CN116757124A - A bearingless helicopter structure dynamic stability analysis method and system - Google Patents

Abstract

The invention discloses a method and a system for analyzing the dynamic stability of a bearingless helicopter structure, and relates to the field of helicopter rotor design, wherein the method comprises the steps of establishing a bearingless rotor blade vibration equation by utilizing a cubic spline function method on the basis of Hamilton principle modeling; establishing a flapping motion equation and a shimmy motion equation in a blade motion coordinate system; establishing a rolling motion equation and a pitching motion equation in a machine body coordinate system; establishing a dynamic inflow equation according to the dynamic inflow model; establishing a motion equation set of a rotor/organism coupling system according to a bearingless rotor blade vibration equation, a flapping motion equation, a shimmy motion equation, a rolling motion equation, a pitching motion equation and a dynamic inflow equation; and determining the stability of the bearingless rotor blade vibration equation and the motion equation system of the rotor/organism coupling system by adopting a eigenvalue analysis method combining a finite element method and a transmission matrix method. The invention can improve the accuracy of dynamic analysis of the bearingless rotor system.

Description

A bearingless helicopter structure dynamic stability analysis method and system

Technical field

The invention relates to the field of helicopter rotor design, and in particular to a bearingless helicopter structure dynamic stability analysis method and system.

Background technique

The rotor hub serves as the connection between the blades and the rotor shaft. Its function is to ensure that the blades can achieve kinematic functions and transfer the load on the blades to the rotor shaft. The propeller hub is closely related to the flight performance, flight quality and maintenance support of the helicopter. Its structural form has been continuously improved. It has experienced a development process from complex to simple while ensuring the fatigue strength and bearing operating cycle life. The emergence of hinges solved the problems of helicopter stability and maneuverability. From the 1940s to the 1960s, articulated rotors were the main rotor type. In the long-term continuous practice, we have accumulated a lot of technical experience. However, this type is far from ideal due to its complex structure, many moving parts, high manufacturing cost, poor reliability, heavy maintenance workload, low operating efficiency and low angle damping.

Since the 1950s, scientific researchers have begun research work on the simplification of rotor hubs. Initially, elastic bearings or titanium alloy flexible parts were used to replace the metal hinges of the past. Blackhawk is a relatively representative one. Since then, researchers have used composite materials to replace metal deformation parts, and spherical flexible and star-shaped flexible propeller hubs have emerged, and have achieved good application results, such as the SA365 "Dolphin" helicopter equipped with a star-shaped flexible propeller hub and a spherical flexible propeller hub. EC155. These rotors have a simple structure and a greatly extended life cycle. After long-term research and experimental work, the propeller hub configuration was further simplified. The bearings were directly replaced by elastic deformation, and only the axial hinge for pitch change was retained. The bearingless rotor came into being.

Bearingless rotors are becoming a trend in helicopter development, and the China Helicopter Research Institute is also stepping up the pace of developing bearingless rotors. In May 2015, the Z-11 demonstrator equipped with a bearingless rotor successfully achieved hovering flight, which marked a phased breakthrough in China's bearingless rotor research and development.

The coupling between the degrees of freedom of motion of the bearingless rotor blades is serious, and the dynamic instability problem of the rotor/body coupling is prominent. The flapping and oscillating motions of the articulated type are realized by rotation around the real hinge, while the bearingless type is realized by the flexural deformation of the rotor. The flapping and oscillation motions of the articulated rotor occur at the horizontal hinge and the vertical hinge respectively, while the variable pitch hinge is generally outside these two hinges, and the bending moment exerted by the blade is generally small, so the elastic coupling is very small. . However, for bearingless rotors, it is difficult to arrange the remaining variable pitch hinges outside the parts where large deflection deformation occurs, resulting in serious coupling between the blade's degrees of freedom, which has an important impact on the structural dynamics of the rotor. In general, bearingless rotors have the advantages of simple structure, low maintenance workload, large control efficiency and high angular velocity damping. Especially for armed helicopters, they are very promising. However, along with the simplification of the propeller hub structure, dynamic instability problems such as multi-degree-of-freedom coupling of the blades and rotor/body coupling have also appeared. These are key technologies for bearing-less rotors that need to be tackled and further studied.

The research scope of dynamic stability problem is whether the vibration motion of the system converges or diverges after the initial disturbance, or whether it is stable or unstable. If the motion after the disturbance is unstable (divergent), it will often bring serious consequences to the helicopter. In the history of helicopter development, helicopters have been destroyed more than once in the air or on the ground due to dynamic instability phenomena such as rotor flutter and helicopter "ground resonance".

Dynamic instability phenomena mostly come from the coupling between vibrations of two different degrees of freedom, and are mostly related to the effect of aerodynamic forces. Unfortunately, the rotor has too much freedom of movement. In this way, there are many possible dynamic instability problems. When designing the rotor, it must be ensured that no dynamic instability occurs within the helicopter's working range (rotation speed range, flight speed range, propeller pitch range, etc.) Stability phenomenon, or to try to move the critical conditions for unstable motion outside the working range. Such a requirement has a great impact on the structure of the rotor and other parts of the helicopter in some aspects.

The dynamic instability problems related to the rotor on the helicopter can be roughly divided into two types: one is the dynamic instability problem caused by the coupling of various degrees of freedom of the rotor itself, such as flutter; the other is the coupling between the rotor and the body. Dynamic instability problems arise. The dynamic instability problem of the rotor itself comes from the fact that the rotor blades have three degrees of freedom of movement: flapping, oscillation and torsion, and each movement is the superposition of different orders of vibrations. But generally speaking, some classic dynamic stability problems are mainly caused by vibrations of the lowest order vibration, that is to say, flapping and oscillation zero-order (articulated) or first-order (bearingless) vibration, torsion first-order vibration shape. Typical dynamic instability problems include: pitch-flapping flutter, pitch-flapping instability, flapping-flapping instability, and the so-called stall flutter. The dynamic instability problem of the coupling between the rotor and the body is caused by the coupling of the oscillation motion of the blades and the vibration of the body. The oscillation motion of the blades is based on its lowest-order vibration shape, and the phase relationship between each blade constitutes a receding vibration shape. The vibration shape of the airframe refers to the vibration shape that drives the center of the propeller hub to produce horizontal displacement. Unstable motion can occur while the helicopter is on the ground, known as ground resonance, or in the air, known as airborne resonance.

Helicopter ground and air resonance is a dynamic instability problem of the rotor/body coupling. It is one of the basic dynamics problems well-known in the helicopter community. They are dynamically unstable motions of the rotor and body coupling, and the main self-excited vibrations The source is the coupling of the rotor oscillation retreat mode and the airframe mode with horizontal displacement of the rotor hub center. The occurrence of ground resonance and air resonance in helicopters will lead to serious consequences. Therefore, accurate analysis, calculation and testing must be carried out during helicopter development, and corresponding design measures must be taken to prevent their occurrence. The key is to establish accurate rotor/body coupling dynamic stability. The calculation model and the selection of corresponding analysis methods with a certain accuracy.

Many scholars have conducted research on the dynamic stability of the helicopter rotor/body coupling system. Coleman and Feingold (Coleman P.R., Feingold A.M.. Theory of self-excitedmechanical oscillations of helicopter rotors with hinged blades[R]. NACAReport 1351,1958) established the dynamic equations of each blade in the rotating coordinate system and in the fixed coordinate system Establish the dynamic equations of the body in . Through multi-blade coordinate transformation (Coleman transformation), the degree of freedom of movement of the blade in the rotating coordinate system is converted into a fixed coordinate system, and then converted into the characteristic equation of the system and then analyzed for its eigenvalue. This approach was adopted by later scholars. It is widely used, and on this basis, a classic ground resonance analysis model that does not take into account the flapping degrees of freedom and aerodynamic forces has been established. The rotor/body coupling dynamic model established by Hu Guocai (Hu Guocai. Research on the nonlinear characteristics of the sway reducer and its impact on the coupled dynamic stability of the helicopter rotor/body [D]. Beijing: Beihang University, 2004.) can satisfy the requirements of the helicopter suspension. Dynamic stability analysis in different states of stopping, ground and forward flight. The blade model is an equivalent hinge rigid blade, and the aerodynamic model selects the extended Pitt/Peters dynamic inflow model, taking into account the flapping degrees of freedom and the pitch and roll motion of the body. An implicit multi-blade coordinate conversion method is proposed, which eliminates the need for small perturbation assumptions and dimensional analysis processes, and uses the Floquet transfer matrix method to complete the solution of differential equations.

In the study of dynamic stability of bearingless rotor helicopters, Hermite interpolation and linear interpolation are mostly used in the literature to represent the vibration displacement on each unit of the rotor blade. Such interpolation functions will lead to a complicated process of solving vibration equations and often lead to poor calculation accuracy. insufficient.

Based on the above problems, there is an urgent need to provide a bearingless helicopter structure dynamic stability analysis method or system that takes into account the complex motion of the blades, which can improve the accuracy of dynamic stability analysis.

Contents of the invention

The purpose of the present invention is to provide a bearingless helicopter structure dynamic stability analysis method and system, which can improve the accuracy of the bearingless rotor system dynamic analysis.

In order to achieve the above objects, the present invention provides the following solutions:

A bearingless helicopter structure dynamic stability analysis method, including:

Based on Hamilton's principle modeling, the cubic spline function method is used to establish the vibration equation of the bearingless rotor blade;

Establish the flapping motion equation and the oscillation motion equation of the blade within the blade motion coordinate system;

Establish the rolling motion equation and pitching motion equation of the body in the body coordinate system;

Establish the dynamic inflow equation based on the dynamic inflow model;

Establish a set of motion equations of the rotor/body coupled system based on the bearingless rotor blade vibration equation, flapping motion equation, oscillation motion equation, roll motion equation, pitching motion equation and dynamic inflow equation;

The eigenvalue analysis method combined with the finite element method and the transfer matrix method is used to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/body coupled system.

Optionally, based on Hamilton's principle modeling, the cubic spline function method is used to determine the bearingless rotor blade vibration equation, which specifically includes the following formula:

;

in, is the inertia matrix,/> is the damping matrix,/> is the stiffness matrix,/> is the external load vector,/> and/> are respectively the first-order and second-order time derivatives of the blade node vibration displacement,/> is the vibration displacement of the blade node under the global degree of freedom.

Optionally, establishing the flapping motion equation and the oscillating motion equation of the blade within the blade motion coordinate system specifically includes the following formulas:

;

;

in, ,/> ,/> and/> are the inertial moment, root binding moment, structural damping moment and aerodynamic moment of the blade in the flapping direction, respectively./> ,/> ,/> and/> They are the inertia moment, root binding moment, structural damping moment and aerodynamic moment of the blade on the oscillation hinge respectively.

Optionally, establishing the rolling motion equation and pitching motion equation of the body in the body coordinate system specifically includes the following formulas:

;

;

in, and/> are respectively the moment of inertia around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> are respectively the damping coefficients around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> are respectively the constraint stiffness around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> Respectively:/> The rolling moment and pitching moment of the blade on the aircraft body,/> is the number of helicopter blades,/> and/> are the body roll angle and the body pitch angle respectively,/> and/> is the body roll angle/> The first derivation and the second derivation of ,/> and/> is the body roll angle/> The first and second derivatives of , subscript/> and/> are the horizontal and vertical coordinates of the body coordinate system respectively.

Optionally, the dynamic inflow equation is established based on the dynamic inflow model, specifically including the following formula:

;

in, and/> are the air explicit mass matrix and the inflow gain matrix respectively,/> ,/> and/> Respectively represent the total aerodynamic lift coefficient of the rotor, the aerodynamic rolling moment coefficient and the aerodynamic pitching moment coefficient relative to the center of the propeller hub,/> is the average induced speed of the propeller disk,/> and/> are the induced speeds caused by changes in the rotor aerodynamic rolling moment and pitching moment, respectively,/> is the first time derivative of the average induced velocity of the propeller disk,/> and/> are the first time derivatives of the induced velocity caused by changes in the rotor aerodynamic roll moment and pitching moment respectively, subscript/> Indicates aerodynamic coefficient, superscript/> Represents transposition.

Optionally, the motion equation set of the rotor/body coupling system is determined based on the vibration equation of the bearingless rotor blade, the flapping motion equation, the oscillation motion equation, the roll motion equation, the pitching motion equation and the dynamic inflow equation, specifically including the following formula:

;

in, is the set of motion equations of the rotor/body coupled system,/> ,/> are the flapping angle and oscillation angle of each blade respectively.

Optionally, the eigenvalue analysis method using a combination of finite elements and transfer matrix methods is used to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/body coupling system, specifically including:

Use formula Determine the steady-state response equation for the bearingless rotor blade vibration equation;

When the steady-state response equation is a steady state, use The formula determines the linearized small perturbation motion equation;

Solve the inherent modal characteristics and perform modal conversion based on the linearized small perturbation motion equation to determine the stability of the bearingless rotor blade vibration equation and the motion equation set of the rotor/body coupled system;

in, and/> are respectively the blade node displacement and external load vector in the hovering state,/> , and/> is the inertia matrix, damping matrix and stiffness matrix in the steady state,/> It is a small disturbance vibration,/> and/> are the first-order and second-order time derivatives of small perturbation vibration respectively.

A bearingless helicopter structure dynamic stability analysis system, including:

The bearingless rotor blade vibration equation establishment module is used to establish the bearingless rotor blade vibration equation using the cubic spline function method based on Hamilton's principle modeling;

The flapping motion equation and the oscillating motion equation establishment module is used to establish the flapping motion equation and the oscillating motion equation of the blade within the blade motion coordinate system;

The roll motion equation and pitch motion equation establishment module is used to establish the roll motion equation and pitch motion equation of the body in the body coordinate system;

The dynamic inflow equation establishment module is used to establish the dynamic inflow equation based on the dynamic inflow model;

The module for building a set of motion equations for the rotor/body coupling system is used to establish the rotor/body coupling system based on the bearingless rotor blade vibration equation, flapping motion equation, oscillation motion equation, roll motion equation, pitching motion equation and dynamic inflow equation. The system of equations of motion;

The stability determination module is used to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/airframe coupling system using the eigenvalue analysis method combined with the finite element method and the transfer matrix method.

According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

The invention provides a bearingless helicopter structure dynamic stability analysis method and system to stabilize the vibration equations of the bearingless rotor blades and the motion equations of the rotor/body coupled system including complex aerodynamics and structural dynamics. Based on the linear analysis model, a dynamic stability analysis method of bearing-less helicopter structures considering the complex motion of the blades is provided. Firstly, a bearingless rotor blade vibration equation and a set of motion equations of the rotor/airframe coupling system that accurately reflect the helicopter rotor structure and flight aerodynamic environment are established; the eigenvalue analysis method combining finite element and transfer matrix methods is used to solve the bearingless rotor blade vibration. Equations and stability of a system of equations of motion for a coupled rotor/airframe system. Based on the characteristics of severe coupling between the degrees of freedom of motion of bearing-less helicopter rotor blades, this invention uses a cubic spline interpolation function to represent the vibration displacement between nodes, and uses an eigenvalue analysis method that combines finite element and transfer matrix methods. It greatly improves the accuracy of the dynamic analysis of the bearingless rotor system. For bearingless rotor blades, the coupling between the degrees of freedom of motion is serious, the dynamic instability problem of the rotor/body coupling is prominent, and the dynamic stability analysis and research are difficult. Questions are instructive.

Description of the drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the drawings needed to be used in the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some of the drawings of the present invention. Embodiments, for those of ordinary skill in the art, other drawings can also be obtained based on these drawings without exerting creative efforts.

Figure 1 is a schematic flow chart of a bearingless helicopter structure dynamic stability analysis method provided by the present invention.

Figure 2 is a schematic diagram of the selection of the coordinate system of the bearingless rotor blade system of the present invention.

Figure 3 is a schematic diagram of the global degrees of freedom and unit local coordinates of the bearingless rotor blade system of the present invention.

Figure 4 is a schematic diagram of the physical model and coordinate system of the helicopter rotor/body coupling system of the present invention.

Figure 5 is a schematic diagram of the variation curve of the steady-state response at the propeller tip with the rotation speed according to the embodiment of the present invention.

Figure 6 is a schematic diagram of the root locus of the fundamental mode in three directions when the rotor speed increases according to the embodiment of the present invention.

Figure 7 is a schematic diagram of the rotor steady-state response of each part of the rotating blade at the reference speed in the coupling system according to the embodiment of the present invention.

Figure 8 is a schematic diagram of the fundamental vibration mode frequencies of each degree of freedom of the coupling system according to the embodiment of the present invention.

Figure 9 is a schematic diagram of the fundamental-order vibration mode damping of each degree of freedom of the coupling system according to the embodiment of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of the present invention.

The purpose of the present invention is to provide a bearingless helicopter structure dynamic stability analysis method and system, which can improve the accuracy of the bearingless rotor system dynamic analysis.

In order to make the above objects, features and advantages of the present invention more obvious and understandable, the present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments.

As shown in Figure 1, a bearingless helicopter structure dynamic stability analysis method provided by the present invention includes:

S101, based on Hamilton's principle modeling, use the cubic spline function method to establish the vibration equation of the bearingless rotor blade.

Determine the vibration equation of the bearingless rotor blade, that is, establish the bearingless rotor dynamic model. When only considering the vibration characteristics of the bearingless rotor system, it can be assumed that the center of the propeller hub is fixed and the rotor blades rotate around the propeller hub. The coordinate system is selected as shown in Figure 2. The meaning of each coordinate system in Figure 2 is as follows:

: Fixed coordinate system (propeller hub coordinate system), the origin is at the center of the propeller hub, no rotation,/> Up is positive.

: Rotor rotation coordinate system (blade undeformed coordinate system), the origin is at the center of the propeller hub, and it rotates with the rotor around the center of the propeller hub, /> There is an angle between the axis and the plane of rotation/> , that is, the pre-cone angle;/> for/> tensile displacement in the direction.

: Coordinate system after blade deformation (blade motion coordinate system), /> The axis coincides with the tangent of the elastic axis of the deformed blade,/> Within the blade cross-section and perpendicular to each other,/> Pointing toward the leading edge of the blade is positive.

Represents the point on the rotor before deformation,/> Represents the deformed point on the rotor,/> Indicates rotor speed,/> for/> Displacement in direction,/> for/> Displacement in direction.

According to Hamilton's principle:

(1)

in, ,/> ,/> represent the variation of strain energy and kinetic energy of the system and the virtual work done by the external force, respectively. ,/> They are the upper and lower bounds of the integral interval of Hamilton's principle respectively. Due to/> ,/> ,/> are independent of torsional vibration displacement and/> The differential of the vibration displacement in the direction with respect to time, where,/> ,/> ,/> are the torsional vibration displacement and/> vibration displacement in the direction. Therefore Hamilton's principle can be written as:

(2)

in, It is the difference between the strain energy of the system and the sum of kinetic energy and virtual work of external forces. The elastic beam that the blade is simplified into has vibrations in three directions: flapping vibration, shimmy vibration and torsional vibration. Divide the beam into several units, take the vibration displacement (and slope) of the two end points of each unit as degrees of freedom, and divide the blade into equal parts/> Segments, the length of each segment is/> , taking the endpoint of each blade section as a node, then we have/> nodes, as shown in Figure 3. Use numerical analysis method to obtain the cubic spline function, and take the vibration displacement of all nodes of the blade under the global degree of freedom as/> ,Right now:

(3)

in, ,/> ,/> are the torsional vibration displacement and/> vibration displacement in the direction. In Figure 3,/> ,/> ,/> Respectively:/> torsional vibration displacement at nodes and/> Vibration displacement in the direction,/> Indicates the first/> The position of the node on the x-axis,/> Indicates rotor speed. Paddle No./> Vibration displacements for three vibration degrees of freedom on segment elements are available/> The vibration displacement of nodes is expressed as:

(4)

Among them, the row vectors are:

(5)

(6)

(7)

in, For the first/> coordinates of nodes,/> is the unit length,/> is the unit column vector (/> dimension), ,/> ,/> is the shape function matrix of the propeller blade. /> Represent matrix/> of/> OK; the rest is similar. superscript/> Represents the transpose of a matrix, here/> represents a column vector, then/> Just means/> Transposed row vector. The subscript represents the node coordinates, such as/> Express/> of/> elements, and because/> is a column vector, and/> Indicates its/> OK, that’s the first/> elements. In formula (5), (6), (7)/> .

According to the three-slope method formula:

.

in, for/> The square matrix,/> for/> Column vector of matrix.

make:

.

.

.

in, ,/> and/> All/> matrix.

Substituting equations (4), (5), (6), and (7) into Hamilton's principle, we can get:

(8)

The boundary condition is fixed support, and finally the vibration equation of the blade can be obtained:

(9)

in, is the inertia matrix,/> is the damping matrix, is the stiffness matrix,/> is the external load vector,/> and/> are respectively the first-order and second-order time derivatives of the blade node vibration displacement,/> is the vibration displacement of the blade node under the global degree of freedom.

S102, in the blade motion coordinate system Establish the flapping motion equation and the oscillating motion equation of the blade.

No. The flapping motion equation and the oscillation motion equation of the blade are:

(10)

(11)

in, ,/> ,/> and/> are the inertial moment, root binding moment, structural damping moment and aerodynamic moment of the blade in the flapping direction, respectively./> ,/> ,/> and/> They are the inertia moment, root binding moment, structural damping moment and aerodynamic moment of the blade on the oscillation hinge respectively.

S103. Establish the rolling motion equation and pitching motion equation of the body in the body coordinate system.

S103 in the body coordinate system The rolling motion equations and pitching motion equations of the body established in the machine specifically include the following formulas:

and/> (12)

and/> are respectively the moment of inertia around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> are respectively the damping coefficients around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> are respectively the constraint stiffness around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> Respectively:/> The rolling moment and pitching moment of the blade on the aircraft body,/> is the number of helicopter blades,/> and/> are the body roll angle and the body pitch angle respectively, and/> is the body roll angle/> The first derivation and the second derivation of ,/> and/> is the body roll angle/> The first and second derivation of .

S104. Establish a dynamic inflow equation based on the dynamic inflow model.

Use the corresponding dynamic inflow model to describe the effect of unsteady aerodynamic forces, and obtain the dynamic inflow equation:

(13)

in, and/> are the air explicit mass matrix and the inflow gain matrix respectively,/> ,/> and/> Respectively represent the total aerodynamic lift coefficient of the rotor, the aerodynamic rolling moment coefficient and the aerodynamic pitching moment coefficient relative to the center of the propeller hub,/> is the average induced speed of the propeller disk,/> and/> are the induced speeds caused by changes in the rotor aerodynamic rolling moment and pitching moment, respectively,/> is the first time derivative of the average induced velocity of the propeller disk,/> and/> are the first time derivatives of the induced velocity caused by changes in the rotor aerodynamic roll moment and pitching moment respectively, subscript/> Indicates aerodynamic coefficient, superscript/> Represents transposition.

S105, establish a set of motion equations of the rotor/body coupling system based on the bearingless rotor blade vibration equation, flapping motion equation, oscillation motion equation, roll motion equation, pitching motion equation and dynamic inflow equation; that is, establishing the helicopter rotor/body coupling system system model.

The physical model of the helicopter rotor and body is shown in Figure 4. : The body coordinate system coincides with the equilibrium position, and the origin is located at the center of gravity of the helicopter;/> : Rotor coordinate system, the origin coincides with the center of the propeller hub before deformation, and is firmly connected to the body;/> : Propeller hub coordinate system, the coordinate origin is fixed at the center of the propeller hub and does not rotate;/> : Propeller hub rotation coordinate system, rotating with the propeller hub;/> ：The undeformed coordinate system of the propeller blade;/> : Paddle motion coordinate system, /> The axis coincides with the blade pitch axis; : Height from the center of the propeller hub to the center of gravity of the helicopter;/> : Rotor speed;/> : Blade azimuth angle;/> : Any point on the blade;/> ：/> Distance from point to blade root;/> : Waving angle;/> : Oscillation angle.

S105 specifically includes the following formulas:

(14)

in, is the set of motion equations of the rotor/body coupled system,/> ,/> are the flapping angle and oscillation angle of each blade respectively.

Then it can be written as:

(15)

in, is the azimuth angle of the blade motion,/> ,/> is the coefficient matrix of the equation of motion,/> is a system variable including the flapping angle, oscillation angle, body roll angle, body pitch angle and induced speed component of each blade,/> and/> Respectively, system variables/> The first and second time derivatives of .

S106, use the eigenvalue analysis method combined with the finite element method and the transfer matrix method to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/body coupled system.

S106 specifically includes:

Before solving for stability, the steady-state response of the blade in the hovering state must be obtained first. Omitting the time-related terms in formula (9), the steady-state response equation can be obtained:

(16)

Formula (16) is a one-dimensional linear non-homogeneous equation. The Newton algorithm can be used to iterate the solution of the odd-order equation corresponding to the initial value selection equation, that is , the iteration termination condition is/> ,/> For a given tolerance,/> is the external load vector. /> Only periodic aerodynamic forces are included.

When the steady-state response equation is a steady state, it is assumed that there is a small disturbance vibration near the steady-state corresponding ,Right now , brought into formula (9) and simplified, the linearized small perturbation motion equation can be obtained:

(17)

Solve the inherent modal characteristics and perform modal conversion according to the linearized small perturbation motion equation, that is, omit the damping term in formula (17) , and ignoring the aerodynamic force, the control equation under the natural vibration of the blade can be obtained:

(18)

The coefficient matrix of the control equation is a symmetric matrix, and its eigenvalues are real numbers, which are the natural frequencies of oscillation, flapping and torsion, and the corresponding eigenvectors are the natural modes.

In order to reduce the number of degrees of freedom when calculating system stability, the modal conversion method is used to convert the global degrees of freedom into modal space. Take the front part of formula (18) Eigenvectors form the modal space/> (/> order), let:

(19)

Substituting into formula (18) and multiplying on the left Available:

(20)

Convert formula (20) into the following state equation:

(twenty one)

in, ,/> . Coefficient matrix/> Eigenvalues/> is a complex number, the real part represents the modal damping of the system, and the imaginary part represents the modal frequency of the system. If the real part is negative, the system is stable. /> and/> are respectively the blade node displacement and external load vector in the hovering state,/> ,/> and/> is the inertia matrix, damping matrix, and stiffness matrix in the steady state,/> It is a small disturbance vibration,/> ,/> ,/> They are the inertia matrix, damping matrix and stiffness matrix,/> ,/> are the first-order and second-order time derivatives of small disturbance vibration respectively,/> is the small perturbation vibration column vector after mode conversion,/> and/> respectively/> The first and second time derivatives of .

The application of the present invention will be described in detail below with reference to specific embodiments and the accompanying drawings.

(1) Numerical calculation example of isolated rotor

A bearingless rotor blade is selected as an example for numerical calculation. The fundamental natural vibration frequencies corresponding to the three vibration directions of the blade model are: ,/> ,/> . The basic parameters of the blade are as shown in Table 1 for the structure and inertial parameters of the bearingless rotor blade model and the geometric and aerodynamic parameters of the bearingless rotor blade model in Table 2. The parameters in Table 1 below are the values after non-dimensionalization.

Table 1

Table 2

The data in Table 3 is a comparison between the results obtained by the present invention and the theoretical values. It can be seen that the results obtained by the present invention are in good agreement with the theoretical values.

table 3

Table 4 shows the numerical results of the fundamental natural frequency obtained when using different numbers of units. The relative differences between the results obtained when the number of units is 30 and 25 are less than 0.05% in the three vibration directions.

Table 4

As the rotor speed increases, the oscillation displacement and flapping displacement in the steady-state response at the blade tip tend to gradually decrease, while the torsional displacement first changes rapidly to the opposite direction and then gradually decreases, as shown in Figure 5.

Table 5 shows the bearingless rotor system model blades obtained after segmentation. The steady-state response of each node is the oscillation, flapping and torsional displacement at the blade tip. The iteration termination condition in the calculation is/> .

table 5

When the number of units is 30, the steady-state response at the propeller tip (Table 4) and the fundamental natural frequency (Table 2) are both highly accurate. Considering the amount of calculation and calculation time, the number of units in the numerical calculation process is is 30.

During the mode conversion process, the natural mode number will affect the accuracy of the stability calculation results. Table 6 shows the real parts of the basic-order vibration mode eigenvalues in the three vibration directions obtained when different mode numbers are used. It can be seen from the data in Table 6 that the stability results obtained by taking the first six natural modes have quite high accuracy. The base-order vibration of the rotor model in the flapping direction and torsion direction is stable, but the swing direction is slightly unstable. The stability can be ensured by changing the rotor structural parameters.

Table 6

The fundamental vibration has the greatest influence on the system vibration. During the change of rotation speed, the stability of the fundamental vibration mode of the blade will change. As shown in Figure 6, when the rotation speed is 0.4 ~2/> Within the range of It is in the unstable zone, but the instability tends to decrease,/> is the rotor speed after dimensionless transformation.

(2) Numerical Calculation Example of Ground Resonance of Bearingless Rotor Helicopter

The body parameters and other main parameters of the coupling system are shown in Table 7. The eigenvalue analysis method is now used to calculate the helicopter ground resonance stability.

Table 7

Helicopter ground resonance occurs when the helicopter is driving or taxiing on the ground, and its body system is supported on the ground by the landing gear. This embodiment shows how the modal damping and modal frequency of the system change with the rotation speed when the collective pitch angle is 0.

Figure 7 shows the steady-state response values of the casing, flexible beam and blade parts of the rotor blade in the coupled system at the reference speed. As can be seen from Figure 7, the root tensile displacement , shimmy displacement/> and swing displacement/> Mainly borne by flexible beams, while torsional displacement/> It is mainly borne by the casing, which is consistent with the actual situation of the bearingless rotor.

Figure 8 shows the fundamental vibration mode frequencies of each degree of freedom of the coupled system, where LR: oscillation retreat mode, LP: oscillation forward mode, FR: flailing retreat mode, FP: flailing forward mode state. at starting speed Previously, the oscillation retreat mode had an intersection with the aircraft body roll mode, but due to the relatively low rotor speed, the possibility of dynamic instability in the system was relatively small; in/> There is an intersection point between the front and rear rotor oscillation retreat mode and the body pitch mode frequency curve, indicating that the system is prone to ground resonance near this speed; at/> to/> The oscillation retreat mode within the rotational speed has an intersection point with the body roll mode, and ground resonance is more likely to occur in this rotational speed range.

Figure 9 shows the basic-order vibration mode damping of each degree of freedom of the coupled system. It can be seen that in Near the rotation speed, there is a slight coupling effect between the oscillation forward mode and the body roll mode, which improves the dynamic stability of the body roll mode, while the stability of the rotor oscillation forward mode decreases. You can also see it at/> Near the rotation speed, the stability of the flapping forward mode is significantly enhanced, while at the same time the stability of the swinging backward mode decreases.

It has been verified by the examples that the error between the base-order frequency of the isolated rotor obtained by the present invention and the exact solution is very small, and the calculated modal analysis of the steady-state response of the rotor and the ground resonance is consistent with the actual situation of the bearing-less rotor, proving that the results of the present invention are The correctness of the simplified beam model adopted verifies the effectiveness of the method used, and it has a complete theoretical analysis process.

As another specific embodiment, a bearingless helicopter structure dynamic stability analysis system provided by the present invention includes:

The bearingless rotor blade vibration equation establishment module is used to establish the bearingless rotor blade vibration equation using the cubic spline function method based on Hamilton's principle modeling.

The module for establishing the flapping motion equation and the oscillating motion equation is used to establish the flapping motion equation and the oscillating motion equation of the blade within the blade motion coordinate system.

The roll motion equation and pitch motion equation establishment module is used to establish the roll motion equation and pitch motion equation of the body in the body coordinate system.

The dynamic inflow equation establishment module is used to establish the dynamic inflow equation based on the dynamic inflow model.

The module for building a set of motion equations for the rotor/body coupling system is used to establish the rotor/body coupling system based on the bearingless rotor blade vibration equation, flapping motion equation, oscillation motion equation, roll motion equation, pitching motion equation and dynamic inflow equation. system of equations of motion.

The stability determination module is used to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/airframe coupling system using the eigenvalue analysis method combined with the finite element method and the transfer matrix method.

Each embodiment in this specification is described in a progressive manner. Each embodiment focuses on its differences from other embodiments. The same and similar parts between the various embodiments can be referred to each other. As for the system disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple. For relevant details, please refer to the description in the method section.

Specific examples are used in the present invention to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea; at the same time, for those of ordinary skill in the art, based on this The idea of the invention will be subject to change in the specific implementation and scope of application. In summary, the contents of this description should not be construed as limitations of the present invention.

Claims (8)

1. A bearingless helicopter structure dynamic stability analysis method, which is characterized by including: Based on Hamilton's principle modeling, the cubic spline function method is used to establish the vibration equation of the bearingless rotor blade; Establish the flapping motion equation and the oscillation motion equation of the blade within the blade motion coordinate system; Establish the rolling motion equation and pitching motion equation of the body in the body coordinate system; Establish the dynamic inflow equation based on the dynamic inflow model; Establish a set of motion equations of the rotor/body coupled system based on the bearingless rotor blade vibration equation, flapping motion equation, oscillation motion equation, roll motion equation, pitching motion equation and dynamic inflow equation; The eigenvalue analysis method combined with the finite element method and the transfer matrix method is used to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/body coupling system. 2. A bearingless helicopter structure dynamic stability analysis method according to claim 1, characterized in that, based on the Hamilton principle modeling, the cubic spline function method is used to determine the bearingless rotor blades. Vibration equation, specifically including the following formulas: ; in, is the inertia matrix,/> is the damping matrix,/> is the stiffness matrix,/> is the external load vector,/> and/> are respectively the first-order and second-order time derivatives of the blade node vibration displacement,/> is the vibration displacement of the blade node under the global degree of freedom. 3. A method for analyzing the dynamic stability of a bearingless helicopter structure according to claim 2, characterized in that the flapping motion equation and the oscillation motion equation of the blade are established in the blade motion coordinate system, specifically including: The following formula: ; ; in, ,/> ,/> and/> are the inertial moment, root binding moment, structural damping moment and aerodynamic moment of the blade in the flapping direction, respectively./> ,/> ,/> and/> They are the inertia moment, root binding moment, structural damping moment and aerodynamic moment of the blade on the oscillation hinge respectively. 4. A bearingless helicopter structure dynamic stability analysis method according to claim 3, characterized in that the rolling motion equation and the pitching motion equation of the body are established in the body coordinate system, specifically including the following formulas: ; ; in, and/> are respectively the moment of inertia around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> are respectively the damping coefficients around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> are respectively the constraint stiffness around the instantaneous rotation axis of the body based on the horizontal and vertical coordinates of the body coordinate system,/> and/> Respectively:/> The rolling moment and pitching moment of the blade on the aircraft body,/> is the number of helicopter blades,/> and/> are the body roll angle and the body pitch angle respectively, and/> is the body roll angle/> The first derivation and the second derivation of ,/> and/> is the body roll angle/> The first and second derivatives of , subscript/> and/> are the horizontal and vertical coordinates of the body coordinate system respectively. 5. A method for analyzing the dynamic stability of a bearingless helicopter structure according to claim 4, characterized in that the dynamic inflow equation is established according to the dynamic inflow model, specifically including the following formula: ; in, and/> are the air explicit mass matrix and the inflow gain matrix respectively,/> ,/> and/> Respectively represent the total aerodynamic lift coefficient of the rotor, the aerodynamic rolling moment coefficient and the aerodynamic pitching moment coefficient relative to the center of the propeller hub,/> is the average induced speed of the propeller disk,/> and/> are the induced speeds caused by changes in the rotor aerodynamic rolling moment and pitching moment, respectively,/> is the first time derivative of the average induced velocity of the propeller disk,/> and/> are the first time derivatives of the induced velocity caused by changes in the rotor aerodynamic roll moment and pitching moment respectively, subscript/> Indicates aerodynamic coefficient, superscript/> Represents transposition. 6. A method for analyzing the dynamic stability of a bearingless helicopter structure according to claim 5, characterized in that the bearingless rotor blade vibration equation, flapping motion equation, shimmy motion equation, and rolling motion equation are , the pitching motion equation and the dynamic inflow equation determine the motion equations of the rotor/body coupling system, which specifically include the following formulas: ; in, is the set of motion equations of the rotor/body coupled system,/> ,/> are the flapping angle and oscillation angle of each blade respectively. 7. A bearingless helicopter structure dynamic stability analysis method according to claim 6, characterized in that the eigenvalue analysis method combining finite element and transfer matrix methods is used to determine the bearingless rotor blade vibration equation and The stability of the motion equations of the rotor/body coupled system, including: Use formula Determine the steady-state response equation for the bearingless rotor blade vibration equation; When the steady-state response equation is a steady state, use The formula determines the linearized small perturbation motion equation; Solve the inherent modal characteristics and perform modal conversion based on the linearized small perturbation motion equation to determine the stability of the bearingless rotor blade vibration equation and the motion equation set of the rotor/body coupled system; in, and/> are respectively the blade node displacement and external load vector in the hovering state,/> ,/> and/> is the inertia matrix, damping matrix and stiffness matrix in the steady state,/> It is a small disturbance vibration,/> and are the first-order and second-order time derivatives of small perturbation vibration respectively. 8. A bearingless helicopter structure dynamic stability analysis system, which is characterized by including: The bearingless rotor blade vibration equation establishment module is used to establish the bearingless rotor blade vibration equation using the cubic spline function method based on Hamilton's principle modeling; The flapping motion equation and the oscillating motion equation establishment module is used to establish the flapping motion equation and the oscillating motion equation of the blade within the blade motion coordinate system; The roll motion equation and pitch motion equation establishment module is used to establish the roll motion equation and pitch motion equation of the body in the body coordinate system; The dynamic inflow equation establishment module is used to establish the dynamic inflow equation based on the dynamic inflow model; The module for building a set of motion equations for the rotor/body coupling system is used to establish the rotor/body coupling system based on the bearingless rotor blade vibration equation, flapping motion equation, oscillation motion equation, roll motion equation, pitching motion equation and dynamic inflow equation. The system of equations of motion; The stability determination module is used to determine the stability of the vibration equations of the bearingless rotor blades and the motion equations of the rotor/airframe coupling system using the eigenvalue analysis method combined with the finite element method and the transfer matrix method.