30) 2 2 2 2 ðΩ ω Þ2 þ ð2ζωΩÞ2 θðtÞ 

ϕ β 2

combining these results we evaluate the complex amplitude of the beating signal as 2 3  

 

Ω  ω ϕ þ β 1 6 7 ϕβ d φðtÞ ¼ jω4Ce  ζωt cos 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5ejð 2 Þ t 2 2 2 2 2 ðΩ  ω2 Þ þ ð2ζωΩÞ

(31)

(32)

As discussed in Section 2, based on relation (16) we can express the slowly varying complex forcing amplitude of the IMO modeling the IMF (28) as _ þ λφωðtÞ ΛðtÞ  2½φðtÞ

(33)

where λ is the damping factor of the IMO equation (following the notation of relation (15)). Neglecting the first term in (33) since it quickly decays to zero, the above complex amplitude can be approximated as 2 3  

 

6 Ω  ωd ϕþβ 6  ζωt 7 ΛðtÞ  λω6Ce  1 þ ½ðΩ2  ω2 Þ2 þð2ζωΩÞ2   1=2 5 t cos (34) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Particular solution component Homogeneous solution component

at this point we denote the ratio, RðtÞ, of the amplitudes of the homogeneous and particular components in (27) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RðtÞ ¼ Ce  ζωt ðΩ2  ω2 Þ2 þð2ζωΩÞ2

(35)

it turns out that when damping is weak this ratio is very slowly varying, so it can be regarded as being approximately constant, RðtÞ  R. Taking the natural logarithm of the expression (34) and accounting for the approximation (36), we obtain the relation  

 

       Ω  ωd ϕþβ ln ΛðtÞ  ln λω Ce  ζωt cos  1 þ RCe  ζωt  t (36) 2 2 at this point (and without loss of generality) we assume a zero phase difference for the sake of simplicity, ϕ ¼ 0. Finally, taking the derivative with respect of time of both sides of relation (36), we obtain   d  e  tζω ζω þe  tζω Rζω e  tζω Rζλω2 cos ðtεω=2Þ  ð1=2Þe  tζω Rζλω2 sin ðtεω=2Þ ln ΛðtÞ  dt e  tζω ð1  RÞ þ e  tζω Rλω cos ðtεω=2Þ

(37)

We will be mainly interested in two specific features of the approximate analytical expression (37). The first feature concerns the behavior of the slope of ln jΛðtÞj when t approaches the time instant t node corresponding to a nodal point of the beating signal     d ω½2ð1 RÞζ þRελω Rελω2 ln ΛðtÞ  ¼ ζω þ (38) limt-tnode dt 2ðR  1Þ 2ðR  1Þ note that expression (38) is dependent on the ratio R. Indeed, when R ¼ 1 the above limit reaches 1, whereas when R ¼ 0 (which means that there is no beating) the limit equals ζω which is in agreement with the previous analysis based on relation (20). In between these limiting values of R the derivative of the slope is dependent on R; ε; λ and ω. In general, we should expect very steep slopes between the aforementioned limiting cases since R will usually be close to unity for a beating signal.

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The second feature of interest in (37) is the behavior of the slope of ln jΛðtÞj at the maximum values of the beating signal. Assuming that at t ¼ t max a maximum of the beating phenomenon is reached, we obtain the limiting expression     d ζω þRζω  Rζλω2 ln ΛðtÞ ¼ ¼ ζω (39) limt-t max dt 1  R þ Rλω hence, in the vicinity of the maxima in the beating signal, the slope of ln jΛðtÞj approaches the value  ζω, which is identical to the corresponding slope of ln jΛðtÞj in the case of the nearly monochromatic signal examined in Section 3.3 [see expression (20)]. The limiting expressions (38) and (39) provide the necessary framework for extending the nonlinear system identification methodology outlined in Section 3 to the case of time series with beats. Before we proceed to formulate this extension, we provide a numerical example that highlights the previous findings. In the following example we consider the beating signal composed of two harmonics with closely spaced frequencies given by yðtÞ ¼ e  ζωt ½ cos ωt þ cos ωð1 þ εÞt with ω ¼ 100 rad=s; ζ ¼ 0:01 and ε ¼ 0:01. In Fig. 11 we plot ln jΛðtÞj derived from this signal, and the predicted line with slope ζω. We see that, in accordance with the previous analytical findings, the slope of ln jΛðtÞj tends to 1 at nodal points of the beat (since R ¼ 1 in this particular example). In addition, the slope of ln jΛðtÞj in the neighborhoods of the maxima of the beating approximates ζω, again in accordance to the previous analysis. 5.2. Nonlinear system identification of beating signals Here, the analysis of the previous section for the SDOF oscillator (26) will be applied to the IMFs computed in Section 3 for the strongly nonlinear case. It was observed in Section 3 that, due to strong stiffness nonlinearity at the end of the cantilever, the lower frequency modes (i.e., the higher order IMFs) exhibit nonlinear beating phenomena. As discussed in Section 4, these nonlinear beats are generated by internal nonlinear resonances which lead to nonlinear mode mixing and strongly nonlinear modal interactions. In Fig. 12 we consider the first four IMFs of the beam with strongly nonlinear stiffness employing the NSI methodology for nonlinear beats discussed in the previous section. As indicated, these IMFs correspond to the highest frequency ð2Þ oscillatory modes that are embedded in the measured time series. From Fig. 12a and b we deduce that IMFs cð1Þ Tip ðtÞ and cTip ðtÞ exhibit nearly linear behavior and do not appear to be significantly affected by the presence of the strong stiffness ð1Þ ð2Þ nonlinearity. This should be clear from the fact that the corresponding plots of ln jΛTip ðtÞj and ln jΛTip ðtÞj almost perfectly ð2Þ match the straight lines with slopes equaling  ζð1Þ ω and ζ ω , respectively, which is the prediction for linear behavior Tip 1 Tip 2 [as given in (20)]. Discrepancies from linear behavior are observed only when the amplitudes of the IMFs decrease below the noise level. Moreover, we note absence of nonlinear beats in these two IMFs. Considering now the 3rd IMF depicted in Fig. 12c, we note the occurrence of nonlinear beats, as evidenced by the ð3Þ deviation of the plot of ln jΛTip ðtÞj from the line with the slope  ζ ð3Þ ω . In addition, we observe that the line with slope Tip 3 ζ ð3Þ ω crosses the maxima of the plot of ln jΛð3Þ Tip 3 Tip ðtÞj corresponding to nonlinear beats between 0.2 and 0.4 s, as predicted by relation (39). The reason for drawing multiple lines with the predicted slope comes from the fact that there occur multiple nonlinear beats with different beating frequency; indeed, recalling the methodology derived in Section 5.1 we note that relation (39) was derived under the assumption that the nonlinear beats possess the same beating frequency. The occurrence of multiple beating phenomena will be further analyzed below when IMF cð8Þ Tip ðtÞ is considered. For the 4th IMF

Fig. 11. Transient variation of ln jΛðtÞj (logarithm of the modulus of the complex forcing of the IMO) of a beating signal, and its comparison to the line with slope  ζω; ————ln jΛðtÞj, predicted linear line, and -o- nodal points.

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Fig. 12. Application of the NSI technique to beating signals in the (a) 1st IMF, (b) 2nd IMF, (c) 3rd IMF, and (d) 4th IMF of the strongly nonlinear system; ———— cðmÞ ðtÞ, ln jΛðmÞ ðtÞj, and – . – . – predicted linear slope. Tip Tip

Fig. 13. Application of the NSI technique to beating signals in the (a) 5th IMF and (b) 6th IMF of the strongly nonlinear system; ———— cðmÞ ðtÞ, Tip . . ln jΛðmÞ Tip ðtÞj, and – – – predicted linear slope. ð4Þ cTip ðtÞ, we note that this mode has the highest modal damping (cf. Table 1) and the line with slope ζ ð4Þ ω crosses the Tip 4 maxima of the plot of ln jΛð4Þ ðtÞj only in the beginning of the time span. Tip In Fig. 13 we analyze the 5th and 6th IMFs of the strongly nonlinear system. From the plot of Fig. 13a the nonlinear beating in the early phase of the 5th IMF is evident, so relation (38) can be applied in this case. Indeed, we note that in accordance with the theoretical prediction the maxima of the plot of ln jΛð5Þ ðtÞj that correspond to the maxima of the Tip nonlinear beats, are approximately bounded by the predicted linear line with slope equal to  ζð5Þ Tip ω5 . Also, considering the 6th IMF depicted in Fig. 13b, it is clearly observed that this mode is dominated by nonlinear beats ð6Þ caused by the strong nonlinearity. The line with predicted slope  ζ Tip ω6 crosses the maxima of the nonlinear beats in the beginning of the time span; however, after approximately 0.1 s, this mode becomes dominated by nonlinear effects and

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Fig. 14. Application of the NSI technique to beating signals in the (a) 7th IMF, and (b) 8th IMF of the strongly nonlinear system (cf. Fig. 7b); ———— cðmÞ ðtÞ, Tip ln jΛðmÞ ðtÞj, and – . – . – predicted linear slope. Tip

traces of the linear mode disappear, which can be deduced by looking at the completely different damping behavior of that mode. In Fig. 14 we consider the beating signals in the 7th and 8th IMFs. For the 7th IMF (at 798 rad/s or 127 Hz) the line with ð7Þ slope ζ ð7Þ ω crosses the maxima of ln jΛTip ðtÞj as predicted by (38) until 0.6 s, where a different type of beating occurs. Tip 7 Moving on to the 8th IMF depicted in Fig. 14b, we noted previously that for the linear case this IMF has a fast frequency equaling 409 rad/s (or 65 Hz), whereas for the strongly nonlinear case that frequency is nearly 534 rad/s (or 85 Hz). Until around 0.5 s, strong nonlinearity governs this mode, whereas linear behavior is recovered afterwards. It is observed that the 8th IMF possesses multiple beating frequencies, so application of (39) requires the detection of the maxima of the beats ð8Þ corresponding to the same beating frequency. In Fig. 14b the maxima of ln jΛTip ðtÞj, denoted by horizontal ellipses, vertical ellipses and rectangles correspond to maxima of beats with identical beating frequencies. We note at this point that, since this represents a completely new mode that is generated by the strong stiffness nonlinearity (as it has no analog in the linear case – that is, the fast frequency of this mode is different from that of the corresponding linear mode), we draw separate parallel lines with the predicted slope ζ ð8Þ Tip ω8 , where the damping and frequency values no longer correspond to ð8Þ those of the linear mode. We observe that each of these lines crosses the maxima of the plot of ln jΛTip ðtÞj separately, as predicted in (39). 6. Conclusions In this paper, we study a linear cantilever beam with a strongly nonlinear grounding stiffness at its free boundary. The nonlinear stiffness has no linear component, so it can be characterized as essentially nonlinear (since it is nonlinearizable). Due to this strong local nonlinearity and the fact that no cantilever modes have nodes at the free end of the beam, we anticipate that all modes will be affected to a certain extent by the nonlinearity, although as this work shows, it is the lowerfrequency cantilever modes that are most affected by the strong stiffness nonlinearity. In addition, we observed strongly nonlinear beat phenomena, which upon further analysis we attributed to nonlinear modal interactions arising from internal resonances between cantilever modes. It is interesting to note that these internal resonances involve cantilever modes whose linearized natural frequencies are not necessarily close or even related by rational ratios; rather, conditions of internal resonances in the present system are realized due to the strong energy-dependencies of the frequencies of the participating cantilever modes as they are affected by the strong local stiffness nonlinearity. These essentially nonlinear beat phenomena generate ‘nonlinear mode mixing,’ through which a cantilever modes is ‘mixed’ with another mode, ultimately being transformed to the lateral mode as energy varies. This is in marked contrast to the linear case (i.e., in the system with either no spring or with a linear spring attached), where beat phenomena can only be generated by modal interactions of closely spaced modes. We note that this type of nonlinear mode mixing is generic in nonlinear systems, and its realization in the frequency–energy domain depends on the specific system configuration and the specific system parameters. We demonstrate the relation between nonlinear beat phenomena and internal resonances by constructing a reduced order model of the full finite-element dynamic model using Guyan reduction and studying the dynamics of the reducedorder system in the frequency–energy plane. By superimposing the wavelet transform spectra of damped transitions onto the Hamiltonian frequency–energy plot of the periodic orbits of the reduced system, we show that nonlinear beats are caused by excitation of certain subharmonic tongues that are generated by nonlinear internal resonances involving wellseparated beam modes. This provides an explanation of the nonlinear beat phenomena that are observed in the transient responses of the full system (beam with attached nonlinear stiffness). Based on our understanding of the nonlinear dynamics of this system, we develop a nonlinear system identification method capable of identifying strongly nonlinear modal interactions and their effects on the cantilever modes participating in these interactions. The system identification method is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, as explained in Section 2.2, which is extended to multi-frequency

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beating signals. In particular, our work extends an earlier nonlinear system identification approach developed for nearly monofrequency (monochromatic) signals, and applies it to the identification of the complex nonlinear dynamics of the cantilever beam with the strong end nonlinearity. Important features of the beating nonlinear dynamics are recovered using the proposed approach, including the nodal positions, the positions of the maxima, the modal damping factors and the beating frequencies for each participating modal response. The methodologies and results presented in this work pave the way for identifying unexpected strongly nonlinear dynamical phenomena (such as the nonlinear beats considered herein) and the properties of the modes participating in these phenomena in flexible structures with local stiffness nonlinearities. Acknowledgment This work was supported in part by the National Science Foundation through Grant 09-27995 ARRA. This support is gratefully acknowledged. 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