CN111814255A - Method for obtaining physical period of aeroelastic system based on confusion-removing harmonic balancing method - Google Patents

Method for obtaining physical period of aeroelastic system based on confusion-removing harmonic balancing method Download PDF

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CN111814255A
CN111814255A CN202010688387.XA CN202010688387A CN111814255A CN 111814255 A CN111814255 A CN 111814255A CN 202010688387 A CN202010688387 A CN 202010688387A CN 111814255 A CN111814255 A CN 111814255A
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confusion
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谢丹
陈翛然
冀春秀
车驰
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Northwestern Polytechnical University
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Abstract

The invention discloses a method for obtaining the physical period of a pneumatic elastic system based on a confusion removal harmonic balancing method, which adopts a simple and effective confusion removal scheme to shift higher harmonic coefficients to correct low-order positions and properly scale the frequency, thereby greatly improving the probability of obtaining the understanding of a real object; the method solves the nonlinear algebraic equation set by adopting a scalar homotopy method, overcomes the defect that a classical Newton iteration method is too sensitive to an initial value, does not need to invert a Jacobian matrix, and improves the stability and the accuracy; the invention provides an effective and high-precision method DHB-GOIA (confusion removal harmonic balance method combined with a scalar same-wheel method) for solving the periodic object understanding of an aeroelastic system (autonomous system).

Description

Method for obtaining physical period of aeroelastic system based on confusion-removing harmonic balancing method
Technical Field
The invention relates to the field of solving of aeroelastic systems of aircrafts, in particular to a method for obtaining a physical period of an aeroelastic system based on a confusion-removing harmonic balancing method.
Background
The period solution is solved for the nonlinear two-degree-of-freedom power system, the traditional approximate method basically adopts a harmonic balance method, and a high-dimensional harmonic balance method is provided on the basis of the harmonic balance method to be suitable for complex response.
The harmonic balancing method is suitable for analyzing simple response and can obtain a high-precision solution with only a few harmonics. However, the number of harmonics involved in analyzing complex responses increases, and the expansion of the nonlinear terms into a fourier series requires a large amount of computation. The development of symbol calculation software enables a high-order harmonic balance method to be feasible when complex response is solved. The single harmonic balancing method and the fast harmonic balancing method are used for analyzing the two-degree-of-freedom aeroelastic dynamic system. The harmonic balancing method is effective for non-autonomous systems subjected to external periodic forces. However, a mathematical confusion may arise in solving autonomous systems.
High-dimensional harmonic balancing can effectively improve computational efficiency, but can produce non-physical understanding due to physical confusion. The existing technology significantly suppresses the generation of non-matter understanding and improves the accuracy by introducing means such as a frequency filtering technology, a spectral viscosity operator, a time domain coordinate method and the like.
Disclosure of Invention
The invention aims to overcome the defects, provides a method for obtaining the physical period of the aeroelastic power system based on a confusion-removing harmonic balancing method, and achieves the aim of effectively inhibiting mathematical confusion on the premise of meeting the requirements of calculation efficiency, accuracy and stability so as to obtain the period solution of the aeroelastic power system.
In order to achieve the above object, the present invention comprises the steps of:
constructing a nonlinear algebraic equation set of a two-degree-of-freedom airfoil aeroelastic system based on a harmonic balance method to obtain a harmonic balance algebraic system;
step two, constructing a confusion removal scheme for the harmonic balance algebraic system;
and step three, solving a nonlinear algebraic equation system of the harmonic balance algebraic system to obtain the physical period of the dynamic elastic system.
The specific method for constructing the nonlinear algebraic equation system in the first step is as follows:
and converting the two-degree-of-freedom airfoil aeroelastic system into a nonlinear ordinary differential equation set, and performing trial solution in a Fourier expansion form to obtain a harmonic balance algebraic system.
In the second step, the method for constructing the confusion removing scheme is as follows:
firstly, applying a non-linear algebraic equation system solver to a non-linear algebraic equation system to obtain a result
Figure BDA0002588447200000021
If it is
Figure BDA0002588447200000022
If the primary main harmonic exists, the solution is a real solution; otherwise, carrying out the second step;
secondly, calculating a harmonic amplitude vector and searching a first non-zero element;
thirdly, appointing x as the initial value of the non-linear algebraic equation system solver to calculate new
Figure BDA0002588447200000023
If it is not
Figure BDA0002588447200000024
Is the primary harmonic, then
Figure BDA0002588447200000025
Is a true solution; otherwise, repeating the second step and the third step until a true solution is obtained.
The concrete method of the third step is as follows:
converting the nonlinear algebraic equation system into Newton homotopy functions, and obtaining generalized scalar Newton homotopy functions by the aid of the drawn-up virtual time function Newton homotopy functions;
taking a constructed equivalent power system as a consistency condition, differentiating a generalized scalar Newton homotopy function relative to time t, converting the generalized scalar Newton homotopy function into an ordinary differential equation set, then separating the power system, and discretizing the ordinary differential equation set into a time dynamic system through a forward Euler method to obtain a scalar equation;
and determining the convergence quantity of the iterative Newton homotopy method to obtain the physical period of the dynamic elastic system.
Compared with the prior art, the method adopts a simple and effective confusion removing scheme, shifts the higher harmonic coefficient to a correct low-order position and properly scales the frequency, and greatly improves the probability of obtaining the understanding of a real object; the method solves the nonlinear algebraic equation set by adopting a scalar homotopy method, overcomes the defect that a classical Newton iteration method is too sensitive to an initial value, does not need to invert a Jacobian matrix, and improves the stability and the accuracy; the invention provides an effective and high-precision method DHB-GOIA (confusion removal harmonic balance method combined with a scalar same-wheel method) for solving the periodic object understanding of an aeroelastic system (autonomous system).
Drawings
FIG. 1 is a two degree of freedom airfoil geometry schematic;
FIG. 2 is a flow diagram of an implementation of a defrobbing scheme;
FIG. 3 is a response curve of frequency versus flow rate calculated by the speed boost routine using the fourth-order Runge-Kutta method;
FIG. 4 is a graph of pitch angle versus time for flow velocity near the principal bifurcation;
FIG. 5 is a phase plot of pitch angle versus time for flow velocity near the principal bifurcation;
FIG. 6 is a schematic diagram of frequency ω obtained by a fourth-order Runge-Kutta method, a third-order harmonic balancing method, and a third-order confusion-free harmonic balancing method;
FIG. 7 is a drawing showing
Figure BDA0002588447200000031
Then, the Monte Carlo simulation results of the third-order harmonic balance method and the third-order confusion-removing harmonic balance method; wherein (a) is HB3, and (b) is DHB 3;
FIG. 8 is a drawing showing
Figure BDA0002588447200000032
Then, the Monte Carlo simulation results of the third-order harmonic balance method and the third-order confusion-removing harmonic balance method; wherein (a) is HB3, and (b) is DHB 3;
FIG. 9 is a graph of the iterative residuals of the NR method and the OVDA method;
FIG. 10 is a graph of the iterative residual of the NR method
FIG. 11 is a schematic view of
Figure BDA0002588447200000033
Phase diagram of time.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1, the two-degree-of-freedom airfoil aeroelastic system aimed by the invention can be used for parameter analysis and energy capture problems of airfoil profiles. The dimensionless motion equation of the two-freedom-degree airfoil in subsonic airflow is as follows:
Figure BDA0002588447200000041
wherein: u is the incoming flow velocity, rαIs the radius of gyration about the elastic axis, alpha is the pitch angle, beta, gamma is the coefficient of the pitch and heave cubic springs, S is the static moment of the wing about the elastic axis, mu-m/pi ρ b2Is the wing-to-gas mass ratio, ω is the fundamental circular frequency of motion, ωξAnd ωαThe natural frequencies of pitch and heave motions respectively,
Figure BDA0002588447200000042
∈1and e2Is the Vainer equation constant, ζαAnd ζξIs the pitch and heave viscous damping ratio, U*＝U/bωαIs dimensionless speed, tau is dimensionless time, xi is h/b is dimensionless sinking and floating deflection,
Figure BDA0002588447200000046
is the dimensionless linear flutter velocity.
(& gtThe) represents the derivative to dimensionless time tau. P (τ) and Q (τ) are the drive force and moment, respectively. The nonlinear pitch and heave stiffness term is M (alpha) ═ alpha + beta alpha3，G(ξ)＝ξ+γξ3. Wagner equation
Figure BDA0002588447200000043
Wherein psi1,2＝0.165,0.335。
The invention only considers periodic motion and assumes no external force, and the nonlinearity of the pitching degree of freedom is far stronger than the sinking and floating degree of freedom, therefore, the invention only considers the nonlinear condition of the pitching motion, and the system (1) can be converted into a nonlinear ordinary differential equation set:
Figure BDA0002588447200000044
the method comprises the following steps: constructing a nonlinear algebraic equation system based on a harmonic balance method;
first, a trial solution in the form of a fourier expansion is set. The approximate solution of the six variables in equation (2) is:
Figure BDA0002588447200000045
all coefficient variables are written in vector form:
Figure BDA0002588447200000051
substituting the trial solution (3) into the control equation (2) and balancing the fourier coefficients of the first N harmonics yields:
Figure BDA0002588447200000052
wherein I is an identity matrix, A and MαGiven by:
Figure BDA0002588447200000053
will be provided with
Figure BDA0002588447200000057
Substituting into the first two equations in (5) to eliminate
Figure BDA0002588447200000054
Obtaining:
Figure BDA0002588447200000055
solving Q from the first equation of equation (7)ξAnd substituting the harmonic balance algebraic system into a second equation:
Figure BDA0002588447200000056
for periodic response, the phase of the first harmonic of the pitching motion can be fixed, and alpha is specified in order to obtain a defined system1＝0。
Step two: constructing a confusion removal scheme;
the harmonic balance algebraic system (8) can be written as a general formula
F(x)＝0 (9)
Wherein F ∈ R2N+1，x∈R2N+2And x isi+1＝αi-11, 2N + 1. Assuming (α, ω) is a real solution satisfying equation (9), the harmonic amplitude AkShould have a downward trend wherein
Figure BDA0002588447200000061
The assumed approximate solution should have the form:
Figure BDA0002588447200000062
m is marked as a scale factor for frequency. When m is 2, the solution x of the nonlinear algebraic equation system is:
Figure BDA0002588447200000063
the steps to eliminate the mathematical confusion solution are (see fig. 2):
the method comprises the following steps: non-linear generationApplication of numerical equation system solver to nonlinear algebraic equation system to obtain result
Figure BDA0002588447200000064
If it is
Figure BDA0002588447200000065
With a primary main harmonic, i.e.
Figure BDA0002588447200000066
Then it is the true solution; otherwise, go to step two.
Secondly, the step of: the harmonic magnitude vector is calculated and the first non-zero element is found. Assuming that the first non-zero element is the ith element, the scale factor m-i, when p-1, 2
Figure BDA0002588447200000067
Is assigned to alpha2p-1And alpha2pWherein N is determined so that 2Nm-1 is equal to or less than
Figure BDA0002588447200000068
Length of (d). Unassigned elements in x, i.e. alpha2p+1，α2p+2，…，α2NIs set to zero. Then will be
Figure BDA0002588447200000069
Is assigned to the value of omega,
Figure BDA00025884472000000610
is assigned to alpha0。
③: assigning x as the initial value of the non-linear algebraic equations solver to compute a new
Figure BDA00025884472000000611
If it is not
Figure BDA00025884472000000612
Is the primary harmonic, then
Figure BDA00025884472000000613
Is a true solution;otherwise, repeating the step (II) and the step (III) until a real solution is obtained.
Step three: solving a nonlinear algebraic equation system;
the newton homotopy function can be written as follows:
Figure BDA0002588447200000071
equation (13) holds for t ∈ [0,1 ]. Then introducing the formulated virtual time function Q (t) into equation (13) to obtain a generalized scalar Newton homotopy function:
Figure BDA0002588447200000072
where the time function Q (t), t ∈ [0, ∞) must satisfy Q (0) ═ 1, Q (t) is a monotonically increasing function, Q (∞) ∞. Using the virtual time function q (t), when the virtual time t is 0, it can be obtained
Figure BDA0002588447200000073
When t is infinity, can be obtained
Figure BDA0002588447200000074
To construct an equivalent power system, equation (14) is differentiated with respect to time t and x ═ x (t) is obtained as a consistency condition
Figure BDA0002588447200000075
To convert the original nonlinear algebraic equation system into an ordinary differential equation system, assume:
Figure BDA0002588447200000076
where λ is a scalar. Then separating the power system, and discretizing into a time dynamics system through a forward Euler method:
Figure BDA0002588447200000077
where β ═ q (t) Δ t,
Figure BDA0002588447200000078
f is then derived over t to yield:
Figure BDA0002588447200000079
wherein A ═ BBT. Similarly, equation (18) is integrated using the forward Euler format:
Figure BDA00025884472000000710
consider that equation (15) is a constant manifold in time and makes
Figure BDA0002588447200000081
Can obtain
Figure BDA0002588447200000082
The following scalar equation can be obtained by integration:
a0β2-2β+1-s＝0 (21)
wherein
Figure BDA0002588447200000083
s can be used as a quantity to evaluate the convergence of the iterative newton homotopy method. To ensure 1- (1-s) a0Greater than or equal to 0, order
Figure BDA0002588447200000084
The following algorithm can be obtained
Figure BDA0002588447200000085
Where 0< γ <1 is a parameter that needs to be specified. Equation (24) is a general form of scalar homotopy.
Example (b):
and carrying out numerical simulation of pitching and sinking-floating vibration aiming at the two-degree-of-freedom subsonic wing dynamic model. The parameters are shown in Table 1:
TABLE 1 System parameters
Figure BDA0002588447200000086
For the parameters in Table 1, linear flutter velocity
Figure BDA0002588447200000087
Flow rate U relative to U*And (6) standardizing. FIG. 3 is a response curve of frequency versus flow rate calculated using RK 4. FIG. 3 shows the forward propulsion
Figure BDA0002588447200000088
In and propelled backwards
Figure BDA0002588447200000089
There is a secondary bifurcation, which means a hysteresis region [1.84,2.35 ]]Are present. There are two stable solutions to the flow rate in the lag region. Under external disturbances, there is a possibility that a jump phenomenon occurs.
Fig. 4 and 5 show the pitch angle versus time curves and phase diagrams of the flow velocity near the main bifurcation.
Figure BDA0002588447200000091
The amplitude of the pitching motion approaches zero in 1000 time units.
Figure BDA0002588447200000092
The pitch response decreases very slowly.
Figure BDA0002588447200000093
In time, a small amplitude limit ring oscillation occurs. First of allThe bifurcation is a supercritical Hopf bifurcation with a bifurcation value between 1.00 and 1.01. The second branch is a subcritical Hopf bifurcation, in which a jump phenomenon and a hysteresis phenomenon occur.
1. Suppression of mathematical confusion solutions
In fig. 6, HBn represents the nth order harmonic balancing method, and DHBn represents the nth order antialiasing harmonic balancing method. Monte carlo simulations were used in the calculations. Illustrated by the bifurcation diagram (RK4 calculation) in FIG. 6
Figure BDA0002588447200000094
And 2.0 there are one and two stable solutions, respectively.
Figure BDA0002588447200000095
The upper solution of the three solutions of (a) is a stable real solution (ω ═ 0.0724). The middle solution (ω ═ 0.0387) and the bottom solution (ω ═ 0.0258) are mathematically confused solutions of unstable solutions, with a frequency ω of 0.0774. RK4 only produces a stable response, whereas harmonic balancing can predict all possible steady state solutions.
Figure BDA0002588447200000096
Solutions of (c) are ω ═ 0.0670 (true solution), 0.0379 (confusion solution) and 0.0253 (confusion solution). The high-dimensional harmonic balancing method can produce both biological and non-biological understanding. The antialiasing harmonic balancing method only produces a physical understanding, which may be a steady-state solution and a non-steady-state solution, but may suppress a mathematical confusion solution.
And (3) verifying the effect of the confusion removing scheme by using a third-order confusion removing harmonic balance method. FIG. 7 is a drawing showing
Figure BDA0002588447200000097
Results of monte carlo simulations of the methods HB3 and DHB 3. As shown in fig. 7(a), "stagnation", "ω 1", "ω 2", "ω 3", the probabilities of the four possibilities are 9.2%, 48.0%, 24.3% and 18.5%, respectively, where ω 3 is the true solution. Fig. 7(b) shows the results of the third-order confusion-free harmonic balance method "stagnation" and "ω 3", with probabilities of 4.4% and 95.6%, respectively. FIG. 8 is
Figure BDA0002588447200000098
The result shows that the third-order confusion removal harmonic balance method only generates a real solution, and the third-order harmonic balance method generates a real solution and a confusion solution.
Convergence of DHB-SHM
The performance of the antialiasing harmonic balancing method in combination with the newton iteration method (DHB-NR) and the antialiasing harmonic balancing method in combination with the scalar homotopy method (DHB-SHM) was evaluated by numerical simulations. Shown in FIG. 9 is
Figure BDA0002588447200000099
Residual errors generated by iteration steps of the DHB6-NR and DHB6-OVDA methods are corrected to a stop criterion of 10-8. The OVDA method is one of scalar homologies. DHB6-OVDA globally converges, satisfying the convergence condition, while DHB6-NR fails to converge. FIG. 10 shows the convergence history of DHB6-NR on the linear abscissa axis. It can be seen that the robustness of the DHB-SHM method is superior to that of the DHB-NR method.
Accuracy of DHB-SHM
As shown in fig. 11, is
Figure BDA0002588447200000101
Phase plane diagrams obtained by RK4, DHB3-GOIA, DHB6-GOIA and DHB 9-GOIA. The integration time step Δ τ is 0.01. The stopping criterion for DHB-GOIA is 10-8. As can be seen in FIG. 11(a), the DHB3-GOIA curve approximately coincides with the RK4 curve. However, at | α | ∈ [2 °,8 °]A significant difference occurred in the area. FIG. 11(b) shows that the DHB6-GOIA curve closely matches the reference curve, with only minor differences around + -4 deg.. Increasing the harmonic number to 9, it can be seen from FIG. 11(c) that the DHB9-GOIA and RK4 curves are completely identical.
Table 2 lists the fundamental frequencies calculated by different methods, the results representing the difference of DHB-GOIA with respect to RK4 as the baseline, where "+" and "-" represent greater and less than the baseline, respectively. It can be seen that DHB-GOIA calculation
Figure BDA0002588447200000102
The accuracy of the method is the highest,
Figure BDA0002588447200000103
in the next place, the first step is,
Figure BDA0002588447200000104
the accuracy is lowest. Since more and more higher harmonics are involved in the dynamic response as the flow rate increases. DHB9-GOIA pair
Figure BDA0002588447200000105
The relative errors of 0.00238% and 0.0317% for 2.0, respectively, indicate that the DHB-GOIA method has extremely high accuracy.
TABLE 2 fundamental frequencies calculated by RK4 and DHB-GOIA methods
Method of producing a composite material Fundamental frequency at 1.6 Fundamental frequency at 1.8 Fundamental frequency in excess of 2.0
RK4 0.07118950 0.06667925 0.06188501
DHB3-GOIA 0.00118696 0.00272926 0.00513004
DHB6-GOIA 0.00010873 0.00042775 0.00126445
DHB9-GOIA 0.00000071 0.00000189 0.00001963
While a higher number of harmonics may result in a more accurate solution to DHB-GOIA, a DHB-GOIA method with nine harmonics is sufficient to provide a highly accurate solution for the system studied by the present invention.

Claims (4)

1. The method for obtaining the physical period of the aeroelastic system based on the confusion-removing harmonic balance method is characterized by comprising the following steps:
constructing a nonlinear algebraic equation set of a two-degree-of-freedom airfoil aeroelastic system based on a harmonic balance method to obtain a harmonic balance algebraic system;
step two, constructing a confusion removal scheme for the harmonic balance algebraic system;
and step three, solving a nonlinear algebraic equation system of the harmonic balance algebraic system to obtain the physical period of the dynamic elastic system.
2. The method for obtaining the physical period of the aeroelastic system based on the confusion-removing harmonic balancing method as claimed in claim 1, wherein the specific method for constructing the nonlinear algebraic equation system in the first step is as follows:
and converting the two-degree-of-freedom airfoil aeroelastic system into a nonlinear ordinary differential equation set, and performing trial solution in a Fourier expansion form to obtain a harmonic balance algebraic system.
3. The method for obtaining the physical period of the aeroelastic system based on the confusion-removing harmonic balancing method according to claim 1, wherein in the second step, the method for constructing the confusion-removing scheme is as follows:
firstly, applying a non-linear algebraic equation system solver to a non-linear algebraic equation system to obtain a result
Figure FDA0002588447190000011
If it is
Figure FDA0002588447190000012
If the primary main harmonic exists, the solution is a real solution; otherwise, carrying out the second step;
secondly, calculating a harmonic amplitude vector and searching a first non-zero element;
thirdly, appointing x as the initial value of the non-linear algebraic equation system solver to calculate new
Figure FDA0002588447190000013
If it is not
Figure FDA0002588447190000014
Is the primary harmonic, then
Figure FDA0002588447190000015
Is a true solution; otherwise, repeating the second step and the third step until a true solution is obtained.
4. The method for obtaining the physical period of the aeroelastic system based on the confusion-removing harmonic balancing method according to claim 1, wherein the specific method in the third step is as follows:
converting the nonlinear algebraic equation system into Newton homotopy functions, and obtaining generalized scalar Newton homotopy functions by the aid of the drawn-up virtual time function Newton homotopy functions;
taking a constructed equivalent power system as a consistency condition, differentiating an semantic scalar Newton homotopy function relative to time t, converting the differential function into an ordinary differential equation set, then separating the power system, and discretizing the ordinary differential equation set into a time dynamic system through a forward Euler method to obtain a scalar equation;
and determining the convergence quantity of the iterative Newton homotopy method to obtain the physical period of the dynamic elastic system.
CN202010688387.XA 2020-07-16 2020-07-16 Method for obtaining physical period of aeroelastic system based on confusion-removing harmonic balancing method Pending CN111814255A (en)

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Application publication date: 20201023