Rotorcraft dynamic state dynamic characteristic analysis method based on time propulsion

Abstract

The invention provides a rotor aircraft dynamic state dynamic characteristic analysis method based on time propulsion. The method comprises the following steps: constructing a first dynamic equation of a rotor; constructing a propeller hub rotating coordinate system XYZ and a fixed inertial coordinate system XIYIZI, converting the degree of freedom of the propeller blade from the propeller hub rotating coordinate system XYZ to the fixed inertial coordinate system XIYIZI, and writing the first kinetic equation of the rotor wing as a second kinetic equation of the rotor wing; establishing a rotor system motion equation in the fixed inertial coordinate system XIYIZI, and calculating to obtain a second kinetic equation of a rotor; giving an initial response, and iteratively solving the second kinetic equation of the rotor wing to obtain an aeroelastic response convergence solution; based on the aeroelastic response convergence solution, obtaining a propeller hub load at the moment; and according to the propeller hub load and the aeroelastic response convergence solution at the moment, constructing and solving a second kinetic equation of the rotor again, and gradually propelling to finally obtain a rotor aircraft dynamic state dynamic characteristic result of the whole time history.

Description

Technical Field

[0001] The present application belongs to the technical field of transient dynamics of rotorcraft, and in particular relates to a method for analyzing the dynamic characteristics of rotor maneuvering state based on time advancement. Background Art

[0002] When studying the dynamic characteristics of rotors considering hub motion, existing dynamic methods require establishing a rotor / fuselage (or rotor / wing, rotor / wing / fuselage) coupling system and taking the impact of system motion on the rotor into account in the theoretical model. Therefore, the system equations of the rotor system need to be re-derived, which requires a large amount of computational work.

[0003] At the same time, the existing dynamic calculation method cannot take into account the impact of the time-varying flight parameters of the maneuvering state on the rotor system. In essence, it can only calculate the steady-state process of the rotor and cannot reflect the dynamic characteristics of the rotor at the moment of transition from the steady state to the maneuvering state. That is, the motion state of the coupled system does not change continuously, which is inconsistent with the actual situation.

[0004] Therefore, the current dynamic calculation method cannot accurately reflect the dynamic characteristics of the rotor in maneuvering state. Summary of the Invention

[0005] Purpose of the invention: To provide a time-advancing-based rotor maneuvering state dynamics calculation method that can accurately simulate the continuous motion process of the rotor maneuvering state and solve the dynamic response of the rotor maneuvering state from it.

[0006] The present application provides a method for analyzing the dynamic characteristics of a rotor maneuvering state based on time advancement, the method comprising:

[0007] Construct the first dynamic equation of the rotor;

[0008] Construct the blade rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , transform the blade degrees of freedom from the blade hub rotating coordinate system XYZ to the fixed inertial coordinate system X I Y I Z I , writing the first dynamic equation of the rotor as the second dynamic equation of the rotor;

[0009] In the fixed inertial coordinate system X I Y I Z I The motion equation of the rotor system is established and the second dynamic equation of the rotor is obtained by calculation;

[0010] Given the initial response, the Newton-Raphson nonlinear iteration is used to solve the second dynamic equation of the rotor and obtain the converged solution of the aeroelastic response.

[0011] Based on the converged solution of the aeroelastic response, the blade hub load at this moment is obtained;

[0012] According to the converged solution of the blade load and aeroelastic response at that moment, the second dynamic equation of the rotor is reconstructed and solved again, and the results of the dynamic characteristics of the rotor maneuvering state throughout the entire time history are finally obtained.

[0013] Preferably, the first dynamic equation for constructing the rotor includes:

[0014] The isolated rotor is composed of multiple blades, so the first dynamic equation of the rotor can be obtained by superimposing multiple blades together, and its expression can be written as:

[0015]

[0016] Where M i is the mass matrix of the i-th blade; C i is the damping matrix of the i-th blade; K i is the stiffness matrix of the i-th blade; F i is the load column vector on the i-th blade; F is the overall load column vector on the rotor; q i is the generalized coordinate of the i-th blade, Respectively represent the first and second derivatives with respect to time; N b is the number of blades on the isolated rotor.

[0017] Preferably, the blade hub rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , transform the blade degrees of freedom from the blade hub rotating coordinate system XYZ to the fixed inertial coordinate system X I Y I Z I ,include:

[0018] For an isolated rotor system, the hub center is assumed to be a fixed point. In this case, the non-rotating coordinate system of the blade is equivalent to the fixed inertial coordinate system.

[0019] Construct the blade rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , perform coordinate transformation on the rotor blade coordinates, transform the blade coordinates into the fixed inertial coordinate system, then the coordinate q of the i-th blade in the rotating coordinate system is i The relationship between the rotor's overall coordinate ξ and the fixed inertial coordinate system is:

[0020]

[0021] Where m is the total number of degrees of freedom of a single blade;

[0022] For A i Element a in i ,have

[0023]

[0024] Where, ψ i represents the azimuth angle of the i-th blade;

[0025] For the element ξ in ξ (k) , which represents all N b The k-th degree of freedom displacement of the blade is a column vector in the inertial coordinate system, with dimension N b ×1, writing

[0026]

[0027] Wherein, n is defined as above; ξ r (i) represents the coordinate of the k-th degree of freedom displacement of the i-th blade in the rotating coordinate system;

[0028] q i =A i The time derivative of ξ is

[0029]

[0030] Preferably, the second dynamic equation of the rotor is expressed as:

[0031]

[0032] Where,

[0033] Preferably, in the fixed inertial coordinate system X I Y I Z I The motion equation of the rotor system is established and the second dynamic equation of the rotor is calculated, including:

[0034] Based on Hamilton's principle and moderate deformation beam theory, the motion equation of the rotor system is established in a fixed inertial coordinate system, and its expression is as follows:

[0035] δ∏=∫(δU-δT-δW)dt=0

[0036] Where Π is the potential energy of the system; U is the strain energy of the system; T is the kinetic energy of the system; W is the virtual work of the external force of the system; t is the time integration domain; δ is the variation symbol.

[0037] Preferably, the method further comprises:

[0038] According to δU, δT, and δW in the rotor system motion equation, the M, C, K, and F matrices in the second dynamic equation of the rotor are calculated.

[0039] Preferably, obtaining the hub load at this moment based on the converged solution of the aeroelastic response includes:

[0040] Based on the converged solution of the aeroelastic response, the aerodynamic load and structural load distribution on the rotor are obtained by overall calculation. By integrating the aerodynamic load and structural load at the center of the blade hub, the blade hub load at that moment is obtained.

[0041] Preferably, the second dynamic equation of the rotor is re-constructed and solved based on the converged solution of the blade load and the aeroelastic response at that moment, and the process is gradually advanced to finally obtain the dynamic characteristic results of the rotor maneuvering state for the entire time history, including:

[0042] According to the converged solution of the blade load and aeroelastic response at that moment, the rotor control, flow field, and attitude input at the next moment are updated. According to a certain time step, the second dynamic equation of the rotor is reconstructed and solved again, and the solution is gradually advanced to finally obtain the dynamic characteristics of the rotor maneuvering state throughout the entire time history.

[0043] Beneficial technical effects of this application:

[0044] This invention solves the problem of calculating the dynamic characteristics of a rotor in a maneuvering state. Using the method described in this invention, the second dynamic equation for a rotor in a maneuvering state can be established to calculate the dynamic characteristics of the rotor in any maneuvering state. Furthermore, the rotor speed, attitude, and incoming airflow can all be arbitrarily set, thereby fully reflecting the dynamic characteristics of the rotor in a maneuvering state in a single calculation.

[0045] The method proposed in this paper modifies the traditional Hamiltonian method, significantly reducing the computational workload compared to re-deriving the rotor dynamics model for rotor-to-airframe motion. Furthermore, because the rotor maneuvering state can be continuously varied during the calculation process, the resulting rotor dynamics model is closer to reality and more consistent with actual rotor maneuvering conditions than the traditional Hamiltonian method. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1A flow chart of rotor maneuvering dynamics calculation based on time advancement provided in an embodiment of the present application;

[0047] Figure 2 This is the time history curve of blade tip flapping displacement during the rotor forward flight with variable speed calculated by this method;

[0048] Figure 3 This is the time history curve of the rotor blade torque during the variable incoming flow process calculated by this method;

[0049] Figure 4 This is the time history curve of the vertical force on the blade during the rotor variable flow process calculated by this method. DETAILED DESCRIPTION

[0050] See also Figures 1-4 This application provides a time-advanced dynamic calculation method for rotor maneuvering states. Aiming at the dynamic problems in the rotor maneuvering state, a time-step-advanced dynamic calculation method is developed to study the dynamic characteristics of the rotor in the maneuvering state, and provide a theoretical basis and analysis tools for the study of the dynamic characteristics of the rotor in the maneuvering state.

[0051] A method for calculating rotor maneuvering dynamics based on time advancement is provided, the method comprising:

[0052] 1) The isolated rotor is composed of multiple blades, so the first dynamic equation of the rotor can be obtained by superimposing multiple blades together, and its expression can be written as

[0053]

[0054] Where M i is the mass matrix of the i-th blade; C i is the damping matrix of the i-th blade; K i is the stiffness matrix of the i-th blade; F i is the load column vector on the i-th blade; F is the overall load column vector on the rotor; q i is the generalized coordinate of the i-th blade, Respectively represent the first and second derivatives with respect to time; N b is the number of blades on the isolated rotor.

[0055] 2) For an isolated rotor system, assume that the hub center is equivalent to a fixed point. In this case, the non-rotating coordinate system of the blade is equivalent to the fixed inertial coordinate system. Construct the blade rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , perform coordinate transformation on the rotor blade coordinates, transform the blade coordinates into the fixed inertial coordinate system, then the coordinate q of the i-th blade in the rotating coordinate system is iThe relationship between the rotor's overall coordinate ξ and the fixed inertial coordinate system is:

[0056]

[0057] Where m is the total number of degrees of freedom of a single blade;

[0058] For A i Element a in i ,have

[0059]

[0060] Where, ψ i represents the azimuth angle of the i-th blade.

[0061] For the element ξ in ξ (k) , which represents all N b The k-th degree of freedom displacement of the blade is a column vector in the inertial coordinate system, with dimension N b ×1, writing

[0062]

[0063] Wherein, n is defined as above; ξ r (i) Represents the coordinate of the k-th degree of freedom displacement of the i-th blade in the rotating coordinate system.

[0064] q i =A i The time derivative of ξ is

[0065]

[0066] After transforming the blade's degrees of freedom from the blade hub's rotating coordinate system to the fixed inertial coordinate system, the first dynamic equation of the rotor can be written as

[0067]

[0068] Where,

[0069] 3) Construct the equation of motion of the rotor system. Based on Hamilton's principle and the theory of moderate deformation beams, the equation of motion of the rotor system is established in a fixed inertial coordinate system. Its expression is as follows:

[0070] δ∏=∫(δU-δT-δW)dt=0

[0071] Where Π is the system potential energy; U is the system strain energy; T is the system kinetic energy; W is the virtual work of the system external force; t is the time integration domain; δ is the variation symbol.

[0072] According to the equation of motion of the rotor system, δU, δT, and δW are M 、 C 、 K 、 F The contribution of the array is used to construct and calculate the second dynamic equation of the rotor; given the initial response, the Newton-Raphson nonlinear iteration is used to solve the converged solution of the aeroelastic response.

[0073] 4) Based on the converged solution of the aeroelastic response, the overall calculation is used to obtain the aerodynamic load and structural load distribution on the rotor. By integrating the aerodynamic load and structural load at the center of the blade, the blade load at that moment is obtained.

[0074] 5) Based on the blade load and aeroelastic response at that moment, the rotor control, flow field, attitude and other inputs at the next moment are updated. At a certain time step, the second dynamic equation of the rotor is constructed and solved again. This is gradually advanced to finally obtain the dynamic characteristics of the rotor maneuvering state throughout the entire time history.

[0075] It should be noted that:

[0076] 1) When constructing and calculating the second dynamic equation of the rotor, it is necessary to consider the influence of the hub motion on the second dynamic equation of the rotor. For the maneuvering state, the T term in the dynamic equation includes not only the speed of the blade relative to the hub center, but also the speed of the rotor blade relative to the hub center. Also includes the blade motion speed caused by the blade motion The system kinetic energy in the maneuvering state is deduced and the system kinetic energy T can be written as

[0077] 2) For the rotor dynamics analysis in the maneuvering state, the rotor is analyzed as a whole, and the aeroelastic response and blade load of all blades at each moment are solved simultaneously.

[0078] 3) For the rotor maneuvering process, the initial rotor control, rotor attitude, and aeroelastic response are input, and the rotor system motion equation and equilibrium differential equation are constructed and solved. The system is advanced according to a certain time step to obtain the blade load and aeroelastic response at each moment. Therefore, the system motion equation can continuously reflect the rotor dynamic characteristics at any time and in any attitude, thereby performing dynamic simulation of the rotor maneuvering state.

[0079] In an embodiment of the present application, a method for calculating the dynamics of a rotor maneuvering state based on time advancement is provided to solve the problem of difficulty in calculating the dynamic response of a rotor maneuvering state. Specifically, the method includes the following steps:

[0080] Step 1: Input rotor parameters, including blade configuration, physical characteristics, geometric dimensions, number of rotors, rotor solidity, number of blades, and blade radius; input blade structural parameters, blade structural unit division, aerodynamic unit division, and airfoil C81 table, and perform dimensionless processing on blade structural parameters; at the same time, determine the input parameters at the initial moment: rotor speed Ω, rotational acceleration Azimuth angle ψ; blade total pitch, lateral cyclic pitch and longitudinal cyclic pitch θ0,θ c ,θ s ; Pull coefficient, pitch moment coefficient and rolling moment coefficient C T ,C L ,C M ; The time-marching method calculates the step length dt in the maneuvering state; the rotor maneuvering translation speed Roll, pitch, and yaw angular velocity According to the input given C T ,C L ,C M , the rotor flow field λ is solved iteratively by the fourth-order Runge-Kutta method;

[0081] Step 2: Assume that the previous moment is t n , the initial aeroelastic response of each blade i in the rotating coordinate system Perform coordinate transformation to convert the response of each blade i in the rotating coordinate system into Converted into the initial aeroelastic response of the rotor in a fixed inertial coordinate system by and For example, the conversion relationship is

[0082]

[0083] Where m is the total number of degrees of freedom of a single blade;

[0084] For A i Element a in i ,have

[0085]

[0086] Where, ψ i represents the azimuth angle of the i-th blade.

[0087] for Elements in It represents all N b The k-th degree of freedom displacement of the blade is a column vector in the inertial coordinate system, with dimension N b ×1, writing

[0088]

[0089] Wherein, n is defined as above; Represents the coordinates of the k-th degree of freedom displacement of the i-th blade in the rotating coordinate system. Next, solve for the current time t after the dt step. n+1 =t n +dt rotor aeroelastic response

[0090] Step 3: Based on the current rotor motion state, in the fixed inertial coordinate system X I Y I Z I The rotor system motion equation at this moment is constructed in δ∏=∫(δU-δT-δW)dt=0; based on the motion state of each blade on the rotor, the system motion equation of each blade is obtained δ∏i=∫(δU i -δT i -δW i )dt=0; According to the finite element unit division, the control equation δ∏ of each unit j on each blade i is obtained discretized i j =∫(δU i j -δT i j -δW i j )dt=0; For each finite element j on each blade i, the control equation is calculated δT i j , δW i j For the dynamic equation F ij The contribution of the matrix is ​​finally obtained, and the mass matrix of the finite element is obtained. Damping Matrix Stiffness matrix and the load column vector F i j , it should be noted that, when calculating the kinetic energy term δT i j In addition to considering the blade's speed relative to the center of the blade hub, the blade's associated motion speed caused by the blade hub motion must also be considered. and According to the corresponding relationship between the node coordinates of the finite element matrix and the unit matrix on each blade i, the assembly of the finite element matrix is ​​completed, and the damping and stiffness of the blade structure are adjusted to C according to different blade configurations. i , K i 、F iTaking into account the influence of the rotor aerodynamic force on F i The influence of the matrix is ​​obtained, and the overall matrix M of each blade i in the second dynamic equation of the rotor at the current moment is obtained. i 、C i , K i 、F i ; The overall matrix M of each blade i i 、C i , K i 、F i Assemble according to the following rules to finally obtain the rotor's overall array M, C, K, and F arrays.

[0091]

[0092] Where,

[0093] Step 4: Using the Newton-Raphson nonlinear method, perform multiple rounds of aeroelastic iterative calculations on the second dynamic equation of the rotor with t n Time response is the initial response, solve for t n+1 Time response The difference between the two is Δξ; let t n+1 The velocity and displacement at the moment are expressed as follows:

[0094]

[0095] In the formula, α and β are empirical coefficients; therefore, t n+1 The acceleration at the moment is

[0096]

[0097] After reverse substitution, we can get

[0098]

[0099] Next, we calculate Δξ. If the iterative calculation is the first iteration, we can solve Δξ by the following formula and substitute it into the above formula to solve Then we get t n+1 Time response

[0100]

[0101] A·Δξ=B

[0102] If the number of iterative calculations is not the first time, solve Δξ, Then we get t n+1 Time response

[0103]

[0104] A·Δξ=B

[0105]

[0106] Step 5: Solve to get t n+1 Time response After that, calculate Is it less than the preset tolerance ε1 (| | represents the Euclidean norm), if it is greater than the tolerance, the calculated Assign to Repeat step 4 until you meet So far, it is considered that the obtained t n+1 The response at time is convergent; since what is ultimately needed is t n+1 Blade response in the moment-rotating coordinate system Therefore, it is necessary to use the inverse transformation to convert the coordinates in the fixed inertial coordinate system The result of transforming back to the blade rotation coordinate system

[0107] Step 6: Based on the response convergence results of step 5 Calculate t n+1 Structural load F at time struct and aerodynamic load F aero The results of the spanwise distribution of the blades are shown. The aerodynamic loads are solved using a quasi-steady aerodynamic model. By calculating the velocity and angle of attack of each section and combining the C81 table of the airfoil, the corresponding lift, drag and moment coefficients are obtained, and finally the lift, drag and moment distribution of each section is obtained.

[0108] Step 7: Place the n+1 Structural load F at time struct and aerodynamic load F aero Integrate to the propeller root and get t n+1 The blade root load F in the undeformed coordinate system at the moment root , and then through the transformation matrix of the non-rotating coordinate system of the hub and the non-deformed coordinate system of the blade, we can get t n+1 The propeller load F at the moment hub :

[0109]

[0110]

[0111] Where, F x ,F y ,F z is the force in three directions, M x ,My ,M z are the moments in three directions;

[0112] Step 8: According to t n+1 The output result of the blade load at the moment is updated and the input given C T ,C L ,C M , calculate dt time step after t n+1 +dt time rotor inflow λ1; by interpolation, get the next time t n+1 +dt rotor speed Ω, rotational acceleration Azimuth angle ψ; blade total pitch, lateral cyclic pitch and longitudinal cyclic pitch θ0,θ c ,θ s , and rotor maneuver translation speed Roll, pitch, and yaw angular velocity Re-construct and solve the first dynamic equation of the rotor and the second dynamic equation of the rotor;

[0113] Step 9: Repeat steps 2 to 8 until the calculation of the entire rotor maneuvering state history is completed. The entire calculation process is as follows: Figure 1 As shown, the final output is the dynamic characteristics of the entire maneuvering process, including the aeroelastic response in the blade coordinate system. Blade load F hub , the rotor flow field λ and other time history, among which the blade tip flapping displacement time history curve of the rotor forward flight speed change process is as follows Figure 2 As shown in the figure, the time history curve of the blade torque and the time history curve of the blade vertical force during the rotor flow change process are as follows: Figure 3 、 Figure 4 shown.

[0114] In summary, the present invention proposes a method for calculating rotor maneuvering dynamics based on time marching. This method uses time marching to calculate rotor dynamics in a maneuvering state. The method takes into account the effects of varying rotor speed, attitude, and incoming airflow during the calculation of any maneuvering state. This method can continuously reflect the rotor dynamics in any maneuvering state, achieving the goal of accurately calculating the rotor dynamics in a maneuvering state.

Claims

1. A method for analyzing the dynamic characteristics of rotor maneuvering state based on time marching, characterized in that: The method comprises: Construct the first dynamic equation of the rotor; Construct the blade rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , transform the blade degrees of freedom from the blade hub rotating coordinate system XYZ to the fixed inertial coordinate system X I Y I Z I , writing the first dynamic equation of the rotor as the second dynamic equation of the rotor; In the fixed inertial coordinate system X I Y I Z I The motion equation of the rotor system is established and the second dynamic equation of the rotor is obtained by calculation; Given the initial response, the second dynamic equation of the rotor is solved iteratively to obtain a converged solution of the aeroelastic response; Based on the converged solution of the aeroelastic response, the blade hub load at this moment is obtained; According to the converged solution of the blade load and aeroelastic response at that moment, the second dynamic equation of the rotor is reconstructed and solved again, and the results of the dynamic characteristics of the rotor maneuvering state throughout the entire time history are finally obtained.

2. The method according to claim 1, characterized in that The first dynamic equation for constructing the rotor includes: The isolated rotor is composed of multiple blades, so the first dynamic equation of the rotor can be obtained by superimposing multiple blades together, and its expression can be written as: Where M i is the mass matrix of the i-th blade; C i is the damping matrix of the i-th blade; K i is the stiffness matrix of the i-th blade; F i is the load column vector on the i-th blade; F is the overall load column vector on the rotor; q i is the generalized coordinate of the i-th blade, Respectively represent the first and second derivatives with respect to time; N b is the number of blades on the isolated rotor.

3. The method according to claim 2, characterized in that The blade rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , transform the blade degrees of freedom from the blade hub rotating coordinate system XYZ to the fixed inertial coordinate system X I Y I Z I ,include: For an isolated rotor system, the hub center is assumed to be a fixed point. In this case, the non-rotating coordinate system of the blade is equivalent to the fixed inertial coordinate system. Construct the blade rotating coordinate system XYZ and the fixed inertial coordinate system X I Y I Z I , perform coordinate transformation on the rotor blade coordinates, transform the blade coordinates into the fixed inertial coordinate system, then the coordinate q of the i-th blade in the rotating coordinate system is i The relationship between the rotor's overall coordinate ξ and the fixed inertial coordinate system is: Where m is the total number of degrees of freedom of a single blade; For A i Element a in i ,have Where, ψ i represents the azimuth angle of the i-th blade; For the element ξ in ξ (k) , which represents all N b The k-th degree of freedom displacement of the blade is a column vector in the inertial coordinate system, with dimension N b ×1, writing Wherein, n is defined as above; ξ r (i) represents the coordinate of the k-th degree of freedom displacement of the i-th blade in the rotating coordinate system; q i =A i The time derivative of ξ is 4. The method according to claim 3, characterized in that The expression of the second dynamic equation of the rotor is: Where, 5. The method according to claim 4, characterized in that In the fixed inertial coordinate system X I Y I Z I The motion equation of the rotor system is established and the second dynamic equation of the rotor is calculated, including: Based on Hamilton's principle and moderate deformation beam theory, the motion equation of the rotor system is established in a fixed inertial coordinate system, and its expression is as follows: δ∏=∫(δU-δT-δW)dt=0 Where Π is the system potential energy; U is the system strain energy; T is the system kinetic energy; W is the virtual work of the system external force; t is the time integration domain; δ is the variation symbol.

6. The method according to claim 5, characterized in that The method further comprises: According to δU, δT, and δW in the rotor system motion equation, the M, C, K, and F matrices in the second dynamic equation of the rotor are calculated.

7. The method according to claim 6, characterized in that The blade hub load at this moment is obtained based on the converged solution of the aeroelastic response, including: Based on the converged solution of the aeroelastic response, the aerodynamic load and structural load distribution on the rotor are obtained by overall calculation. By integrating the aerodynamic load and structural load at the center of the blade hub, the blade hub load at that moment is obtained.

8. The method according to claim 7, characterized in that Based on the converged solution of the blade load and aeroelastic response at that moment, the second dynamic equation of the rotor is re-constructed and solved, and the results of the dynamic characteristics of the rotor maneuvering state over the entire time history are finally obtained, including: According to the converged solution of the blade load and aeroelastic response at that moment, the rotor control, flow field, and attitude input at the next moment are updated. According to a certain time step, the second dynamic equation of the rotor is reconstructed and solved again, and the solution is gradually advanced to finally obtain the dynamic characteristics of the rotor maneuvering state throughout the entire time history.

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