A method and system for analyzing dynamic characteristics of a rotating beam system based on a finite difference reduced multi-body system transfer matrix method

Abstract

The application discloses a kind of based on finite difference reduction multibody system transfer matrix method of rotating beam system dynamic characteristic analysis method and system.Method includes: the physical parameter model of rotating beam system is established, and the structural parameter of rotating beam is acquired;Rotating beam system dynamics control differential equation is established;Rotating beam system is discretized using finite difference method;The transfer matrix of the discretized multibody system model is constructed, and the transfer matrix is processed using reduction equation, and the reduced difference point reduction transfer matrix recursive relationship is obtained;The reduced state vector recursive equation is established, and the dynamic characteristic parameter of rotating beam system is calculated and output.The application is combined with finite difference method by multibody system transfer matrix method, and rotating beam element is discretized difference, and the differential term of rotating beam differential equation is differentially approximated to deduce transfer matrix, retains the characteristics of low matrix order involved in transfer matrix method, reduces the calculation scale, improves the success rate of numerical calculation result, and can effectively solve rotating beam and other dynamics problems.

Description

Technical Field

[0001] This invention belongs to the field of computational structural dynamics and multibody system dynamics, and in particular, it is a method and system for analyzing the dynamic characteristics of a rotating beam system based on the finite difference reduced multibody system transfer matrix method. Background Technology

[0002] Flexible beams are important components in many engineering structures, such as space robotic arms, wind turbine blades, helicopter rotors, and machine tool spindles. The vibration characteristics of rotating flexible beams, caused by centrifugal forces and other factors resulting from rotation, significantly impact the system's accuracy, performance, and reliability.

[0003] The general steps for analyzing flexible beams are as follows: ① Determine the flexible beam model, such as the Euler-Bernoulli beam model or the Timoshenko beam model; ② Establish the dynamic equations of the flexible beam, which can be achieved using methods such as Newton-Euler's method, Hamilton's principle, or the finite element method; ③ Analyze the vibration characteristics of the flexible beam, with typical methods including numerical methods and semi-analytical methods. Numerical methods include the direct difference method, the Rayleigh-Ritz method, the Galerkin method, and various algorithms developed based on the finite element concept. Semi-analytical methods, by setting an analytical structure of an approximate function and combining it with boundary conditions, determine the vibration response characteristics of the system. These mainly include the Frobenius method or the differential transformation method. Based on semi-analytical methods, spectral finite element methods, dynamic stiffness methods, and transfer matrix methods have also been developed.

[0004] When analyzing the vibration characteristics of a rotating beam using the transfer matrix method for linear multibody systems, it is difficult to define an approximate function to solve the problem when the dynamic equations contain coupling terms and integral terms. The finite difference method can directly approximate the differential terms in the differential equations through difference, thereby transforming the differential equations into a system of algebraic equations for solution. However, the calculation requires a large number of difference points to ensure accuracy, resulting in a large computational scale and the potential for calculation failure. Summary of the Invention

[0005] The purpose of this invention is to address the deficiencies or shortcomings of the existing technology by providing a method and system for analyzing the dynamic characteristics of a rotating beam system based on the finite difference reduced multibody system transfer matrix method.

[0006] The technical solution to achieve the objective of this invention is as follows: On the one hand, a method for analyzing the dynamic characteristics of a rotating beam system based on the finite difference reduced multibody system transfer matrix method is provided, the method comprising the following steps:

[0007] Step 1: Establish a physical parameter model of the rotating beam system and obtain the structural parameters of the rotating beam;

[0008] Step 2: Based on the physical and structural parameters, establish the dynamic control differential equations of the rotating beam system;

[0009] Step 3: Discretize the rotating beam system using the finite difference method to obtain a discretized multibody system model;

[0010] Step 4: Construct the transfer matrix of the discretized multibody system model, and process the transfer matrix using the reduction equation to obtain the recursive relationship of the reduced difference point reduced transfer matrix.

[0011] Step 5: Establish the reduced state vector recursive equation and calculate the dynamic characteristic parameters of the rotating beam system, including natural frequencies and mode shapes;

[0012] Step 6: Output the dynamic characteristic parameters.

[0013] Furthermore, the structural parameters mentioned in step 1 include at least geometric parameters, material parameters, boundary conditions, and rotational speed.

[0014] Further, step 2 includes: obtaining the general rotational Euler-Bernoulli beam vibration differential equation according to Hamilton's theorem; for the non-uniformly rotating rotational beam differential equation, using Fourier series to transform it into an ordinary differential equation in the frequency domain that only concerns spatial variables for solution.

[0015] Furthermore, step 3 specifically includes:

[0016] Step 3.1, discretize on the rotating beam The node at the node Perform a difference transform on the nth node to obtain the nth node. The first four order differential difference schemes of each node, where ;

[0017] Step 3.2, Processing the boundary points of the rotating beam: Perform a difference transformation on the boundary point and three nearby points to obtain the differential difference scheme of the input point.

[0018] Furthermore, step 4 specifically includes:

[0019] Step 4.1: Substitute the first four orders of differential difference schemes for the discrete nodes of the rotating beam into the fourth order differential equation of the rotating beam, and obtain the... The equation in the th equation The difference schemes at each node are vectorized to obtain the transitive relationships between the state vectors of the difference schemes;

[0020] Step 4.2: Substitute the intermediate point and boundary point state vectors into the component model to obtain the relationship between the differential state vector and the component state vector, and derive the transfer equation of component j from this.

[0021] Step 4.3: Divide the state vector into two column vectors according to the boundary conditions, and divide the transfer matrix of element i into corresponding blocks;

[0022] Step 4.4 introduces a reduction transformation to describe the two parts of the state vector at the input of element i, obtains the relationship between the two parts of the state vector at the output of element i, and derives the general recursive formula of the reduction transfer matrix.

[0023] Step 4.5: From the general recursive relation of the reduced transfer matrix and the transfer equation of element j, obtain the recursive relation of the differential point reduced transfer matrix of the element.

[0024] Furthermore, step 5 specifically includes:

[0025] Step 5.1: Considering the boundary conditions of the components, the characteristic equation of the system is obtained by reducing the transfer matrix of the component's difference point and solving it using the bisection method to obtain the system's natural frequency.

[0026] Step 5.2: Substitute the system's natural frequency into the system output transfer equation to obtain the system output state vector;

[0027] Step 5.3: Then, recursively calculate the state vector of each connection point from the output end to the input end, and extract the displacement and angular displacement in the modal coordinates to obtain the mode shape of the system.

[0028] On the other hand, a dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method is provided, the system comprising:

[0029] The first module is configured to obtain the physical parameters, structural parameters, and boundary conditions of the rotating beam system.

[0030] The second module is connected to the first module and is configured to obtain the system characteristic equations.

[0031] The third module, connected to the second module, is configured to output dynamic response characteristics.

[0032] Furthermore, the second module is configured to implement:

[0033] Establish the dynamic differential equations of the rotating beam system;

[0034] The dynamic differential equation of the rotating beam is discretized using the finite difference method to construct the transfer matrix;

[0035] The constructed transfer matrix is ​​reduced using modal reduction equations to obtain the system characteristic equations.

[0036] Furthermore, the third module is configured to implement:

[0037] Based on the output of the second module, calculate the dynamic characteristic parameters, including: searching for the natural frequency of the system characteristic equation using the bisection method, and inversely solving for the state vector of the discrete points to confirm the system mode shape;

[0038] Output the dynamic characteristic parameters.

[0039] On the other hand, a computer device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the dynamic characteristic analysis method for a rotating beam system based on the finite difference reduced multibody system transfer matrix method.

[0040] Compared with the prior art, the significant advantages of this invention are:

[0041] (1) The present invention proposes a dynamic characteristic analysis method and system for a rotating beam system based on a finite difference reduction multibody system transfer matrix method. By combining the multibody system transfer matrix method with the finite difference method, the rotating beam element is discretized and the differential terms of the rotating beam differential equation are derived by difference approximation. The transfer matrix is ​​retained by the low order of the matrix involved in the transfer matrix method, the computational scale is reduced, the success rate of numerical calculation results is improved, and the dynamic problems of rotating beams are solved.

[0042] (2) This method can be used to analyze rotating beams with spatial vibration and rotating beams with complex shapes. In addition, this method can be extended to plates and warping elements to form a two-dimensional finite difference method for the transfer matrix of multibody systems. At the same time, this method also provides a theoretical basis for combining other discretized numerical methods with the transfer matrix method of multibody systems.

[0043] (3) The frequency error between the numerical calculation results obtained by this method and the calculation results in the literature is within 0.3%, which verifies the correctness of the method.

[0044] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0045] Figure 1 This is a flowchart of a method for analyzing the dynamic characteristics of a rotating beam system based on the finite difference reduced multibody system transfer matrix method in one embodiment.

[0046] Figure 2 This is a schematic diagram of a uniformly rotating Euler-Bernoulli beam (rigid body radius is 0) in one embodiment.

[0047] Figure 3 This is a schematic diagram of the transmission of element state vectors and differential points in one embodiment.

[0048] Figure 4This is a schematic diagram of the component state vector and differential point reduction transmission in one embodiment. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0050] It should be noted that if the embodiments of the present invention involve directional indicators (such as up, down, left, right, front, back, etc.), the directional indicators are only used to explain the relative positional relationship and movement of the components in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indicators will also change accordingly.

[0051] Furthermore, if the embodiments of this invention involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

[0052] In one embodiment, combined Figure 1 This paper presents a method for analyzing the dynamic characteristics of a rotating beam system based on the finite difference reduced multibody system transfer matrix method. The method includes the following steps:

[0053] Step 1: Establish a physical parameter model of the rotating beam system and obtain the structural parameters of the rotating beam;

[0054] The structural parameters include at least geometric parameters, material parameters, boundary conditions, and rotational speed;

[0055] Step 2: Based on the physical and structural parameters, establish the dynamic control differential equations of the rotating beam system;

[0056] Step 3: Discretize the rotating beam system using the finite difference method to obtain a discretized multibody system model;

[0057] Step 4: Construct the transfer matrix of the discretized multibody system model, and process the transfer matrix using the reduction equation to obtain the recursive relationship of the reduced difference point reduced transfer matrix.

[0058] Step 5: Establish the reduced state vector recursive equation and calculate the dynamic characteristic parameters of the rotating beam system, including natural frequencies and mode shapes;

[0059] Step 6: Output the dynamic characteristic parameters.

[0060] Further, in one embodiment, step 2 includes: obtaining the general rotational Euler-Bernoulli beam vibration differential equation according to Hamilton's theorem; for the non-uniformly rotating rotational beam differential equation, using Fourier series to transform it into an ordinary differential equation in the frequency domain that only concerns spatial variables for solution.

[0061] Furthermore, in one embodiment, step 3 specifically includes:

[0062] Step 3.1, discretize on the rotating beam The node at the node Perform a difference transform on the nth node to obtain the nth node. The first four order differential difference schemes of each node, where ;

[0063] Step 3.2, Processing the boundary points of the rotating beam: Perform a difference transformation on the boundary point and three nearby points to obtain the differential difference scheme of the input point.

[0064] Preferably, in some embodiments, step 3 specifically includes:

[0065] According to Hamilton's theorem, the differential equation for the vibration of a rotating Euler-Bernoulli beam is established as follows:

[0066]

[0067] In the formula, Let ω be the angular velocity of the rigid body motion. For rigid body motion, angular acceleration; It is the elastic modulus of the rotating beam; It is the moment of inertia of the cross section of the rotating beam; It is the centrifugal force acting on the rotating beam; It is the linear density of the rotating beam;

[0068] Among them, rigid body motion angular acceleration for:

[0069]

[0070] In the formula, For external torque, It is the moment of inertia of the beam about the axis of rotation; It is the moment of inertia of a rigid body about its axis of rotation; It is the radial distance from the center of rotation to the fixed end of the rotating beam, which is the hub radius; It is the length of the rotating beam;

[0071] When the Euler-Bernoulli beam rotates at a constant speed, the equation will be... Simplified to:

[0072]

[0073] When the Euler-Bernoulli beam rotates non-uniformly, the equation is:

[0074]

[0075] Using Fourier series to express The problem is transformed into solving an ordinary differential equation in the frequency domain that depends only on spatial variables. A sufficiently long time sample T is selected, which includes all modal periods of the flexible beam. At this point, a series of sequences are defined. Its unit can be represented as ( The sequence contains subsets. Its unit can be represented as ( ), , , Steady-state vibration response The Middle The first modal period. Steady-state vibration response The Middle The first modal period. Wherein It is the first The positive angular frequencies of the first mode; Is with The corresponding negative angular frequency; Represents the spatial coordinates along the axis of the rotating beam; express A subset containing the modal frequencies that actually occur in steady-state vibrations. ; Represent natural numbers; This represents the set of natural numbers. Therefore, the steady-state vibration of a flexible beam can be composed of a series of modal combinations through linear superposition, i.e.:

[0076]

[0077] In the formula, , and It is a flexible beam deformation The conjugate real part of the variable.

[0078] External torque This can be extended by Fourier series as follows:

[0079]

[0080] In the formula, Here are the Fourier coefficients, and ;

[0081] Based on the orthogonality of Fourier series in T, the following relationship exists:

[0082]

[0083] in, Indicates the first The frequency of the first mode;

[0084] When the flexible beam is in steady-state vibration, the rotational velocity of the central rigid body It is also a periodic function. Therefore, the rotational speed and axial force It can be replaced by the average value over a sampling period T:

[0085]

[0086] In the formula, This indicates the mass density of the rotating beam material; This represents the cross-sectional area of ​​the rotating beam;

[0087] The formula Japanese style Substitution At the same time Considering the orthogonality of the series, integrating over time [0, T], we can obtain the equation. Rewritten as:

[0088]

[0089] In the formula, According to the orthogonality of series, the external torque and The relevant value should be 0.

[0090] definition , for example Take respectively and Adding these equations together, we obtain the vibration equation for the Euler-Bernoulli beam, a rigid body with a non-uniformly rotating center:

[0091]

[0092] Discretized on the rotating beam The node at the node Perform a difference transform on the nth node to obtain the nth node. The first four order differential difference schemes of the nodes are expressed as follows:

[0093]

[0094] In the formula, , , , The first The first four differential differences of each node ; It is a function exist Taylor expansion at the point.

[0095] Handling the boundary points of the rotating beam: To prevent index overflow, consider performing a difference transformation on the boundary point and three nearby points to obtain the input point. The differential difference scheme is as follows:

[0096]

[0097] Solve for the output point The differential difference scheme is as follows:

[0098]

[0099] Furthermore, in one embodiment, step 4 specifically includes:

[0100] Step 4.1: Substitute the first four orders of differential difference schemes for the discrete nodes of the rotating beam into the fourth order differential equation of the rotating beam, and obtain the... The equation in the th equation The difference schemes at each node are vectorized to obtain the transitive relationships between the state vectors of the difference schemes;

[0101] Step 4.2: Substitute the intermediate point and boundary point state vectors into the component model to obtain the relationship between the differential state vector and the component state vector, and derive the transfer equation of component j from this.

[0102] Step 4.3: Divide the state vector into two column vectors according to the boundary conditions, and divide the transfer matrix of element i into corresponding blocks;

[0103] Step 4.4 introduces a reduction transformation to describe the two parts of the state vector at the input of element i, obtains the relationship between the two parts of the state vector at the output of element i, and derives the general recursive formula of the reduction transfer matrix.

[0104] Step 4.5: From the general recursive relation of the reduced transfer matrix and the transfer equation of element j, obtain the recursive relation of the differential point reduced transfer matrix of the element.

[0105] Preferably, in some embodiments, step 4 specifically includes:

[0106] Substituting the first four order differential difference schemes of the discrete nodes of the rotating beam obtained in step 3 into the fourth order differential equation of the rotating beam, we can obtain the... The equations in The difference format at the point is:

[0107]

[0108] In the formula, For discrete points Taylor expansion at point All These are all coefficients of the discrete linear equations, determined by the differential equations and the differential difference point scheme.

[0109] Let the state vector of the difference scheme be:

[0110]

[0111] In the formula, For discrete points The corresponding difference scheme state vector;

[0112] The transitive relationships between the state vectors in the difference scheme are obtained:

[0113]

[0114] In the formula, This is the transfer matrix in differential format.

[0115] Substituting the intermediate point and boundary point difference schemes into the component model, the relationship between the difference state vector and the component state vector can be obtained as follows:

[0116]

[0117] In the formula, , , Determined by the component model, it is a discrete point. A matrix relating the state vectors of its location to the state vectors of its location; It is the state vector corresponding to the input point; Discrete points The corresponding state vector; It is the state vector corresponding to the output point.

[0118] From this, the relationship between the state vectors of the input and output terminals of component j can be derived, that is, the transfer equation of component j:

[0119]

[0120] In the formula, Let j be the state vector at the input terminal of element j. Let j be the output state vector of component j. yes The corresponding transfer matrix, The transfer matrix for element j:

[0121]

[0122] Will The state vector of dimension is divided into orders of 1 and 2 based on the boundary conditions. The two column vectors, namely:

[0123]

[0124] In the formula, , For components The two orders are The column vectors can be set to zero column vectors according to the boundary conditions; For components Input state vector, For components Output state vector;

[0125] Transfer matrix for element i By dividing the data into blocks, we can obtain:

[0126]

[0127] In the formula, the matrix , , and The order of all are , , yes The corresponding transfer matrix block; , yes The corresponding transfer matrix block.

[0128] The transfer equation for element i can be rewritten as:

[0129]

[0130] The following reduction transformation is introduced to describe the relationship between the two state vectors at the input of element i:

[0131]

[0132] In the formula, These are the two state vectors at the input of component i, respectively, and are unknown. This is called the reduced transfer matrix at the input of element i.

[0133] The formula Substitution Zhongde:

[0134]

[0135] In the formula, These are the two state vectors at the output of component i;

[0136] Organizing This yields the relationship between the two state vectors at the output of component i, namely:

[0137]

[0138] In the formula, This is the reduced transfer matrix for the output of component i.

[0139] The reduced transfer matrix of the input and output terminals of element i+1 is:

[0140]

[0141] In the formula, This is called the reduced transfer matrix at the input of element i+1. This is called the reduced transfer matrix at the output of element i+1;

[0142] From the formula Japanese style The recursive formula for the differential point reduction transfer matrix of the component can be obtained as follows:

[0143]

[0144] In the formula, and These are the input point and the output point, respectively. Discrete points Reduced transfer matrix at the location; , Is the input point and The matrix blocks of correspondence; , Is the input point and The matrix blocks of correspondence; , Is the output point and The matrix blocks of correspondence; , Is the output point and The matrix blocks of correspondence; , Discrete points and The matrix blocks of correspondence; , Discrete points and A matrix block representing the correspondence.

[0145] Furthermore, in one embodiment, step 5 specifically includes:

[0146] Step 5.1: Considering the boundary conditions of the components, the characteristic equation of the system is obtained by reducing the transfer matrix of the component's difference point and solving it using the bisection method to obtain the system's natural frequency.

[0147] Step 5.2: Substitute the system's natural frequency into the system output transfer equation to obtain the system output state vector;

[0148] Step 5.3: Then, recursively calculate the state vector of each connection point from the output end to the input end, and extract the displacement and angular displacement in the modal coordinates to obtain the mode shape of the system.

[0149] Preferably, in some embodiments, step 5 specifically includes:

[0150] When considering the symmetry of the boundary conditions of the components, substitute the boundary conditions into the equation. We can obtain:

[0151]

[0152] In the formula, This is the reduced transfer matrix for the component output.

[0153] From the above equation, the characteristic equation of the system is:

[0154]

[0155] In the formula, gn(*) represents the sign function. , det(*) represents the determinant of *;

[0156] When the boundary conditions are asymmetric, substitute the boundary conditions into the equation. We can obtain:

[0157]

[0158] The system characteristic equation can be written as:

[0159] (31)

[0160] System characteristic equation or The natural frequency of the system can be obtained by using a root-searching method such as the bisection method.

[0161] When the system boundary conditions are symmetric, the solved natural frequencies are substituted into the transfer equations at the system output. The state vector at the system output can be obtained. Then, the recursive process is performed from the output to the input in reverse order, with the following recursive relationship:

[0162]

[0163] in, , Representing discrete points The difference state vector Column vector; , Represents the differential state vector of the input point Column vectors.

[0164] Similarly, when the boundary conditions are asymmetric, the natural frequencies obtained from the solution are substituted into the transfer equation at the system output. The recursive relationship is as follows:

[0165]

[0166] in, , Representing discrete points The difference state vector Column vector; , Represents the differential state vector of the input point Column vectors.

[0167] The state vector of each connection point can be calculated using the above recursive relationship. The mode shape of the system can be obtained by extracting the displacement and angular displacement in the modal coordinates.

[0168] In one embodiment, a dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method is provided, the system comprising:

[0169] The first module is configured to obtain the physical parameters, structural parameters, and boundary conditions of the rotating beam system.

[0170] The second module is connected to the first module and is configured to obtain the system characteristic equations.

[0171] The third module, connected to the second module, is configured to output dynamic response characteristics.

[0172] Furthermore, in one embodiment, the second module is configured to implement:

[0173] Establish the dynamic differential equations of the rotating beam system;

[0174] The dynamic differential equation of the rotating beam is discretized using the finite difference method to construct the transfer matrix;

[0175] The constructed transfer matrix is ​​reduced using modal reduction equations to obtain the system characteristic equations.

[0176] Furthermore, in one embodiment, the third module is configured to implement:

[0177] Based on the output of the second module, calculate the dynamic characteristic parameters, including: searching for the natural frequency of the system characteristic equation using the bisection method, and inversely solving for the state vector of the discrete points to confirm the system mode shape;

[0178] Output the dynamic characteristic parameters.

[0179] Specific limitations regarding the dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method can be found in the limitations of the dynamic characteristic analysis method for a rotating beam system based on the finite difference reduced multibody system transfer matrix method mentioned above, and will not be repeated here. Each module in the aforementioned dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device, or stored in the memory of a computer device as software, so that the processor can call and execute the operations corresponding to each module.

[0180] In one embodiment, a computer device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements:

[0181] Step 1: Establish a physical parameter model of the rotating beam system and obtain the structural parameters of the rotating beam;

[0182] Step 2: Based on the physical and structural parameters, establish the dynamic control differential equations of the rotating beam system;

[0183] Step 3: Discretize the rotating beam system using the finite difference method to obtain a discretized multibody system model;

[0184] Step 4: Construct the transfer matrix of the discretized multibody system model, and process the transfer matrix using the reduction equation to obtain the recursive relationship of the reduced difference point reduced transfer matrix.

[0185] Step 5: Establish the reduced state vector recursive equation and calculate the dynamic characteristic parameters of the rotating beam system, including natural frequencies and mode shapes;

[0186] Step 6: Output the dynamic characteristic parameters.

[0187] For specific limitations on each step, please refer to the limitations on the dynamic characteristic analysis method of the rotating beam system based on the finite difference reduced multibody system transfer matrix method mentioned above, which will not be repeated here.

[0188] In one embodiment, a computer-readable storage medium is provided having a computer program stored thereon, the computer program being implemented when executed by a processor:

[0189] Step 1: Establish a physical parameter model of the rotating beam system and obtain the structural parameters of the rotating beam;

[0190] Step 2: Based on the physical and structural parameters, establish the dynamic control differential equations of the rotating beam system;

[0191] Step 3: Discretize the rotating beam system using the finite difference method to obtain a discretized multibody system model;

[0192] Step 4: Construct the transfer matrix of the discretized multibody system model, and process the transfer matrix using the reduction equation to obtain the recursive relationship of the reduced difference point reduced transfer matrix.

[0193] Step 5: Establish the reduced state vector recursive equation and calculate the dynamic characteristic parameters of the rotating beam system, including natural frequencies and mode shapes;

[0194] Step 6: Output the dynamic characteristic parameters.

[0195] For specific limitations on each step, please refer to the limitations on the dynamic characteristic analysis method of the rotating beam system based on the finite difference reduced multibody system transfer matrix method mentioned above, which will not be repeated here.

[0196] As a specific example, the invention will be further described in detail in one embodiment.

[0197] This invention provides a dynamic characteristic analysis method for a rotating beam system based on the finite difference method for reducing the transfer matrix of a multibody system. The core of this method is to discretize the complex continuous rotating beam system into a multibody system composed of multiple difference points, and to reduce the scale of its dynamic equations using an improved finite difference method. Finally, the method of transfer matrix is ​​combined to efficiently solve the natural frequencies and mode shapes of the system.

[0198] like Figure 2 As shown, this invention takes a uniformly rotating planar Euler-Bernoulli beam as an example, with a beam length of l. = 1m; linear density of mass = 1kg / m; Bending stiffness The radius of the central rigid body is [value missing]; the rotational speed of the beam is 0~10 [units missing]. Furthermore, 100 points are discretized from the rotating beam, and a coordinate system is defined with the center of rotation of the principal axis as the origin. Establish a global coordinate system , The shaft is along the axis of the rotating main shaft. The axis perpendicular to the plane of rotation and pointing upwards is positive. The axis is determined by the right-hand rule.

[0199] As specific steps:

[0200] Step 1: Establish a physical parameter model of the rotating beam system and obtain the geometric parameters, material parameters, boundary conditions, and rotation speed of the rotating beam.

[0201] Step 2: Based on the physical and structural parameters, establish the dynamic control differential equations of the rotating beam system.

[0202] According to Hamilton's theorem, the general differential equation for the vibration of a rotating Euler-Bernoulli beam is obtained:

[0203] (34)

[0204] When the Euler-Bernoulli beam rotates at a constant speed, equation (34) simplifies to:

[0205] (35)

[0206] Step 3: Discretize the rotating beam system using the finite difference method to obtain a discretized multibody system model.

[0207] function exist nearby ( )and ( The Taylor expansion at () is:

[0208] (36)

[0209] In the formula, ;

[0210] From equation (36), we obtain the first-order differential forward difference scheme and backward difference scheme (error truncated to first order):

[0211] (37)

[0212] From equation (37), we can see that the first-order differential central difference scheme (error truncated to first order) is:

[0213] (38)

[0214] From equation (36), we can see that the second-order differential central difference scheme (error truncation is second order) is:

[0215] (39)

[0216] Continuing to derive the higher-order differential difference scheme with second-order error truncation, in and Performing a Taylor expansion at this point, we obtain:

[0217] (40)

[0218] The method of undetermined coefficients is used to derive the following from equations (36) to (40):

[0219] (41)

[0220] (42)

[0221] in, , These are undetermined coefficients;

[0222] The first-order differential term in equation (41) should be 0, therefore Then the third-order differential difference scheme is:

[0223] (43)

[0224] The second-order differential term in equation (42) should be 0, therefore Then the fourth-order differential difference scheme is:

[0225] (44)

[0226] Therefore, 100 nodes are discretized on the rotating beam, and at the 100th node ( Perform a difference transform. The expressions for the first four order differential difference schemes of the nodes are as follows:

[0227] (45)

[0228] To handle the boundary points of the rotating beam and prevent index overflow, Taylor expansion is considered at the boundary point and three nearby points.

[0229] For the input points, consider the Taylor expansion of the function at x2, x3, and x4:

[0230] (46)

[0231] The differential difference scheme of the input point obtained from equation (46) is:

[0232] (47)

[0233] For the output point, consider the function at... , , Taylor expansion form at:

[0234] (48)

[0235] From equation (48), the differential difference scheme of the output point is obtained as follows:

[0236] (49)

[0237] Step 4: Construct the transfer matrix of the discretized multibody system model, and process the transfer matrix using the reduction equation to obtain the recursive relationship of the reduced difference point reduced transfer matrix.

[0238] make Substituting into equation (35), we get:

[0239] (50)

[0240] The boundary conditions are:

[0241] (51)

[0242] Substituting the difference scheme into equation (50) yields:

[0243] (52)

[0244] Summarized as follows:

[0245] (53)

[0246] make Then, the following transitive relation can be derived from the difference scheme:

[0247] (54)

[0248] In the formula, .

[0249] According to the constitutive relation of Euler-Bernoulli beams,

[0250] (55)

[0251] in, It is a rotating beam section around The rotation angle of the shaft; It is the winding on the cross section of the rotating beam. Axial bending moment; It is the upper edge of the rotating beam section Shear force in the direction; It is the axial force acting on the cross section;

[0252] Substituting the boundary point difference scheme into equation (55), the input and output state vectors of the beam satisfy:

[0253] (56)

[0254] (57)

[0255] ,

[0256]

[0257] in, Representing discrete points Axial force at the location; Representing discrete points Moment of inertia of the cross section at the location; express The element in the 4th row and i-th column of the matrix; express The element in the 4th row and i-th column of the matrix;

[0258] according to Figure 3 The schematic diagram of the state vectors of the intermediate components and the differential point transmission allows us to derive the relationship between the state vectors at the input and output ends of the rotating beam, i.e., the transmission equation of the rotating beam:

[0259] (58)

[0260] In the formula, The transfer matrix for the rotating beam:

[0261] (59)

[0262] Will The state vector of dimension is divided into orders of 1 and 2 based on the boundary conditions. The two column vectors, namely:

[0263] (60)

[0264] By dividing the transfer matrix of the rotating beam into blocks, we get:

[0265] (61)

[0266] Components The transfer equation can be rewritten as:

[0267] (62)

[0268] Introduce the following reduction transformation:

[0269] (63)

[0270] Substituting equation (63) into equation (62), we get:

[0271] (64)

[0272] By rearranging formula (64), the components can be obtained. The relationship between the two state vectors at the output terminal is as follows:

[0273] (65)

[0274] In the formula, This is the reduced transfer matrix at the output end of the rotating beam. Then the element... The reduced transfer matrix at the input and output terminals is:

[0275] (66)

[0276] like Figure 4 As shown, from equations (66) and (59), the recursive formula for the difference point reduction transfer matrix of the rotating beam can be obtained:

[0277] (67)

[0278] Step 5: Establish the reduced state vector recursive equation and calculate the dynamic characteristic parameters of the rotating beam system, including natural frequencies and mode shapes.

[0279] Considering the asymmetric boundary conditions of the rotating beam in the example, substituting the boundary conditions into equation (65), we get:

[0280] (68)

[0281] The system characteristic equation can be written as:

[0282] (69)

[0283] The natural frequencies of the system are obtained by using a root-searching method such as the bisection method. Substituting the obtained natural frequencies into the system output transfer equation (68), the recursive relationship is as follows:

[0284] (70)

[0285] Based on the above method, the present invention constructs a dynamic characteristic analysis system for a rotating beam system, which inputs the physical information of a uniformly rotating planar vibrating beam into the parameter input module.

[0286] The results of the variation of the first two frequencies of the uniformly rotating planar vibrating beam output by the system with the rotational speed are compared with the results in the literature (ZHAO Z, LIU CS, MA W. Characteristics of steady vibration in a rotating hub–beam system [J]. Journal of Sound and Vibration, 2016, 363: 571-83.) as shown in Table 1.

[0287] Table 1. Variation of the first two frequencies of a uniformly rotating planar vibrating beam with rotational speed (dimensionless units)

[0288]

[0289] As shown in Table 1, the frequencies calculated by this method are highly consistent with the literature values ​​under all rotational speed conditions. The maximum error of the first-order frequency is only 0.26%; the error of the second-order frequency remains stable between 0.10% and 0.11%, indicating that this method has excellent calculation accuracy. With increasing rotational speed, both the first and second-order frequencies of the system gradually increase, which is consistent with the physical law of the stiffening effect caused by the rotational centrifugal force, indicating that this method accurately captures the dynamic characteristics of the rotating beam. Furthermore, whether in a static or high-speed rotating state, this method maintains a stable and consistent level of accuracy, demonstrating its applicability to the analysis of rotating beam systems under a wide range of operating conditions.

[0290] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention without departing from its spirit and scope should be included within the protection scope of the present invention.

Claims

1. A method for analyzing the dynamic characteristics of a rotating beam system based on the finite difference reduced multibody system transfer matrix method, characterized in that, The method includes the following steps: Step 1: Establish a physical parameter model of the rotating beam system and obtain the structural parameters of the rotating beam; Step 2: Based on the physical and structural parameters, establish the dynamic control differential equations of the rotating beam system; Step 3: Discretize the rotating beam system using the finite difference method to obtain a discretized multibody system model; Step 4: Construct the transfer matrix of the discretized multibody system model, and process the transfer matrix using the reduction equation to obtain the recursive relationship of the reduced difference point reduced transfer matrix. Step 5: Establish the reduced state vector recursive equation and calculate the dynamic characteristic parameters of the rotating beam system, including natural frequencies and mode shapes; Step 6: Output the dynamic characteristic parameters.

2. The method for dynamic characteristic analysis of a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 1, characterized in that, The structural parameters mentioned in step 1 include at least geometric parameters, material parameters, boundary conditions, and rotational speed.

3. The method for dynamic characteristic analysis of a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 1, characterized in that, Step 2 includes: obtaining the general rotational Euler-Bernoulli beam vibration differential equation according to Hamilton's theorem; for the non-uniformly rotating rotating beam differential equation, using Fourier series to transform it into an ordinary differential equation in the frequency domain with respect to spatial variables only for solution.

4. The method for dynamic characteristic analysis of a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 1, characterized in that, Step 3 specifically includes: Step 3.1, discretize on the rotating beam The node at the node Perform a difference transform on the nth node to obtain the nth node. The first four order differential difference schemes of each node, where ; Step 3.2, Processing the boundary points of the rotating beam: Perform a difference transformation on the boundary point and three nearby points to obtain the differential difference scheme of the input point.

5. The method for dynamic characteristic analysis of a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 1, characterized in that, Step 4 specifically includes: Step 4.1: Substitute the first four orders of differential difference schemes for the discrete nodes of the rotating beam into the fourth order differential equation of the rotating beam, and obtain the... The equation in the th equation The difference schemes at each node are vectorized to obtain the transitive relationships between the state vectors of the difference schemes; Step 4.2: Substitute the intermediate point and boundary point state vectors into the component model to obtain the relationship between the differential state vector and the component state vector, and derive the transfer equation of component j from this. Step 4.3: Divide the state vector into two column vectors according to the boundary conditions, and divide the transfer matrix of element i into corresponding blocks; Step 4.4 introduces a reduction transformation to describe the two parts of the state vector at the input of element i, obtains the relationship between the two parts of the state vector at the output of element i, and derives the general recursive formula of the reduction transfer matrix. Step 4.5: From the general recursive relation of the reduced transfer matrix and the transfer equation of element j, obtain the recursive relation of the differential point reduced transfer matrix of the element.

6. The method for dynamic characteristic analysis of a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 1, characterized in that, Step 5 specifically includes: Step 5.1: Considering the boundary conditions of the components, the characteristic equation of the system is obtained by reducing the transfer matrix of the component's difference point and solving it using the bisection method to obtain the system's natural frequency. Step 5.2: Substitute the system's natural frequency into the system output transfer equation to obtain the system output state vector; Step 5.3: Then, recursively calculate the state vector of each connection point from the output end to the input end, and extract the displacement and angular displacement in the modal coordinates to obtain the mode shape of the system.

7. A dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claims 1 to 6, characterized in that, The system includes: The first module is configured to obtain the physical parameters, structural parameters, and boundary conditions of the rotating beam system. The second module is connected to the first module and is configured to obtain the system characteristic equations. The third module, connected to the second module, is configured to output dynamic response characteristics.

8. The dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 7, characterized in that, The second module configuration is used to achieve: Establish the dynamic differential equations of the rotating beam system; The dynamic differential equation of the rotating beam is discretized using the finite difference method to construct the transfer matrix; The constructed transfer matrix is ​​reduced using modal reduction equations to obtain the system characteristic equations.

9. The dynamic characteristic analysis system for a rotating beam system based on the finite difference reduced multibody system transfer matrix method according to claim 7, characterized in that, The third module configuration is used to achieve: Based on the output of the second module, calculate the dynamic characteristic parameters, including: searching for the natural frequency of the system characteristic equation using the bisection method, and inversely solving for the state vector of the discrete points to confirm the system mode shape; Output the dynamic characteristic parameters.

10. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the method according to any one of claims 1 to 6.

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