A mathematical modeling method for a spatially arbitrary arrangement vibration isolation system
Patent Information
- Application Number
- CN202611409733.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-09-11
- Publication Date
- 2026-10-30
AI Technical Summary
[0004]针对当前光电设备任意空间分布的隔振系统设计分析难以构建准确数学模型、通过有限元仿真方法分析效率较低的问题，本发明提出一种空间任意布置的隔振系统数理建模方法，能够快速分析隔振系统特性，实现光电设备快速迭代设计优化的目标
首先，本发明无需进行繁琐的有限元网格划分与整体三维建模，直接利用各弹性元件和阻尼元件的空间位置坐标、刚度/阻尼参数以及负载的质量和转动惯量，即可快速构建系统动力学方程并求得完整的六自由度隔振传递函数，大幅提高了计算效率，能够有效支撑光电设备隔振系统的快速迭代设计；其次，本发明能够完整表征隔振系统的线振动传递特性、角振动传递特性以及线振动与角振动之间的耦合传递特性，解决了传统有限元方法难以求解角传递函数的局限；此外，本发明基于坐标系映射和矢量变换的方式，能够精确适应隔振元件在空间中任意非对称布置的工程实际，有效拓宽了数理建模方法的适用范围和工程实用性。
Smart Images
Figure CN123013479A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a mathematical modeling method for a spatially arbitrarily arranged vibration isolation system, which is particularly suitable for mathematical modeling of vibration isolation platforms composed of elastic and damping components of optoelectronic equipment, and belongs to the field of optical technology. Background Technology
[0002] With the generational improvement in the precision of optoelectronic equipment, the operating environment conditions are becoming increasingly stringent, and the impact of environmental vibration on the precision of optoelectronic equipment cannot be ignored. Introducing vibration isolation systems to attenuate and suppress the effects of environmental vibration has become a mainstream trend in optoelectronic equipment. An optoelectronic equipment vibration isolation system consists of three parts: first, a vibration isolation base, which is fixed to the external environment and vibrates along with it during environmental vibrations; second, the vibration isolation system itself, composed of vibration isolation elements, which provide different vibration attenuation effects through the spatial arrangement of these elements and differences in their elasticity and damping performance, attenuating the external vibration excitation transmitted from the base before transferring it to the vibration isolation load; and third, the vibration isolation load, mainly consisting of precision components such as optical sensors, which are easily affected by vibration excitation, leading to image quality degradation.
[0003] Because the vibrations in the working environment of optoelectronic equipment are caused by the combined excitation of three directions (lateral, longitudinal, and vertical), the vibration isolation system needs to simultaneously meet the multi-degree-of-freedom and multi-directional vibration isolation performance requirements. Constrained by the size and layout of optoelectronic equipment, the vibration isolation elements are arbitrarily arranged in space and are usually asymmetrical, making it difficult to establish a mathematical model to analyze the characteristics of the vibration isolation system. Currently, the finite element method is used to solve the characteristics of spatially arbitrary vibration isolation systems, but this method typically has the following shortcomings: (1) The finite element simulation method is to solve the stress field analysis. The simulation model is divided into meshes. For complex models, the simulation time is long and the analysis efficiency is low due to the limited computer processing power. (2) Finite element simulation method requires overall modeling of optoelectronic equipment, which takes more time for equipment with complex models; (3) The finite element simulation method can solve the linear transfer function characteristics of the vibration isolation system by assigning equivalent values to the elastic element and the damping element, but it is difficult to solve the angular transfer function characteristics. Summary of the Invention
[0004] To address the challenges of constructing accurate mathematical models for the design and analysis of vibration isolation systems with arbitrary spatial distribution of optoelectronic equipment, and the low efficiency of analysis using finite element simulation methods, this invention proposes a mathematical modeling method for vibration isolation systems with arbitrary spatial arrangement. This method enables rapid analysis of the characteristics of the vibration isolation system, achieving the goal of rapid iterative design optimization of optoelectronic equipment.
[0005] To solve the above technical problems, the present invention provides the following technical solution: In a first aspect, the present invention provides a mathematical modeling method for a spatially arbitrarily arranged vibration isolation system, wherein the vibration isolation system includes a vibration isolation base, a vibration isolation assembly composed of elastic elements and damping elements, and a vibration isolation load, and the method includes: Establish a reference coordinate system corresponding to the vibration isolation base, a motion reference coordinate system with the center of mass of the vibration isolation load as the origin and the coordinate axes parallel to the reference coordinate system, and a follower coordinate system with the center of mass of the vibration isolation load as the origin and fixed to the vibration isolation load. Obtain the mass, moment of inertia, and center of mass of the vibration isolation load in the reference coordinate system; obtain the performance parameters of each elastic element and each damping element, as well as the positions of the endpoints of each elastic element and each damping element connected to the vibration isolation base and the vibration isolation load in the reference coordinate system. Based on the position of the endpoints of each elastic element and each damping element in the reference coordinate system, determine the initial vectors of each elastic element and each damping element in the motion reference coordinate system before vibration excitation; The vibration isolation load is excited by the vibration of the six-degree-of-freedom base and generates a six-degree-of-freedom pose response. Based on the six-degree-of-freedom pose response, the coordinate mapping relationship from the reference coordinate system to the follower coordinate system under vibration excitation is determined. Combining the position of the center of mass of the vibration isolation load in the reference coordinate system and the position of each endpoint in the reference coordinate system, the deformation vector of each elastic element and each damping element under vibration excitation is determined. Based on the performance parameters of each elastic element and each damping element and the corresponding initial vector and deformation vector, determine the force and moment of each elastic element and each damping element acting on the vibration isolation load, and construct the system stiffness matrix and system damping matrix based on the force and moment. The system mass matrix is established based on the mass and moment of inertia of the vibration isolation load. Based on the system mass matrix, system stiffness matrix, and system damping matrix, the effective load dynamic equilibrium equation is established using the Newton-Euler equation. Finally, the vibration transfer function model of the vibration isolation system is obtained.
[0006] Following the above technical solution, the reference coordinate system takes any point on the vibration isolation base as its origin, and the coordinate axes are parallel to the three orthogonal directions of the external vibration excitation; the follower coordinate system and the motion reference coordinate system only have relative rotation about the center of mass.
[0007] Following the above technical solution, a vector relationship between the elastic element and the damping element in the reference coordinate system is established. Based on the positional relationship between the reference coordinate system and the follower coordinate system, the position vectors of the elastic element and the damping element in the follower coordinate system are calculated. as follows: ; In the formula, Let be the position vector of any endpoint of the elastic or damping element in the reference coordinate system. The position vector of the origin of the moving coordinate system in the reference coordinate system. This is the coordinate mapping matrix from the reference coordinate system to the follower coordinate system, where the superscripts and subscripts 0 and 2 represent the reference coordinate system and the follower coordinate system, respectively.
[0008] Following the above technical solution, if the six-degree-of-freedom pose response generated by the vibration isolation load is expressed as... ,but and The expression is: ; In the formula, The coordinates of the center of mass of the vibration isolation load in the reference coordinate system at the initial moment are given.
[0009] Following the above technical solution, the dynamic equilibrium equations of the effective load are established in the servo coordinate system using Newton's and Euler's equations as follows: ; In the formula, the left side represents the inertial force on the vibration isolation load. y The load displacement vector contains The displacement vector of the six-degree-of-freedom base vibration excitation. M The system quality matrix; , Let be the generalized elastic force and damping force acting on the vibration isolation load, respectively, and their expressions are: Then the above expression becomes: ; in, , These are the system stiffness matrix and the system damping matrix, respectively.
[0010] Following the above technical solution, by performing a Laplace transform on both sides of the equation, the vibration transfer function model of the vibration isolation system is obtained as follows: ; In the formula, s For the Lagrange operator, The transfer function matrix characterizes the linear vibration, angular vibration, and coupling characteristics between the two in the vibration isolation system.
[0011] Following the above technical solution, a system stiffness matrix is established based on the elastic force and elastic moment of each elastic element, including: Suppose that the elastic element is connected at both ends A and B to the vibration isolation load and the vibration isolation base, respectively. Let the position vectors of points A and B in the reference coordinate system be... and They are represented as follows: ; Then the vector of the elastic element ABIn the moving reference coordinate system, it is represented as: ; In the formula, For the vector of the elastic element AB Representation in the moving reference coordinate system For the vector of the elastic element AB Representation in the reference coordinate system; Let A be the position coordinates of point A in the reference coordinate system. Let B be the position coordinates of point B in the reference coordinate system; the superscripts 0, 1 and 2 represent the reference coordinate system, the motion reference coordinate system and the follower coordinate system, respectively. The vibration isolation load is excited by the vibration of the six-degree-of-freedom base to generate a six-degree-of-freedom pose response. hour, A The relative positions of the points in the moving coordinate system remain unchanged, that is... In the formula The coordinates of the centroid of the vibration isolation load in the reference coordinate system at the initial moment; B The position of the point changes in the moving coordinate system, therefore the vector of the elastic element... The expression for the deformation vector is as follows: ; In the formula, For the vector of the elastic element AB Representation in a moving coordinate system; and Let be the position vectors of points A and B in the moving coordinate system; The position vector of the origin of the moving coordinate system in the reference coordinate system. This is the coordinate mapping matrix from the reference coordinate system to the follower coordinate system; Then the elastic force after the elastic element deforms The calculation is as follows: ; In the formula, For the stiffness of the elastic element, , The calculation results of the displacement scalar of the elastic element in the follower coordinate system and the motion reference coordinate system; Substituting the vector of the elastic element into the above formula, the detailed expression is as follows: ; In the formula: ; in, All were small quantities; The force and torque exerted by the elastic element on the center of mass of the vibration isolation load are calculated as follows: ; In the formula, ; Combining like terms in the above equation yields the stiffness matrix of the elastic element. The stiffness matrix represents the contribution of each independent elastic element to the system stiffness; therefore, the system stiffness matrix is determined based on the stiffness matrices of all elastic elements. Similarly, the system damping matrix is established based on the damping force and damping torque of each damping element.
[0012] Following the above technical solution, the vibration isolation load is an optoelectronic load.
[0013] In a second aspect, the present invention provides an electronic device including a processor and a memory, wherein the memory stores a computer program, and when the computer program is executed by the processor, the electronic device performs the method described in the first aspect.
[0014] Thirdly, the present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the method described in the first aspect.
[0015] In summary, compared with the prior art, the above-described technical solutions conceived by this invention can achieve the following beneficial effects: First, this invention eliminates the need for cumbersome finite element mesh generation and overall 3D modeling. It directly utilizes the spatial coordinates, stiffness / damping parameters, and the mass and moment of inertia of each elastic and damping element to quickly construct the system dynamic equations and obtain the complete six-degree-of-freedom vibration isolation transfer function, significantly improving computational efficiency and effectively supporting the rapid iterative design of optoelectronic equipment vibration isolation systems. Second, this invention can fully characterize the linear vibration transmission characteristics, angular vibration transmission characteristics, and coupling transmission characteristics between linear and angular vibrations of the vibration isolation system, overcoming the limitation of traditional finite element methods in solving angular transfer functions. Furthermore, based on coordinate system mapping and vector transformation, this invention can accurately adapt to engineering realities where vibration isolation elements are arbitrarily asymmetrically arranged in space, effectively broadening the applicability and engineering practicality of mathematical modeling methods. Attached Figure Description
[0016] Figure 1 This is a schematic diagram of the initial state layout and coordinate system definition of the optoelectronic device according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the coordinate system layout of a motion-state optoelectronic device according to an embodiment of the present invention. Detailed Implementation
[0017] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. All other embodiments obtained by those skilled in the art based on the embodiments provided by this invention without inventive effort are within the scope of protection of this invention.
[0018] Obviously, the accompanying drawings described below are merely some examples or embodiments of the present invention. Those skilled in the art can apply the present invention to other similar scenarios based on these drawings without any inventive effort. Furthermore, it is understood that although the efforts made in this development process may be complex and lengthy, for those skilled in the art related to the content disclosed in this invention, modifications to design, manufacturing, or production based on the technical content disclosed in this invention are merely conventional technical means and should not be construed as insufficient disclosure of the present invention.
[0019] In this invention, the reference to "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of the invention. The appearance of this phrase in various places in the specification does not necessarily refer to the same embodiment, nor is it a mutually exclusive, independent, or alternative embodiment. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described in this invention may be combined with other embodiments without conflict.
[0020] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "a," "an," "an," "the," and similar words used in this invention do not indicate quantity limitation and may indicate singular or plural. The terms "comprising," "including," "having," and any variations thereof used in this invention are intended to cover non-exclusive inclusion; for example, a process, method, system, product, or device that includes a series of steps or modules (units) is not limited to the listed steps or units, but may also include steps or units not listed, or may include other steps or units inherent to these processes, methods, products, or devices. The terms "connected," "linked," "coupled," and similar words used in this invention are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "A plurality" used in this invention refers to two or more. "And / or" describes the relationship between related objects, indicating that three relationships may exist; for example, "A and / or B" can represent: A alone, A and B simultaneously, and B alone. The character " / " generally indicates that the preceding and following objects have an "or" relationship. The terms "first," "second," and "third" used in this invention are merely to distinguish similar objects and do not represent a specific ordering of the objects.
[0021] To address the challenges of constructing accurate mathematical models for the design and analysis of vibration isolation systems with arbitrary spatial distribution of optoelectronic equipment, and the low efficiency of analysis using finite element simulation methods, this invention proposes a mathematical modeling method for vibration isolation systems with arbitrary spatial arrangement. This method enables rapid analysis of the characteristics of the vibration isolation system, achieving the goal of rapid iterative design optimization of optoelectronic equipment.
[0022] The method includes: First, determining the components of the optoelectronic equipment and dividing the whole into three parts: vibration isolation base, vibration isolation system, and vibration isolation load; Second, establishing the reference coordinate system, motion reference coordinate system, and follow-up coordinate system of the vibration isolation base; Third, determining the characteristics of the vibration isolation load, including parameters such as load mass, moment of inertia, and spatial coordinates of the center of mass; Fourth, determining the spatial coordinates of the elastic and damping elements of the vibration isolation system; Fifth, determining the stiffness coefficient of the elastic element and the damping coefficient of the damping element of each vibration isolation element; Sixth, establishing a dynamic model to calculate and solve the transfer function of the vibration isolation system.
[0023] The specific steps are as follows: (1) Divide the components of optoelectronic equipment, take the mechanical structure that is fixed to the outside and does not have vibration isolation performance as the vibration isolation base part, take the elastic or damping element with vibration isolation performance as the vibration isolation system part, and take the components such as optical sensors that need to attenuate vibration as the vibration isolation load part. (2) Establish a reference coordinate system for the vibration isolation base. The reference coordinate system takes any point on the base as its origin, and the coordinate axes are aligned with the external three-dimensional vibration excitation (lateral, longitudinal, and vertical). Establish a motion reference coordinate system. The reference coordinate system takes the center of mass of the vibration isolation load as its origin, and the coordinate axes are aligned with the direction of the vibration isolation reference coordinate system. The coordinate axes are parallel to each other. Establish a follower coordinate system for the vibration isolation load. The follower coordinate system takes the position of the center of mass of the load as its origin. The coordinate system is fixed to the load and changes with the position of the load. (3) Measure the vibration isolation load parameters, obtain the vibration isolation load mass, moment of inertia, and center of mass coordinates in the reference coordinate system. Measure the performance parameters of the vibration isolation components, including component stiffness, damping coefficient, and coordinates of each vibration isolation component in the reference coordinate system; (4) Establish the Newton-Euler dynamic equations and construct the vibration isolation transfer function model; (5) Stiffness matrix and damping matrix analysis and calculation, and further calculation of the transfer function curve of the vibration isolation system.
[0024] Compared to traditional finite element method for analyzing vibration isolation systems, this invention offers faster simulation and solution speeds, which is beneficial for design iteration. It eliminates the need for finite element 3D modeling and setup, allowing for rapid modification of vibration isolation element parameters and layout to solve for characteristics such as the vibration isolation transfer function of the vibration isolation system.
[0025] like Figure 1 and Figure 2 As shown, the mathematical modeling method for a spatially arbitrarily arranged vibration isolation system according to an embodiment of the present invention includes six steps: dividing the components of the optoelectronic equipment, establishing a reference coordinate system for the vibration isolation base, determining the vibration isolation load characteristics, determining the spatial coordinates of the elastic and damping elements of the vibration isolation system, determining the stiffness coefficient of the elastic element and the damping coefficient of the damping element of each vibration isolation element, and establishing a dynamic model to calculate and solve the transfer function of the vibration isolation system. The details are as follows: Step 1: Divide the equipment into its components.
[0026] Vibration isolation system model as follows Figure 1 As shown, based on the component connection method, location and function, it can be divided into three parts: vibration isolation base, vibration isolation element, and vibration isolation load (the component being isolated).
[0027] Step 2: Establish the reference coordinate system, motion reference coordinate system, and vibration isolation load follow-up coordinate system for the vibration isolation base.
[0028] Take any point on the vibration isolation base as the origin of the base's reference coordinate system, and establish the reference coordinate system. The center of mass of the vibration isolation load is taken as the origin of the motion reference coordinate system, and the three axes of the motion reference coordinate system are parallel to the three axes of the base reference coordinate system. The origin of the vibration isolation load follower coordinate system is located at the center of mass of the effective load and is fixedly connected to the photoelectric load. Therefore, the vibration isolation load follower coordinate system is established. The vibration isolation load's servo coordinate system and the moving reference coordinate system only exhibit relative rotation about the center of mass. Initially, the coordinate axes of the servo coordinate system completely coincide with those of the moving reference coordinate system.
[0029] Step 3: Calculate the vibration isolation load parameters and vibration isolation element parameters.
[0030] The load performance parameters are calculated based on the model, including load mass, moment of inertia about the center of mass, and coordinates of the load's center of mass in the reference coordinate system; the stiffness parameters of the elastic element are calculated, including the coordinates of the connection point between the elastic element and the base, and the connection point between the elastic element and the load in the reference coordinate system; the damping parameters of the damping element are calculated, including the coordinates of the connection point between the damping element and the base, and the connection point between the damping element and the load in the reference coordinate system. Vectors of the elastic element and the damping element in the reference coordinate system are established. Based on the positional relationship between the reference coordinate system and the follower coordinate system, the position vectors of the elastic element and the damping element in the follower coordinate system are calculated. as follows: ; In the formula, Let be the position vector of any endpoint of the elastic / damped element in the reference coordinate system. The vector representing the position of the origin (center of mass of the load) of the moving coordinate system in the reference coordinate system is constant. This is the mapping matrix from the reference coordinate system to the follower coordinate system, where the superscripts and subscripts 0 and 2 represent the reference coordinate system and the follower coordinate system, respectively.
[0031] The displacement vector generated by the photoelectric load in six degrees of freedom is represented as follows: If each component is a small amount, then and The expression is: ; In the formula, Let be the initial coordinates of the center of mass of the photoelectric load in the reference coordinate system.
[0032] Step 4: Dynamic modeling of the Newton-Euler equations.
[0033] The dynamic equilibrium equations of the effective load are established using the Newton-Euler equations in a coordinate system where the center of mass moves astray, as follows: ; In the equation, the left side represents the inertial force acting on the photoelectric load. y The load displacement vector contains The platform's basic six-degree-of-freedom excitation displacement vector, M For the load quality matrix, , Let be the generalized elastic force and damping force acting on the photoelectric load, respectively, and their expressions are: Then the above formula can be transformed into: ; in, M The system quality matrix, , These are the system stiffness matrix and damping matrix, respectively, which are given based on the definitions of elastic force and damping force.
[0034] Applying the Laplace transform to both sides of the equation, we obtain the following model of the system vibration transfer function: ; In the formula, s For Lagrange operators; The 6×6 transfer function matrix characterizes the linear vibration, angular vibration, and coupling characteristics between the two in the vibration isolation system.
[0035] Step 5: Calculation and analysis of stiffness matrix and damping matrix.
[0036] The system stiffness matrix is obtained through elastic force / moment analysis. K Suppose that the elastic element (axial stiffness only) has its two ends A and B connected to the photoelectric load to be applied and the base, respectively. Figure 1 As shown. The position vectors of points A and B in the reference coordinate system are represented as follows: ; Then the spring unit vector AB In the reference coordinate system, it is represented as: .
[0037] like Figure 2 As shown, under the vibration excitation of the carrier base, the center of mass generates displacement in six degrees of freedom. hour, A The relative positions of the points in the moving coordinate system remain unchanged, that is... In the formula Let be the initial position coordinates of the centroid of the photoelectric load in the reference coordinate system. B The position of the point changes in the moving coordinate system. Therefore, the spring element vector... The transformed vector expression is as follows: ; The elastic force after the spring unit deforms is calculated as follows: ; In the formula, For the stiffness of the elastic element, , This represents the scalar displacement calculation results of the elastic element in the following coordinate system and the motion reference coordinate system.
[0038] Spring unit vector Substituting the vectors before and after deformation into the above formula, the detailed expression is as follows: ; In the formula: ; in, All were in small quantities.
[0039] The elastic force and torque of the spring unit acting on the center of mass are calculated as follows: ; In the formula, .
[0040] Combining like terms in the above expression yields... AB Spring element stiffness matrix .
[0041] This stiffness matrix represents the contribution of independent elastic elements to the system stiffness; therefore, the system stiffness matrix is: ; In the formula, n The number of spring units. See Table 1 for details of each element of the matrix.
[0042] The derivation process of the damping force and damping moment coefficient matrix is similar to that of the stiffness matrix. The damping matrix coefficients are shown in Table 2.
[0043] Table 1. Stiffness Matrix Solution Table
[0044] Table For the first j Each flexible unit connection point A The initial position vector components in the centroid-connected coordinate system For the first j Each elastic unit is located at the connection point of the outer frame. B The initial state of the position vector components in the centroid-connected coordinate system. As an intermediate variable, the expression is as follows: ; In the formula, For the stiffness of the elastic element, .
[0045] Table 2 Solution table for damping matrix
[0046] Table For the damping unit connection point A The initial position vector components in the centroid-connected coordinate system For the damping unit at the connection point of the outer frame B The initial state of the position vector components in the centroid-connected coordinate system. As an intermediate variable, its expression is: ; In the formula, Let j be the viscous damping coefficient of the damping element. As an intermediate variable, its expression is: .
[0047] In addition, the present invention also provides an electronic device, including a processor and a memory, wherein the memory stores a computer program, and when the computer program is executed by the processor, the electronic device performs the mathematical modeling method for spatially arbitrarily arranged vibration isolation systems described in the above method embodiments.
[0048] The present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the mathematical modeling method for spatially arbitrarily arranged vibration isolation systems as described in the above method embodiments.
[0049] It should be noted that, depending on the implementation needs, the various steps / components described in this invention can be broken down into more steps / components, or two or more steps / components or parts of the operation of steps / components can be combined into new steps / components to achieve the purpose of this invention.
[0050] Those skilled in the art will readily understand that the above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A mathematical modeling method for a spatially arbitrarily arranged vibration isolation system, characterized in that, The vibration isolation system includes a vibration isolation base, a vibration isolation assembly composed of elastic elements and damping elements, and a vibration isolation load. The method includes: Establish a reference coordinate system corresponding to the vibration isolation base, a motion reference coordinate system with the center of mass of the vibration isolation load as the origin and the coordinate axes parallel to the reference coordinate system, and a follower coordinate system with the center of mass of the vibration isolation load as the origin and fixed to the vibration isolation load. Obtain the mass, moment of inertia, and center of mass of the vibration isolation load in the reference coordinate system; obtain the performance parameters of each elastic element and each damping element, as well as the positions of the endpoints of each elastic element and each damping element connected to the vibration isolation base and the vibration isolation load in the reference coordinate system. Based on the position of the endpoints of each elastic element and each damping element in the reference coordinate system, determine the initial vectors of each elastic element and each damping element in the motion reference coordinate system before vibration excitation; The vibration isolation load is excited by the vibration of the six-degree-of-freedom base and generates a six-degree-of-freedom pose response. Based on the six-degree-of-freedom pose response, the coordinate mapping relationship from the reference coordinate system to the follower coordinate system under vibration excitation is determined. Combining the position of the center of mass of the vibration isolation load in the reference coordinate system and the position of each endpoint in the reference coordinate system, the deformation vector of each elastic element and each damping element under vibration excitation is determined. Based on the performance parameters of each elastic element and each damping element and the corresponding initial vector and deformation vector, determine the force and moment of each elastic element and each damping element acting on the vibration isolation load, and construct the system stiffness matrix and system damping matrix based on the force and moment. The system mass matrix is established based on the mass and moment of inertia of the vibration isolation load. Based on the system mass matrix, system stiffness matrix, and system damping matrix, the effective load dynamic equilibrium equation is established using the Newton-Euler equation. Finally, the vibration transfer function model of the vibration isolation system is obtained.
2. The mathematical modeling method for spatially arbitrarily arranged vibration isolation systems according to claim 1, characterized in that, The reference coordinate system takes any point on the vibration isolation base as its origin, and the coordinate axes are parallel to the three orthogonal directions of the external vibration excitation. The follower coordinate system and the motion reference coordinate system only have relative rotation about the center of mass.
3. The mathematical modeling method for a spatially arbitrarily arranged vibration isolation system according to claim 1, characterized in that, Establish the vectors of the elastic element and the damping element in the reference coordinate system. Based on the positional relationship between the reference coordinate system and the follower coordinate system, the position vectors of the elastic element and the damping element in the follower coordinate system are calculated. as follows: ； In the formula, Let be the position vector of any endpoint of the elastic or damping element in the reference coordinate system. The position vector of the origin of the moving coordinate system in the reference coordinate system. This is the coordinate mapping matrix from the reference coordinate system to the follower coordinate system, where the superscripts and subscripts 0 and 2 represent the reference coordinate system and the follower coordinate system, respectively.
4. The mathematical modeling method for spatially arbitrarily arranged vibration isolation systems according to claim 3, characterized in that, If the six-degree-of-freedom pose response generated by the vibration isolation load is expressed as ,but and The expression is: ； In the formula, The coordinates of the center of mass of the vibration isolation load in the reference coordinate system at the initial moment are given.
5. The mathematical modeling method for a spatially arbitrarily arranged vibration isolation system according to claim 1, characterized in that, The dynamic equilibrium equations of the effective load are established in the moving coordinate system using Newton's and Euler's equations as follows: ； In the formula, the left side represents the inertial force on the vibration isolation load. y The load displacement vector contains The displacement vector of the six-degree-of-freedom base vibration excitation. M The system quality matrix; , Let be the generalized elastic force and damping force acting on the vibration isolation load, respectively, and their expressions are: Then the above expression becomes: ； in, , These are the system stiffness matrix and the system damping matrix, respectively.
6. The mathematical modeling method for a spatially arbitrarily arranged vibration isolation system according to claim 5, characterized in that, Applying the Laplace transform to both sides of the equation, we obtain the vibration transfer function model of the vibration isolation system as follows: ； In the formula, s For the Lagrange operator, The transfer function matrix characterizes the linear vibration, angular vibration, and coupling characteristics between the two in the vibration isolation system.
7. The mathematical modeling method for a spatially arbitrarily arranged vibration isolation system according to claim 1, characterized in that, The system stiffness matrix is established based on the elastic force and elastic moment of each elastic element, including: Suppose that the elastic element is connected at both ends A and B to the vibration isolation load and the vibration isolation base, respectively. Let the position vectors of points A and B in the reference coordinate system be... and They are represented as follows: ； Then the vector of the elastic element AB In the moving reference coordinate system, it is represented as: ； In the formula, For the vector of the elastic element AB Representation in the moving reference coordinate system For the vector of the elastic element AB Representation in the reference coordinate system; Let A be the position coordinates of point A in the reference coordinate system. Let B be the position coordinates of point B in the reference coordinate system; the superscripts 0, 1 and 2 represent the reference coordinate system, the motion reference coordinate system and the follower coordinate system, respectively. The vibration isolation load is excited by the vibration of the six-degree-of-freedom base to generate a six-degree-of-freedom pose response. hour, A The relative positions of the points in the moving coordinate system remain unchanged, that is... In the formula The coordinates of the centroid of the vibration isolation load in the reference coordinate system at the initial moment; B The position of the point changes in the moving coordinate system, therefore the vector of the elastic element... The expression for the deformation vector is as follows: ； In the formula, For the vector of the elastic element AB Representation in a moving coordinate system; and Let be the position vectors of points A and B in the moving coordinate system; The position vector of the origin of the moving coordinate system in the reference coordinate system. This is the coordinate mapping matrix from the reference coordinate system to the follower coordinate system; Then the elastic force after the elastic element deforms The calculation is as follows: ； In the formula, For the stiffness of the elastic element, , The calculation results of the displacement scalar of the elastic element in the follower coordinate system and the motion reference coordinate system; Substituting the vector of the elastic element into the above formula, the detailed expression is as follows: ； In the formula: ； in, All were small quantities; The force and torque exerted by the elastic element on the center of mass of the vibration isolation load are calculated as follows: ； In the formula, ; Combining like terms in the above equation yields the stiffness matrix of the elastic element. The stiffness matrix represents the contribution of each independent elastic element to the system stiffness; therefore, the system stiffness matrix is determined based on the stiffness matrices of all elastic elements. Similarly, the system damping matrix is established based on the damping force and damping torque of each damping element.
8. The mathematical modeling method for a spatially arbitrarily arranged vibration isolation system according to claim 1, characterized in that, The vibration isolation load is an optoelectronic load.
9. An electronic device, characterized in that, The device includes a processor and a memory, wherein the memory stores a computer program that, when executed by the processor, causes the electronic device to perform the method according to any one of claims 1 to 8.
10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the method according to any one of claims 1 to 8.