A recursive convolution method for calculating rotor-body coupling response and stability

Abstract

The present invention provides a recursive convolution method for calculating the coupling response and stability of a rotor-airframe, comprising: establishing a rotor-airframe coupling dynamics analysis model; deriving a differential equation based on the rotor-airframe coupling dynamics analysis model, linearizing the coefficient matrix of the differential equation with a zero response, forming a linear equation and reducing the order to a first-order standard equation form; obtaining a homogeneous solution and a forced response steady-state solution based on the first-order standard equation; calculating a state transition matrix at one time point based on a periodic integral of a state transition matrix; calculating an eigenvalue of the state transition matrix, and judging the stability of the coupling between the rotor and the airframe based on the eigenvalue; if the eigenvalue of the state transition matrix is ​​less than 1 and the coupling response of the rotor and the airframe meets the convergence requirement, calculating a periodic value of a right-end excitation term, and then calculating the convolution integral response of the first cycle; and calculating the convolution integral response of the second cycle by adopting a recursive convolution integral calculation method based on the response of the first cycle of convolution integral, the state transition matrix and the periodicity of the excitation term.

Description

Technical Field

[0001] The invention belongs to the technical field of helicopter dynamics analysis, and in particular relates to a recursive convolution method for calculating rotor-body coupling response and stability. Background Art

[0002] Estimating the rotor blade vibration load and body vibration response of the helicopter in flight, as well as the rotor-body coupling stability is an important part of helicopter dynamics design. Currently, in the analysis technology of helicopter rotor blade vibration load and body vibration response, the rotor blade vibration load and body vibration response are usually separated, that is, coupling is not considered, and coupling stability is classified as ground resonance and air resonance analysis technology. Commonly used methods for estimating rotor blade vibration loads include direct integration method, time finite element method, harmonic balancing method, etc.

[0003] Direct integration of rotor blade dynamics differential equations to calculate rotor blade vibration loads is very easy to diverge if the equation order is high. In addition to low integration step size, the calculation time is long, and in most cases, even a small step size or some techniques will not converge.

[0004] The time finite element method utilizes the periodic characteristics of the rotor vibration load to discretize the rotor blade dynamics differential equation into algebraic equations corresponding to the time nodes of one rotation, and solves the response of the time nodes of one rotation of the rotor blade (thus calculating the load). This method has a large demand on computer memory, and the demand is even greater when the time nodes and blade modal orders are high. The calculation time is also long, and nonlinear iterations increase the amount of calculation even more.

[0005] The harmonic balancing method has a relatively small calculation scale, but has low accuracy and cannot effectively include nonlinear effects.

[0006] In summary, to calculate the vibration response of some parts of the airframe structure, the rotor load is taken as input and directly loaded onto the right side of the airframe structural dynamics equation. The response of the airframe is solved at one time and cannot be fed back to the impact on the rotor blade load, that is, there is no coupling. Summary of the invention

[0007] In response to the above technical problems, the present invention provides a recursive convolution method for calculating the rotor-airframe coupling response and stability, which is used for calculating and analyzing the rotor blade vibration load, airframe vibration response, and rotor-airframe coupling stability under helicopter flight conditions. Compared with conventional analysis models and methods, it can provide more accurate results that meet model design requirements.

[0008] The present invention provides a recursive convolution method for calculating rotor-body coupling response and stability, the method comprising:

[0009] Establish a rotor-airframe coupled dynamics analysis model;

[0010] Based on the rotor-body coupled dynamics analysis model, a differential equation is derived, and the coefficient matrix of the differential equation is linearized with a zero response to form a linear equation and reduced to a first-order standard equation form, wherein the equation includes a coefficient matrix: a mass matrix M(Ψ), a damping matrix C(Ψ), a stiffness matrix K(Ψ) and a right-hand excitation term F(Ψ);

[0011] Based on the first-order standard equation, a homogeneous solution and a forced response steady-state solution are obtained; wherein the homogeneous solution is expressed in the form of a state transfer matrix, and the forced response steady-state solution is expressed in the form of a convolution integral;

[0012] Calculate the state transition matrix at one week time point based on the periodic integration of the state transition matrix;

[0013] Calculating the eigenvalue of the state transition matrix, and judging the stability of the coupling between the rotor and the fuselage based on the eigenvalue;

[0014] If the eigenvalue of the state transfer matrix is ​​less than 1 and the coupling response between the rotor and the fuselage meets the convergence requirements, calculate the value of one cycle of the right-hand excitation term and then calculate the convolution integral response of the first cycle;

[0015] Based on the response of the first cycle of convolution integral, the periodicity of the state transition matrix and the excitation term, the recursive convolution integral calculation method is used to calculate the response of the second cycle of convolution integral.

[0016] Preferably, the establishing of the rotor-body coupling dynamics analysis model comprises:

[0017] Establish the rotor blade structural dynamics and aerodynamic model;

[0018] Establish the structural dynamics and aerodynamic model of the aircraft;

[0019] The rotor blade structural dynamics and aerodynamic model is coupled with the airframe structural dynamics and aerodynamic model to obtain the rotor airframe coupled dynamics analysis model.

[0020] Preferably, the calculating the eigenvalue of the state transition matrix and judging the stability of the coupling between the rotor and the body based on the eigenvalue comprises:

[0021] Based on whether the eigenvalue is greater than 1, judging whether the iterative response of the coupling between the rotor and the body can converge;

[0022] The stability of the coupling between the rotor and the airframe is determined based on whether the iterative response of the coupling between the rotor and the airframe can converge.

[0023] Preferably, the method further comprises:

[0024] The convolution integral response of the third cycle is calculated, and so on, recursively iterating the convolution integral response of the nth cycle.

[0025] Preferably, the method further comprises:

[0026] Compare the response errors at time T of the two cycles before and after to determine whether the response calculation meets the convergence requirements.

[0027] Preferably, the method further comprises:

[0028] Based on the nonlinear coefficient matrix in the differential equation, the coefficient matrix and the right-hand side excitation term in the differential equation are updated.

[0029] Preferably, the method further comprises:

[0030] For the updated equation, recalculate the state transfer matrix and recursive convolution integral response calculation according to the above method until the convolution integral response twice before and after meets the error requirement, and save the response, recorded as Y Φ2 .

[0031] Preferably, the method further comprises:

[0032] According to the convolution integral response calculated after the coefficient matrix is ​​updated (the state transfer matrix is ​​updated), it is determined whether the convolution integral response results after the state transfer matrix is ​​updated twice before and after meet the error requirements. If the error requirements are not met, the above solution steps are repeated. If the error requirement ε is met, the calculation is terminated and the convolution integral response result is output, which is the coupling response value of the rotor and the fuselage.

[0033] Beneficial effects of the present invention:

[0034] The present invention provides a recursive convolution method for calculating the rotor-airframe coupling response and stability, which uses a unified model analysis to obtain the rotor blade load, airframe response, and stability of the rotor-airframe coupling. It is only necessary to integrate and calculate the state transfer matrix and the first week of convolution calculation once, and calculate the integral convolution of each subsequent week of time points using a recursive iteration method, which greatly reduces the amount of calculation. By linearizing the coefficient matrix, updating the coefficient matrix and the excitation term with the steady-state response, the structural and aerodynamic nonlinear effects are effectively taken into account. Compared with conventional analysis models and methods that can only perform linear analysis, it can provide more accurate results that meet the model design requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is a calculation flow chart involved in the present invention;

[0036] Figure 2 The present invention relates to the first to sixth order modal displacement responses of the machine body;

[0037] Figure 3 The first-order swing modal displacement response of the five blades involved in the present invention;

[0038] Figure 4 The first-order flapping modal displacement response of the five blades involved in the present invention;

[0039] Figure 5 The present invention relates to the second-order flapping modal displacement response of five blades. DETAILED DESCRIPTION

[0040] This application divides the end time of the convolution integral into several weeks by calculating a state transition matrix for one cycle, and uses the periodic characteristics of the coefficients of the rotor-airframe coupling dynamic equation and the excitation load. After calculating a convolution response for one cycle, the response after any number of cycles is recursively calculated according to the cycle duration. It is no longer necessary to calculate the convolution integral at each moment from time 0 to the specified duration, which greatly saves the convolution calculation time. After the response converges, the nonlinear characteristics of the dynamic equation are linearized, the calculated state transition matrix is ​​updated, and the recursive convolution response calculation is repeated. The condition for the convergence of the recursive convolution calculation response is that the modulus of the eigenvalues ​​of the state transition matrix at the moment of period T is less than 1. This condition directly determines the stability of the coupling between the rotor and the airframe. If there is a modulus of the eigenvalue that is not less than 1, it indicates that there is instability in the coupling between the rotor and the airframe, that is, there is ground resonance or air resonance. At this time, the rotor blade and airframe response calculation does not need to be performed, because there is instability and the vibration response will diverge, indicating that there is a problem with the design and the design needs to be improved.

[0041] In an embodiment of the present application, a recursive convolution method for calculating rotor-body coupling response and stability includes:

[0042] (1) This method solves the problem of calculating the rotor blade load, airframe response and the stability of the rotor-airframe coupling using the same analysis model, which is the rotor-airframe coupling dynamics analysis model. The differential equation derived based on this dynamics model can be expressed as:

[0043]

[0044] Where M(ψ), C(ψ), and K(ψ) are all periodic functions containing ψ, ψ is the dimensionless time after linearization, ψ=2π is the angle of one rotation of the rotor, and is also the period T; F(ψ) is the right-hand excitation term, including the steady-state term after linearization of the aerodynamic and structural nonlinear terms based on the steady-state response.

[0045] (2) Reduce equation (1) to the first-order standard equation form:

[0046]

[0047] In the formula,

[0048]

[0049] The homogeneous solution of equation (2) can be expressed by the state transition matrix:

[0050] {Y(ψ)}=[Φ(ψ)]{Y0(ψ)} (3)

[0051] Then, the forced response steady-state solution of equation (2) is expressed as the following convolution integral:

[0052]

[0053] (3) From the convolution integral expression of equation (4), it can be seen that to obtain the response at time ψ, it is necessary to integrate from 0 to ψ. When ψ is very long, the convolution integral is both time-consuming and repetitive. Therefore, the periodic characteristics of the excitation term and the following characteristics of the state transition matrix are used:

[0054]

[0055] Assuming that a week is divided into NT integration points, the response of the first week is calculated first, and then based on the response of the first week, the characteristics of the state transition matrix and the periodicity of the excitation term, the following recursive convolution integral calculation method is derived:

[0056] The first week responses are recorded as:

[0057] [Y1]=[{Y(ψ1)},{Y(ψ2)},...,{Y(T)}] (6)

[0058] Then, the result of the second week response recursive convolution integral is:

[0059] [Y2]=[Y1(T)]+[Φ(T)][Y1]=[Y1(T)]+[Z1] (7)

[0060] The result of the third week response recursive convolution integral is:

[0061] [Y3]=[Y2(T)]+[Φ(T)] 2 [Y1]=[Y2(T)]+[Φ(T)][Z2] (8)

[0062] Therefore, the convolution integral result of the nth week can be deduced as follows:

[0063] [Y n ]=[Y n-1 (T)]+[Φ(T)][Z n-1 ][Z n ]=[Φ(T)][Z n-1 ] (9)

[0064] The above formulas are the recursive expressions of the recursive convolution method.

[0065] (4) Since the response is periodic, the error between the values ​​at time T two weeks before and after is compared to determine whether the response calculation meets the convergence requirements. When the response converges, save the response and record it as Yφ1. Then use the steady-state response to linearize the nonlinear coefficient matrix in equation (1), update the coefficient matrix of equation (1) and the right-hand excitation term, recalculate the state transfer matrix and recursive convolution calculation until the response Yφ2 after two updates of the state transfer matrix meets the error requirements, and the response calculation ends.

[0066] From formula (9), we can see that if the eigenvalues ​​of are all less than 1, the recursive convolution response will converge. Therefore, after each update of the state transfer matrix, the eigenvalue is calculated. Judging the eigenvalue is not only to determine whether the response iteration can converge, but also to analyze the coupling stability of the rotor and the body. This is the characteristic of this method.

[0067] The following is combined with Figure 1-5 The recursive convolution method for calculating the rotor-body coupling response and stability involved in the present invention is further described in detail.

[0068] Step 1: Based on the unified dynamic analysis model of rotor blade load, airframe response and rotor-airframe coupling stability, derive the differential equation (1);

[0069] Step 2: Linearize the coefficient matrix of equation (1) with zero response to form the linear equation (1) and the right-hand side excitation term;

[0070] Step 3: Integrate and calculate the state transition matrix of one week time point: [Φ(ψ1)], [Φ(ψ2)], ....., [Φ(T)];

[0071] Step 4: Calculate the eigenvalues, determine whether the response iteration can converge, and analyze the coupling stability between the rotor and the fuselage;

[0072] Step 5: If the convergence requirement is met, enter the recursive convolution calculation iteration and calculate {F A (ψ)} week value, and then calculate the first week response to obtain [Y1], [Z1];

[0073] Step 6: Iterate and calculate the convolution response of the nth cycle according to formula (9);

[0074] Step 7: Determine the nth and n-1th response errors until the error requirements are met;

[0075] Step 8: Use the steady-state response to linearize the nonlinear coefficient matrix in equation (1), and update the coefficient matrix and the right-hand side excitation term of equation (1);

[0076] Step 9: Repeat steps (3) to (8);

[0077] Step 10: Determine whether the response results after updating the state transfer matrix twice meet the error requirements. If not, repeat steps (3) to (9). If yes, end the calculation and output the response and eigenvalue calculation results.

[0078] Once the vibration response of the rotor blade is obtained, it can be substituted back into the blade structural unit array to obtain the load on the blade section and the composite rotor hub load. Figure 1 It is a calculation flow chart.

[0079] Figures 2 to 5 It is the body modal response and the first-order swing and first- and second-order flapping responses of the five blades. Tables 1 and 2 are the characteristic index of the state transfer matrix Φ(T) and the characteristic value of the rotor-body coupling stability. After each update of the state transfer matrix, only 4 to 6 recursive calculations are required to reach convergence (if the order of the equation is high, the number of iterations will increase accordingly), and with the update of the state transfer matrix, the recursive calculation converges faster, and the number of iterations of the state transfer matrix update is generally about 5 times. The error requirement given in the legend shown is less than 5%. The error value is the sum of the squares of the difference between the two response values ​​before and after divided by the sum of the squares of the responses, and then the square root needs to be less than 5%, which is quite accurate.

[0080] Figures 2 to 5 The static displacement components are removed from the responses of the fuselage and blades, so the values ​​are very small, indicating that the vibration response of this flight state is small. There are non-periodic components in the response, which is a free response. Although the excitation load is periodic, the state transfer matrix changes according to the inherent characteristics within a week, and non-periodic free response components appear. Therefore, this method can not only estimate the steady-state periodic response, but also analyze the free response. The given example only considers the first and second order flapping and first order swing of the blades and the six modes of the fuselage. The amplitude of their periodic response (without this) is very small, so the curve in the figure does not show obvious periodicity, which focuses on the stability analysis of the coupling between the rotor and the fuselage. If you focus on the vibration response of the coupling between the rotor and the fuselage, you need to select more fuselage and blade modes, as well as high-order aerodynamic loads.

[0081] Table 1 Characteristic index of state transition matrix Ф(T)

[0082]

[0083] Table 2 Characteristic values ​​of rotor-body coupling stability

[0084]

[0085] The modulus of the characteristic indexes listed in Table 1 is less than 1. From the characteristic values ​​of the rotor-body coupling stability in Table 2, it can be seen that the real part of the characteristic values ​​is less than 0, indicating that the flight state is stable. The analysis results also show that it is still stable when considering the structural and aerodynamic nonlinearity, especially the nonlinear characteristics of the damper, which is more in line with the actual state than considering only the pure linear situation. Tables 3 to 7 are the responses of a blade section load, hub load, control rod load and the hub point of the body.

[0086] Table 3 Loads at blade section r = 0.3R

[0087] Order FU(N) FV(N) FW(N) MQ(NM) MY(NM) MZ(NM) Tier 0 129352 -30420.2 18473.7 0.473516 16513.2 -3653.61 1st level COS 0.675429 355.609 -10.1207 -4.54E-03 -16.7008 14.4133 1st order SIN -7.11E-02 -157.756 9.49439 1.95E-03 -35.7325 0.495051 2nd level COS 0.343188 171.645 -4.55107 -2.20E-03 -11.1157 7.44478 2nd order SIN 6.21E-02 -27.7681 3.27782 3.24E-04 -20.2542 2.31711 3rd level COS 0.151512 72.8677 -1.81628 -9.34E-04 -5.76551 3.32764 3rd order SIN 3.40E-03 -35.082 2.53169 4.29E-04 -11.5701 0.688777 4th level COS 3.50E-02 15.0742 -0.287858 -1.94E-04 -1.91408 0.803582 4th order SIN 2.44E-03 -25.5337 1.83894 3.12E-04 -8.4013 0.498157 Level 5 COS 1.87E-02 6.43752 -4.52E-02 -8.40E-05 -1.50802 0.453455 5th-order SIN -6.20E-03 -24.4189 1.57506 3.01E-04 -6.4384 0.221529

[0088] Table 4 Loads from the left node of the blade root unit acting on the hub

[0089] Order FU(N) FV(N) FW(N) MQ(NM) MY(NM) MZ(NM) Tier 0 -323305 39876.4 -15759.3 -0.797251 -0.160434 2.96E-02 1st level COS -0.89636 -475.269 5.90686 7.67E-03 2.26E-04 1.87E-04 1st order SIN 0.135458 213.055 -23.7004 -3.32E-03 7.10E-04 -2.12E-04 2nd level COS -0.452558 -229.245 1.38237 3.71E-03 1.67E-04 8.09E-05 2nd order SIN -6.19E-02 38.219 -10.9796 -5.59E-04 3.87E-04 -7.93E-05 3rd level COS -0.198855 -97.2667 8.20E-02 1.58E-03 9.03E-05 3.12E-05 3rd order SIN 8.09E-03 47.5657 -7.04116 -7.32E-04 2.26E-04 -5.81E-05 4th level COS -4.52E-02 -20.0846 -0.340408 3.27E-04 3.23E-05 4.29E-06 4th order SIN 5.92E-03 34.6186 -5.11346 -5.33E-04 1.64E-04 -4.22E-05 Level 5 COS -2.37E-02 -8.54177 -0.479981 1.41E-04 2.67E-05 -2.42E-07 5th-order SIN 1.55E-02 33.0252 -4.11214 -5.13E-04 1.27E-04 -3.56E-05

[0090] Table 5 Pitch rod load

[0091] Order Fcont(N) Tier 0 -16471.1 1st level COS 9.91478 1st order SIN -3.41545 2nd level COS 4.87131 2nd order SIN -0.272478 3rd level COS 2.0963 3rd order SIN -0.676905 4th level COS 0.446788 4th order SIN -0.49201 Level 5 COS 0.208454 5th-order SIN -0.508134

[0092] Table 6 Hub load (5Ω)

[0093] FX(N)COS FX(N)SIN FY(N)COS FY(N)SIN FZ(N)COS FZ(N)SIN -7.914447 -25.76986 74.65322 165.1785 2.399903 20.56070 MQ(NM)COS MX(NM)SIN MY(NM)COS MY(NM)SIN MZ(NM)COS MZ(NM)SIN -1.494012 -0.1899854 -0.7907712 -7.524183 15.05914 -58.22200

[0094] Table 7 Harmonic value of body node (5Ω) response (g)

[0095] Node Number Harmonic order gx gy gz 1 5 7.56E-05 1.58E-04 5.08E-04

[0096] This method can take into account higher-order airframe modes and blade modes, quickly estimate the responses of rotor blades and airframe under the coupled state, and determine the rotor vibration load by back-calculating the rotor blade responses.

Claims

1. A recursive convolution method for calculating rotor-body coupling response and stability, characterized in that: The method comprises: Establish a rotor-airframe coupled dynamics analysis model; Based on the rotor-body coupled dynamics analysis model, a differential equation is derived, and the coefficient matrix of the differential equation is linearized with a zero response to form a linear equation and reduced to a first-order standard equation form, wherein the equation includes a coefficient matrix: a mass matrix M(Ψ), a damping matrix C(Ψ), a stiffness matrix K(Ψ) and a right-hand excitation term F(Ψ); Based on the first-order standard equation, a homogeneous solution and a forced response steady-state solution are obtained; wherein the homogeneous solution is expressed in the form of a state transfer matrix, and the forced response steady-state solution is expressed in the form of a convolution integral; Calculate the state transition matrix at one week time point based on the periodic integration of the state transition matrix; Calculating the eigenvalue of the state transition matrix, and judging the stability of the coupling between the rotor and the fuselage based on the eigenvalue; If the eigenvalue of the state transfer matrix is ​​less than 1 and the coupling response between the rotor and the fuselage meets the convergence requirements, calculate the value of one cycle of the right-hand excitation term and then calculate the convolution integral response of the first cycle; Based on the response of the first cycle of convolution integral, the periodicity of the state transition matrix and the excitation term, the recursive convolution integral calculation method is used to calculate the response of the second cycle of convolution integral; Wherein, the establishment of the rotor-body coupling dynamics analysis model includes: Establish the rotor blade structural dynamics and aerodynamic model; Establish the structural dynamics and aerodynamic model of the aircraft; The rotor blade structural dynamics and aerodynamic model is coupled with the airframe structural dynamics and aerodynamic model to obtain the rotor airframe coupled dynamics analysis model.

2. The method according to claim 1, characterized in that The calculating the characteristic value of the state transition matrix and judging the stability of the coupling between the rotor and the body based on the characteristic value include: Based on whether the eigenvalue is greater than 1, judging whether the iterative response of the coupling between the rotor and the body can converge; The stability of the coupling between the rotor and the airframe is determined based on whether the iterative response of the coupling between the rotor and the airframe can converge.

3. The method according to claim 1, characterized in that The method further comprises: The convolution integral response of the third cycle is calculated, and so on, recursively iterating the convolution integral response of the nth cycle.

4. The method according to claim 3, characterized in that The method further comprises: Compare the response errors at time T of the two cycles before and after to determine whether the response calculation meets the convergence requirements.

5. The method according to claim 4, characterized in that The method further comprises: Based on the nonlinear coefficient matrix in the differential equation, the coefficient matrix and the right-hand side excitation term in the differential equation are updated.

6. The method according to claim 5, characterized in that The method further comprises: For the updated equation, recalculate the state transfer matrix and recursive convolution integral response calculation according to the above method until the convolution integral response twice before and after meets the error requirement, and save the response, recorded as Y Φ2 .

7. The method according to claim 6, characterized in that The method further comprises: According to the convolution integral response calculated after the coefficient matrix is ​​updated, it is determined whether the convolution integral response results after the state transfer matrix is ​​updated twice before and after meet the error requirements. If the error requirements are not met, the above solution steps are repeated. If the error requirement ε is met, the calculation is terminated and the convolution integral response result is output, which is the coupling response value of the rotor and the fuselage.

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