Helicopter tail rotor and tail beam coupling modeling method based on time-varying mixed interface modal synthesis method

Abstract

The invention provides a helicopter tail rotor and tail beam coupling modeling method based on a time-varying mixed interface modal synthesis method, and the method comprises the steps: firstly considering the rotation influence of a blade, and building a blade structure kinetic equation under a physical coordinate system; a blade structure kinetic equation under the modal coordinate system is obtained through modal coordinate transformation, tail beam modal information is obtained, and a tail beam structure kinetic equation under the modal coordinate system is established. Thirdly, a time-varying effect brought by rotation of the blade is considered, a quadratic coordinate transformation formula of modal speed and modal acceleration is obtained through conversion according to the displacement boundary condition between the blade and a tail beam position structure, and quadratic coordinate transformation is conducted on a blade structure kinetic equation under a modal coordinate system based on the quadratic coordinate transformation formula; and finally, assembling the blades and the tail beam according to a number-to-number seating principle to obtain a tail rotor and tail beam coupling time-varying structure kinetic equation under the modal coordinate system. The method is suitable for the coupling dynamics problem of the uniformly and non-uniformly arranged tail rotor and the tail beam, and the calculation efficiency in the modeling process is remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of structural dynamics technology, specifically relating to a coupling modeling method for helicopter tail rotor and tail boom based on time-varying hybrid interface modal synthesis. Background Technology

[0002] As a unique type of aircraft, the flight performance and safety of helicopters are highly dependent on the complex dynamic coupling characteristics between their various components. The tail rotor, as a critical control and balancing component of the helicopter, primarily functions to counteract the anti-torque generated by the main rotor and provide directional control. Therefore, the dynamic characteristics and aeroelasticity of the tail rotor are both a key focus and a challenge in helicopter tail rotor dynamics research.

[0003] With advancements in aviation technology, helicopter structural design is showing a significant trend towards lightweight construction. As the support structure for the tail rotor, the tail boom's stiffness is relatively reduced, while its elastic effect becomes increasingly pronounced. The elastic deformation of the tail boom directly alters the dynamic boundary conditions of the tail rotor, while the dynamic loads on the tail rotor react on the tail boom, creating a strong "tail rotor / tail boom" coupled dynamic phenomenon. This coupling effect significantly impacts the vibration level, stability, and fatigue life of the entire tail system. Therefore, accurately considering the coupling effect between the tail rotor and tail boom in modeling and analysis is crucial for the design and development of modern helicopters.

[0004] In the prior art, there are several modeling methods for such coupling problems. For example, Chinese patent application CN119442474A discloses a "direct synthesis modeling method for coupling substructures of helicopter tail rotor / tail boom". This method aims to construct a complete coupling model by directly establishing the dynamic equations of the tail rotor and tail boom in the physical coordinate system and considering their motion in different directions. The advantage of this method is that it does not perform modal coordinate transformation, the model building process is simple and direct, and theoretically it can be applied to various tail rotor configurations, such as uniform arrangement (e.g., cross rotor) and non-uniform arrangement (e.g., scissor rotor).

[0005] However, this existing technical solution has significant limitations and is difficult to meet the actual needs of modern engineering. The tail boom structure in real-world engineering projects is typically very complex, containing numerous detailed features, resulting in a massive finite element model. The aforementioned direct synthesis method for substructures, modeling and coupling in the physical coordinate system, leads to an extremely large number of degrees of freedom in the final tail rotor and tail boom model, resulting in low computational efficiency and excessively long computation time, failing to meet the engineering requirements for rapid design. Summary of the Invention

[0006] To address the issue that existing direct synthesis modeling methods for coupled tail rotor and tail boom substructures in helicopters cannot balance computational efficiency and model accuracy for large-scale complex structures, this invention provides a coupled modeling method for helicopter tail rotor and tail boom based on time-varying hybrid interface modal synthesis. This method not only considers the rotational dynamics of the blades, improving the reliability of the model, but also significantly improves the computational efficiency of the model by reducing the order of the tail boom and tail rotor model. It can efficiently handle various complex tail boom configurations, including scissor rotors.

[0007] To achieve the above objectives, this invention provides a coupled modeling method for helicopter tail rotor and tail boom based on time-varying hybrid interface modal synthesis, comprising the following steps:

[0008] Step 1: For each blade of the tail rotor, establish the blade structural dynamics equations in the modal coordinate system; including the following sub-steps:

[0009] Step 1.1: For each blade, establish the blade structure dynamics equation in the physical coordinate system.

[0010] Step 1.2: For each blade, the blade structure dynamics equation in the physical coordinate system is transformed by the pre-established modal coordinate transformation formula to obtain the blade structure dynamics equation in the modal coordinate system for each blade.

[0011] Step 2: Determine the modal information of the tail beam, and establish the structural dynamics equations of the tail beam in the modal coordinate system based on the modal information of the tail beam;

[0012] Step 3: Based on the blade structural dynamics equations in the modal coordinate system, the tail boom structural dynamics equations in the modal coordinate system, and the physical displacement boundary conditions between the blades and the tail boom structure, establish a coupled time-varying structural dynamics model of the tail rotor-tail boom in the modal coordinate system; including the following sub-steps:

[0013] Step 3.1, Boundary condition mode transformation;

[0014] The physical displacement boundary conditions between each blade and the tail boom structure are transformed into the modal coordinate system to obtain the displacement boundary conditions between the blade and the tail boom in the modal coordinate system.

[0015] Step 3.2, Solve the quadratic transformation equation:

[0016] The displacement boundary conditions of the blade and tail boom in the modal coordinate system are solved to obtain the displacement quadratic coordinate transformation formula;

[0017] Step 3.3, Derivation of the time-varying derivative:

[0018] Based on the time-varying effect of blade rotation, the derivative of the displacement quadratic coordinate transformation formula is obtained to obtain the corresponding quadratic coordinate transformation formulas of blade modal velocity and blade modal acceleration.

[0019] Step 3.4, Secondary transformation of blade equations and parameter extraction:

[0020] By using the quadratic coordinate transformation formulas of blade modal velocity and blade modal acceleration, the dynamic equations of blade structure in the modal coordinate system are transformed into quadratic coordinates to obtain the generalized stiffness matrix, generalized damping matrix, generalized mass matrix and generalized force vector of the blade; and in the process of quadratic coordinate transformation, the additional stiffness matrix and additional damping matrix generated by the time-varying rotation of the blade are derived.

[0021] Step 3.5, Model Assembly:

[0022] The dynamic equations of each blade structure after the second coordinate transformation are combined with the dynamic equations of the tail boom in the modal coordinate system to obtain the time-varying dynamic equations of the tail rotor-tail boom coupled in the modal coordinate system.

[0023] Further, in step 1.1, the dynamic equations of the blade structure in the physical coordinate system are established, including the following steps:

[0024] Step 1.1.1 Discretize the established blade structure model using the finite element method, divide it into multiple beam elements, and obtain the beam element stiffness matrix, beam element damping matrix, beam element mass matrix, and beam element generalized force vector of each beam element in the physical coordinate system.

[0025] Step 1.1.2: Based on the position of each beam element in the overall blade structure, according to the matching rule, assemble the beam element stiffness matrix, beam element damping matrix, beam element mass matrix and beam element generalized force vector obtained in Step 1.1.1 to obtain the overall stiffness matrix, overall damping matrix, overall mass matrix and overall generalized force vector of a single blade in the physical coordinate system.

[0026] Step 1.1.3: Based on the overall stiffness matrix, overall damping matrix, overall mass matrix and overall generalized force vector obtained in Step 1.1.2, establish the structural dynamics equations of a single blade in the physical coordinate system.

[0027] Furthermore, step 1.2 includes the following sub-steps:

[0028] Step 1.2.1: Based on the Craig-Bampton method in the fixed-interface modal synthesis method, construct the coordinate transformation relationship between the physical coordinates and modal coordinates of the blade. { q β } = [ ϕ β ]{ p β } ,in, , , [ ϕ β ] These represent the physical coordinates, modal coordinates, and mode shape set of the blade, respectively. The mode set of the blade adopts the Craig-Bampton assumption mode set and is obtained by solving the blade structure dynamics equations in the physical coordinate system.

[0029] Step 1.2.2: The coordinate transformation relationship between the physical coordinates and modal coordinates of the blade established in Step 1.2.1 is divided into blocks, and the transformation relationship after block processing is defined as the modal coordinate transformation formula.

[0030] The modal coordinate transformation formula is:

[0031] { { q i β } { q j β } } = [ [ φ I β ] [ φ ic β ] [ φ jk β ] [ φ jc β ] ] { { p k β } { p c β } } ; ;

[0032] In the formula, Physical coordinates representing the degrees of freedom within the blade structure. Physical coordinates representing the degrees of freedom at the blade structure connection. Represents the coordinates of the principal modes within the blade structure. Represents the blade constraint modal coordinates. This represents the principal mode shape corresponding to the internal degrees of freedom of the blade structure. These represent the constrained mode shapes corresponding to the internal degrees of freedom of the blade structure. This represents the principal mode shape at the connection point of the corresponding blade structure. This represents the constrained mode shape corresponding to the degree of freedom at the blade structure connection.

[0033] Step 1.2.3: Using the modal coordinate transformation formula determined in Step 1.2.2, perform coordinate transformation on the blade structure dynamics equations in the physical coordinate system to obtain the blade structure dynamics equations in the modal coordinate system as follows:

[0034]

[0035] In the formula, , , These represent the modal displacement, modal velocity, and modal acceleration of the blade, respectively. , , , These are the generalized mass matrix, generalized stiffness matrix, generalized damping matrix, and generalized force vector of the blade in the modal coordinate system; they are obtained by transforming the corresponding matrices in the physical coordinate system using the modal coordinate transformation formula.

[0036] in:

[0037] [ m ¯ ] β = [ φ β ] T [ M ] β [ φ β ] [ k ¯ ] β = [ φ β ] T [ K ] β [ φ β ] [ c ¯ ] β = [ φ β ] T [ C ] β [ φ β ] [ F ¯ ] β = [ φ β ] T [ F ] β [ φ β ]

[0038] In the formula, , , and These represent the overall stiffness matrix, overall damping matrix, overall mass matrix, and overall generalized force vector of the blade in the physical coordinate system, respectively.

[0039] Furthermore, step 2 includes the following sub-steps:

[0040] Step 2.1: Obtain low-order modal information of the tail boom using commercial finite element software or ground modal testing; the modal information should include at least the front... First natural frequency ,forward First-order modal damping ratio ,forward First-order modal mass and tail beam mode shape set [ ϕ α ] ;in, , The selected number of natural modes of the tail beam;

[0041] Step 2.2: Using the Shou-nin-Hou method in the free interface modal synthesis method, combined with the low-order modal information of the tail beam, the dynamic equations of the tail beam structure in the natural modal coordinate system are established as follows:

[0042] [ m ¯ ] α { p ¨ α } + [ c ¯ ] α { p ˙ α } + [ k ¯ ] α { p α } − [ F ¯ ] α = [ 0 ]

[0043] in:

[0044] [ m ¯ ] α = [ m 1 0 0 0 ⋱ 0 0 0 m u ] , [ c ¯ ] α = [ c 1 0 0 0 ⋱ 0 0 0 c u ] ;

[0045] [ k ¯ ] α = [ ω 1 2 * m 1 0 0 0 ⋱ 0 0 0 ω u 2 * m u ] , [ F ¯ ] α = [ 0 ] ;

[0046] In the formula, , , and Let represent the generalized mass matrix, generalized damping matrix, generalized stiffness matrix, and generalized force vector of the tail beam in the natural modal coordinate system, respectively. , , These represent the modal displacement, modal velocity, and modal acceleration of the tail beam, respectively.

[0047] Furthermore, in step 3.1, the method for obtaining the displacement boundary conditions of the blade and tail boom in the modal coordinate system is as follows:

[0048] First, obtain the physical displacement boundary conditions between the blades and the tail boom structure, including:

[0049] { q j β n } = [ L ] α − β n { q j α }

[0050] [ L ] α − β m = [ [ T θ ] [ 0 ] [ 0 ] [ T θ ] ]

[0051] [ T θ ] = [ because θ that θ 0 − that θ because θ 0 0 0 1 ]

[0052] In the formula, the matrix [ L ] α − β n Indicates the first Physical displacement boundary conditions between the blades and the tail boom structure. For the first Physical coordinates of the degrees of freedom at the blade connection point. Physical coordinates representing the degrees of freedom at the tail beam structure connection. [ T θ ] This is the displacement transformation matrix. Indicates the azimuth angle between the tail boom and the blade;

[0053] Then, using the coordinate transformation relationship between physical coordinates and modal coordinates { q } = [ ϕ ]{ p } The physical displacement boundary conditions between the blade and the tail boom structure are transformed to the modal coordinate system, resulting in the following displacement boundary conditions for the blade and tail boom in the modal coordinate system:

[0054] [ [ 0 ] [ I ] ] { { p k β n } { p c β n } } = [ L ] α − β n [ [ ϕ jk α ] [ ϕ jc α ] ] { { p k α } { p c α } }

[0055] In the formula, These are physical coordinates. These are modal coordinates. [ ϕ ] It is a modality set; For the first Principal mode coordinates inside the blade structure For the first Blade constrained modal coordinates, The coordinates of the principal modes inside the tail beam structure are as follows. The coordinates of the tail beam constrained mode are as follows: [ ϕ jk α ] This represents the principal mode shape at the connection point of the corresponding tail beam structure. [ ϕ jc α ] This represents the constrained mode shape at the connection point of the corresponding tail beam structure. [ I ] for 3D identity matrix;

[0056] Finally, the expression for the displacement boundary conditions of the blade and tail boom in the modal coordinate system is expanded as follows:

[0057] { p c β n } = [ L ] α − β n [ ϕ jk α ] { p k α } + [ L ] α − β n [ ϕ jc α ] { p c α } .

[0058] Furthermore, in step 3.2, the expression for the displacement boundary conditions of the blade and tail boom in the expanded modal coordinate system is solved, yielding the displacement quadratic coordinate transformation formula:

[0059] { { p k β n } { p c β n } } = [ T ] α − β n { { p k α } { p c α } { p k β n } }

[0060] [ T ] α − β n = [ [ 0 ] [ 0 ] [ I ] [ L ] α − β n [ ϕ jk α ] [ L ] α − β n [ ϕ jc α ] [ 0 ] ]

[0061] In the formula, [ T ] α − β n This is the coordinate transformation matrix for the second mode of the propeller blade. for 3D identity matrix The selected tail beam natural mode number, For the first Principal mode coordinates inside the blade structure For the first Blade constrained modal coordinates, The coordinates of the principal modes inside the tail beam structure are as follows. The coordinates are the constraint modal coordinates of the tail beam.

[0062] Further, in step 3.3, the derivative of the displacement quadratic transformation equation is taken to obtain the corresponding quadratic coordinate transformation equations for the blade modal velocity and the blade modal acceleration; where:

[0063] The quadratic transformation formula for the blade modal velocity is:

[0064]

[0065] In the formula, Indicates the first The dominant mode velocity inside the blade structure, Indicates the first Blade constrained modal velocities, This indicates the principal mode velocity within the tail beam structure. Indicates the tail beam constrained modal velocity. [ T ˙ ] α − β n The time-dependent transformation matrix of the blade's second-mode coordinates The first derivative;

[0066] The quadratic transformation formula for the blade modal acceleration is:

[0067]

[0068] In the formula, Indicates the first Principal mode acceleration within the blade structure, Indicates the first Blade-constrained modal acceleration, This indicates the principal modal acceleration within the tail beam structure. This represents the tail beam constrained modal acceleration. [ T ¨ ] α − β n The time-dependent transformation matrix of the blade's second-mode coordinates The second derivative.

[0069] Furthermore, step 3.4 includes the following procedures:

[0070] Step 3.4.1: Using the quadratic coordinate transformation formulas for the blade modal velocities and modal accelerations, a quadratic coordinate transformation is performed on the blade structural dynamics equations in the modal coordinate system to eliminate non-independent modal coordinates. This yields the blade's generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector, respectively:

[0071]

[0072] in, , , and The first two coordinates are obtained after two coordinate transformations. The generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade; [ T ] α − β m T [ c ¯ ] β m [ T ˙ ] α − β m and [ T ] α − β m T [ m ¯ ] β m [ T ¨ ] α − β m To add a stiffness matrix, 2 [ T ] α − β m T [ m ¯ ] β m [ T ˙ ] α − β m For the addition of a damping array;

[0073] Step 3.4.2 involves performing a block expansion on the generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade obtained through the second transformation, resulting in:

[0074] [ [ M ¯ ] yes β n [ M ¯ ] okay β n [ M ¯ ] I β n [ M ¯ ] kk β n ] { { p ¨ o α } { p ¨ k β n } } + [ [ C ¯ ] yes β n [ C ¯ ] okay β n [ C ¯ ] I β n [ C ¯ ] kk β n ] { { p ˙ o α } { p ˙ k β n } } + [ [ K ¯ ] yes β n [ K ¯ ] okay β n [ K ¯ ] I β n [ K ¯ ] kk β n ] { { p o α } { p k β n } } =− [ [ R ¯ ] o β n [ R ¯ ] k β n ] ;

[0075] in:

[0076] ; [ [ R ¯ ] o β n [ R ¯ ] k β n ] = [ R ¯ ] β n ;

[0077] [ [ M ¯ ] yes β n [ M ¯ ] okay β n [ M ¯ ] I β n [ M ¯ ] kk β n ] = [ M ¯ ] β n ; [ [ C ¯ ] yes β n [ C ¯ ] okay β n [ C ¯ ] I β n [ C ¯ ] kk β n ] = [ C ¯ ] β n ; [ [ K ¯ ] yes β n [ K ¯ ] okay β n [ K ¯ ] I β n [ K ¯ ] kk β n ] = [ K ¯ ] β n ;

[0078] In the formula, , , These represent the sets of all modal coordinates, modal velocities, and modal accelerations in the tail beam structure, respectively. [ M ¯ ] yes β n , [ M ¯ ] okay β n , [ M ¯ ] I β n , [ M ¯ ] kk β n The th coordinates after the second transformation are respectively blade generalized mass array [ M ¯ ] β n The block submatrix, [ C ¯ ] yes β n , [ C ¯ ] okay β n , [ C ¯ ] I β n , [ C ¯ ] kk β n The th coordinates after the second transformation are respectively bladed generalized damping array [ C ¯ ] β n The block submatrix, [ K ¯ ] yes β n , [ K ¯ ] okay β n , [ K ¯ ] I β n , [ K ¯ ] kk β n The th coordinates after the second transformation are respectively blade generalized stiffness array [ K ¯ ] β n The block submatrix, [ R ¯ ] o β n , [ R ¯ ] k β n The th coordinates after the second transformation are respectively blade generalized force vector [ R ¯ ] β n The block submatrix.

[0079] The advantages of this invention are:

[0080] This invention, based on a time-varying hybrid interface modal synthesis method, constructs a coupled dynamic model of the tail rotor and tail boom, achieving significant improvements in modeling efficiency, applicability, and analytical capabilities. Specifically, this is reflected in the following aspects:

[0081] (1) This invention transforms the dynamic equations of the blade structure into the modal coordinate system through modal coordinate transformation and quadratic coordinate transformation, effectively eliminating the non-independent modal coordinates, and obtaining the additional stiffness matrix and additional damping matrix generated by the time-varying effect of blade rotation. Thus, while ensuring the accuracy of the model, the computational scale and solution time are greatly reduced, and the modeling efficiency is significantly improved.

[0082] (2) The method of the present invention is adaptable to complex tail rotor configurations and can effectively handle unconventional tail rotor structures, such as scissor tail rotors and non-uniformly arranged tail rotors. It breaks through the limitations of traditional modeling methods in terms of configuration adaptability and provides a universal and efficient coupling modeling means for the coupling design of various non-uniformly arranged tail rotors and tail booms.

[0083] (3) The modeling process of this invention is simplified. Only the modal information of the tail beam structure needs to be obtained to complete the modeling. It does not need to rely on complex structural details. It is easy to implement quickly by combining commercial software or ground modal test data. It has good engineering applicability and promotion.

[0084] (4) The method of this invention has broad application prospects. The model established by this method is applicable to key issues such as dynamic characteristic solving and stability analysis of tail rotor / tail boom coupled models, and provides a new analytical approach for the dynamic study of multibody coupled systems such as helicopter tail rotor / tail boom and even rotor / fuselage. Attached Figure Description

[0085] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0086] Figure 1 This is a flowchart of the helicopter tail rotor and tail boom coupling modeling method based on the time-varying hybrid interface modal synthesis method of the present invention;

[0087] Figure 2 This is a schematic diagram of discretizing the beam element model of the blade in this invention;

[0088] Figure 3 This is a schematic diagram of the verification working condition of the scissor tail rotor-tail boom coupling model in the embodiment;

[0089] Figure 4 This is a comparison chart of the response calculation results of the scissor tail rotor-tail boom coupling model obtained by the method of this invention and Adams multibody dynamics software; wherein, Figure 4 a represents a comparison of the paddle tip flapping response results. Figure 4 b represents a comparison of the propeller hub flapping response results;

[0090] Figure 5 This is a comparison chart of the response calculation results of the scissor tail rotor-tail boom coupling model obtained by the method of this invention and the direct synthesis method of substructure; wherein, Figure 5a represents a comparison of the paddle tip flapping response results. Figure 5 b represents a comparison of the propeller hub flapping response results. Detailed Implementation

[0091] The embodiments of the present invention are described in detail below. These embodiments are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0092] Reference Figure 1 This embodiment specifically illustrates the process of the helicopter tail rotor and tail boom coupling modeling method based on the time-varying hybrid interface modal synthesis method:

[0093] Step 1: For each blade of the tail rotor, establish the blade structural dynamics equations in modal coordinates. This includes the following process:

[0094] Step 1.1: Establish the blade structure dynamics equations for each blade of the tail rotor in the physical coordinate system.

[0095] First, taking a single blade as the research object, considering the influence of inertial force caused by rotational speed, based on Hamilton's equation and the theory of moderately deformable beams, the variational expressions of the kinetic energy and strain energy of a single blade are obtained, and a structural model of a single blade is established.

[0096] Then, the blade structure model is discretized using the finite element method, dividing it into multiple beam elements, such as... Figure 2 As shown, the stiffness matrix, damping matrix, mass matrix, and generalized force vector expression of each beam element in the physical coordinate system are obtained, resulting in the structural dynamic equations for each beam element as follows:

[0097] (1)

[0098] in, It is the first The mass matrix of beam elements of a beam. It is the first The stiffness matrix of the beam element of a beam. It is the first Damping matrix of beam elements of a beam, It is the first The generalized force vector of a beam element. It is the first The displacement vector of each beam.

[0099] Next, based on the position of each beam element in the overall blade structure, all the obtained beam element structural dynamic equations are assembled according to the principle of "matching the numbers to their corresponding positions" to obtain the overall mass matrix of a single blade in the physical coordinate system. Overall damping array Overall stiffness matrix and the overall generalized force vector Based on the overall mass matrix, overall damping matrix, overall stiffness matrix, and overall generalized force vector matrix, the substructure dynamic equations of a single blade are obtained, as shown below:

[0100] (2)

[0101] in, This represents all physical degrees of freedom after the blades are discretized. This represents the velocity of all physical degrees of freedom after the blades are discretized. This represents the acceleration of all physical degrees of freedom after the blades are discretized.

[0102] In this embodiment, it is assumed that the tail beam is... Structure, blades are Structure, subscripts in the formula are The relevant parameters of the tail beam are represented by the subscripts in the formula. The parameters are represented as those of the blade.

[0103] Step 1.2: For each blade, the blade structural dynamics equations in the physical coordinate system are transformed using a pre-established modal coordinate transformation formula to obtain the blade structural dynamics equations in the modal coordinate system for each blade. This specifically includes the following sub-steps:

[0104] Step 1.2.1: Based on the Craig-Bampton method (CB method) in the fixed interface modal synthesis method, construct the coordinate transformation relationship between the physical coordinates and modal coordinates of the blade:

[0105] { q β } = [ ϕ β ]{ p β } (3)

[0106] in, , , [ ϕ β ] These represent the physical coordinates, modal coordinates, and mode shape set of the blade, respectively. The mode set of the blade adopts the Craig-Bampton assumption mode set (i.e., the CB assumption mode set). The CB assumption mode set is obtained by solving the blade structure dynamics equations in the physical coordinate system.

[0107] Step 1.2.2: The coordinate transformation relationship between the physical coordinates and modal coordinates of the blade established in Step 1.2.1 is processed in blocks to obtain the modal coordinate transformation formula:

[0108] { { q i β } { q j β } } = [ [ φ I β ] [ φ ic β ] [ φ jk β ] [ φ jc β ] ] { { p k β } { p c β } } ; (4)

[0109] In the formula, Physical coordinates representing the degrees of freedom within the blade structure. Physical coordinates representing the degrees of freedom at the blade structure connection. Represents the coordinates of the principal modes within the blade structure. Represents the blade constraint modal coordinates. This represents the principal mode shape corresponding to the internal degrees of freedom of the blade structure. These represent the constrained mode shapes corresponding to the internal degrees of freedom of the blade structure. This represents the principal mode shape at the connection point of the corresponding blade structure. This represents the constrained mode shape at the connection point of the corresponding blade structure.

[0110] Step 1.2.3: Using the modal coordinate transformation formula determined in Step 1.2.2, perform coordinate transformation on the blade structure dynamics equations in the physical coordinate system to obtain the blade structure dynamics equations in the modal coordinate system as follows:

[0111] (5)

[0112] In the formula, , , These represent the modal displacement, modal velocity, and modal acceleration of the blade, respectively. , , , These are the generalized mass matrix, generalized stiffness matrix, generalized damping matrix, and generalized force vector of the blade in the modal coordinate system, respectively; they are obtained by transforming the corresponding matrices in the physical coordinate system using the modal coordinate transformation formula expressed by formula (4);

[0113] in:

[0114] [ m ¯ ] β = [ φ β ] T [ M ] β [ φ β ] [ k ¯ ] β = [ φ β ] T [ K ] β [ φ β ] [ c ¯ ] β = [ φ β ] T [ C ] β [ φ β ] [ F ¯ ] β = [ φ β ] T [ F ] β [ φ β ] (6)

[0115] In the formula, , , and These represent the overall stiffness matrix, overall damping matrix, overall mass matrix, and overall generalized force vector of the blade in the physical coordinate system, respectively.

[0116] Step 2: Determine the modal information of the tail beam, and establish the structural dynamics equations of the tail beam in the modal coordinate system based on the modal information.

[0117] Before establishing the structural dynamics equations of the tail beam in the modal coordinate system, the inventors of this application conducted the following analysis of the tail beam structure:

[0118] First, the coordinate transformation relationship between the physical coordinates and modal coordinates of the tail beam was established:

[0119] { q α } = [ ϕ α ]{ p α } (7)

[0120] in, Represents the physical coordinates of the tail beam. Represents the modal coordinates of the tail beam. [ ϕ α ] This represents the set of low-order natural modes of the tail beam.

[0121] Then, the coordinate transformation relationship between the physical coordinates and modal coordinates of the tail beam is processed by dividing it into blocks. The physical coordinates are divided into internal degrees of freedom physical coordinates and connection degree of freedom physical coordinates, and the modal coordinates are divided into internal principal modal coordinates and constraint modal coordinates. The block division is as follows:

[0122] ; { { q i α } { q j α } } = [ [ φ I α ] [ φ ic α ] [ φ jk α ] [ φ jc α ] ] { { p k α } { p c α } } (8)

[0123] In the formula, This represents the physical coordinates of the internal degrees of freedom of the tail beam structure. This represents the physical coordinates of the degrees of freedom at the connection point of the tail beam structure. This represents the coordinates of the principal modes within the tail beam structure. Represents the constraint modal coordinates of the tail beam. This represents the principal mode shape of the corresponding tail beam structure's internal degrees of freedom. This represents the constrained mode shape corresponding to the internal degrees of freedom of the tail beam structure. This represents the principal mode shape at the connection point of the corresponding tail beam structure. This represents the constrained mode shape at the connection point of the tail beam structure.

[0124] As can be seen from the above modal analysis theory, in the natural modal coordinate system, the structural dynamics equations can be directly established from the modal analysis results such as modal frequencies, modal damping ratios, and modal masses. Therefore, compared to the tail rotor, the tail boom structure is simpler and can be modally analyzed using commercial finite element software such as Patran / Nastran, Abaqus, and Ansys, and the tail boom structural dynamics equations can be established based on the modal analysis results; alternatively, tail boom modal information can be obtained through ground modal experiments, and then the tail boom structural dynamics equations in the natural modal coordinate system can be established based on the tail boom modal information.

[0125] Based on the above analysis results, this embodiment completes the establishment of the structural dynamics equations of the tail beam in the modal coordinate system according to the following steps.

[0126] Step 2.1: Obtain low-order modal information of the tail boom using commercial finite element software or ground modal experiments; the modal information includes at least the front... First natural frequency ,forward First-order modal damping ratio ,forward First-order modal mass and tail beam mode shape set [ ϕ α ] ;in, , The selected number of natural modes of the tail beam;

[0127] Step 2.2: Using the Shou-nin-Hou method in the free interface modal synthesis method, combined with the low-order modal information of the tail beam, the dynamic equation of the tail beam structure in the intrinsic modal coordinate system is established as follows:

[0128] [ m ¯ ] α { p ¨ α } + [ c ¯ ] α { p ˙ α } + [ k ¯ ] α { p α } − [ F ¯ ] α = [ 0 ] (9)

[0129] in:

[0130] [ m ¯ ] α = [ m 1 0 0 0 ⋱ 0 0 0 m u ] ;

[0131] [ c ¯ ] α = [ c 1 0 0 0 ⋱ 0 0 0 c u ] ;

[0132] [ k ¯ ] α = [ ω 1 2 * m 1 0 0 0 ⋱ 0 0 0 ω u 2 * m u ] ;

[0133] [ F ¯ ] α = [ 0 ] ;

[0134] In the formula, , , and Let represent the generalized mass matrix, generalized damping matrix, generalized stiffness matrix, and generalized force vector of the tail beam in the natural modal coordinate system, respectively. , , These represent the modal displacement, modal velocity, and modal acceleration of the tail beam, respectively.

[0135] Step 3: Based on the blade structural dynamics equations in the modal coordinate system, the tail boom structural dynamics equations in the modal coordinate system, and the physical displacement boundary conditions between the blades and the tail boom structure, establish a coupled time-varying structural dynamics model of the tail rotor-tail boom in the modal coordinate system. This includes the following sub-steps:

[0136] Step 3.1, Boundary condition mode transformation: Transform the physical displacement boundary conditions between each blade and the tail boom structure to the modal coordinate system to obtain the displacement boundary conditions between the blade and the tail boom in the modal coordinate system.

[0137] First, the physical displacement boundary conditions between the blade and the tail boom structure are obtained as follows:

[0138] { q j β n } = [ L ] α − β n { q j α }

[0139] [ L ] α − β m = [ [ T θ ] [ 0 ] [ 0 ] [ T θ ] ]

[0140] [ T θ ] = [ because θ that θ 0 − that θ because θ 0 0 0 1 ] (10)

[0141] In the formula, the matrix Indicates the first Physical displacement boundary conditions between the blades and the tail boom structure. Indicates the first Physical coordinates of the degrees of freedom at the blade connection point. Physical coordinates representing the degrees of freedom at the tail beam structure connection. [ T θ ] This is the displacement transformation matrix. This indicates the azimuth angle between the tail boom and the propeller blade.

[0142] Then, the physical displacement boundary conditions between the blades and the tail boom structure are transformed to the modal coordinate system. The transformation method is as follows:

[0143] Using the coordinate transformation relationship between physical coordinates and modal coordinates { q } = [ ϕ ]{ p } The physical displacement boundary conditions between the blade and the tail boom structure are transformed to the modal coordinate system, resulting in the following displacement boundary conditions for the blade and tail boom in the modal coordinate system:

[0144] [ [ 0 ] [ I ] ] { { p k β n } { p c β n } } = [ L ] α − β n [ [ ϕ jk α ] [ ϕ jc α ] ] { { p k α } { p c α } } (11)

[0145] In the formula, These are physical coordinates. These are modal coordinates. [ ϕ ] It is a modality set; For the first Principal mode coordinates inside the blade structure For the first Blade constrained modal coordinates, The coordinates of the principal modes inside the tail beam structure are as follows. The coordinates of the tail beam constrained mode are as follows: [ ϕ jk α ] This represents the principal mode shape at the connection point of the corresponding tail beam structure. [ ϕ jc α ] This represents the constrained mode shape at the connection point of the corresponding tail beam structure. [ I ] for 3D identity matrix.

[0146] Finally, the expression for the displacement boundary conditions of the blade and tail boom in the modal coordinate system is expanded as follows:

[0147] { p c β n } = [ L ] α − β n [ ϕ jk α ] { p k α } + [ L ] α − β n [ ϕ jc α ] { p c α } (12)

[0148] As can be seen from formula (10-12), the physical coordinates between the blade and the tail boom are coupled and not independent. Therefore, there is also a coupling relationship between the modal coordinates, and a secondary coordinate transformation is needed to eliminate the non-independent modal coordinates.

[0149] Step 3.2, Solving the quadratic transformation equation: Solve the expression for the displacement boundary conditions of the blade and tail beam in the modal coordinate system shown in formula (12), and obtain the displacement quadratic coordinate transformation equation as follows:

[0150] { { p k β n } { p c β n } } = [ T ] α − β n { { p k α } { p c α } { p k β n } } (13)

[0151] [ T ] α − β n = [ [ 0 ] [ 0 ] [ I ] [ L ] α − β n [ ϕ jk α ] [ L ] α − β n [ ϕ jc α ] [ 0 ] ] (14)

[0152] In the formula, [ T ] α − β n This is the coordinate transformation matrix for the second mode of the propeller blade. for 3D identity matrix The selected tail beam natural mode number, For the first Principal mode coordinates inside the blade structure For the first Blade constrained modal coordinates, The coordinates of the principal modes inside the tail beam structure are as follows. The coordinates are the constraint modal coordinates of the tail beam.

[0153] Step 3.3, Derivation of time-varying derivatives: Based on the time-varying effect of blade rotation, the derivative of the displacement quadratic coordinate transformation formula is obtained to obtain the corresponding quadratic coordinate transformation formulas of blade modal velocity and blade modal acceleration.

[0154] Due to the rotational motion of the propeller blades, the azimuth angle between the propeller blades and the tail boom changes accordingly. It can be seen from formula (10) that the azimuth angle between the propeller blades and the tail boom... The change in the time-varying parameters causes the displacement boundary conditions between the blade and the tail boom to change over time. Therefore, the time-varying effect needs to be considered in the quadratic coordinate transformation of the blade. To account for the time-varying effect caused by blade rotation, the time derivatives of equations (10), (13), and (14) are taken sequentially to obtain the quadratic coordinate transformation equations for the blade modal velocity and blade modal acceleration, where:

[0155] The quadratic transformation formula for the blade modal velocity is:

[0156] (15)

[0157] In the formula, Indicates the first The dominant mode velocity inside the blade structure, Indicates the first Blade constrained modal velocities, This indicates the principal mode velocity within the tail beam structure. Indicates the tail beam constrained modal velocity. [ T ˙ ] α − β n The time-dependent transformation matrix of the blade's second-mode coordinates The first derivative.

[0158] The quadratic transformation formula for the blade modal acceleration is:

[0159] (16)

[0160] In the formula, Indicates the first Principal mode acceleration within the blade structure, Indicates the first Blade-constrained modal acceleration, This indicates the principal modal acceleration within the tail beam structure. This represents the tail beam constrained modal acceleration. [ T ¨ ] α − β n The time-dependent transformation matrix of the blade's second-mode coordinates The second derivative.

[0161] The three terms shown in the dashed boxes in formulas (15) and (16) are all additional terms caused by the rotation of the blades and contain time variables.

[0162] Step 3.4, Blade Equation Quadratic Transformation and Parameter Extraction: Using the quadratic coordinate transformation formulas of the blade modal velocity and the blade modal acceleration obtained in Step 3.3, a quadratic coordinate transformation is performed on the blade structural dynamics equations in the modal coordinate system to obtain the generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade; and in this quadratic coordinate transformation process, the additional stiffness matrix and additional damping matrix generated by the time-varying rotation of the blade are derived.

[0163] After a quadratic coordinate transformation, the generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade are as follows:

[0164] (17)

[0165] in, [ K ¯ ] β n , [ C ¯ ] β n , [ M ¯ ] β n and [ R ¯ ] β n The first two coordinates are obtained after two coordinate transformations. The generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade; [ T ] α − β n T [ c ¯ ] β n [ T ˙ ] α − β n and [ T ] α − β n T [ m ¯ ] β n [ T ¨ ] α − β n To add a stiffness matrix, 2 [ T ] α − β n T [ m ¯ ] β n [ T ˙ ] α − β n For the additional damping array.

[0166] Next, the generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade obtained through the second transformation are expanded in blocks to obtain:

[0167] [ [ M ¯ ] yes β n [ M ¯ ] okay β n [ M ¯ ] I β n [ M ¯ ] kk β n ] { { p ¨ o α } { p ¨ k β n } } + [ [ C ¯ ] yes β n [ C ¯ ] okay β n [ C ¯ ] I β n [ C ¯ ] kk β n ] { { p ˙ o α } { p ˙ k β n } } + [ [ K ¯ ] yes β n [ K ¯ ] okay β n [ K ¯ ] I β n [ K ¯ ] kk β n ] { { p o α } { p k β n } } =− [ [ R ¯ ] o β n [ R ¯ ] k β n ] (18)

[0168] Among them, for the first The generalized mass matrix of a single blade after a quadratic coordinate transformation. [ M ¯ ] β n Generalized damping array [ C ¯ ] β n Generalized stiffness array [ K ¯ ] β n and generalized force vector [ R ¯ ] β n Represented in a block-based manner:

[0169] ; [ [ R ¯ ] o β n [ R ¯ ] k β n ] = [ R ¯ ] β n ;

[0170] [ [ M ¯ ] yes β n [ M ¯ ] okay β n [ M ¯ ] I β n [ M ¯ ] kk β n ] = [ M ¯ ] β n ; [ [ C ¯ ] yes β n [ C ¯ ] okay β n [ C ¯ ] I β n [ C ¯ ] kk β n ] = [ C ¯ ] β n ; [ [ K ¯ ] yes β n [ K ¯ ] okay β n [ K ¯ ] I β n [ K ¯ ] kk β n ] = [ K ¯ ] β n ;

[0171] In the formula, , , These represent the sets of all modal coordinates, modal velocities, and modal accelerations in the tail beam structure, respectively. [ M ¯ ] yes β n , [ M ¯ ] okay β n , [ M ¯ ] I β n , [ M ¯ ] kk β n The th coordinates after the second transformation are respectively blade generalized mass array [ M ¯ ] β n The block submatrix, [ C ¯ ] yes β n , [ C ¯ ] okay β n , [ C ¯ ] I β n , [ C ¯ ] kk β n The th coordinates after the second transformation are respectively bladed generalized damping array [ C ¯ ] β n The block submatrix, [ K ¯ ] yes β n , [ K ¯ ] okay β n , [ K ¯ ] I β n , [ K ¯ ] kk β n The th coordinates after the second transformation are respectively blade generalized stiffness array [ K ¯ ] β n The block submatrix, [ R ¯ ] o β n , [ R ¯ ] k β n The th coordinates after the second transformation are respectively blade generalized force vector [ R ¯ ] β n The block submatrix.

[0172] Step 3.5, Model Assembly:

[0173] Based on the installation positions of each blade on the tail boom, the structural dynamic equations of each blade (including the transformed generalized mass matrix, generalized damping matrix, generalized stiffness matrix, and generalized force vector) after a two-dimensional coordinate transformation are assembled with the structural dynamic equations of the tail boom in the modal coordinate system to obtain the time-varying structural dynamic equations of the tail rotor-tail boom coupling in the modal coordinate system (i.e., the time-varying model of the tail rotor-tail boom coupling in the modal coordinate system). The assembly of the stiffness matrix, damping matrix, and generalized force vector is similar to that of the mass matrix. The assembly process is illustrated below using the assembly of the mass matrix as an example:

[0174] [ M ¯ ] = [ [ M ¯ ] α + [ M ¯ ] yes β 1 + [ M ¯ ] yes β 2 + [ M ¯ ] yes β 3 + [ M ¯ ] yes β 4 [ M ¯ ] okay β 1 [ M ¯ ] okay β 2 [ M ¯ ] okay β 3 [ M ¯ ] okay β 4 [ M ¯ ] I β 1 [ M ¯ ] kk β 1 [ 0 ] [ 0 ] [ 0 ] [ M ¯ ] I β 2 [ 0 ] [ M ¯ ] kk β 2 [ 0 ] [ 0 ] [ M ¯ ] I β 3 [ 0 ] [ 0 ] [ M ¯ ] kk β 3 [ 0 ] [ M ¯ ] I β 4 [ 0 ] [ 0 ] [ 0 ] [ M ¯ ] kk β 4 ] (19)

[0175] In the formula, [ M ¯ ] The tail rotor-tail boom coupled mass array, [ M ¯ ] yes β 1 , [ M ¯ ] okay β 1 , [ M ¯ ] I β 1 , [ M ¯ ] kk β 1 These are the generalized mass matrices of the first blade after a quadratic coordinate transformation. [ M ¯ ] β 1 The block submatrix; [ M ¯ ] yes β 2 , [ M ¯ ] okay β 2 , [ M ¯ ] I β 2 , [ M ¯ ] kk β 2 These are the generalized mass matrices of the second blade after a quadratic coordinate transformation. [ M ¯ ] β 2 The block submatrix; [ M ¯ ] yes β 3 , [ M ¯ ] okay β 3 , [ M ¯ ] I β 3 , [ M ¯ ] kk β 3 These are the generalized mass matrices of the third blade after a quadratic coordinate transformation. [ M ¯ ] β 3 The block submatrix; [ M ¯ ] yes β 4 , [ M ¯ ] okay β 4 , [ M ¯ ] I β 4 , [ M ¯ ] kk β 4 These are the generalized mass matrices of the fourth blade after a quadratic coordinate transformation. [ M ¯ ] β 4 The block submatrix.

[0176] In summary, the method of this invention does not require specific tail rotor layout; it is applicable to both uniformly distributed tail rotors, such as cruciform tail rotors, and non-uniformly distributed tail rotors, such as scissor tail rotors. Furthermore, this method also considers the time-varying effects caused by the tail rotor's rotational motion, thus expanding the application scope of the traditional substructure modal synthesis method.

[0177] Furthermore, as can be seen from the tail boom modeling process, the method of this invention does not require establishing structural dynamic equations in the physical coordinate system of the tail boom. Instead, it can directly utilize the modal information of the tail boom to establish the structural dynamic equations in the modal coordinate system. Therefore, this method avoids the problems of large scale, difficult modeling, and low computational efficiency associated with real tail boom models, reducing the requirements for tail boom modeling in the tail rotor-tail boom coupling model. Based on this invention, the tail rotor-tail boom coupling frequency characteristics can be calculated using eigenvalue analysis.

[0178] The following example uses a scissor rotor-tail boom coupled model to verify the working condition, illustrating the accuracy of the modeling method of this invention by comparing the structural dynamic response calculations. Figure 3 In the operating condition shown, the tail boom is 0.8m long, the tail rotor is 0.24m long, and both the tail rotor and tail boom are made of aluminum alloy. The tail rotor is a cross-shaped tail rotor. With the tail rotor not rotating, the frequency characteristics of the cross-shaped tail rotor-tail boom coupling were calculated using the method of this invention, and the calculation results were compared with those calculated by Patran / Nastran finite element software, as shown in Table 1.

[0179] Table 1 Comparison of calculated natural frequencies of the cross-shaped tail rotor-tail boom coupling when the tail rotor is not rotating.

[0180]

[0181] As can be seen from Table 2, when the tail rotor is not rotating, the method of the present invention is almost consistent with the natural frequencies of each order of the cross tail rotor-tail boom coupling calculated by the Patran / Nastran finite element software.

[0182] To calculate the structural dynamic response, this example assumes that the structural damping of the tail rotor and tail boom is... A simple harmonic excitation force with a period of 2s and an amplitude of 0.001N is applied in the flapping direction at points A, B, C, and D. The Newmark method is used to calculate the coupled dynamic response of the tail rotor-tail boom. The dynamic response in the flapping direction at points A and E is extracted, and the calculation results are compared with those calculated by Adams multibody dynamics software. Figure 4 As shown, where, Figure 4 a represents a comparison of the paddle tip flapping response results. Figure 4 b represents a comparison of the propeller hub flapping response results, from... Figure 4 As can be seen, the waving response results obtained by the two methods are in agreement, verifying the accuracy of the model.

[0183] Comparing the direct substructure synthesis method and this method, under the same working conditions, the dynamic responses of points A and E in the waving direction are extracted and compared. The comparison results are as follows: Figure 5 As shown. Among them, Figure 5 a represents a comparison of the paddle tip flapping response results. Figure 5 b represents a comparison of the propeller hub flapping response results, from... Figure 5 As can be seen, the waving response results obtained by the two methods are consistent, and compared with the direct synthesis method of substructure, the calculation time of this method is reduced from 61.27s to 10.68s, a reduction of 82.58%. The calculation efficiency is improved by nearly 5 times.

[0184] The above comparative analysis confirms that the tail rotor-tail boom coupling model constructed based on the method of this invention is reliable and applicable to solving the dynamic characteristics and stability analysis of actual tail rotor / tail boom coupling models in engineering.

[0185] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in the present invention, and such modifications or substitutions should all be covered within the scope of protection of the present invention.

Claims

1. A coupled modeling method for helicopter tail rotor and tail boom based on time-varying hybrid interface modal synthesis, characterized in that, Includes the following steps: Step 1: For each blade of the tail rotor, establish the blade structure dynamics equations in the modal coordinate system; Includes the following sub-steps: Step 1.1: For each blade, establish the blade structure dynamics equation in the physical coordinate system. Step 1.2: For each blade, the blade structure dynamics equation in the physical coordinate system is transformed by the pre-established modal coordinate transformation formula to obtain the blade structure dynamics equation in the modal coordinate system for each blade. Step 2: Determine the modal information of the tail beam, and establish the structural dynamics equations of the tail beam in the modal coordinate system based on the modal information of the tail beam; Step 3: Based on the blade structure dynamics equations in the modal coordinate system, the tail boom structure dynamics equations in the modal coordinate system, and the physical displacement boundary conditions between the blades and the tail boom structure, establish a time-varying structural dynamics model of the tail rotor-tail boom coupled in the modal coordinate system. Includes the following sub-steps: Step 3.1, Boundary condition mode transformation; The physical displacement boundary conditions between each blade and the tail boom structure are transformed into the modal coordinate system to obtain the displacement boundary conditions between the blade and the tail boom in the modal coordinate system. Step 3.2, Solve the quadratic transformation equation: The displacement boundary conditions of the blade and tail boom in the modal coordinate system are solved to obtain the displacement quadratic coordinate transformation formula; Step 3.3, Derivation of the time-varying derivative: Based on the time-varying effect of blade rotation, the derivative of the displacement quadratic coordinate transformation formula is obtained to obtain the corresponding quadratic coordinate transformation formulas of blade modal velocity and blade modal acceleration. Step 3.4, Secondary transformation of blade equations and parameter extraction: Using the quadratic coordinate transformation formulas of the blade modal velocities and the blade modal accelerations, a quadratic coordinate transformation is performed on the blade structural dynamics equations in the modal coordinate system to obtain the blade's generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector; and during the quadratic coordinate transformation process, the additional stiffness matrix and additional damping matrix generated by the blade's time-varying rotation are derived. Step 3.5, Model Assembly: The dynamic equations of each blade structure after the aforementioned quadratic coordinate transformation are assembled with the dynamic equations of the tail boom in the modal coordinate system to obtain the time-varying dynamic equations of the tail rotor-tail boom coupled in the modal coordinate system.

2. The helicopter tail rotor tail boom coupling modeling method according to claim 1, characterized in that, In step 1.1, establishing the blade structure dynamics equations in the physical coordinate system includes the following steps: Step 1.1.1 Discretize the established blade structure model using the finite element method, divide it into multiple beam elements, and obtain the beam element stiffness matrix, beam element damping matrix, beam element mass matrix, and beam element generalized force vector of each beam element in the physical coordinate system. Step 1.1.2: Based on the position of each beam element in the overall blade structure, according to the matching rule, assemble the beam element stiffness matrix, beam element damping matrix, beam element mass matrix and beam element generalized force vector obtained in Step 1.1.1 to obtain the overall stiffness matrix, overall damping matrix, overall mass matrix and overall generalized force vector of a single blade in the physical coordinate system. Step 1.1.3: Based on the overall stiffness matrix, overall damping matrix, overall mass matrix and overall generalized force vector obtained in Step 1.1.2, establish the structural dynamics equations of a single blade in the physical coordinate system.

3. The helicopter tail rotor tail boom coupling modeling method according to claim 1, characterized in that, Step 1.2 includes the following sub-steps: Step 1.2.1: Based on the Craig-Bampton method in the fixed-interface modal synthesis method, construct the coordinate transformation relationship between the physical coordinates and modal coordinates of the blade. ,in, , , These represent the physical coordinates, modal coordinates, and mode shape set of the blade, respectively. The mode set of the blade adopts the Craig-Bampton assumption mode set and is obtained by solving the blade structure dynamics equations in the physical coordinate system. Step 1.2.2: The coordinate transformation relationship between the physical coordinates and modal coordinates of the blade established in Step 1.2.1 is divided into blocks, and the transformation relationship after block processing is defined as the modal coordinate transformation formula. The modal coordinate transformation formula is as follows: ； ； In the formula, Physical coordinates representing the internal degrees of freedom of the blade structure. Physical coordinates representing the degrees of freedom at the blade structure connection. Represents the coordinates of the principal modes within the blade structure. Represents the blade constraint modal coordinates. This represents the principal mode shape corresponding to the internal degrees of freedom of the blade structure. These represent the constrained mode shapes corresponding to the internal degrees of freedom of the blade structure. This represents the principal mode shape at the connection point of the corresponding blade structure. This represents the constrained mode shape corresponding to the degree of freedom at the blade structure connection. Step 1.2.3: Using the modal coordinate transformation formula determined in Step 1.2.2, perform coordinate transformation on the blade structure dynamics equations in the physical coordinate system to obtain the blade structure dynamics equations in the modal coordinate system as follows: In the formula, , , These represent the modal displacement, modal velocity, and modal acceleration of the blade, respectively. , , , These are the generalized mass matrix, generalized stiffness matrix, generalized damping matrix, and generalized force vector of the blade in the modal coordinate system; they are obtained by transforming the corresponding matrices in the physical coordinate system using the modal coordinate transformation formula. in: In the formula, , , and These represent the overall stiffness matrix, overall damping matrix, overall mass matrix, and overall generalized force vector of the blade in the physical coordinate system, respectively.

4. The helicopter tail rotor tail boom coupling modeling method according to claim 1, characterized in that, Step 2 includes the following sub-steps: Step 2.1: Obtain low-order modal information of the tail boom using commercial finite element software or ground modal experiments; the modal information includes at least the front... First natural frequency ,forward First-order modal damping ratio ,forward First-order modal mass and tail beam mode shape set ;in, , The selected number of natural modes of the tail beam; Step 2.2: Using the Shou-nin-Hou method in the free interface modal synthesis method, combined with the low-order modal information of the tail beam, the dynamic equation of the tail beam structure in the intrinsic modal coordinate system is established as follows: in: ， ； ， ； In the formula, , , and Let represent the generalized mass matrix, generalized damping matrix, generalized stiffness matrix, and generalized force vector of the tail beam in the natural modal coordinate system, respectively. , , These represent the modal displacement, modal velocity, and modal acceleration of the tail beam, respectively.

5. The helicopter tail rotor tail boom coupling modeling method according to claim 4, characterized in that, In step 3.1, the method for obtaining the displacement boundary conditions of the blade and tail boom in the modal coordinate system is as follows: First, obtain the physical displacement boundary conditions between the blades and the tail boom structure, including: In the formula, the matrix Indicates the first Physical displacement boundary conditions between the blades and the tail boom structure. For the first Physical coordinates of the degrees of freedom at the blade connection point. Physical coordinates representing the degrees of freedom at the tail beam structure connection. This is the displacement transformation matrix. Indicates the azimuth angle between the tail boom and the blade; Then, using the coordinate transformation relationship between physical coordinates and modal coordinates The physical displacement boundary conditions between the blade and the tail boom structure are transformed to the modal coordinate system, resulting in the following displacement boundary conditions for the blade and tail boom in the modal coordinate system: In the formula, These are physical coordinates. These are modal coordinates. It is a modality set; For the first Principal mode coordinates inside the blade structure For the first Blade constrained modal coordinates, The coordinates of the principal modes inside the tail beam structure are as follows. The coordinates of the tail beam constrained mode are as follows: This represents the principal mode shape at the connection point of the corresponding tail beam structure. This represents the constrained mode shape at the connection point of the corresponding tail beam structure. for 3D identity matrix; Finally, the expression for the displacement boundary conditions of the blade and tail boom in the modal coordinate system is expanded as follows: 。 6. The helicopter tail rotor tail boom coupling modeling method according to claim 5, characterized in that, In step 3.2, the expression for the displacement boundary conditions of the blade and tail boom in the expanded modal coordinate system is solved to obtain the displacement quadratic coordinate transformation formula: In the formula, This is the coordinate transformation matrix for the second mode of the propeller blade. for 3D identity matrix The selected tail beam natural mode number, For the first Principal mode coordinates inside the blade structure For the first Blade constrained modal coordinates, The coordinates of the principal modes inside the tail beam structure are as follows. The coordinates are the constraint modal coordinates of the tail beam.

7. The helicopter tail rotor and tail boom coupling modeling method according to claim 6, characterized in that, In step 3.3, the derivative of the displacement quadratic transformation equation is taken to obtain the corresponding quadratic coordinate transformation equations for the blade modal velocity and the blade modal acceleration; wherein: The quadratic transformation formula for the blade modal velocity is: In the formula, Indicates the first The dominant mode velocity inside the blade structure, Indicates the first Blade constrained modal velocities, This indicates the principal mode velocity within the tail beam structure. Indicates the tail beam constrained modal velocity. The time-dependent transformation matrix of the blade's second-mode coordinates The first derivative; The quadratic transformation formula for the blade modal acceleration is: In the formula, Indicates the first Principal mode acceleration within the blade structure, Indicates the first Blade-constrained modal acceleration, This indicates the principal modal acceleration within the tail beam structure. This represents the tail beam constrained modal acceleration. The time-dependent transformation matrix of the blade's second-mode coordinates The second derivative.

8. The helicopter tail rotor tail boom coupling modeling method according to claim 7, characterized in that, Step 3.4 includes the following process: Step 3.4.1: Using the quadratic coordinate transformation formulas for the blade modal velocities and modal accelerations, perform a quadratic coordinate transformation on the blade structural dynamics equations in the modal coordinate system to eliminate non-independent modal coordinates, obtaining the generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade as follows: in, , , and The first two coordinates are obtained after two coordinate transformations. The generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade; and To add a stiffness matrix, For the addition of a damping array; Step 3.4.2 involves performing a block expansion on the generalized stiffness matrix, generalized damping matrix, generalized mass matrix, and generalized force vector of the blade obtained through the second transformation, resulting in: ； in: ； ； ； ； ； In the formula, , , These represent the sets of all modal coordinates, modal velocities, and modal accelerations in the tail beam structure, respectively. , , , The th coordinates after the second transformation are respectively blade generalized mass array The block submatrix, , , , The th coordinates after the second coordinate transformation bladed generalized damping array The block submatrix, , , , The th coordinates after the second coordinate transformation blade generalized stiffness array The block submatrix, , The th coordinates after the second coordinate transformation blade generalized force vector The block submatrix.

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