Mathematical Problems in Engineering

Volume 2018, Article ID 3496870, 8 pages

https://doi.org/10.1155/2018/3496870

## Flutter Test Data Processing Based on Improved Hilbert-Huang Transform

School of Power and Energy, Northwestern Polytechnical University, Shaanxi, Xi’an 710072, China

Correspondence should be addressed to Hua Zheng; nc.ude.upwn.liam@hz_aesi

Received 29 May 2018; Revised 22 July 2018; Accepted 30 July 2018; Published 12 August 2018

Academic Editor: Arkadiusz Zak

Copyright © 2018 Hua Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Flutter tests are conducted primarily for the purpose of modal parameter estimation and flutter boundary prediction, the accuracy of which is severely affected by the acquired data quality, structural modal density, and nonstationary conditions. An improved Hilbert-Huang Transform (HHT) algorithm is presented in this paper which mitigates the typical mode mixing effect via modulation. The algorithm is validated by theory, by numerical simulation, and per actual flight flutter test data. The results show that the proposed method could extract the flutter model parameters and predict the flutter speed more accurately, which is feasible for the current flutter test data processing.

#### 1. Introduction

“Flutter” is the self-excited vibration of an elastic structure under the coupling of aerodynamic force, elastic force, and inertial force; it is often accompanied by catastrophic structural damage [1]. Flutter analysis is a crucial aspect in the design of new or modified aircraft. Wind tunnel tests, aeroelastic models, and flight flutter tests are important components of flutter analysis. Such tests are risky, time-consumptive, and costly to conduct. To this effect, accurate and effective flutter test data processing techniques are in high demand. One of the most popular flutter boundary prediction (FBP) methods currently is the damping-based method [2, 3], which serves to extrapolate a curve fitted by the estimated structural modal damping factor against the airspeed to the abscissa while zero damping is considered the critical flutter criterion. The key to this type of FBP method is the accurate identification of modal damping factors.

Many algorithms have been previously developed for identifying flutter test modal parameters, including fast Fourier transform- (FFT-) based methods [4, 5], Random Decrement Technique (RDT) [6], natural excitation technique combined with the eigensystem realization algorithm (NExT-ERA) [7], time series analysis based on the Autoregressive (AR) model [8, 9], and Stochastic Subspace Identification (SSI) [10, 11]. Although some of these algorithms are effective, none is ideal; for instance, nonstationary measured data and low signal to noise ratio (SNR) affect the Fourier-based methods. The AR model requires an appropriate model order and algorithm to function properly; RDT functionality is limited by the number of the main modals contained in the structure [12]. SSI and NExT-ERA are problematic in terms of their spurious mode [13, 14]. The Hilbert-Huang Transform (HHT) proposed in 1998 is an adaptive scheme well-suited to nonlinear, nonstationary time series analysis; however, the mode mixing effect [15] which emerges when dealing with signals over multifrequencies in each frequency band severely limits its application to flutter test data processing due to the inherent density of modal problems. This paper proposes an improved HHT which applies to flutter test modal parameter identification.

#### 2. Hilbert-Huang Transform Theory

HHT is a relatively new nonstationary and nonlinear signal processing approach which is not limited under linear and stationary spectral analysis theory based on the Fourier transform. HHT has two basic steps. First, the original signal is decomposed into a series of Intrinsic Mode Function (IMF) components via Empirical Mode Decomposition (EMD); second, a Hilbert transform is performed on each IMF component to obtain instantaneous amplitude and phase information.

According to classical Fourier theory, the local frequency only can be defined when there is at least one complete sine or cosine oscillation. This strict definition is not suitable for nonstationary signals which have ever-changing frequencies. A Hilbert transformation for the original signal is performed as follows to reveal the instantaneous frequency of the signal at any moment:where denotes the Cauchy principal value; the transformation is available for all classes. According to this definition, when a complex conjugate is formed by and , the analytical signal iswhere denotes the instantaneous amplitude and denotes the instantaneous phase.

In this context, the instantaneous frequency can be defined as follows:

In a sense, the local properties of are emphasized because the Hilbert transform is defined as a convolution between and under (1). The local characteristics are further expressed as a polar coordinate in (2), which is the optimal approximation of the trigonometric functions with variable amplitude and phase.

Although the definition of the instantaneous frequency and phase is given, some extra conditions are required to ensure the instantaneous frequencies have physical significance. The signals must be symmetrical with a local mean of zero and must have the same zero crossings and extreme points. A general signal can be described by the Hilbert transform in the frequency domain only after EMD.

The EMD decomposition is based on the assumption that any signal is comprised of a series of different IMF components. The envelope of each IMF is defined by the local maximum and the minimum is symmetrical about the abscissa. The number of extreme points and zero crossing points should be equal or no more than one in the entire data sequence. Ideally, each IMF component contains a single modal and the number of IMFs equals the modal number of the original signal.

The EMD steps to decompose any signal are as follows.(a)Seek all the maximum and minimum points.(b)Use cubic spline interpolation (CSI) to interpolate every maxima and minima point sequence to obtain the upper and lower envelopes and .(c)Calculate the average envelope line by and extract the details by .(d)Determine whether meets the two IMF conditions given above. If so, is an IMF, as . Otherwise, let and repeat steps (a)-(d) until the conditions are met.(e)Let the residual be the new signal to be decomposed, and repeat steps (a)-(d) for other IMFs. Repeat the steps above until the residual is a monotonic signal or falls below a certain threshold; then the decomposition is complete.

The EMD procedure is also shown in Figure 1. The IMF, per its theoretical definition, meets the conditions of the Hilbert transform as the instantaneous frequencies have physical significance. Thus, the HHT can be used for complex nonstationary signal analysis.