# Identification of physical nonlinearities of a hybrid aeroelastic–pressure balance

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## Abstract

This study has presented an improved method for determining physical nonlinearities of weakly nonlinear spring-suspension system and successfully applied to a novel hybrid aeroelastic–pressure balance (HAPB) system used in wind tunnel, which can be used for simultaneously obtaining the unsteady wind pressure and aeroelastic response of a test model. A nonlinear identification method of equivalent linearization approximation was firstly developed on the basis of the averaging method of Krylov–Bogoliubov to model the physical nonlinearity of a weakly nonlinear system. Subsequently, the nonlinear physical frequency and damping were identified using a modified Morlet wavelet transform method and a constant variant method. Using these methods, the physical nonlinear frequency and damping of the HAPB system with a vertical test model were determined and were validated by a time domain method and the *Newmark*-\(\beta \) method. Finally, the nonlinear mechanical frequency and damping of the HAPB system with inclined test models were determined in a similar way. This study has not only provided an identification method for determining physical nonlinearities of weakly nonlinear system, but presented the detail for developing a hybrid aeroelastic–pressure balance used in wind tunnel.

## Keywords

Physical nonlinearities Equivalent linearization approximation Modified Morlet wavelet transform Hybrid aeroelastic–pressure balance## 1 Introduction

Conventionally, a linear mechanical vibration model is utilized to determine the physical nonlinearities (nonlinear damping and stiffness) of a system from a free decay test in wind and offshore engineering [1, 2, 3, 4]. Using the values identified by the linear model, responses of various engineering objects are predicted. However, as pointed out by Staszewski [5], all physical and engineering systems inevitably exhibit in practice nonlinear behaviors which may arise from structural, geometric and material properties. A linear model applied to a nonlinear system would result in significant inaccuracies in response predictions of the system. Though the nonlinear frequency and damping of a weakly nonlinear system may vary slowly with oscillating amplitude, and the physical nonlinearities of the system are small in quantity, their presence would lead to different dynamic behaviors, such as the long-duration behaviors of energy dissipation and phase modulation due to the time-varying nonlinearities of the system. Since a nonlinear system can display complex phenomenon that a linear system cannot, the distinction between a linear and a nonlinear system should be well concerned.

The most common method to identify the physical nonlinearities of a system is the Hilbert transform (HT) method, which was developed in 1980s [6]. It utilizes the transit amplitude and frequency of the impulse response function to obtain the backbone curve and quantitative information about the nonlinear behavior of the system [5]. Due to its convenience, the method has received much attention and been widely utilized to many systems [7, 8, 9]. Despite its success, it has some limitations. For example, it is valid only for asymptotic signals and the ‘end effect’ or ‘Gibbs effect’ may be significant due to the incomplete data periodicity in the discrete HT [10, 11]. Combining the HT method and an empirical mode decomposition method, the Hilbert-Huang transform (HHT) method was proposed and has been utilized to analyze nonlinear and nonstationary time series [12]. The HHT method is effective in signal decomposition and time-frequency domain analysis, but it could not separate closely spaced multi-frequency signals into a set of mono-frequency components. The Wigner–Ville distribution [13], Gabor transform [14] and Wavelet transform [15, 16, 17] have also been developed and provided the most effective results for single-degree-of-freedom (SDOF) systems. However, they have been proved to be difficult for nonlinearities of multi-degree-of-freedom (MDOF) systems since the analysis requires bandpass filtration of the signal. To avoid the problems, a few attempts have been made to identify nonlinear parameters of MDOF systems, i.e., the improved HHT [18], the improved Wavelet transform [5, 15], and the harmonic balance method [19]. For weakly nonlinear systems, other methods have also been developed including the equivalent linearization method [20], the force-state mapping method [21], and the restoring force surface method [22], etc. These methods are complicated in use, and the accuracy is not good enough for weakly nonlinear systems.

A high-frequency base balance (HFBB) or synchronous multi-pressure sensing system (SMPSS) test technique is often utilized to obtain aerodynamic performances of bluff bodies, such as building and bridge models, and an aeroelastic test technique is frequently carried out for the evaluation of aeroelastic performances [23, 24]. However, both the HFBB and SMPSS techniques are static force measurements, which means that the techniques cannot consider the effect of structural motion that may have great effect on the evaluation of aerodynamic and aeroelastic performances of a structure. The aeroelastic test is used for obtaining aeroelastic response, but it cannot give aerodynamic forces on a test model simultaneously. To comprehensively study the effect of structural motion, both the aerodynamic force and aeroelastic response measurements of a structure are required. To the author’s best knowledge, very few studies have concentrated on the hybrid aeroelastic–pressure balance used in wind tunnel.

This study aims to (1) propose a method to identify the physical nonlinearity of weakly nonlinear systems; (2) to devise a novel hybrid aeroelastic–pressure balance (HAPB) test system and identify the physical nonlinearities of the system using the proposed method. In Sect. 2, the solution of a weak nonlinear system is derived from the averaging method of Krylov–Bogoliubov and the ELA method. Subsequently, a modified Morlet wavelet transform (MMWT) method is proposed and utilized for the identification of the nonlinearities of a weakly nonlinear system. A constant variant method is also employed to determine the nonlinear damping of the system. In Sect. 3, a HAPB system is devised and the physical damping and stiffness of the system were identified by a conventional linear model. The physical damping and stiffness identified by the linear model are constant, and the linear model has been proved to be not precise enough in the prediction of long-term free decay response due to the ignorance of the slow varying characteristics of the system. In Sect. 4, the physical nonlinearities of the HAPB system with a vertical test model are determined by using the methods developed in Sect. 2. The nonlinearities are validated by comparing the obtained nonlinearities with those obtained from a time domain method and are further verified by comparing the response calculated from the identified nonlinearities with the response directly observed from a free decay test. After verifying the nonlinearities of the HAPB system with a vertical test model, Sect. 5 presents the physical nonlinearities of the system with inclined test models. The present study makes sense in several aspects: (1) the analytical methods can be used for identifying physical nonlinearities of weakly nonlinear systems; (2) a HAPB system that can measure the aeroelastic and aerodynamic performance of structures was devised; (3) the identified nonlinearities of the HAPB system are potential to be employed to the dynamic analysis of a test system, such as the analysis of the wind-induced oscillation of bluff bodies (e.g., building and bridge pylon or deck models).

## 2 Nonlinear identification model

### 2.1 Equivalent linearization approximation

*u*and velocity \({\dot{u}}\). It is well known that the solution of Eq. (1), when \(\varepsilon = 0\) (linear problem), is \(u(t)=A\cos (\omega _0 t+\varphi )\) where

*A*and \(\varphi \) are constants. When \(\varepsilon \ne 0\), the solution of Eq. (1) can be determined by the averaging method of Krylov–Bogoliubov [15, 25], and is expressed as

*A*(

*t*) and \(\varphi (t)\) are the amplitude and phase modulation of the free decay response of a nonlinear vibrating system and they are time-dependent functions. Then, the first-order derivatives of Eq. (2) is expressed as

*D*(

*A*) which is in-phase with oscillation velocity and a restoring force coefficient

*S*(

*A*) which is in-phase with oscillation displacement, defined in Eqs. (22) and (23).

*t*and combining it with Eqs. (24) and (25), we have

From Eq. (33), the damping and frequency of a nonlinear system are approximated by a first-order approximation. Comparing Eq. (33) with the linear model of Eq. (1), the nonlinear damping and frequency of a weakly nonlinear system are amplitude-dependent and would be accurate to model the physical nonlinearities of a spring-suspension weakly nonlinear system.

### 2.2 System identification

To determine the slow varying amplitude-dependent damping and frequency in Eq. (33), a modified Morlet wavelet transform (MMWT) method and a constant variant method are proposed hereunder. These methods will be verified and applied to the identification of the nonlinearities of a weakly nonlinear system.

#### 2.2.1 Modified Morlet wavelet transform and wavelet entropy

*u*(

*t*) is given by a series of sampled values \(\{u(n)\}\), where \(n=1,2,\ldots N\). In the wavelet multi-resolution analysis of the time series \(\{u(n)\}\), the energy for each scale \(a_i \) is

*i*. The distribution of \(E_{pi} \) is considered as a time-scale density and the Shannon entropy [15, 31] is a useful tool to analyze and compare the distribution. Based on the Shannon entropy, the time-varying wavelet entropy is defined as

#### 2.2.2 The ridge and skeleton of the modified Morlet wavelet transform

*A*(

*t*) varies slowly compared to the variations of the phase \(\varphi (t)\). From the definition, the signal is expressed as \(u_a (t)=A(t)e^{i\varphi (t)}\) and the time-varying angular frequency is \(\omega (t) = {\dot{\varphi }}(t)\). Then, the continuous wavelet transform of an asymptotic signal

*u*(

*t*) is obtained by asymptotic techniques and is expressed as [15]

*b*, a value of

*a*is determined such as \(\left| {W_\lambda [u](a(b),b)} \right| =\hbox {max}_a \left| {W_\lambda [u](a,b)} \right| \). To obtain the ridge, the dilatation parameter, \(a=a(b)=\omega _0 /{\dot{\varphi }}(b)\), has to be determined by maximizing the \(\hat{{\lambda }}[a{\dot{\varphi }}(b)]\) using the MMWT. We obtain

*A*(

*t*) and instantaneous angular frequency \(\omega _\mathrm{e} (A)={\dot{\varphi }}(t)\). It should be noted that since the identified circular frequency \(\omega _\mathrm{e} (A)\) varies slowly with the variable

*A*(

*t*), only the long-term trend of \(\omega (t)\) is considered in the function of \(\omega _\mathrm{e} (A)\). However, the \(\omega (t)\) of a weakly nonlinear system always contains significant fast-varying components that are large in magnitude. The fast components can be eliminated by removing in advance the fast component contained in \(\varphi (t)\) through a least square polynomial fitting on the identified data.

#### 2.2.3 Determination of instantaneous damping ratio

The instantaneous damping is dependent on energy dissipation of a vibration system and is often different from case to case. It is generally difficult to determine the instantaneous damping from responses of vibration. In this study, a constant variant method is applied to identify the nonlinear damping ratio of a weakly nonlinear system.

*B*(

*t*) is linear with time and the damping ratio is the slope of the linear curve of

*B*(

*t*).

*B*(

*t*) varies nonlinearly with time, and the slope of

*B*(

*t*) is the amplitude-dependent damping ratio \(\xi _\mathrm{e} (A)\) that can be determined by the instantaneous amplitude

*A*(

*t*) identified by the MMWT given in Eq. (46). The amplitude-dependent damping ratio is expressed by

## 3 A hybrid aeroelastic–pressure balance (HAPB)

### 3.1 Development of HAPB system

### 3.2 System identification: a linear identification

## 4 Verification of the identification for the physical nonlinearities of the HAPB system

The nonlinearities of the HAPB system with a vertical test model (\(\alpha =0^{\circ }\), in Fig. 3) will be identified by the proposed method (Sect. 2), and will be validated by a time domain method and the *Newmark*-\(\beta \) method. Then, the nonlinearities of the HAPB system with inclined test models will be identified in a similar way.

### 4.1 Analytical process using the modified Morlet wavelet transform

*A*(

*t*) of the signal is determined. The comparisons of the envelope predicted by the MMWT and that directly observed are given in Fig. 9. It shows that the envelope predicted by the MMWT for \(f_\mathrm{b} =49\) is in better agreement with the directly observed than that for \(f_\mathrm{b} ={2}\). Moreover, the plots of the envelope predicted by the MMWT for \(f_\mathrm{b} ={49}\) and the directly observed coincide well with each other apart from the beginning of the data due to significant ‘end effect’. This effect has been studied in a number of previous studies [35, 36, 37]. Note that the effect can be reduced by adding zeros or by adding negative values of the signal at the end region [35], or it could be fitted out as long as the signal is long enough. We adopt the latter method to reduce the effect, and the identification of nonlinearities using this method will be verified.

*B*(

*t*) given by Eq. (47). It is observed that the slope of

*B*(

*t*) varies slowly with time, indicating that the damping ratio identified by Eq. (49) would also varies slowly with time. Using the LS method,

*B*(

*t*) is weighted by a polynomial, which is readily utilized to obtain the damping ratio of the signal.

### 4.2 Nonlinear physical frequency and damping ratio

Figure 13 plots the nonlinear damping of the HAPB system with a vertical test model. It is noted that the nonlinear damping increases gradually with amplitude of oscillation. The damping identified from the linear model presented in “Appendix A” is linear, and can only describe approximately the dissipative behaviors of the nonlinear oscillating HAPB system. Furthermore, the nonlinear damping identified by a time domain method is also plotted and compared with the nonlinear damping identified by the MMWT method. Figure 13 shows that the nonlinear damping identified by the MMWT agrees well with that obtained from the time domain method defined in “Appendix B” (Figs. 16, 17). There exist slight discrepancies at large oscillations (i.e., \({u}/{D}=12--20\%)\), which may be attribute to the errors caused by curve fitting, such as the errors in the curve fitting of *B*(*t*), \(f_\mathrm{e}\), etc. But the discrepancy is so small (the maximum is around 1.27%) that it is negligible. This suggests that the proposed identification method is reliable and the identified nonlinear damping of the HAPB system, \(\xi =-\,0.004914 A(t)^{-0.05913}+0.0142\), could be utilized for further analysis.

### 4.3 Verification of the identified nonlinear physical frequency and damping ratio

To further validate the nonlinear frequency (Fig. 12) and nonlinear damping (Fig. 13) of the HAPB system with a vertical test model, identified by the MMWT method, the free decay response of the model is numerically calculated by the *Newmark-*\(\beta \) method and compared with the directly observed in Fig. 5.

*Newmark-*\(\beta \) method, the numerically calculated response of the displacement, velocity and acceleration of the HAPB system at time step \(t_{i+1} \) is given as

The comparison of the time-history free decay response computed from Eq. (50) and that directly observed is presented in Fig. 14. It demonstrates that the computed response based on the identified nonlinear frequency and damping is coincide with the directly measured, suggesting that the nonlinearities of the HAPB system identified by the MMWT are reliable and accurate. Comparing the result in Fig. 14 with that in Fig. 4 indicates that the relatively poor agreement using the linear model is attribute to the ignorance of the nonlinearities of the HAPB system.

## 5 Nonlinearities of the HAPB system: with inclined test models

## 6 Concluding remarks

This study has proposed an improved method for determining physical nonlinearities of weakly nonlinear spring-suspension system and successfully applied to a novel hybrid aeroelastic–pressure balance (HAPB) system used in wind tunnel, which can be used for simultaneously obtaining the unsteady wind pressure and aeroelastic response of a test model. The identification was verified through comparing the identifications with those identified by a time domain method. The main findings are concluded: (1) The frequency and damping of the HAPB system identified by a linear model are constant and will lead to large discrepancies in response predictions due to the ignorance of the slow varying characteristics of the system. The nonlinearities of the system have to be considered; (2) the proposed identification method and the analytical scheme are precise and reliable in identifying the nonlinearities of the system. They are able to predict the long-duration free decay response of the system; (3) the identified nonlinearities of the system with inclined test models vary with the inclination of the models, which may be attribute to the different physical sources induced by the inclination.

## Notes

### Acknowledgements

The work described in this paper was supported by 111 project of China (Grant No.: B18062), Natural Science Foundation of Chongqing, China (Grant No.: cstc2019jcyj-msxm0639), Fundamental Research Funds for the Central Universities (Project No.: 2019CDXYTM0032), Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 16242016). Also, the authors appreciate the use of the testing facility, as well as the technical assistance provided by the CLP Power Wind/Wave Tunnel Facility at the Hong Kong University of Science and Technology. The authors would also like to express our sincere thanks to the Design and Manufacturing Services Facility (Electrical and Mechanical Fabrication Unit) of the Hong Kong University of Science and Technology for their help in manufacturing the test rig of the pressure–aeroelastic hybrid wind tunnel test system.

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