Modal identification from turbulence response based on improved frequency domain decomposition

Abstract

Turbulence excitation is an unavoidable form of excitation in flutter flight tests, and it is also a necessary and effective excitation method in high-speed flights, dives, other high-risk test, and high-frequency modal information mining. However, because the turbulence excitation signals are unmeasurable in the time domain, although the turbulence response contains rich and valuable flutter test information, the randomness and low quality of the data often cause difficulties in modal analysis. Therefore, an improved frequency domain decomposition algorithm for turbulence response processing is proposed in this paper. First, due to the irrelevancy of the atmospheric excitation, the power spectral density function matrix of the multi-channel turbulence response is subjected to singular value decomposition. There is a mathematical relationship between the maximum singular value curve and the system frequency response function. Second, the modal assurance criteria are used to calculate the maximum singular value of a single-degree-of-freedom system. Finally, an orthogonal polynomial method is applied to fit the maximum singular value curve, and the system identification is performed directly in the frequency domain. The simulated data and a certain type of aircraft flutter flight test data are used to verify the proposed method, and the results confirm the effectiveness and engineering applicability of the method developed in this work.

Keywords

Turbulence response signal improved frequency domain decomposition frequency response function basis function expansion

Introduction

The flutter test represents a crucial strategy for verifying the flutter design during the development of new types of aircraft. The accurate and effective characterization analysis of the flutter response is an important task for processing the data from such flutter tests. In general, the implementation of artificial excitation sources (frequency sweep of the rudder surface, eccentric wheel excitation, rudder pedaling, small rocket excitation, etc.) in the flutter test can effectively improve the quality of the flutter response signals, thereby improving the reliability of the subsequent data processing. However, the active excitation method is often severely restricted by the conditions of the flutter test itself. For example, the wind tunnel test is limited by the size and structure of the aeroelastic model, and it is not convenient to implement active excitation. Meanwhile, the flutter flight test requires the installation of excitation equipment to modify the aircraft, which increases the risk of flight test. However, the high-frequency modal information in the structure cannot easily be fully excited because of the narrow-band characteristics of the active excitation signal. This phenomenon affects the analysis of the higher-frequency modal parameters. Turbulence excitation is an inevitable form of excitation in flutter flight tests. It does not require modifying the aircraft/aeroelastic model, so it is a necessary and effective incentivized approach in the analysis of high-risk flight tests and wind tunnel tests. Although the turbulence response signal contains rich and valuable flutter test information, the randomness and low quality of this data often complicate modal analysis and affect the conclusion of the flight test. Therefore, determining how to process relevant data for turbulence response signals, especially the identification of flutter structure modal parameters, is a research area with clear engineering application value.

In recent years, many algorithms have been proposed to extract modal parameters from turbulence response signals, including Fast Fourier Transform (FFT)-based methods,^1,2 Random Decrement Technique (RDT),³ Natural Excitation Technology (NExT) combined with eigenfunctions,^4–6 and Stochastic Subspace Identification (SSI) methods.^7,8 However, owing to the randomness and complexity of the turbulence response, these strategies all face certain limitations.^9–11 The Frequency Domain Decomposition (FDD) method is a relatively stable approach that has been used to estimate the modal parameters of the turbulence response. The fundamental principle involves extracting the Frequency Response Function (FRF) of the system by considering the irrelevance of the noise excitation. The advantage mainly lies in the fact that the independent modal shape that can be used to distinguish different mode.^12,13 Brincker et al.¹⁴ calculated the Power Spectrum Density (PSD) matrix of the turbulence response at multiple measurement channels based on the periodogram method. The utility of this approach was illustrated for the output-only identification of the Great Belt Bridge. Aiming to resolve the spectrum calculations of the PSD matrix, Malekjafarian¹⁵ estimated the bridge modal shape using the short time FDD of the responses measured in a passing vehicle. In the applied algorithm, the obtained system FRF was identified in the time domain. Furthermore, the auto-correlation function of the impulse response was calculated via inverse Fourier transform, and the logarithmic decrement method of the corresponding Single-Degree-Of-Freedom (SDOF) auto-correlation function was used to estimate the modal parameters.¹⁶ Brincker et al.^17,18 analyzed the modal parameters of the civil engineering structure based on the FDD. Danowsky et al.¹⁹ achieved the open-loop analysis and system identification of the flutter suppression system on a small flexible flying-wing aircraft based on the FDD. Schulze et al.²⁰ analyzed the modal parameter and system identification of the response signal of the F/A-18C flutter flight test based on the Curve fitting Frequency Domain Decomposition (CFDD). In fact, since the FRF itself is the result of frequency domain calculations, it is a more concise and effective approach to directly perform system identification in the frequency domain. Therefore, this report describes an improved frequency decomposition method, which employs the Singular Value Decomposition (SVD) to form a reference system FRF. In pursuit of multi-mode separation, the basis function expansion method is applied to directly form the frequency domain to identify the modal parameters. Ultimately, an improved frequency decomposition method for modal parameter estimation of the turbulence response is proposed. The contributions can be summarized as follows:

• The SVD of the PSD matrix at each spectral pin is achieved, the maximum singular value can approximately represent the FRF of the system.

• According to the similarity of the modal shape vector, the singular value curve of the SDOF system is calculated based on the modal assurance criterion.

• Considering the frequency resolution of the SDOF spectral density function, and the basis function expansion method is applied to identify the modal parameters in the frequency domain.

Related methods

Frequency domain decomposition

Herein, we describe how the FRF of the system is estimated considering the turbulence responses of multiple channels. The estimated multi-mode FRF is used for the separation of the FRF of the SDOF system and the calculations of modal parameters. Considering the convolution of the excitation and response in the linear time-invariant system, the expression in equation (1) describes the relationship between the atmospheric turbulence excitation x(t) and the measurable turbulence response y(t)

G_{y y} (j ω) = \bar{H} (j ω) G_{x x} (j ω) {H (j ω)}^{T}

(1)

where

G_{x x} (j ω)

is an r × r PSD matrix of the excitation signal x(t), r is the channel number of the excitation signal,

G_{y y} (j ω)

is an m × m PSD matrix of the response signal, m is the channel number of the response signal, H(jω) is an m × r FRF,

\bar{H} (j ω)

is the complex conjugate of H(jω), and

{H (j ω)}^{T}

is the matrix transpose of H(jω). When the excitation signal is Gaussian white noise, its PSD matrix is a constant matrix, that is,

G_{x x} (j ω) = C

. Therefore, equation (1) can be rewritten as equation (2)

G_{y y} (j ω) = \sum_{k = 1}^{n} \sum_{s = 1}^{n} [\frac{R_{k}}{j ω - λ_{k}} + \frac{{\bar{R}}_{k}}{j ω - {\bar{λ}}_{k}}] C {[\frac{R_{s}}{j ω - λ_{s}} + \frac{{\bar{R}}_{s}}{j ω - {\bar{λ}}_{s}}]}^{H}

(2)

This formula can be obtained through Heaviside’s partial fraction expansion theory, as shown in equation (3)

G_{y y} (j ω) = \sum_{k = 1}^{n} \frac{A_{k}}{j ω - λ_{k}} + \frac{{\bar{A}}_{k}}{j ω - {\bar{λ}}_{k}} + \frac{B_{k}}{- j ω - λ_{k}} + \frac{{\bar{B}}_{k}}{- j ω - {\bar{λ}}_{k}}

(3)

where

A_{k}

represents the residual matrix of the k^th modal of the response signal PSD matrix, which is a Hermitian matrix. Furthermore, the PSD function of the multi-channel response

{\hat{G}}_{y y}

is subjected to SVD at discrete spectral pin

ω = ω_{i}

, according to equation (4)

{\hat{G}}_{y y} = U_{i} S_{i} U_{i}^{T}

(4)

where

U_{i} = [u_{i 1}, u_{i 2}, \cdot \cdot \cdot, u_{i m}]

is a unit matrix composed of singular value vectors

u_{i j}

, and

S_{i}

is a diagonal matrix composed of singular values

S_{i j}

. Comparing equations (3) and (4), the first singular value vector

u_{i 1}

represents the modal shape. Moreover, the singular value of the PSD function

{\hat{G}}_{y y}

of the response can be expressed as the PSD function of the corresponding system. Thus, the PSD function of the SDOF system can be distinguished by the modal shape.

Modal assurance criteria

Following the described SVD of the PSD function matrix of the response, and the singular value vector represents the modal shape, the singular value of the SDOF system can be separated according to the Modal Assurance Criteria (MAC). The MAC is defined as shown in equation (5)

{M A C}_{i} = \frac{| u_{r}^{T} u_{i} |}{‖ u_{r} ‖ ‖ u_{i} ‖}

(5)

where

‖ ‖

represents the two-norm of the vector,

u_{r}

is the left singular value vector of the peak spectral line, and

u_{i}

represents the left singular value vector corresponding to the point

ω_{i}

near the peak spectral line. When

{M A C}_{i} > Ω

(typically set as Ω = 0.85∼0.95),²¹ the spectral pin

ω_{i}

is considered to be the same mode as the peak spectral line point, otherwise it is removed.

Basis function expansion

The basis function expansion is an orthogonal polynomial method that is used to perform polynomial fitting on the singular value curve of the SDOF system based on the FDD. Furthermore, orthogonal polynomial fitting based on basis function expansion is used to calculate the modal parameters of the FRF of the system.

The transfer function for a linear time-invariant system can be expressed as shown in equation (6)

H (s) = \frac{a_{m} s^{m} + \cdot \cdot \cdot + a_{2} s^{2} + a_{1} s {+ a}_{0}}{b_{n} s^{n} + \cdot \cdot \cdot + b_{2} s^{2} + b_{1} s {+ b}_{0}}

(6)

where m<n. Dividing the numerator and denominator of equation (6) by

b_{n}

gives

p_{m} (s) = s^{m}, \cdot \cdot \cdot, p_{1} (s) = s, p_{0} (s) = 1

q_{n} (s) = s^{n}, \cdot \cdot \cdot, q_{1} (s) = s, q_{0} (s) = 1

which leads to equation (7)

H (s) = \frac{\sum_{k = 0}^{m} c_{k} p_{k} (s)}{\sum_{k = 0}^{n} d_{k} q_{k} (s)}, where d_{n} = 1

(7)

In practical engineering, it is often impossible to measure the transfer function covering the entire S-plane, so only the data covering the entire frequency axis is considered. This data is generally known as the FRF, that is,

s = j ω

. In this case, equation (7) can be further expressed as shown in equation (8)

H (j ω) = \frac{\sum_{k = 0}^{m} c_{k} p_{k} (j ω)}{\sum_{k = 0}^{n} d_{k} q_{k} (j ω)}

(8)

where the coefficients

c_{k}

and

d_{k}

are the quantities that need be calculated. To use the orthogonal polynomial to fit the FRF, it is necessary to simplify the calculation using the characteristics of the orthogonal polynomial and the FRF. The concept of negative frequency is introduced in the calculation process, so that ω = [

ω_{- L}

, ···,

ω_{- 1}

ω_{1}

, ···

ω_{L}

] over a total of 2L points. If we let

ω_{- i} = - ω_{i}

, then

H (j ω_{- i}) = H (- j ω_{i}) = H^{*} (j ω_{i})

, where i = 1, 2, ···, L, and the measured value of the FRF

\hat{H} (j ω_{i}) = S (j ω_{i})

and

S (j ω_{i})

is the singular value curve

S (j ω_{- i}) = S (- j ω_{i}) = S^{*} (j ω_{i}) (i = 1, 2, \cdot \cdot \cdot, L)

(9)

The error between the theoretical model of the FRF (i.e., a rational fraction of equation (8)) and the actual measurement result can be express as shown in equation (10)

\begin{array}{l} l_{i} = H (j ω_{i}) - S (j ω_{i}) \\ = \frac{\sum_{k = 0}^{m} c_{k} p_{k} (j ω_{i})}{\sum_{k = 0}^{n} d_{k} q_{k} (j ω_{i})} - S (j ω_{i}) \end{array}

(10)

To linearize the coefficients $c_{k}$ and $d_{k}$ , equation (10) can be rewritten as equation (11)

\begin{array}{l} e_{i} = l_{i} \sum_{k = 0}^{n} d_{k} q_{k} (j ω_{i}) \\ = \sum_{k = 0}^{m} c_{k} p_{k} (j ω) - S (j ω_{i}) [\sum_{k = 0}^{n - 1} d_{k} q_{k} (j ω_{i}) + q_{n} (j ω_{i})] \end{array}

(11)

Furthermore, the error is defined by equation (12)

J = \sum_{i = - L}^{L} e_{i}^{*} e_{i} = E^{H} E

(12)

where E = (

e_{- L}

, ···,

e_{- 1}

e_{1}

,···

e_{L}

)^T, and E^H is a conjugate transpose of E. Considering equations (11) and (12), the vector of the error is defined by equation (13)

E = P C - Q D - W

(13)

where

P = [\begin{array}{l} P^{-} \\ P^{+} \end{array}] = [\begin{array}{l} p_{0} (j ω_{- L}) & \dots & p_{m} (j ω_{- L}) \\ ⋮ & ⋱ & ⋮ \\ p_{0} (j ω_{L}) & \dots & p_{m} (j ω_{- L}) \end{array}]

Q = [\begin{array}{l} Q^{-} \\ Q^{+} \end{array}] = [\begin{array}{l} {S (j ω_{- L}) q}_{0} (j ω_{- L}) & \dots & {S (j ω_{- L}) q}_{n - 1} (j ω_{- L}) \\ ⋮ & ⋱ & ⋮ \\ {S (j ω_{L}) q}_{0} (j ω_{L}) & \dots & {S (j ω_{L}) q}_{n - 1} (j ω_{L}) \end{array}] C = {(c_{0}, c_{1}, \cdot \cdot \cdot, c_{m})}^{T}

D = {(d_{0}, d_{1}, \cdot \cdot \cdot, d_{n - 1})}^{T}

W = [\begin{array}{l} W^{-} \\ W^{+} \end{array}] = {({S (j ω_{- L}) q}_{n} (j ω_{- L}), \cdot \cdot \cdot, {S (j ω_{L}) q}_{n} (j ω_{L}))}^{T}

To determine the minimum value of the error J, the partial derivative of J with respect to the vectors C and D can be set to zero, as shown in equation (14)

\frac{\partial J}{\partial C} = 0, \frac{\partial J}{\partial D} = 0

(14)

According to equations (13) and (14), the matrix relationship in equation (15) can be defined

[\begin{array}{l} Y & X \\ X^{T} & Z \end{array}] [\begin{array}{l} C \\ D \end{array}] = [\begin{array}{l} G \\ F \end{array}]

(15)

where

Y = \frac{1}{2} (P^{H} P + P^{T} P^{*})

Z = \frac{1}{2} (Q^{H} Q + Q^{T} Q^{*})

G = R e (P^{H} W)

F = R e (Q^{H} W)

In equation (15), if

p (j ω)

and

q (j ω)

meet the conditions in equations (16a, b)

\begin{array}{l} \sum_{i = - L}^{L} p_{s}^{*} (j ω_{i}) p_{t} (j ω_{i}) = \sum_{i = - L}^{L} p_{s} (j ω_{i}) p_{t}^{*} (j ω_{i}) \\ = {\begin{cases} 1 s = t \\ 0 s \neq t \end{cases} s, t = 0, 1, \cdot \cdot \cdot, m \end{array}

(16a)

\begin{array}{l} \sum_{i = - L}^{L} {| S (j ω_{i}) |}^{2} q_{s}^{*} (j ω_{i}) q_{t} (j ω_{i}) = \sum_{i = - L}^{L} {| S (j ω_{i}) |}^{2} q_{s} (j ω_{i}) q_{t}^{*} (j ω_{i}) \\ = {\begin{cases} 1 s = t \\ 0 s \neq t \end{cases} s, t = 0, 1, \cdot \cdot \cdot, m \end{array}

(16b)

then the Y and Z matrices in equation (15) create the unit matrix, and the final form is shown in equation (17)

[\begin{array}{l} I_{1} & X \\ X^{T} & I_{2} \end{array}] [\begin{array}{l} C \\ D \end{array}] = [\begin{array}{l} G \\ F \end{array}]

(17)

The C and D matrices can be obtained by expanding equation (17) to obtain equation (18)

\begin{array}{l} (I - X^{T} X) D = F - X^{T} G \\ C = G - X D \end{array}

(18)

Satisfying the condition of equation (16) makes $p_{k} (j ω)$ and $q_{k} (j ω)$ an orthogonal polynomial, which in turn makes the diagonal of the coefficient matrix in equation (17) an identity matrix. Then, equation (18) can be used to solve for the coefficients C and D separately.

Considering the derivation described, the Forsythe complex orthogonal polynomial can be used to reconstruct $p_{k} (j ω)$ and $q_{k} (j ω)$ , and the recurrence relationship is further introduced as equation (19)

\begin{array}{l} \bar{p_{0}} (j ω) = 1 \\ \bar{p_{1}} (j ω) = j ω - u_{1} \\ \bar{p_{2}} (j ω) = (j ω - u_{2}) \bar{p_{1}} (j ω) - V_{1} \bar{p_{0}} (j ω) \\ \bar{p_{K}} (j ω) = (j ω - u_{K}) \bar{p_{K - 1}} (j ω) - V_{K - 1} \bar{p_{K - 2}} (j ω) \end{array}

(19)

which clarifies equation (20)

\begin{array}{l} u_{K} = {\sum_{i = - L}^{L} j ω_{i} | \bar{p_{K - 1}} (j ω_{i}) |}^{2} ρ_{i}^{2} / D_{e K - 1} \\ V_{K - 1} = \sum_{i = - L}^{L} j ω_{i} \bar{p_{K - 1}} (j ω_{i}) {\bar{p_{K - 2}}}^{*} (j ω_{i}) ρ_{i}^{2} / D_{e K - 2} = - \frac{D_{e K - 1}}{D_{e K - 2}} \\ D_{e K} = {\sum_{i = - L}^{L} | \bar{p_{K}} (j ω_{i}) |}^{2} ρ_{i}^{2} \end{array}

(20)

where

ρ_{i}

is the weighting factor, the numerator polynomial

\bar{p_{K}} (j ω)

is taken as

ρ_{i} = 1

, and the denominator polynomial

\bar{q_{K}} (j ω)

is taken as

ρ_{i}^{2} = {| S (j ω_{i}) |}^{2}

. The orthogonal polynomial

\bar{p_{K}} (j ω)

is regularized according to equation (21)

p_{K} (j ω) = \frac{\bar{p_{K}} (j ω)}{{[\sum_{i = - L}^{L} {| \bar{P_{K}} (j ω_{i}) |}^{2} ρ_{i}^{2}]}^{\frac{1}{2}}} = \frac{\bar{p_{K}} (j ω)}{D_{e K}^{\frac{1}{2}}}

(21)

Substituting equation (19) into equation (21) gives equation (22)

\begin{array}{l} u_{K} = 0 \\ D_{e K} = 2 \sum_{i = 1}^{L} {| \bar{P_{K}} (j ω_{i}) |}^{2} ρ_{i}^{2} \end{array}

(22)

This equation relies on the definition in equation (23)

\bar{P_{K}} (j ω) = j^{K} \bar{R_{K}} (ω) (K = 0, 1, \cdot \cdot \cdot)

(23)

where

\bar{R_{K}} (ω)

is a real number that can be computed using the recurrence relation in equation (24)

\begin{array}{l} \bar{R_{0}} (ω) = 1 \\ \bar{R_{1}} (ω) = ω \\ \bar{R_{K}} (ω) = ω \bar{R_{K - 1}} (ω) + V_{K - 1} \bar{R_{K - 2}} (ω) \end{array}

(24)

which depends on the relationships in equation (25)

\begin{array}{l} V_{K - 1} = - \frac{D_{e K - 1}}{D_{e K - 2}} \\ D_{e K} = 2 \sum_{i = - 1}^{L} {| \bar{R_{K}} (j ω_{i}) |}^{2} ρ_{i}^{2} \end{array}

(25)

After $\bar{R_{K}} (ω)$ is determined, $R_{K} (ω)$ can be calculated following regularization, and then $p_{K} (j ω)$ can be further calculated using $R_{K} (ω)$ (equations (26) and (27))

R_{K} (ω) = \frac{\bar{R_{K}} (ω)}{D_{e K}^{\frac{1}{2}}}

(26)

p_{K} (j ω) = j^{K} R_{K} (ω) (K = 0, 1, \cdot \cdot \cdot)

(27)

These formulas contribute the solution step of the numerator in the complex orthogonal polynomial. Similarly, the denominator component of the complex orthogonal polynomial can be obtained according to the same steps. Specifically, the column vectors C and D are obtained using equation (28)

\begin{array}{l} (I - X^{T} X) D = F - X^{T} G \\ C = G - X D \end{array}

(28)

where

\begin{array}{l} X = - R e (P^{H} Q) = - 2 R e (P^{+ H} Q^{+}) \\ G = R e (P^{H} W) = 2 R e (P^{+ H} W^{+}) \end{array}

It is worth noting that the C and D are polynomial coefficients after regularization, and further computations are required before calculating the modal parameters.

According to the MAC condition, the maximum singular value curve of the SDOF system is identified, and the poles $λ$ and $\bar{λ}$ of the transfer function $H (ω)$ are calculated. The corresponding modal frequency and modal damping formulas are shown in equations (29) and (30), respectively

f = \frac{‖ λ ‖}{2 π}

(29)

ζ = \frac{- R e (λ)}{‖ λ ‖}

(30)

where

‖ λ ‖

represents the complex modulus of

λ

, and

R e (λ)

is the real part of

λ

Simulation data verification

To verify the proposed method of the improved FDD, the simulation data were used to modal parameter estimation. The impulse response of the structural system is described by equation (31)

h (t) = \sum_{i = 1}^{N} A_{i} e^{- ξ_{i} ω_{i} t} \sin (ω_{i} t + φ_{i})

(31)

where N is the modal number of the system,

A_{i}

represents the amplitude of the i^th modal,

ξ_{i}

represents the damping of the i^th modal,

ω_{i}

represents the frequency of the i^th modal, and

φ_{i}

represents the phase corresponding to the i^th modal, which is used to simulate the measured signal from a different sensor. Finally, the turbulence response of the system is determined based on equation (32)

y (t) = h (t) * x (t)

(32)

where

y (t)

represents the turbulence response,

h (t)

is the impulse response of the structural system, and

x (t)

is Gaussian white noise in the simulation. Furthermore, the time series and PSD of the turbulence response are shown in Figure 1 (signal #1) and Figure 2. The sampling rate was set to 512 Hz, and the sampling time was 60 s in the simulation signal. The modal frequency of the simulation signal is 17.13 Hz in Figure 1. Figure 3 shows the curve corresponding to the first singular value of the single-side PSD matrix in the spectral pin from 0 Hz to 256 Hz. The top plot in Figure 4 shows the PSD of the turbulence response at measurement point #1 in the specified analytical interval. The blue curve in the bottom plot of Figure 4 shows the singular value curve in the specified analytical interval. The red curve in the bottom of Figure 4 represents the transfer function of the SDOF fitted with orthogonal polynomials.

Figure 1.

Time series of turbulence response.

Figure 2.

PSD of turbulence response.

Figure 3.

Maximum singular value.

Figure 4.

Orthogonal polynomial fitting curve.

Comparing the frequency of various measuring points in Figure 2 and the singular value curve with the frequency in Figure 3, it becomes clear that the singular value curve in Figure 3 generally reflects the frequency information of the structural system. First, the PSD of the turbulence response was calculated based on the periodogram method, and then SVD was performed on the PSD matrix at each frequency pin. Therefore, the frequency resolution of the singular value curve is consistent with the PSD. The bottom plot in Figure 4 also shows that the fitting result of the orthogonal polynomial is consistent with the result from the singular value curve. Consistent with the simulation signal with the 17 Hz modal frequency, the modal frequency of the other simulated signal (signal #2) was set to 5.3 Hz, as shown in Figure 5. The PSD function of the corresponding signal is shown in Figure 6. The singular value curve and the fitting results of the corresponding analytical interval are presented in Figures 7 and 8, respectively. In addition, the top plot in Figure 8 shows the PSD of the turbulence response (signal #2) at measurement point #1.

Figure 5.

Time series of turbulence response.

Figure 6.

PSD of turbulence response.

Figure 7.

Maximum singular value.

Figure 8.

Orthogonal polynomial fitting curve.

Table 1 summarizes the comparison between the modal parameters estimated from two sets of simulation signals. These results indicate that the singular value curve of the SDOF system is separated based on the MAC, making it feasible to estimate the modal parameters by the orthogonal polynomial method.

Table 1.

Results of modal parameter estimations.

Simulation signal	True value		Our method result
Simulation signal	Frequency (Hz)	Damping	Frequency (Hz)	Damping
Signal #1 (17 Hz)	17.14	0.005	17.1169	0.0046
Signal #2 (5 Hz)	5.3	0.010	5.2904	0.0138

Physical test verification

Modal parameter estimations

Physical tests verified the proposed method using the acceleration signals of the four channels at the wing and at the horizontal tail in a certain type of aircraft flutter flight test. The sample rate of physical test signal is 512 Hz, sample time is 30 s. Single-side band PSD of the four channels’ acceleration signals at the wing are shown in Figures 9, 10, 11 and 12.

Figure 9.

Time series and PSD of turbulence response from channel 1 of the wing.

Figure 10.

Time series and PSD of turbulence response from channel 2 of the wing.

Figure 11.

Time series and PSD of turbulence response from channel 3 of the wing.

Figure 12.

Time series and PSD of turbulence response from channel 4 of the wing.

Based on the theoretical analysis, the maximum singular value curve of the PSD of the four channels is shown in Figure 13. The system transfer function was calculated based on the orthogonal polynomial. According to the relationship between the system transfer function and the maximum singular value curve, the SDOF system is separated from the maximum singular value curve by MAC, and the SDOF system corresponding to the three modal frequencies (3.4, 7, and 18 Hz) is obtained (Figures 14, 15, and 16, blue dotted lines). Then, based on the orthogonal polynomial for the SDOF system, a curve fitting method was applied to obtain the corresponding system identification results (Figures 14, 15, and 16, red solid lines).

Figure 13.

Singular value curve of the wing.

Figure 14.

3.4 Hz modal of the wing.

Figure 15.

7 Hz modal of the wing.

Figure 16.

18 Hz modal of the wing.

The modal parameter analytical results regarding the wing acceleration signal of the flutter flight testing data are shown in Table 2 including the Ground Vibration Test (GVT), traditional FDD and proposed method. Compared with the frequency of the GVT and the traditional FDD method, and considering the difference between the GVT and the flutter flight test procedures, the error of the modal parameter estimation results fell within an allowable range using the proposed method. Because the GVT cannot obtain the damping ratio, the method described in this paper can only visually compare the singular value fitting results with the calculated results from the engineering designer.

Table 2.

Modal parameter estimation results regarding the turbulence response from the wing.

GVT	Traditional FDD		Method developed herein
Frequency (Hz)	Frequency (Hz)	Damping ratio (%)	Frequency (Hz)	Damping ratio (%)
3.451	3.232	10.64	3.500	15.11
7.387	6.534	5.65	6.958	8.34
18	17.852	8.63	17.932	9.80

In addition, the turbulence response from the front and rear edges of the horizontal tail and the wingtip of the horizontal tail were used to analyze the modal parameters. The time series and PSD function of the turbulence response are shown in Figures 17, 18, 19, and 20, respectively.

Figure 17.

Time series and PSD of turbulence response from channel 1 of the horizontal tail.

Figure 18.

Time series and PSD of turbulence response from channel 2 of the horizontal tail.

Figure 19.

Time series and PSD of turbulence response from channel 3 of the horizontal tail.

Figure 20.

Time series and PSD of turbulence response from channel 4 of the horizontal tail.

The SVD was performed on the data from multiple channels of the horizontal tail to calculate the singular value curve (Figure 21), and the singular value of the SDOF system was obtained according to the MAC conditions. The blue dotted lines in Figures 22, 23, and 24 represent the maximum singular value curves of the SDOF system at modal frequencies of 10.5, 26.5, and 44.5 Hz, respectively. The red solid lines in Figures 22, 23, and 24 are the results of system identification based on orthogonal polynomials. By considering the turbulence excitation interference, the system identification of the separated SDOF system can be realized.

Figure 21.

Singular value curve of the horizontal tail.

Figure 22.

10.5 Hz modal of the horizontal tail.

Figure 23.

26.5 Hz modal of the horizontal tail.

Figure 24.

44.49 Hz modal of the horizontal tail.

These plots present the fitting results in an intuitive manner. Furthermore, the turbulence response modal parameters are compared by GVT, traditional FDD and proposed method (Table 3). Using this method, the frequency identification error is small, and because the damping ratio parameters cannot be obtained in the GVT, this method was not considered in the comparison. Therefore, the damping ratio identification results of the measured flutter flight test data depended only on the data itself to complete the identification. The fitting effect is presented intuitively through the fitting results using the orthogonal polynomial, and the final result is consistent with the phenomena of the flutter test process.

Table 3.

Modal parameter estimation results from the horizontal tail.

GVT	Tradition FDD		Method developed herein
Frequency (Hz)	Frequency (Hz)	Damping ratio (%)	Frequency (Hz)	Damping ratio (%)
10.51	10.325	3.02	10.450	2.40
27.01	25.352	4.64	26.518	6.17
44.97	43.524	5.96	44.493	7.58

Flutter boundary prediction

The Flutter Boundary Prediction (FBP) is a crucial part of the flutter flight test because it represents an important task to finally push out the flight envelope of the aircraft. This prediction is based on the flight test data from the horizontal tail of a certain type of aircraft. The developed method in this article involves conducting the modal parameter estimation of the turbulence response for the multiple velocity steps, and then analyzes the changes in the damping ratio with velocity.

The FBP is based on the turbulence response of the horizontal tail of multiple test velocity steps in the modal parameter analysis. The 10.5 Hz mode of the horizontal tail was taken as an example to predict the flutter boundary, and the velocity-frequency diagram (Figure 25) and velocity-damping ratio diagram (Figure 26) were obtained, and this FBP result is consistent with the flight test phenomenon.

Figure 25.

Modal frequency curve regarding the velocity from the horizontal tail.

Figure 26.

Damping ratio curve regarding the velocity form the horizontal tail.

Conclusions

Due to the randomness of atmospheric turbulence, the maximum singular value is calculated via singular value analysis of the PSD matrix. This is accomplished by considering the relationship between the maximum singular value curve and the system FRF. The identification of the SDOF system can be realized through orthogonal polynomials. Compared with the traditional FDD method, the modal frequency result of SDOF systems determined using the developed method is closer to the GVT result. Based on the trend of the FBP, the identification results of the modal frequencies are relatively stable, and the modal damping has a downward trend, which confirms the stability of the improved FDD method. In addition, as a basis for further research, it is possible to directly perform frequency domain fitting based on orthogonal polynomials over the entire frequency band.

Supplemental Material

Supplemental Material - Modal identification from turbulence response based on improved frequency domain decomposition

Supplemental Material for Modal identification from turbulence response based on improved frequency domain decomposition by Shiqiang Duan, Hua Zheng, Jiangtao Zhou and Zhenglong Wu in Journal of Low Frequency Noise, Vibration and Active Control

Footnotes

Acknowledgments

This work is supported by the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant no. CX2021077) and the Fundamental Research Funds for the Central Universities (Grant no. 31020190MS702).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the the Fundamental Research Funds for the Central Universities (Grant no. 31020190MS702), the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant no.CX2021077).

ORCID iD

Hua Zheng

Data availability statement

The flutter flight testing data used in this study are included within the supplementary information files.*

Supplemental Material

The supplementary material file contains the flutter flight testing data, which we used to validate the proposed method (Section 2). The data were sourced from an acceleration sensor in an aeroelastic model subjected to physical testing. The data is given in. txt format of the supplementary materials.

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