CN111814256A - RK4Henon method-based wing aeroelastic system chaotic response analysis method - Google Patents

RK4Henon method-based wing aeroelastic system chaotic response analysis method Download PDF

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CN111814256A
CN111814256A CN202010688393.5A CN202010688393A CN111814256A CN 111814256 A CN111814256 A CN 111814256A CN 202010688393 A CN202010688393 A CN 202010688393A CN 111814256 A CN111814256 A CN 111814256A
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rk4henon
wing
chaos
transient
chaotic
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谢丹
车驰
冀春秀
陈翛然
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Northwestern Polytechnical University
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/15—Vehicle, aircraft or watercraft design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06F2119/12—Timing analysis or timing optimisation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention discloses a method for analyzing wing aeroelastic system chaos response based on a RK4Henon method, which combines a RK4 numerical integration method with a Henon method which is good at handling discontinuous problems, provides the RK4Henon method capable of capturing chaos and transient chaos better, can perform numerical integration with smaller time step, improves the calculation efficiency and has higher accuracy. The RK4Henon method, the conventional RK4 method and the conventional RP can be approximated to have satisfactory solving precision on limit cycle motion and chaotic motion, but when transient chaotic motion occurs in the system, only the RK4Henon method can be accurately captured, and the invention discovers that the maximum Lyapunov exponent which rapidly decreases along with time is an effective method for accurately and efficiently judging whether transient chaos exists in the system.

Description

RK4Henon method-based wing aeroelastic system chaotic response analysis method
Technical Field
The invention belongs to the field of solving of nonlinear aeroelastic systems, and particularly relates to a method for analyzing wing aeroelastic system chaotic response based on a RK4Henon method.
Background
Gap nonlinearity is one of the concentrated nonlinearities often encountered in engineering, of which the aeroelastic wing is a typical representative. A large number of researches show that the binary wing with gap nonlinearity can present nonlinear aeroelastic responses such as limit ring oscillation (LCO), chaos, transient chaos and the like, and the flutter critical value of the binary wing is often lower than the linear flutter critical value, so that the accurate solving of the nonlinear aeroelastic response of the wing is a necessary premise for nonlinear flutter analysis of the wing.
The above studies mainly adopt an approximate analysis method and a numerical integration method as the research theories. Numerical integration methods, such as the fourth order Runge Kutta method (RK4), are very effective in solving the nonlinear dynamics problem. Compared with a harmonic balance method, a perturbation method and a time domain coordination method, the RK4 method is simpler and easier to use and can be used for analyzing chaotic response, and other methods are only limited to solving periodic response. However, for kinetic systems with discontinuous non-linearity, the RK method may not be able to accurately capture the discontinuous points caused by gap non-linearity. While the Henon method can locate the position of the switching point in the gap model. Another strategy to deal with discontinuous non-linearities is to fit the discontinuous non-linearities with an appropriate Rational Polynomial (RP). However, the existing research has not been able to determine whether the RP method can be used to accurately capture transient chaotic responses.
Disclosure of Invention
The invention aims to overcome the defects and provides a method for analyzing the chaotic response of a wing aeroelastic system based on an RK4Henon method, wherein a RK4Henon method capable of accurately capturing periodic, chaotic and transient chaotic responses is formed by combining a classical RK4 method and the Henon method, and the capacity of analyzing the transient chaotic by the RP method is explored by comparing an RP approximation method with the RK4Henon method.
In order to achieve the above object, the present invention comprises the steps of:
step one, dimensionless processing is carried out on a motion equation of a wing, and a binary wing section mathematical model is established;
step two, adopting a RK4Henon method to solve a binary wing section mathematical model to complete the integral of the whole time sequence;
and thirdly, calculating the maximum Lyapunov index in the time sequence, and predicting the long-period transient chaos through the evolution process of the maximum Lyapunov index in the time domain.
In the first step, the constraint springs in the pitching direction of the wings are gap nonlinear, the sinking and floating directions are linear springs, the pitching stiffness coefficient is determined, so that linear aerodynamic force and aerodynamic moment are obtained, dimensionless motion equations of the wings are converted into a set of second-order ordinary differential equations, and then the two complementary equations are combined to form a binary wing section mathematical model.
The specific method of the second step is as follows:
step one, continuously integrating a binary wing section mathematical model by adopting a Henon method until a linear subdomain changes;
secondly, taking time as an independent variable, and integrating a new binary wing section mathematical model in a new time domain by adopting an RK4 method based on known initial conditions until a discontinuous point is encountered again;
and thirdly, repeating the first step and the second step until the integration of the whole time series is completed.
The RK4Henon method is an integration method with variable step sizes, which are changed each time a switching point is crossed.
Compared with the prior art, the RK4 numerical integration method is combined with the Henon method which is good at processing discontinuous problems, the RK4Henon method which can better capture chaos and transient chaos is provided, numerical integration can be carried out in a smaller time step, the calculation efficiency is improved, and the accuracy is higher. The RK4Henon method, the conventional RK4 method and the conventional RP can be approximated to have satisfactory solving precision on limit cycle motion and chaotic motion, but when transient chaotic motion occurs in the system, only the RK4Henon method can be accurately captured, and the invention discovers that the maximum Lyapunov exponent which rapidly decreases along with time is an effective method for accurately and efficiently judging whether transient chaos exists in the system.
Drawings
FIG. 1 is a schematic representation of a binary wing section geometry;
FIG. 2 is a gap non-linearity diagram of a pitch stiffness coefficient;
FIG. 3 illustrates the mechanism of error generation by conventional integration;
FIG. 4 is a drawing showing
Figure BDA0002588448740000031
Comparison of RK4 method under conditions and the method of the invention; wherein, (a) is a time-course diagram, and (b) is a phase plane diagram;
FIG. 5 is a drawing showing
Figure BDA0002588448740000032
Time profiles of different Δ τ under the conditions;
FIG. 6 is a schematic view of
Figure BDA0002588448740000033
Comparison of RK4 with RK4Henon under conditions; wherein, (a) is a phase diagram, and (b) is a poincare mapping diagram;
FIG. 7 is a schematic view of
Figure BDA0002588448740000034
Schematic representation of the RK4Henon method after complete elimination of transients under conditions; wherein, (a) is a phase diagram; (b) is a poincare map;
FIG. 8 is a schematic view of
Figure BDA0002588448740000035
Under the condition, after transient is completely eliminated, obtaining an LLE time-varying curve by an RK4Henon method;
FIG. 9 is a schematic view of
Figure BDA0002588448740000036
A limit cycle response map of pitch motion under conditions; wherein, (a) is a time-course diagram of an RP approximation method, (b) is a phase diagram of the RP approximation method, (c) is a time-course diagram of an RK4Henon method, and (d) is a phase diagram of the RK4Henon method;
FIG. 10 is a schematic view of
Figure BDA0002588448740000037
A chaotic response map of pitching motion under the condition; wherein, (a) is a time-course diagram, (b) is a phase diagram, and (c) isPoincare map, (d) is amplitude-frequency response curve;
FIG. 11 is a schematic view of
Figure BDA0002588448740000038
Transient chaotic response graph of pitching motion under the condition; wherein, a) is a time-course diagram, (b) is a phase diagram, and (c) is a poincare mapping diagram;
FIG. 12 is a drawing showing
Figure BDA0002588448740000039
Under the condition, a time step convergence analysis chart of an RK4Henon method of delta tau being 0.01,0.1,0.2 and 0.4; wherein, (a) is a time-course diagram, and (b) is a phase diagram;
fig. 13 is a diagram of a pitching motion bifurcation corresponding to embodiment 1 in table 1;
FIG. 14 is a drawing showing
Figure BDA00025884487400000310
Schematic under conditions; wherein, (a) is Poincare mapping, (b) is amplitude spectrum, and (c) is LLE time variation graph;
FIG. 15 is a drawing showing
Figure BDA0002588448740000041
A time course diagram of pitching motion under the condition; wherein (a) and (b) are RK4 process and (c) and (d) are RK4Henon process.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The method adopts the RK4Henon method to carry out chaotic response analysis on the wing aeroelastic system containing gap nonlinearity, and finds the transient chaotic phenomenon in the system for the first time. The solution analysis chaos and transient chaos response of the RP approximation method and the RK4Henon method are compared. In view of the characteristics of transient chaos, a method for simply and efficiently judging whether transient chaos exists based on the time response process of the maximum Lyapunov exponent (LLE) is introduced. Finally, the influence of the RK4 method and the RK4Henon method on the periodic response solving precision is analyzed through comparison.
Fig. 1 is a schematic diagram of a dual-element wing geometry with simultaneous pitch and heave motions and gap nonlinearity in the pitch degree of freedom, wherein the pitch angle α is positive for head-up and the heave displacement h is positive downward.
Step one, modeling a binary wing 6-ODE with gap nonlinearity;
considering the gap nonlinearity in the pitch direction, the equation of motion for the wing has the following dimensionless form:
Figure BDA0002588448740000042
Figure BDA0002588448740000043
wherein, the restraining spring in the pitching direction is a gap nonlinear spring, and the sinking and floating direction is a linear spring. Therefore, the spring rate coefficients M (α) and G (ξ) thereof are expressed as
Figure BDA0002588448740000044
α>αf+
In fact, the nonlinearity exhibited by the pitch stiffness coefficient M (α) in equation 2 is generally referred to as double nonlinearity, as shown in fig. 2. Here, in order to ensure the symmetry of the gap, take
Figure BDA0002588448740000051
Two auxiliary variables were introduced:
Figure BDA0002588448740000052
Figure BDA0002588448740000053
linear aerodynamic force CL(τ) and aerodynamic moment CM(τ) can be expressed as:
Figure BDA0002588448740000054
Figure BDA0002588448740000055
thus, by adding the variable y1And y2The system of equation 1 is converted into a set of second order Ordinary Differential Equations (ODE). To ensure the integrity of this system, two complementary equations should be constructed. The relationship of differential equations 3a and 3b to τ satisfies:
Figure BDA0002588448740000056
Figure BDA0002588448740000057
equation 5 is two complementary first order ODEs. Thus, the equation of motion for the original binary wing consists of two second order ODEs and two first order ODEs. Introduction of x1＝α,
Figure BDA0002588448740000058
x3＝ξ,
Figure BDA0002588448740000059
x5＝y1And x6＝y2It is expressed as a state space form:
Figure BDA00025884487400000510
thus obtaining the binary wing section mathematical model consisting of six first-order ODEs.
Step two, adopting an RK4Henon method to solve the 6-ODE system;
as shown in fig. 3, the discontinuous non-linearity due to the numerical error caused by the intersection of the integration steps limits the application of the classical RK4 method in periodic motion analysis. Therefore, the RK4Henon method is adopted to solve the 6-ODE equation of the binary wing section.
First, the current system is continuously integrated using the Henon method until the linear sub-domain changes (a discontinuity is encountered). Since the extent of the new subdomain in the integrated system is known, the exact discontinuity can be extrapolated from the current position of the system by simply exchanging the dependent variable α and the independent variable τ. Then, time is again used as an argument and the classical RK4 method is used to integrate the new subsystem in the new time domain based on the known initial conditions until the discontinuity is encountered again. And repeating the above processes to complete the integration in the whole time domain range.
The original system (6) can be written in the form of a state space:
Figure BDA0002588448740000061
in the process of applying the Henon method, the independent variable tau and the marked dependent variable x are involved1The exchange is carried out by the following specific method: each equation in equation (7) is first divided by dx1/dτ＝f1(x) Replacing the first equation with d τ/dx1＝1/f1(x) In that respect Thus, in x1The new system as an argument can be expressed as:
Figure BDA0002588448740000062
note that the new system is only aware of the variable x1An integration step is carried out immediately if the specified value is exceeded. The RK4Henon method is essentially a step-variable integration method, with step-changes occurring whenever a switching point is crossed.
Firstly, comparing and analyzing a classical RK4 method and an RK4Henon method;
(1) comparative analysis of LCO
FIG. 4 shows
Figure BDA0002588448740000063
Under the condition, the RK4Henon method and the RK4 method are respectively adopted to analyze the LCO movement of the wing, and the displacement time-course diagram and the phase plan diagram obtained by the two methods are well matched. However, in a close-up view in FIG. 4b,the phase trajectory obtained by the RK4 method contains many loops in a limited frequency band, whereas the phase trajectory obtained by the RK4Henon method falls in only one of the single loops. This indicates that the RK4Henon method is numerically more stable and accurate than the RK4 method. In addition, in the numerical integration process, the integration time step τ of RK4 method must be small enough to produce more accurate results. Moreover, the value of τ can only be determined if the solutions obtained from τ and τ/2, respectively, are essentially the same, which is a cumbersome process that would not be possible in case of solving for chaotic responses. Thus, for limit cycle analysis, the RK4Henon method represents a significant advantage over classical RK4 in both accuracy and computational efficiency.
(2) Chaotic response contrast analysis
A quantitative method to identify chaotic motion is to calculate the Largest Lyapunov Exponent (LLE) in the time series. A positive LLE indicates that the motion is chaotic, while a non-positive LLE indicates that the system is moving regularly.
FIG. 5 shows the method of RK4Henon and the method of RK4 with different integration steps
Figure BDA0002588448740000071
Time-course diagram obtained under the condition. A short time series of chaotic motion is shown, which, although exhibiting similar periodicity over the time range, is still chaotic over the entire time period. The comparison results in FIG. 5 show that the RK4 method gets closer to the calculation result of the RK4Henon method as the time step size decreases. In fact, the RK4 results for Δ τ ═ 0.0001 are closer to RK4Henon results for Δ τ ═ 0.1 (not shown). This indicates that the RK4Henon method with Δ τ ═ 0.1 gives more accurate results than the RK4 method with Δ τ ═ 0.001. Thus, it can be concluded that: compared with the conventional RK4 method, the RK4Henon method has remarkable advantages in accuracy and computational efficiency. It is emphasized that even small numerical errors in the analysis of chaotic motion lead to completely different solutions over a long period of time. Therefore, the RK4Henon method is the optimal choice for chaotic response analysis.
(3) Transient chaos contrast analysis
FIG. 6 shows RK4 and RK4Henon method
Figure BDA0002588448740000072
Phase maps and poincare maps under the conditions. The two subgraphs show that the results of the two methods both show that the wings do chaotic motion. However, by careful observation, the distribution of poincare map points of the RK4Henon method is much more sparse than the distribution of the RK4 method. In fact, the number of integration steps performed by the two methods is the same (2 × 106 integration steps), which means that in the calculation result of the RK4Henon method, a large number of points actually fall repeatedly at one or a few positions, and the characteristic is specific to periodic motion, so the motion is presumed to be a periodic motion with a long transient chaotic process and finally stable to a periodic solution.
FIG. 7 shows the process without the initial 1.5X 104And the phase diagram and the poincare mapping diagram are redrawn after response, so that the influence of transient chaos is completely eliminated. Both the phase diagram of fig. 7a and the poincare diagram of fig. 7b show that the response is a two times periodic LCO. At present, a great deal of research shows that transient chaos is difficult to identify, and due to a long chaotic transient process, the transient chaos is easily mistakenly judged as chaotic motion under the condition of short calculation time. For the transient chaotic motion with a super-long period, the calculation time will increase the calculation cost to a great extent to eliminate the influence of the transient process.
Therefore, the invention provides a more accurate and efficient judgment method, namely, the change trend of the LLE along with the time is utilized to judge whether the response is chaotic or transient chaotic. FIG. 8 shows that
Figure BDA0002588448740000081
Time-varying curves of the pitch LLE calculated by the RK4Henon method under the conditions. It can be seen that the LLE time course curve begins to show a rapid descending trend, and when the calculation time is long enough, LLE finally stabilizes to zero, indicating that the system does periodic motion. The initial rapid downward trend is due to transient chaos corresponding to a positive LLE value, and the final periodic motion corresponds to a zero value, which decreases from positive to zero over time. It was therefore concluded that: gradually decreaseA small LLE time history means that transient chaos exists before the system enters periodic motion. This discovery provides a simple and effective tool for determining whether random-like motion is chaotic or transient chaotic.
Secondly, comparing and analyzing an RP approximation method and an RK4Henon method;
the RP approximation method comprises the following steps: the gap nonlinear stiffness coefficient M (alpha) is firstly approximated by a perfectly-fitted rational polynomial, and then a continuous dynamic system is generated by classical RK4 integration. The invention adopts a high-precision RK4Henon method to directly integrate an original gap-containing nonlinear dynamic system.
And comparing the precision and the efficiency of the two methods respectively from the limit cycle analysis and the chaos analysis and the transient chaos response analysis. FIG. 9 shows that
Figure BDA0002588448740000082
Under the condition, time-course diagrams and phase diagrams are obtained by an RP approximation method and an RK4Henon method. The results are consistent and indicate that the system is in LCO motion. The contrast difference in the figures is only because the wing aeroelastic system itself is a symmetric system, and the same initial conditions are likely to yield two symmetric solutions. Thus, for the analysis of LCO movement, the RP method is consistent with the RK4Henon method. Therefore, the RP approximation is reliable for LCO analysis.
FIG. 10 shows
Figure BDA0002588448740000091
Under the condition, the results of a time-course graph, a phase graph, a Poincare mapping graph and an amplitude-frequency response curve show that the system does chaotic motion, and the judgment conclusion of the two methods on the response form is consistent. Only slightly different in the chaotic response process, because chaotic motion is very sensitive to small changes of the system, the results obtained by two different methods are slightly different and acceptable. Therefore, the RP approximation is also reliable for chaotic response analysis.
FIG. 11 shows
Figure BDA0002588448740000092
ConditionComparison of the next two methods. Very interestingly, the RK4Henon method resulted in a long-lived time-course (up to 3.77X 10)4s) is stabilized to periodic motion after transient chaos, the poincare map of the method comprises two discrete points which represent 2 times periodic motion, and the transient chaos obtained by the RP approximation method is always chaotic response. It was therefore concluded that: the RP approximation cannot be used to capture long-period transient chaos and stabilized periodic motion.
Table 1 examples of system parameters
Figure BDA0002588448740000093
TABLE 2 parameters used in the calculation of LLE
Figure BDA0002588448740000094
Fig. 13 is a diagram of a bifurcation in pitching motion obtained by solving a two-dimensional wing section aeroelastic equation by using the RK4Henon method proposed by the present invention under the condition of example 1 in table 1. The flow rate variation step length is
Figure BDA0002588448740000101
The number of integration steps for each flow rate was 2X 105And the first 20% of the data is discarded to eliminate the effect of transients.
FIG. 14 is a schematic view of
Figure BDA0002588448740000102
In the case of (2), the calculation conditions of the poincare diagram, the amplitude spectrum, and the LLE time-varying diagram obtained by the RK4Henon method are shown in table 2. The poincare mapping is not a plurality of discrete points or a closed curve, the amplitude spectrum is distributed in a certain frequency band, the LLE is converged at about 0.009, and therefore, the system does weak chaotic motion under the condition.
FIG. 15 is a drawing showing
Figure BDA0002588448740000103
Comparison graph of pitch motion time courses obtained by RK4 method and RK4Henon method under the condition. The results show that RK4 method givesIs a chaotic motion, while the RK4Henon method obtains a chaotic motion which goes through a long period (about 1.12 multiplied by 10)4s) periodic motion after transient chaos. It is proved that the RK4Henon method is a necessary choice in capturing transient chaos phenomenon.

Claims (4)

1. The method for analyzing the chaotic response of the wing aeroelastic system based on the RK4Henon method is characterized by comprising the following steps of:
step one, dimensionless processing is carried out on a motion equation of a wing, and a binary wing section mathematical model is established;
step two, adopting a RK4Henon method to solve a binary wing section mathematical model to complete the integral of the whole time sequence;
and thirdly, calculating the maximum Lyapunov index in the time sequence, and predicting the long-period transient chaos through the evolution process of the maximum Lyapunov index in the time domain.
2. The method for analyzing the chaos response of the wing aeroelastic system based on the RK4Henon method as claimed in claim 1, wherein in the step one, the constraint springs in the pitching direction of the wing are nonlinear in clearance, the sinking and floating directions are linear springs, the pitching stiffness coefficient is determined, so as to obtain linear aerodynamic force and aerodynamic moment, the dimensionless motion equation of the wing is converted into a set of second order ordinary differential equations, and then two complementary equations are combined to form a binary wing section mathematical model.
3. The method for analyzing the chaos response of the wing aeroelastic system based on the RK4Henon method according to claim 1, wherein the specific method in the second step is as follows:
step one, continuously integrating a binary wing section mathematical model by adopting a Henon method until a linear subdomain changes;
secondly, taking time as an independent variable, and integrating a new binary wing section mathematical model in a new time domain by adopting an RK4 method based on known initial conditions until a discontinuous point is encountered again;
and thirdly, repeating the first step and the second step until the integration of the whole time series is completed.
4. The method for analyzing the chaos response of the wing aeroelastic system based on the RK4Henon method as claimed in claim 1, wherein the RK4Henon method is an integration method with variable step size, and the step size is changed when a switching point is crossed.
CN202010688393.5A 2020-07-16 2020-07-16 RK4Henon method-based wing aeroelastic system chaotic response analysis method Pending CN111814256A (en)

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Publication number Priority date Publication date Assignee Title
CN113865822A (en) * 2021-08-25 2021-12-31 华北电力大学 Wind tunnel test device and method for simulating aeroelastic response of wind power blade

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CN113865822A (en) * 2021-08-25 2021-12-31 华北电力大学 Wind tunnel test device and method for simulating aeroelastic response of wind power blade

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Application publication date: 20201023