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Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology

Data publikacji: 15 Jul 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 11 Jan 2022
Przyjęty: 12 Mar 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

If you use simple linear equation classification for big data analysis and classification modeling, the work efficiency is low, and the accuracy is also poor. For this reason, the thesis uses nonlinear differential equations to carry out computer-aided unsteady aerodynamic modeling. Based on the perspective of differential equations, the big data classification technology is studied, and the classification model is established. The article constructs the differential classification mathematical model by establishing the differential equation with second-order delay and the constraint conditions of the model specification set. The article identifies and identifies linear parameters such as characteristic time constants in the aerodynamic model. Research shows that the model can accurately predict unsteady aerodynamic characteristics under different maneuvers.

Keywords

MSC 2010

Introduction

The problem of unsteady aerodynamic modeling at high angles of attack is one of the fourth-generation fighter jets and other problems with post-stall maneuverability. Establishing engineering practical unsteady aerodynamic models is the basis for conducting flight dynamics analysis, flight simulation, control law design, and verification research. Earlier studies have shown that the main cause of nonlinear and unsteady aerodynamic phenomena at high angles of attack is the flow separation on the upper surface of the wing and the flow field topology. Some scholars proposed a series of linear response functions, vortex sequence, and linear state space [1]. These studies have achieved certain results in the unsteady aerodynamic modeling of small amplitude motion. The aerodynamic load for the large amplitude test also depends on the phenomenon of the forced oscillation amplitude. The current modeling methods are mainly divided into two categories: continuing to develop the previous methods. We propose the nonlinear response function method and its simplified method, nonlinear state-space method, etc. Another type of thinking is to use intelligent control methods such as neural networks or fuzzy logic to perform nonlinear algebraic fitting modeling directly.

However, the above modeling methods all have certain shortcomings. The response function method, the vortex sequence method, and the flight dynamics equation are combined into a “differential-integral” equation group. These are difficult to perform flight dynamics analysis. Predicting the unsteady aerodynamic force of small-amplitude motion with the nonlinear state-space method based on the results of the large-amplitude oscillating force measurement is far from the actual situation. Neural network and fuzzy logic algebraic fitting model generally need to fit the maneuvering process to constant amplitude oscillation motion. This will lead to the extraction of the reduced frequency as an input parameter, and unclear physical meaning will occur.

In this paper, the dynamic system modeling ideas research and develop nonlinear differential equation modeling and identification methods [2]. The article establishes a nonlinear unsteady aerodynamic model that can accurately predict the movement of small amplitude and large amplitude at different frequencies simultaneously. The aerodynamic model established by this method has a simple form and automatically degenerates into a dynamic derivative model at a small angle of attack.

Unsteady aerodynamic model and mechanism analysis

In the high angle of attack and small amplitude forced oscillation wind tunnel test, the unsteady characteristic is mainly manifested as the time history of aerodynamic load change is closely related to the oscillation frequency. Different dynamic wind tunnel tests show that the dynamic derivative obtained from the test within the range of 20°~50° angle of attack strongly depends on the oscillation frequency. The conventional linear superimposed aerodynamic derivative model based on the real-time motion state cannot reflect this unsteady characteristic [3]. Introducing differential equations to describe the dynamic characteristics of unsteady aerodynamic forces has become one of the main modeling methods.

According to the absolute value of the eigenvalue of the pneumatic system matrix, the response of the linear system can be divided into two groups of dynamic characteristics of high-frequency mode and low-frequency mode. Since the dynamic response process of the high-frequency mode of the pneumatic system has little effect on the body's motion, it is usually only necessary to consider its steady-state output (steady aerodynamic force). The root cause of the unsteady aerodynamic phenomenon lies in a non-negligible low-frequency mode of the pneumatic system coupled with the motion frequency. Its intuitive performance is the hysteresis response of the vortex structure adjustment to the motion state [4]. Therefore, the total aerodynamic force can be simplified into two parts, the steady aerodynamic force, and the unsteady aerodynamic force, directly according to the different time scales of the aerodynamic load in response to the movement change. The steady aerodynamic force is directly determined by the current motion state, reflecting the rapid response to a motion in the aerodynamic load. The unsteady aerodynamic force uses differential equations to describe the dominant low-order model of its dynamic characteristics. It reflects the comprehensive influence of the hysteresis response of the internal vortex topology adjustment of the flow field caused by the motion of the airframe on the aerodynamic load. The choice of the dominant low-order mode order is closely related to the configuration of the aircraft. Usually, only the dynamic influence of the first-order dominant mode needs to be considered near a certain angle of attack to obtain satisfactory unsteady aerodynamic modeling accuracy. Taking longitudinal motion as an example, the total aerodynamic load is decomposed into two parts: steady and unsteady. The aerodynamic coefficient is C(t)=Catt(a)+Cq(a)c¯ 2Vq+Cdyn(t) C(t) = {C_{att}}(a) + {C_q}(a){{\bar c} \over {2V}}q + {C_{dyn}}(t)

Catt(a) is the static aerodynamic coefficient under the assumption of no airflow separation. Cq(a) is the additional steady aerodynamic derivative of the pitching rotating flow field. V is the average aerodynamic chord length. D is the flight speed. Cdyn is the unsteady aerodynamic coefficient produced by the hysteretic motion of the flow field vortex topology structure during the movement of the aircraft. The first two formula items (1) are only related to the current flight state angle of attack α and pitch angle speed q. This reflects the steady aerodynamic load. This article uses the simplest first-order linear differential equation: τdCdyndt=ΔC(a)Cdyn \tau {{d{C_{dyn}}} \over {dt}} = \Delta C(a) - {C_{dyn}}

Describe the hysteretic response of unsteady aerodynamic forces to motion at small amplitudes. τ is the characteristic time constant of unsteady motion, which determines the character of the unsteady dominant mode. t¯ \bar t is the dimensionless time. ΔC(a) = Cst(a) − Catt(a) is the difference between the actual static aerodynamic coefficient and the static aerodynamic coefficient under the assumption of no airflow separation. It reflects the static aerodynamic increment produced by the vortex separation in the steady-state. The unsteady aerodynamic phenomenon also shows that the aerodynamic characteristics change with the amplitude of the forced oscillation in the high angle of attack and large amplitude forced oscillation wind tunnel test [5]. When the unsteady aerodynamic coefficient Cdyn is far from the steady-state equilibrium value ΔC in large-amplitude motion, the comprehensive influence of the vortex system on the aerodynamic load can no longer be described by linear differential equations. This paper introduces a nonlinear term based on a linear differential equation (2) to modify the dynamic characteristics. Furthermore, we construct a nonlinear unsteady aerodynamic model described by a nonlinear differential equation.

Parameter identification of linear differential equation for unsteady aerodynamic load
Identification of the time constant in the unsteady aerodynamic model

The forced oscillation wind tunnel test based on small amplitude and different frequencies can accurately identify the time constant τ in the unsteady model equation (2) and reveal its physical meaning [6]. We will Fourier decomposition of the aerodynamic force obtained from the small amplitude test (center angle of attack a0 and amplitude Δa) according to the dimensionless frequency ω of the test. Only considering the same frequency response of the aerodynamic force to the forced motion, the aerodynamic force obtained in the experiment can be treated as C(t)=C0(a0)+Ca(a0,ω)Δasin(ωt¯)+Ca(a0,ω)ωΔacos(ωt¯) C(t) = {C_0}({a_0}) + {C_a}({a_0},\omega)\,\Delta a\sin (\omega \bar t) + {C_a}({a_0},\omega)\,\omega \Delta a\cos (\omega \bar t)

The phase relationship with the angle of attack in the forced motion, Ca (a0, ω) is defined as the in-phase derivative. Ca (a0, ω) is the out-of-phase derivative [7]. Substitute the change equation a(t)a0Δasin(ωt¯) a(t) - {a_0}\Delta a\sin (\omega \bar t) of the angle of attack in the small amplitude forced oscillation test into the unsteady model equation (1) to linearize and merge the similar terms. We compare it with the in-phase/out-of-phase derivatives in equation (3) and get the following relationship: Ca(a0,ω)=Ca,att(a0)+ΔCa(a0)1+ω2τ2 {C_a}({a_0},\omega) = {C_{a,att}}({a_0}) + {{\Delta {C_a}({a_0})} \over {1 + {\omega ^2}{\tau ^2}}} Ca(a0,ω)=Cq(a0)+τΔCa(a0)1+ω2τ2 {C_a}({a_0},\omega) = {C_q}({a_0}) + {{\tau \Delta {C_a}({a_0})} \over {1 + {\omega ^2}{\tau ^2}}}

From equation (4) and equation (5), it can be seen that the in-phase derivative is not equal to the static derivative, and the out-of-phase derivative is not equal to the dynamic derivative. Both of them add an item containing τ and Δ Ca related to unsteady aerodynamic forces. Only when tends to zero, the in-phase derivative is equal to the static derivative Ca,st(a0) = Ca,att(a0) + ΔCa(a0), and the out-of-phase derivative is equal to the dynamic derivative Cq(a0). We combine formula (4) and formula (5) into Ca(a0,ω)=τCa(a0,ω)+(Cq(a0)+τCa,att(a0)) {C_a}({a_0},\omega) = - \tau {C_a}({a_0},\omega) + ({C_q}({a_0}) + \tau {C_{a,att}}({a_0}))

If the unsteady aerodynamic response approximately has first-order linear dynamic characteristics, the data Ca(a)0, ω), Cȧ(a0, ω)) composed of in-phase/out-of-phase derivatives of different frequencies should fall on the same straight line. The slope of the straight line is the key parameter −τ in the unsteady aerodynamic model. The results of the small-amplitude forced oscillation wind tunnel test of the 65° delta wing model with the front edge of the center body swept-back verify the above inference. Take the pitching moment coefficient Cm as an example [8]. The in-phase/out-of-phase derivative data points of the same central angle of attack and different frequencies within the range of 20°~56° angle of attack all fall near the same straight line (Figure 1). The consistency between the analysis results and the test results in the figure shows that the unsteady model formula (2) is simple in form but can describe the main unsteady phenomena more accurately. The straight line can be directly fitted by the least square method, and the corresponding time constant can be obtained. The greater the value, the greater the negative slope of the straight line in Figure 1. This shows that the unsteady hysteresis effect is more obvious. In the interval of the angle of attack of 30° ~ 48°, the data point spacing is large, and the fitting accuracy is high.

Figure 1

In-phase/out-of-phase derivatives at different center angles of attack and different frequencies

The calculated time constant τ at different center angles of attack (Figure 2). Outside the range of a < 20° and a > 50°, the time constant tends to zero, and the unsteady effect weakens. The above analysis shows that the in-phase derivative is approximately equal to the static derivative, and the out-of-phase derivative is approximately equal to the dynamic derivative [9]. The traditional linear superposition derivative aerodynamic model is established.

Figure 2

The variation curve of the dimensionless time constant with the angle of attack

In areas where the unsteady effect is significant, the time constant is far greater than zero and exhibits nonlinear characteristics as the central angle of attack changes. This reflects that different vortex systems dominate the unsteady aerodynamic load at different angles of attack. When the angle of attack is below 35°, the unsteady effect is mainly caused by the movement of the front body vortex relative to the airframe. In comparison, the unsteady characteristics of the aerodynamic load near the angle of attack of 45° are dominated by the vortex separation movement from the trailing edge [10]. The time constant describes the comprehensive response characteristics of different vortex systems to motion. Although only the dominant first-order mode is considered, it has higher accuracy. This method can avoid the identification of multiple state variables in complex flow fields.

The unsteady model formula (2) has a clear physical meaning. Its value is the dominant mode characteristic time of the unsteady hysteresis. This value quantitatively characterizes the significant degree of unsteady hysteresis. There is no unsteady aerodynamic phenomenon when it approaches zero. At this time, the conventional dynamic derivative aerodynamic model can be applied.

Identification of other parameters in the unsteady aerodynamic model

When the time constant τ (a) curve is obtained, the unknown parameters in equations (1) and (2) of the aerodynamic model have three curves, Catt(a), ΔC(a) and Cq(a). According to the equation ΔC(a) = Cst(a) − Catt(a), the following results can be obtained. Since the static coefficient Cst(a) is known, only one of them needs to be identified in Catt(a), ΔC(a). Since the dynamic derivative Cq(a) produced by the rotating flow field is known outside the range of 20°~56° angle of attack, it can be assumed that its value changes linearly within the range of 20° ~ 56° angle of attack. According to the physical meaning, we can conclude that its value is less than zero because the steady aerodynamic force corresponding to the dynamic derivative in this range of angle of attack is much smaller than the unsteady dynamic aerodynamic force. The above assumptions do not affect the accuracy of the model.

According to the above analysis, the parameters we need to identify are reduced to Catt (a) curves. The method can be divided into two kinds: 1) According to the assumption of non-separation of flow, the curve can be directly obtained by the CFD method or the Polhamus empirical estimation method. 2) Obtained by identifying the derivative Ca,att (a) of the small-amplitude oscillation data concerning the angle of attack and then integrating the angle of attack.

This article uses the 2nd) method. The article uses a gradient-based static optimization algorithm to obtain Ca,att (a). Taking a center angle of attack ai as an example, the corresponding static optimization objective function is J(Ca,att(ai))=ω(Ca(ai,ω)Ca,st(ai)+ω2τ2Ca,att(ai)1+ω2τ2)2min J({C_{a,att}}({a_i})) = \sum\limits_\omega {{{\left({{C_a}({a_i},\omega) - {{{C_{a,st}}({a_i}) + {\omega ^2}{\tau ^2}{C_{a,att}}({a_i})} \over {1 + {\omega ^2}{\tau ^2}}}} \right)}^2} \to \min}

Integrate the obtained Ca,st (ai) at each angle of attack with the angle of attack to obtain the Catt (a), Δ C (a) curve in the unsteady model. Take the pitch moment coefficient curve as an example for calculation. The calculation result is shown in Figure 3.

Figure 3

Identification results of parameters Cm,att (a) and Δ Cm (a)

In the range of angle of attack less than 20°, there is no obvious flow separation phenomenon, and Δ C (a) tends to zero; while in the range of angle of attack from 30° to 50°, the flow field topology changes significantly with the angle of attack. The absolute value of ΔCatt (a) increases rapidly and shows strong aerodynamic nonlinearity [11]. The unsteady aerodynamic effect at this time is also the most significant. When the angle of attack exceeds 50°, the flow field is completely separated and ΔC (a) does not change with the angle of attack.

After we have identified the two curves of Catt (a) and Cq (a), we need to filter the results further. This paper uses reasonable assumptions on the dynamic derivative to reduce the parameters identified to a curve. This method does not need to filter the identified results.

Comparison and verification of aerodynamic model and wind tunnel test data

We compared the predicted value of the aerodynamic model with the wind tunnel test time history data to verify the model's accuracy. When predicting aerodynamic forces, it is necessary to determine the initial value Cdyn (0) of the unsteady aerodynamic forces at the initial moment. This paper uses Cdyn (0) = Δ C (a0) as the initial value for calculation. The initial value Cdyn (0) of the periodic aerodynamic force is obtained by calculating and integrating more than 2 motion cycles by formula (2).

Because the unsteady model is directly based on the small-amplitude forced oscillation test results, the model prediction results are quite consistent with the small-amplitude test data. This paper uses this model to predict the aerodynamic force generated by the large-amplitude forced oscillation harmonic motion and compares it with the experimental data. At the same time, it is compared with the prediction results of the conventional linear superposition derivative model that depends on the angle of attack and the motion frequency to illustrate the effectiveness of the unsteady model. The linear superposition derivative model is Cm(t)=Cm,st(a)+Cma˙(a,ω)c¯ 2Va˙ {C_m}(t) = {C_{m,st}}(a) + {C_{m\dot a}}(a,\omega){{\bar c} \over {2V}}\dot a

Figure 4 compares the pitch aerodynamic moment coefficient test data (TD) of the 40° central angle of attack, 10° amplitude, and 1.0 Hz frequency of the medium amplitude forced oscillating wind tunnel with the predicted results of the dynamic derivative model (SAMR) and the unsteady model The prediction result (UAMR). The article also presents the static test results (STD) and the static aerodynamic parameter identification results (AFIR) under the assumption of vortex non-separation obtained by the identification. It can be seen from Fig. 4 that the unsteady model equation (1) predicts the experimental results well and presents a full hysteresis loop. In contrast, the results predicted by the dynamic derivative model are far from the experimental data. Not only does the hysteresis loop produced by it are slender in shape, but it also produces physically unexplainable spikes.

Figure 4

Comparison results of 10° amplitude value between steady/unsteady model and experiment

Figure 5 shows the comparison results of the model's large-amplitude motion test value with the model's predicted value and the 30° central angle of attack, 20° amplitude, and 1.0 Hz frequency. It can be seen from Figure 5 that the unsteady model equation (1) predicted results are very close to the experimental data during the increase of the angle of attack. However, there is a certain gap between the predicted results and the experimental data in reducing the angle of attack.

Figure 5

Comparison result of 20° amplitude value between steady/unsteady model and experiment

Comparing Figure 4 and Figure 5, it can be seen that the use of linear differential equations to establish an unsteady aerodynamic model better predicts the results of wind-tunnel tests with amplitudes below 10°. However, the prediction accuracy of the large-amplitude test has decreased. This shows that the model does not fully reflect the influence of motion amplitude on unsteady aerodynamic characteristics. When the aerodynamic load Cdyn is far from the equilibrium value ΔC, the comprehensive influence of the vortex system on the aerodynamic force changes. Linear differential equations describe its dynamic characteristics with certain errors.

Nonlinear correction of differential equation of the unsteady aerodynamic model

Based on the linear differential equation model, this paper introduces a nonlinear correction term to construct an unsteady aerodynamic model of the nonlinear differential equation. At the same time, the paper uses the results of large-amplitude oscillating wind tunnel tests to identify the nonlinear term coefficients.

Correction method

We rewrite equation (2) as dCdyndt¯=k1(a)(CdynΔC(a)) {{d{C_{dyn}}} \over {d\bar t}} = {k_1}(a)({C_{dyn}} - \Delta C(a))

In the formula: k1 (a) = − 1 / τ (a). We add the nonlinear third-order term k3 (a) (Cdyn − Δ C(a))3 for referring to the idea of gradual hardening/gradually softening spring on the correction of the linear elastic curve. In this way, the dynamic characteristics of Cdyn is far away from ΔC can be corrected. This method can ensure that the correction does not work when the instantaneous value Cdyn is close to the steady-state equilibrium value ΔC. It will not reduce the prediction accuracy of the unsteady aerodynamic force under the small amplitude of the model.

The longitudinal large-amplitude test shows the obvious difference in aerodynamic characteristics in different directions of the angle of attack. We use a different coefficient k3 (a) in the case of Cdyn ≥ Δ C(a) and Cdyn ≥ Δ C(a) to correct. The corrected differential equation is dCdyndt¯={k1(a)(CdynΔC(a))+k3(a)(CdynΔC(a))3CdynΔC(a)k1(a)(CdynΔC(a))+k3'(a)(CdynΔC(a))3Cdyn<ΔC(a) {{d{C_{dyn}}} \over {d\bar t}} = \left\{{\matrix{{{k_1}(a)({C_{dyn}} - \Delta C(a)) + {k_3}(a)\,{{({C_{dyn}} - \Delta C(a))}^3}{C_{dyn}} \ge \Delta C(a)} \hfill \cr {{k_1}(a)({C_{dyn}} - \Delta C(a)) + k_3^{'}(a)\,{{({C_{dyn}} - \Delta C(a))}^3}{C_{dyn}} < \Delta C(a)} \hfill \cr}} \right.

Global parameter identification based on genetic algorithm

The article uses the minimum sum of squares of the difference between the aerodynamic time data predicted by the model and the test data as the objective function to identify the nonlinear correction term k3 (a) curve. In the identification, we disperse the k3 (a) curve with the angle of attack. The number of independent variable parameters is 26. We use a genetic algorithm to perform global optimization to obtain the best k3 (a) identification result. The article uses a real number vector group as the gene coding method to realize the initialization, crossover, mutation, and evaluation operator based on the real number vector group under the unified framework of the genetic algorithm.

We use four sets of forced oscillation data with different central angles of attack, including 37° central angle of attack, 20° amplitude, and 1.2 Hz, as identification samples. The initial population of the genetic algorithm is 75, and the objective function converges after 20 generations of the genetic algorithm. Compared with the initial random search results, the fitting effect of the genetic algorithm is improved by 36%.

We substitute genetic algorithm identification result k3 (a) into equation (10). We use Δ = Cdyn − Δ C(a), and the angle of attack as dual independent variables, and the nonlinear differential equation (10) at the left end dCdyn/dt¯ d{C_{dyn}}/d\bar t constructs a curved surface (Figure 6). It can be seen from Figure 6 that by selecting the appropriate hybridization and compilation operators, the dynamic surface obtained by the global optimization of the genetic algorithm maintains the global smoothness, and there are no non-physical phenomena such as discontinuities and jumps. The literature established a third-degree polynomial correction model at each central angle of attack and used the local static optimization method to fit separately. It cannot maintain the global smoothness of the entire differential equation surface.

Figure 6

Unsteady aerodynamic dynamic surface

Comparison of results

We compare the unsteady model with the 30° central angle of attack, 20° amplitude, and 1.0 Hz forced oscillation test data that have not been used for modeling. The result is shown in Figure 7. Results with the uncorrected model UAMREq. (1) Compared to the nonlinear correction result UAMREq. (3) The predicted results are more consistent with the experimental data.

Figure 7

Linear/nonlinear unsteady model and steady model and test 20° amplitude value comparison results

Conclusion

(1) The article decomposes the high angle of aerodynamic attack force into two parts: steady and fast change and unsteady and slow change. At the same time, we use nonlinear differential equations to describe the unsteady aerodynamic dynamic characteristics, which can more accurately reflect the unsteady flow characteristics.

(2) The small-amplitude forced oscillation wind tunnel test data can accurately identify the time constant τ and the aerodynamic increment coefficient in the unsteady aerodynamic model of the linear differential equation. At the same time, it can reflect the main characteristics of unsteady aerodynamics. The established aerodynamic model is simple and effective. It can more accurately describe the aerodynamic characteristics under different maneuvering conditions and can be applied to research on flight dynamics analysis, flight simulation, and control law design.

Figure 1

In-phase/out-of-phase derivatives at different center angles of attack and different frequencies
In-phase/out-of-phase derivatives at different center angles of attack and different frequencies

Figure 2

The variation curve of the dimensionless time constant with the angle of attack
The variation curve of the dimensionless time constant with the angle of attack

Figure 3

Identification results of parameters Cm,att (a) and Δ Cm (a)
Identification results of parameters Cm,att (a) and Δ Cm (a)

Figure 4

Comparison results of 10° amplitude value between steady/unsteady model and experiment
Comparison results of 10° amplitude value between steady/unsteady model and experiment

Figure 5

Comparison result of 20° amplitude value between steady/unsteady model and experiment
Comparison result of 20° amplitude value between steady/unsteady model and experiment

Figure 6

Unsteady aerodynamic dynamic surface
Unsteady aerodynamic dynamic surface

Figure 7

Linear/nonlinear unsteady model and steady model and test 20° amplitude value comparison results
Linear/nonlinear unsteady model and steady model and test 20° amplitude value comparison results

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