mechanical systems and signal...

Mechanical Systems and Signal Processing 114 (2019) 628–643

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Mechanical Systems and Signal Processing

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Characterization of a 3DOF aeroelastic system with freeplay andaerodynamic nonlinearities – Part II: Hilbert–Huang transform

https://doi.org/10.1016/j.ymssp.2018.04.0390888-3270/Crown Copyright � 2018 Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (M. Candon), [email protected] (R. Carrese), [email protected] (H. Ogaw

[email protected] (P. Marzocca), [email protected] (C. Mouser), [email protected] (O. Levinski).

Michael Candon a,⇑, Robert Carrese a, Hideaki Ogawa a, Pier Marzocca a, Carl Mouser b,Oleg Levinski b

a School of Engineering (Aerospace and Aviation), RMIT University, Melbourne, Victoria, AustraliabDefence Science and Technology Group, Fishermans Bend, Victoria, Australia

a r t i c l e i n f o

Article history:Received 18 October 2017Received in revised form 12 April 2018Accepted 22 April 2018

Keywords:Nonlinear aeroelasticitySystem identificationHilbert–Huang TransformStructural freeplayTransonic flowNonstationary responseTime–frequency analysis

a b s t r a c t

The Hilbert–Huang Transform is used to analyze the nonlinear aeroelastic response of a 2D3DOF aeroelastic airfoil system with control surface freeplay under transonic flow condi-tions. Both static and dynamic aerodynamic conditions, i:e:, for accelerating freestreamspeed, are considered using a linearized aerodynamic model. The main aim of this paperis to provide an in-depth physical understanding of the observed transition between peri-odic and aperiodic behavior, and the presence of a stable periodic region well below thedomain characterized by stable limit cycles. Physical insights towards the forward andbackward abrupt transition between aperiodic/chaotic and periodic behavior types appearto be the result of an internal resonance (IR) phenomenon between linear modes followedby a lock-in between linear and nonlinear modes. More specifically, initially a 2:1 IRbetween linear modes leads to a shift in the frequency composition and dynamic behaviorof the system. A secondary effect of the IR can be observed immediately after the exactpoint of 2:1 IR such that a nonlinear mode locks into a subharmonic of the linear modewhich in-turn drives a finite stable periodic region.

Crown Copyright � 2018 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Structural and aerodynamic nonlinearities in transonic aeroelastic systems can introduce a range of nonlinear phenom-ena which cause the behavior of the aeroelastic system to vary significantly from that of a linear system. More specifically,the phenomena observed may include bifurcations, chaotic/quasi-period response, limit cycle oscillation (LCO) and reducedflutter instability boundary. All of these phenomena induce cyclic loading on the airframe which can lead to fatigue andhence reduce the operational lifetime of aircraft. Furthermore, in extreme cases these phenomena (LCO in particular) cancause fatigue or catastrophic failure. Airframe vibration can also interfere with the with avionic systems. This can be prob-lematic for smart ordnance and scientific payloads which may contain onboard guidance systems and/or advanced imaging/sensory technology. Finally, intense vibration can lead to poor handling qualities and difficulty for the pilot in reading flightinstrumentation, which is more pertinent to defense based fighter aircraft in atypical maneuvers.

Expanding on the nonlinear phenomenamentioned above, internal resonance (IR) is one of which has been studied exten-sively for nonlinear mechanical systems in general, however, has seen limited attention for nonlinear aeroelastic systems,

a), pier.

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Nomenclature

a pitching displacementb control surface displacementxa pitch natural frequency (coupled)xb control surface natural frequency (coupled)xh plunge natural frequency (coupled)xNL nonlinear modeX pole location in the kernel function of the Hilbert transformcnðtÞ intrinsic modes of yðtÞ for n ¼ 1;2; . . . ;nf frequencyF Fourier transform operatorF�1 inverse Fourier transform operatorh plunging displacement�hðtÞ impulse response functionHðtÞ frequency response function of �hH hilbert transform operatorM1 freestream Mach numberp1, p2, pt pre-flutter periodic regions, t indicates a transition region between regions one and twoPV Cauchy principalrnðtÞ residual signalR2 correlation coefficientt timeV� velocity index_V� first derivative of velocity indexyðtÞ any sample time-seriesYðxÞ Fourier transform of yðtÞ

M. Candon et al. /Mechanical Systems and Signal Processing 114 (2019) 628–643 629

i:e:, when aerodynamic and structural forces are coupled. IR refers to the exchange of energy between two of the systemslinear modes due to the presence of nonlinearity. The amount and mechanism of energy exchange depends upon the typeof nonlinearity and the harmonic relationship between the natural linear modes [1,2]. Gilliatt et al. [3] describe that for IR toexist the natural frequencies of the systemmust be commensurable and that the nonlinearities in the systemmust provide asource of coupling. The notion of commensurability is described as m1x1 þm2x2 þ � � � þmnxn � 0, wheremn are positive ornegative integers and xn are the natural frequencies. Although commensurability is a condition for IR, it does not guaranteeit. Other important factors may include; geometry, types of nonlinearities and boundary conditions. The authors [3] provideone of the only studies which investigates the IR phenomenon in an aeroelastic setting. The study considers a 2DOF aeroe-lastic system with a cubic aerodynamic nonlinearity. The results indicate a large amplitude response as the system passesthrough the 3:1 internal resonance region, which was not predicted by the linear system. No abnormal excitation wasobserved at the 2:1 ratio. Further detail with regard to the IR phenomenon is omitted here for brevity, however, for an excel-lent review of the literature and detailed explanation of the IR phenomenon the readers are refereed to Ref. [3]. It is inter-esting to note that Nayfeh and Balachandran [4] suggest that 2:1 IR may be more likely to occur in the presence of quadraticnonlinearities and 3:1 IR in the presence of cubic nonlinearities. Furthermore it is suggested that if the higher frequencymode is already excited by a principal resonance its amplitude, irrespective of the excitation, should remain constant. How-ever, the IR phenomenon causes an excitation of the lower mode which increases in response amplitude as the amplitude ofthe excitation increases. Bifurcations which induce amplitude and phase modulated motions can be displayed.

While the nonlinear phenomena mentioned above can be detected by an aircraft’s sensory systems (e:g:, strain gauges oraccelerometers), the type of nonlinearity (e:g:, aerodynamic or structural) inducing these phenomena, the location where thenonlinearity is acting, and the associated nonlinear parameters (e:g:, freeplay margin) often remain unknown. Furthermore,aircraft are not typically equipped with sensors which have the capability of providing detail on airframe loading. Hence, thedamage being inflicted on the airframe due to nonlinear behavior is also unknown, without grounding the aircraft and per-forming dedicated inspections. Considering the extent of unknown and unpredictable detail surrounding nonlinear aeroelas-tic phenomena, extensive research has been conducted in the past half century with respect to (i) understanding, (ii)modeling and (iii) identifying nonlinear aeroelastic systems. As the present research surrounds nonlinear behavior as a resultof structural freeplay (i:e:, loosened mechanical linkages between the main wing and control surfaces, main wing and storelink system or within an all-movable horizontal tail), a selection of authoritative literature on the topic for typical sections isprovided in Refs. [5–9]. For a more detailed review of the relevant literature see Part I of this paper [10].

Empirical mode decomposition (EMD) is a modal decomposition technique developed by Huang et al. [11] and is the fun-damental algorithm which drives the Hilbert Huang Transform (HHT) – a system identification capability which provides atime–frequency representation of a nonstationary and nonlinear signal. The HHT can be applied to a broader range of signals

630 M. Candon et al. /Mechanical Systems and Signal Processing 114 (2019) 628–643

than the classical Hilbert transform (HT), as explained in detail in Section 2. The HHT has received widespread attentionacross a variety of disciplines including the analysis of scientific data, structural system identification and mechanical faultdetection [12]. The HHT has been applied recently in aeroelasticity due to the powerful system identification properties fornonstationary data. Brenner and Prazenica [12] and Huang et al. [13] applied the HHT algorithm to aeroelastic flight test datafor the F/A-18 to investigate the application of the methodology for online data processing. The algorithm is used to deter-mine correlations between system input and output, between multiple output sensors, for online stability analysis andmodal identification.

Part I of this paper [10] presents higher-order spectra (HOS) analysis of a 3DOF aeroelastic system with control surfacefreeplay (identical to the structural system used in this Part II). Linearized analysis demonstrates that the system goes intoclassical flutter via instability of the lowest frequency mode, i:e:, airfoil plunge xh. As speed increases, the system is seen totransition between aperiodic, quasi-periodic and limit cycle responses, hence, the analysis is conducted at a range of speedswhich represent each observed behaviors. The major findings include:

� At low speeds the response transitions between aperiodic and periodic responses, and is characterized by a complex set ofinteractions between the linear modes. Conjectures toward the nature of the transitioning periodicity are provided.

� At higher speeds the response is characterized by typical limit cycle behavior and nonlinear aerodynamics (Type-B shockmotion).

� The structural freeplay nonlinearity can be characterized by strong cubic nonlinearities between linear and nonlinearmodes, whereas the Type-B shock motion can be characterized by quadratic nonlinearities, the magnitude of which isproportional to the strength of the aerodynamic nonlinearity.

� The presence of the aerodynamic nonlinearity does not appear to impact the identification of the structural freeplay.

It is suggested that the transition between aperiodic/chaotic, quasi-periodic and periodic states at low-speeds may beattributed to a 2:1 IR phenomenon between the first two natural modes leaving a sole nonlinear mode. It is then conjecturedthat the sole nonlinear mode locks into a lower energy attractor (a subharmonic of the first linear mode), which drives thestable periodic region. A detailed assessment of this observation would require a nonstationary representation of the low-speed response region and time–frequency analysis to facilitate mode tracking – not possible using the linear (PSD) and non-linear (HOS) Fourier based approaches. In this Part II a nonstationary response is generated by utilizing a linearized aerody-namic solver, with dynamic (accelerating) flow conditions. The novel contribution here, comes from further investigation ofthe system dynamics using the HHT to conduct a time–frequency analysis of the nonlinear and nonstationary response,allowing for the modal contributions to be tracked as the system transitions between ordered and disordered aperiodic/chaotic, quasi-periodic and periodic states. Correlations between freestream speed, linear/nonlinear modal relations andthe periodicity of the system are made, essentially in strong support of the suggestions made in Part I. This provides newphysical insights toward the nature of the system dynamics – prior to LCO. Static flow conditions are also considered to

Fig. 1. Example of the sifting process for sinðtÞ þ cosð5tÞ.


demonstrate that the linear frequencies and nonlinear interactions identified via HOS are also identified via HHT, thus con-firming that the identified frequencies are physical and not a mathematical effect. Furthermore, it is shown that additionalfrequencies which are unable to be identified using HOS can be obtained using the HHT approach.

Fig. 3. First four IMFs of the control surface hinge acceleration response at V� ¼ 0:25.

Fig. 2. Bifurcation diagram for the control surface hinge – static aerodynamic loading at each discrete V� .


The remainder of the paper is organized as follows: Section 2 describes the HT, EMD and HHT algorithms, Section 3 pre-sents and discusses the results of the stationary and nonstationary analysis, and Section 4 provides concluding remarks.

It should be noted that in this Part II the computational model and numerical approaches are identical to those of Part Iand hence the readers should refer to [10] for further detail.

2. Hilbert–Huang transform framework

2.1. Hilbert-transform

The Hilbert transform (HT) is a mathematical tool which has significant application in the detection of nonlinearity. It isan integral transform of the same family as the Fourier transform, however, the kernel function is �1=ipðX�xÞ, rather thanthe exponential eixt . Defining the HT operator by H, its actions on functions can be given by

Fig. 4.approareader

HfYðxÞg ¼ ~YðxÞ ¼ �1ip

PVZ 1

�1dX

YðXÞX�x

ð1Þ

where PV denotes that Cauchy principal value of the integral and is required as the integrand is singular, i:e:; has a pole atx ¼ X. For the purpose of nonlinearity detection, the HT works on the following basic principle. Provided that the functionyðtÞ is causal, its Fourier transform YðxÞ should be invariant under the Hilbert transform. Further, the impulse response hðtÞof a linear function is causal and hence the Fourier transform of hðtÞ, the frequency response function (FRF) HðxÞ, should beinvariant in the HT space, i:e:

YðxÞ ¼ HfGðxÞg () F�1fGðxÞg ¼ yðtÞ ¼ 0;8t < 0 ð2Þ
For a nonlinear system this condition does not hold and hence by comparing the FRF HðxÞ and its HTHðxÞ one can deter-
mine whether or not the system whose impulse response is hðtÞ contains nonlinearities. Worden and Tomlinson [14] providefurther information on the HT and its applications. An example of the HT being applied to the detection of nonlinearity inaircraft structures is presented by Wu et al. [15].

First four HS of the control surface hinge acceleration response at V� ¼ 0:25 – black indicates frequencies identified using both HOS and HHTches, and magenta indicates frequencies identified only via the HHT approach. (For interpretation of the references to color in this figure legend, theis referred to the web version of this article.)


The HT expresses a signal as a harmonic with a time dependent frequency and amplitude. Considering this, the Hilberttransform is applicable to nonstationary data. Although powerful, the HT alone is only able to compute a single instanta-neous frequency at any point in time. Hence, when applied to a multi-spectra signal the HT takes a weighted average ofthe frequency components and fails to provide information as to the individual frequencies present. This limits its applica-tion to many fields of research, including nonlinear aeroelasticity where nonlinear signals are typically defined by the pres-ence of multiple modes and interactions between them. Further limitations of the Hilbert transform include signals withnon-zero mean and signals with more extrema than zero crossings yield distorted frequency estimates.

2.2. Empirical mode decomposition

To overcome the shortcomings of the standard HT, Huang et al. [11] developed the empirical mode decomposition withthe motivation of providing a framework by which the time–varying frequency components of multi-spectra, nonlinear andnonstationary signals could be assessed. EMD is a truly empirical method which decomposes a signal in the time domain intoa set of intrinsic mode functions (IMF) each of which contains only a single frequency component at any point in time andmust satisfy two fundamental criteria:

1. Each IMF must have zero mean.2. The number of zero crossings and extrema of each IMF must differ by no more than one.

A signal which possess these properties will admit a well behaved Hilbert transform. The coupling of EMD with the HTconstitutes the Hilbert–Huang Transform (HHT) method.

The sifting process to generate a single IMF is as follows:

1. The local maxima of a time series yðtÞ are connected via a cubic spline and similarly so are the local minima. The regionbetween the upper and lower splines is called the envelope.

2. The mean of the envelope is subtracted from yðtÞ and tested to determine whether it satisfies the IMF criteria.3. If the IMF does not satisfy the criteria, step 2 is repeated until a the criteria are satisfied and the first IMF c1ðtÞ is obtained.

This is known as the sifting process.

Fig. 5. First four IMFs of the control surface hinge acceleration response at V� ¼ 0:35.


4. The sifting process is repeated on the residual signal rnðtÞ ¼ yðtÞ � cnðtÞ to obtain the remainder of the IMFs cnðtÞ until onlythe residual signal rðtÞ remains which represents the trend.

To provide an example of the sifting process, consider the function

Fig. 6.approareader

yðtÞ ¼ sinðtÞ þ cosð5tÞ ð3Þ
Fig. 1 demonstrates that after sifting the initial function once IMF 1 is extracted as defined by c1ðtÞ ¼ cosð5tÞwith the first
residual being r1ðtÞ ¼ yðtÞ � c1ðtÞ. Following this the sifting process is repeated on r1ðtÞ to obtain IMF 2, which is defined byc2ðtÞ ¼ sinðtÞ. As the original function only contains two frequency components only two IMFs are extracted. If the siftingprocess were to be repeated only the final residual function r2ðtÞ ¼ 0 would remain.

After the IMFs have been identified, a time–frequency representation of each IMF is calculated using the HT. The HT’s ofeach of the individual IMFs are called the Hilbert spectra (HS) and it follows that the collection of weighted Hilbert spectraforms the Hilbert spectrum (HSM).

3. Results

In this section the steady state response at a range of discrete velocities (identical to those studied using linear and non-linear Fourier approaches) is analyzed via HHT to confirm that the linear frequencies and nonlinear interactions are consis-tent between the two approaches. The HHT method is then used to extract modal contributions from a nonstationaryrepresentation of the system (accelerating flow conditions), providing an in-depth analysis of the system dynamics as it tran-sitions between aperiodic/chaotic, quasi-periodic and periodic responses.

3.1. Discrete velocities

Fig. 2 presents a bifurcation diagram for the rotation of the control hinge DOF. Several characteristic behaviors areobserved, including:

First four HS of the control surface hinge acceleration response at V� ¼ 0:35 – black indicates frequencies identified using both HOS and HHTches, and magenta indicates frequencies identified only via the HHT approach. (For interpretation of the references to color in this figure legend, theis referred to the web version of this article.)


� Aperiodic response from V� ¼ 0:2� 0:26.� Switching between aperiodic and periodic response from V� ¼ 0:26� 0:3� At V� ¼ 0:3 the aperiodic system shifts to a multi-amplitude periodic response in which several well defined branches canbe observed before returning to aperiodicity at V� ¼ 0:39.

� Aperiodic response from V� ¼ 0:39� 0:57.� A subcritical bifurcation at V� ¼ 0:57, which can be considered as the nonlinear flutter boundary and is characterized byLCO of which the amplitude grows before flutter occurs at V�

f ¼ 0:66.

The discrete velocity index values considered here are identical to those studied using linear and nonlinear Fourierapproaches, which correspond to each of the observed changes in nonlinear behavior, i:e:, V� ¼ 0:25, 0.35, 0.45 and 0.6. Fur-thermore, in Part I [10] it is shown that the onset of periodicity is due to the presence of 2:1 IR (xh ¼ xa=2) at V

� ¼ 0:28,hence, the switching between periodic and aperiodic responses from V� ¼ 0:26� 0:3 can be explained by the values of xh

and xa=2 being in very close in this region.Figs. 3–10 present the IMFs and respective HS for the first four IMFs at each speed. This analysis is conducted for two rea-

sons; firstly to verify that the natural modes and nonlinear frequency interactions identified using HHT agree with thoseidentified using HOS approach; and secondly to identify linear modes and nonlinear frequency interactions which werenot able to be detected using the HOS approach. The R2 values assigned to each IMF indicate the correlation of the IMF withrespect to the original time series, thus this can be used to represent the weighting that each IMF carries.

At V� ¼ 0:25 (aperiodic response) Fig. 4 presents the HS of the first four IMFs at this speed. The HS of IMF 1 indicates thatthe two strongest frequencies are defined by xa and xb �xa. When using linear and nonlinear Fourier approaches [10] it isfound that xa is consistently involved in nonlinear interactions, however, cannot be identified on its own. This immediatelydemonstrates one benefit of using the powerful HHT approach. The fundamental low frequency interactions which were pre-viously identified [10], xNL1 ¼ xh þ 2xa �xb, xNL2 ¼ ðxb � 4xhÞ=2, xNL3 ¼ xb � 2xa and xNL4 ¼ xb � 4xh, are identifiedhere with a weak presence in the HS of IMF 1 and IMF 3. While discussing fundamental low frequency interactions, it is inter-esting to note that two more frequencies are identified in the HS of IMF 4, these are subharmonics of pre-identified inter-actions and are defined according to xNL8 ¼ xNL1=2 and xNL9 ¼ xNL2=2. Other strong interactions include 2xh �xNL2 and2xh �xNL3, which were also identified as being strong interactions using the Fourier approaches [10]. xa=2 and xb=2 are

Fig. 7. (a) First four IMFs of the control surface hinge acceleration response at V � ¼ 0:45.


identified with a moderate-strong prominence. Although xb on its own is not identified, it is encouraging to find the clearpresence of its subharmonic. Finally at this speed, it can be seen that in the HS of IMF 2xh is clearly dominant (as is the caseusing HOS [10]). Two additional nonlinear interactions which were not previously identified in Part I [10] are also found, i:e:,xh þxNL8 and xh �xNL9.

At V� ¼ 0:35 (periodic response), the identified linear and nonlinear modes agree well with those identified via HOS [10].Fig. 6 presents the HS of the first four IMFs at this speed. A single nonlinear mode (xh=3) is identified in the HS of IMF 3. InPart I a conjecture was made towards the presence of xh=3, that a nonlinear modes locks into the low energy attractor,xh=3). This is investigated in-depth in the following section. The HS of IMF 2 demonstrates the strong presence of xh andthat of IMF 1 indicates a range of interactions between xh (or superharmonics of xh) and xh=3.

At V� ¼ 0:45 (aperiodic response) it is encouraging to find thatxa andxb=2 have a clear and strong presence in the HS ofIMF1 Fig. 8, whereas the HOS approach [10] indicated that these frequencies are involved in interactions but not identifiedon their own. The HS of IMF 3 presents the clear identification of xNL5 ¼ ðxb �xa � 2xhÞ=2, xNL6 ¼ ðxb � 2xaÞ=2 andxNL7 ¼ xh=2 (consistent with the linear and nonlinear Fourier approaches [10]). In the HS of IMF 4 two additional fundamen-tal low frequency interactions are identified according to xNL10 ¼ xNL6=2 and xNL11 ¼ xNL7=2. On the other hand, the inter-actions 2xh þxNL5, 2xh �xNL5, 2xh þxNL6, 3xh þxNL5 and 3xh þxNL6 are identified using both HOS and HHT approaches.Finally, in the HS of IMF 2 the dominance ofxh is presented, along with the nonlinear interactionsxh þxNL10 andxh �xNL11

(these are only identified using the HHT method).At V� ¼ 0:6 (limit cycle) there are strong similarities to the results found using the linear and nonlinear Fourier

approaches [10]. Fig. 10 demonstrates the strong presence of 3xh (HS of IMF 1) and xh (HS of IMF 2). The HS of IMF3and IMF4 do not contain any additional information. This is to be expected as at V� ¼ 0:6 the system is in limit cycle andall modes are locked into xh. The identification of additional frequencies which were not identified previously [10] canbe seen in the HS of IMF 1, i:e:, the lower limit of the frequencies in the HS is identified by xb �xa �xh. Further, two mod-erate magnitude frequency bands can be observed atxb �xh andxb �xa=2. This is an important finding as it indicates thateven in limit cycle condition (where all modes are locked into the dynamics of xh) the HHT is able to identify a set of inter-actions, all of which includexb. This finding may suggest to the practitioner that the nonlinearity is contained in the controlsurface hinge, while rigorous studies would be required to confirm this. Again, the HOS approach was unable to extract suchinformation.

Fig. 8. First four HS of the control surface hinge acceleration response at V� ¼ 0:45 – black indicates frequencies identified using both HOS and HHTapproaches, and magenta indicates frequencies identified only via the HHT approach. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)


Table 1 summarizes the linear and low-frequency fundamental modes which have been identified using HOS and HHTapproaches.

At this point it is worth reporting a promising approach which can be used to define the freeplay margin. This is demon-strated by observing the amplitude of the IMFs at each speed (Figs. 3, 5, 7 and 9). It can be seen that pre-LCO (V� ¼ 0:25, 0.35,0.45) the second IMF is bound by b ¼ �0:5�, similarly post-LCO (V� ¼ 0:6) the first IMF is bound by b ¼ �0:5�. Noting that thefreeplay margin is b ¼ �0:5�, this finding would suggest that the EMD process exploits the two structural states of the sys-tem i:e:, control surface fixed and free, and the amplitude of the IMF which characterizes the free state correlates to the free-play margin. Although this is only shown here at four discrete speeds, this trend is found to be true for all V� values withinthe range (0:2 < V� < 0:65). This finding is currently being rigorously investigated as a part of a freeplay detection, sourceand margin estimation framework, which is to be presented elsewhere.

The present section supports the findings of Part I of this paper [10] while providing further detail as to the linear andnonlinear frequency content of the system. The segregation of the HS from IMF 1 – IMF 4 is a logical and simplistic mannerto display the frequencies and nonlinear interactions, which facilitates the identification process for the practitioner.

3.2. Accelerated flight condition

In this section an analysis of the system with accelerating freestream conditions ( _V� ¼ dV�dt ) is conducted to investigate the

behavior of the aeroelastic system in the pre-flutter periodic region. It can be observed for the range 0:28 < V� < 0:40 inFig. 11 – a control surface hinge rotation bifurcation diagram which displays the variation in b as a function of time and

velocity index, developed using the nonstationary analysis of _V�. This is in contrast to the bifurcation diagram which waspresented in the previous section (Fig. 2), developed by assessing the steady-state values at each discrete V� value. Thebehavior is essentially very similar, however, the pre-flutter periodic region is slightly larger, returning to an aperiodicresponse at V� ¼ 0:4 (compared to V� ¼ 0:39 in Part I). This discrepancy is intuitive if we consider the nonstationary natureof the flow conditions, i:e:, there is a lag in the systems response to the accelerating flow. Furthermore, the switchingbetween the aperiodic and periodic states (V� ¼ 0:26� 0:3), which is observed in Fig. 2 and can be explained by the valuesof xh and xa=2 being very close within this region, is not observed in Fig. 11 (nonstationary approach). Rather the response

Fig. 9. First four IMFs of the control surface hinge acceleration response at V � ¼ 0:6.


remains aperiodic from V� ¼ 0:2� 0:28, at which point we can observe an abrupt shift from aperiodic to periodic response,which coincides with the point at 2:1 IR is observed according to xh ¼ xa=2.

A low _V� ¼ 0:0014=s is used to facilitate the tracking of linear and nonlinear frequencies within the region of interest in aseemingly quasi-static manner. This being the case, it is found that the effects of the accelerating flow on the aperiodicregions (pre and post the periodic region of interest) are minimal, i:e:, the system responds similarly, as expected, to the sta-tic condition (Fig. 4 for V� ¼ 0:2� 0:28 and Fig. 8 for V� ¼ 0:40� 0:56). Thus, characterization of the aperiodic regions can beomitted in the present analysis. Furthermore, a steady-state response is obtained at V� ¼ 0:2 prior to engaging the acceler-ating aerodynamics, thus, the initial transient effects of the computational model are omitted.

Fig. 12 presents the first four IMFs for 0:28 < V� < 0:40 for the control surface hinge rotation. Interestingly, a behavioralchange is observed within the this region which is not clearly indicated via inspection of the bifurcation diagram Fig. 11. Itcan be seen that two separate stable regions exist p1 ð0:28 < V� < 0:368Þ and p2 ð0:381 < V� < 0:40Þ, with a region between

Fig. 10. (a) First four HS of the control surface hinge acceleration response at V� ¼ 0:6 – black indicates frequencies identified using both HOS and HHTapproaches, and magenta indicates frequencies identified only via the HHT approach. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

Table 1Natural modes and fundamental low-frequency nonlinear modes at each steady-state speed.

Speed Linear Modes NL Interactions (HOS, HHT) NL Interactions (HHT Only)

0.25 xh 9.77 Hz xNL1 ¼ xh þ 2xa �xb ¼ 2:78 Hz xNL8 ¼ xNL1=2 ¼ 1:39 Hzxa 20.08 Hz xNL2 ¼ ðxb � 4xhÞ=2 ¼ 4:04 Hz xNL9 ¼ xNL2=2 ¼ 2:02 Hzxb 47.15 Hz xNL3 ¼ xb � 2xa ¼ 6:99 Hz

xNL4 ¼ xb � 4xh ¼ 8:07 Hz

0.35 xh 10.24 Hz xh=3 ¼ 3:42 Hzxa 19.9 Hzxb 47.20 Hz

0.45 xh 10.74 Hz xNL5 ¼ ðxb �xa � 2xhÞ=2 ¼ 3:04 Hzxa 19.68 Hz xNL6 ¼ ðxb � 2xaÞ=2 ¼ 3:94 Hz xNL10 ¼ xNL6=2 ¼ 1:97 Hzxb 47.24 Hz xNL7 ¼ xh=2 ¼ 5:37 Hz xNL11 ¼ xNL7=2 ¼ 2:69 Hz

Fig. 11. Bifurcation diagram for the control surface hinge – dynamic aerodynamic loading.

Fig. 12. First four IMFs for 0:28 < V� < 0:40.


p1 and p2, pt ð0:368 < V� < 0:381Þ, where the dynamic behavior shifts between that of p1 and p2 in an irregular manner.Figs. 13–16 present the HS of IMF 1 – IMF 4 for 0:28 < V� < 0:40.

The 2:1 IR according to xh ¼ xa=2 at V� ¼ 0:28 results in xa locking into xh and the tri-modal nonlinear system becom-ing bi-modal causing a shift to a higher state of order. Considering the relationship xh ¼ xa=2, the fundamental low fre-quency nonlinear interactions which are identified pre-p1 (Table 1) can be re-written at the onset of p1 according to;xNL1 ¼ 5xh �xb; xNL2 ¼ ðxb � 4xhÞ=2 and; xNL3 ¼ xNL4 ¼ xb � 4xh. Hence, only one fundamental low-frequency nonlin-ear mode (and its harmonics) needs to be considered in the characterization of all three regions (p1, p2 and pt), defined by

xNLP ¼ xb � 4xh ð4Þ
The HS of IMF 1 and IMF 2, displayed in Figs. 13 and 14 respectively, indicate the clear contrast in frequency content as the
system transitions between p1 and p2. It can be seen that p1 contains a vast range of frequency components which are char-acterized by interactions between high and low superharmonics of xh and harmonics of xNLP . On the other hand, within p2

the highest frequency components are characterized by high superharmonics of xNLP and the only interactions which con-tain xh can be seen in IMF 2 at 2xh �xNLP and xh �xNLP=4. Clear frequency bands are difficult to extract within pt .

Fig. 13. HS of IMF 1 for 0:28 < V� < 0:40.

Fig. 14. HS of IMF 2 for 0:28 < V� < 0:40.


The harmonics (8, 5, 3, 2, 1/4, 1/8)xNLP are identified and interact with xh and its harmonics to formulate all frequencieswhich are identified in the system, although not shown herexNLP/4 andxNLP/8 are identified in isolation for p1 and p2 in IMFs5, 6 and 7. It is interesting to note thatxNLP=2 is not present within the range of identified nonlinear interactions, which sug-gests that it is locking into the lower energy attractor,xh=3, as previously speculated. To investigate this further, Fig. 17 pre-sents regions p1, p2 and pt from the HS of IMFs 3 and 4 with both xNLP=2 and xh=3 plotted. It can be seen that xh=3 isconsistently present within the region and although xNLP=2 is not present, its values are in close proximity to xh=3.xNLP=2 is initially greater than xh=3, the trajectories of the two frequencies then converge until a point of intersection(xh=3 ¼ ðxb � 4xhÞ=2) at V� ¼ 0:374, which is the mid-point of the transition region pt , beyond which the two frequenciesdiverge until V� ¼ 0:4 where the periodic system returns to chaos. This sheds light on the presence of p1, p2 and pt , and thecontrasting frequency content within these regions. These findings strongly support the suggestion that the presence ofxh=3and the fundamental nature of the periodicity within the region 0:28 < V� < 0:40 can be explained by ðxb � 4xhÞ=2 lockinginto the lower energy attractor xh=3, i:e:, i) xNLP=2 is not identified within the interactions but its trajectory lies in closeproximity to that of xh=3, and ii) clear correlations can be drawn between the system dynamics (p1, p2 and pt) and the rela-tionship between the trajectories of ðxb � 4xhÞ=2 and xh=3.

Fig. 15. HS of IMF 3 for 0:28 < V� < 0:40.

Fig. 16. HS of IMF 4 for 0:28 < V� < 0:40.


Interestingly, xNLP can only be identified when considering the nonstationary response, i:e:, if the steady-state response(obtained by applying an impulse to the system at t ¼ 0) is analyzed at any V� within the periodic region (0:28 < V� < 0:4)the HS are characterized byxh and its harmonics only, similar to Fig. 6. To investigate this further the steady-state responsecan be obtained using a nonstationary approach as opposed to applying an impulse to the system at t ¼ 0. To do so, accel-erating flow conditions are considered from 0:2 < V� < 0:35, at which point V� ¼ 0:35 is fixed and the response is allowed torecover to a steady-state. It is found that the frequency content from the steady-state response, obtained from nonstationaryinitial conditions, is to identical to that presented in Fig. 6. Although not shown here, this is consistent for all discrete veloc-ities within the range 0:28 < V� < 0:4. This finding suggests that the presence of xNLP is characteristic of transient phenom-ena only, however, when allowed to recover to a steady-state the preferential attractor xh absorbs the more energetic andcomplex nonlinear mode xNLP (and its harmonics).

4. Concluding remarks

This paper characterizes a 3DOF 2D airfoil system of NACA0012 profile with freeplay in the control surface. Using the HHTapproach, physical insights towards the abrupt transition between periodic and aperiodic states as the nonstationary system(accelerating freestream speed) transitions between periodic and aperiodic states. As this phenomenon is observed at low-

Fig. 17. HS of IMFs 3 and 4 combined for 0:28 < V� < 0:40.


speeds (i:e:, well below limit cycle), the aerodynamic system is shown to be linear and hence linearized solutions to the gen-eralized aerodynamic forces are considered.

Part I of this study demonstrates how higher-order spectra alone can be used as a global nonlinearity diagnosis tool byidentifying the nonlinear form (quadratic, cubic or quadratic-cubic) associated with (i) isolated freeplay (cubic only) and (ii)freeplay combined with nonlinear aerodynamics in the from of large scale shock motion (quadratic-cubic).

Prior to LCO and diverging flutter, the system is found to transition between aperiodic/chaotic and periodic responses. Theaperiodic/chaotic response is driven by the freeplay nonlinearity only and is characterized by a set of highly complex mod-ular frequency interactions.

Providing physical insights toward this phenomenon (i:e:, the transition between aperiodic/chaotic and periodic states) isthe subject of this paper. To investigate this phenomenon the HHT approach is used to analyze a nonstationary response(accelerating flow conditions) within the pre-flutter periodic region. This allows for the linear modes and nonlinear interac-tions to be tracked as the speed increases.

It is shown that the onset of periodicity is related to the systems modality, i:e:, if two prominent modes become harmon-ically related (in this case due to a 2:1 IR phenomenon) the number of physical modes acting decreases and hence the systemshifts to a higher state of order. Although this intuitive finding provides a sound explanation for the onset of periodicity, itdoes not provide insight into the ongoing periodic behavior. To account for the ongoing periodic behavior, it should be con-sidered that once the system has shifted to a periodic behavior type it is preferential for it to remain so, as opposed to shiftingback to an aperiodic/chaotic and more energetic state. With this in mind, an explanation for the continuation of periodicity inthe current work is twofold; (i) the system becomes fundamentally characterized by a single nonlinear mode and its inter-action with the driving linear mode, and (ii) a lock-in phenomenon is observed where a subharmonic of the nonlinear modelocks into that of a linear mode as it is a preferential lower energy attractor. The system remains periodic/quasi-periodicwhile the linear mode remains in a lower energy state that the nonlinear mode. However, it is shown that shortly afterthe energy states switch, i:e:, the linear mode is no longer in a lower energy state the system reverts to disorder.

To better understand this nonlinear mechanism and its general applicability to nonlinear aeroelastic systems with free-play, the findings in this work need to be rigorously investigated and expanded to more complex aeroelastic systems (e:g:, 3Dwings with control surfaces and full aircraft models).

For stationary data the HOS and HHT approaches are complimentary tools which provide detail as to the type and spatialsource of the nonlinearity. Further, the approaches indicate which modes are activated and to what extent. For nonstationarydata analysis the HHT approach is an extremely powerful tool in providing insight into the variation in modal contributionsas system properties change. This paper, in conjuction with the Part I paper [10], provides sufficient findings to suggest thatthe application of the HOS and HHT approaches to real-life flight test data could be beneficial in diagnosing nonlinearity, andmore specifically freeplay nonlinearities appearing in control surfaces. With further development these approaches have thepotential to be applied within health and usage monitoring (HUMS) for aircraft, which is to be investigated in future researchutilizing three-dimensional wing models and flight test data.

Acknowledgments

The authors are grateful for the financial support provided by the Defence Science Institute (DSI – Australia) for: High-Fidelity Modelling of Wing Flutter and Nonlinear Aeroelastic Predictions. WBS: RE-02290. ResearchMaster Code: 0200313955.


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