Modular dynamic modeling method for complex assembly system combined with origin point stiffness correction

Abstract

This invention discloses a modular dynamic modeling method for complex assembled systems that incorporates origin-based dynamic stiffness correction. This method decomposes the complex system into secondary structural units (including supports and multi-excitation device-vibration isolator combinations) and primary structural units (including excitation device-vibration isolator combinations), establishing modular dynamic equations. This enables rapid adaptive adjustment of the dynamic equations when parameters such as the number of vibration isolators and motors change. Furthermore, to improve the response prediction accuracy of the vibration isolator stiffness model, an origin-based dynamic stiffness parameter is introduced to quantify the influence of component dynamic deformation characteristics on vibration transmission, resulting in a corrected dynamic equation. This invention effectively avoids the redundancy problems caused by repetitive derivations in traditional modeling and significantly improves the prediction accuracy of system vibration response, providing a solid theoretical framework for rapid response to structural changes in the system.

Description

Technical Field

[0001] This invention relates to the field of dynamic modeling, specifically a modular dynamic modeling method for complex assembled systems that incorporates origin dynamic stiffness correction. Background Technology

[0002] In modern engineering, dynamic modeling plays a crucial role in analyzing the dynamic behavior of complex mechanical systems and is widely used in many key industries such as robotics, aerospace, vehicle engineering, and precision manufacturing.

[0003] Dynamic modeling provides a deeper understanding of a system's response under dynamic loads, offering strong theoretical support for system design, optimization, and performance improvement. Traditional dynamic modeling methods primarily rely on Newton-Euler equations, Lagrange equations, or Hamilton's principle, deriving differential equations for the entire system step-by-step. This method yields relatively accurate results when dealing with systems with relatively simple structures and fixed parameters.

[0004] However, with the rapid development of modern industrial equipment towards modularity and reconfigurability, the demand for multi-form changes in system topology and parameters (such as the addition or removal of components, adjustment of the number of vibration isolators, and expansion of motor units) is becoming increasingly significant. In such scenarios, traditional modeling methods suffer from inefficiency and insufficient flexibility due to the need to repeatedly construct global equations. This is especially true for multi-degree-of-freedom, strongly coupled assembly systems, where the equation reconstruction process may generate a large amount of computational redundancy.

[0005] This significant resource consumption not only reduces computational efficiency but also severely restricts the effective implementation of dynamic design methods in real-time optimization engineering applications. This is particularly true in industrial scenarios requiring rapid response, where the problem significantly diminishes the practical application value of theoretical models. Secondly, as the complexity and accuracy requirements of industrial systems increase, the limitations of classical dynamic equations become increasingly apparent: their assumptions of linear and static boundary conditions, as well as their simplified treatment of local dynamic parameters, often lead to significant deviations between theoretical predictions and finite element simulations or physical experiments. Summary of the Invention

[0006] The purpose of this invention is to provide a modular dynamics modeling method for complex assembled systems that incorporates origin dynamic stiffness correction, comprising the following steps:

[0007] Step 1) Define the dynamic second-order structural unit and the dynamic first-order structural unit;

[0008] Step 2) Based on the dynamic second-level structural unit, derive the general dynamic equations of the support in three degrees of freedom when all excitation devices are working;

[0009] Step 3) Based on the first-order structural unit of dynamics, derive the general dynamic equations of the three degrees of freedom of the outer shell when all excitation devices are working;

[0010] Step 4) Derive the general dynamic equations of the non-working excitation devices when the first-level structural unit and the second-level structural unit have r and k non-working excitation devices respectively;

[0011] Step 5) Determine the multi-degree-of-freedom system to be modeled, and represent the multi-degree-of-freedom system as one second-order dynamic structural unit and n first-order dynamic structural units, and record the model parameters ( , , ,r, , , , ); m represents the number of working excitation devices in the secondary structural unit, n represents the number of working primary structural units, k represents the number of non-working excitation devices in the secondary structural unit, r represents the number of non-working primary structural units, a represents the number of vibration isolators below the excitation devices in the secondary structural unit, b represents the number of vibration isolators below the supports in the secondary structural unit, c represents the number of vibration isolators below the excitation devices in the primary structural unit, and s represents the number of vibration isolators below the outer shell.

[0012] Step 6) Calculate the origin dynamic stiffness of the components connected at both ends of the vibration isolator in the multi-degree-of-freedom system;

[0013] Step 7) Input the number of first-level structural units, the number of excitation motors and vibration isolators from Step 5), and the stiffness of the vibration isolators from Step 6) into the general dynamic equations of the three degrees of freedom of the support, the general dynamic equations of the three degrees of freedom of the shell, and the general dynamic equations of the non-working excitation device itself to obtain the general dynamic equations of the multi-degree-of-freedom system.

[0014] Furthermore, the dynamic secondary structural unit includes a support, several excitation devices located above the support, and several vibration isolators arranged in the upper and lower layers of the support; the vibration isolators arranged in the upper layer of the support are located between the support and the excitation devices; the vibration isolators arranged in the lower layer of the support are located between the support and the outer shell.

[0015] The model contains several primary structural units, each of which includes an excitation device, and each excitation device is equipped with several vibration isolators.

[0016] The excitation device of the first-level structural unit of dynamics is fixed to the outer shell through vibration isolators.

[0017] Furthermore, the steps to derive the general dynamic equations for the three degrees of freedom of the support include:

[0018] Step 2.1) Construct the general dynamic equations for the three degrees of freedom of the support when the secondary structural unit has only one excitation device;

[0019] Step 2.2) Based on the general dynamic equations of the support with three degrees of freedom when the secondary structural unit has only one excitation device, construct the general dynamic equations of the support with three degrees of freedom when the secondary structural unit has m excitation devices.

[0020] Furthermore, when the number of excitation devices in the secondary structural unit is 1, the general dynamic equations for the three degrees of freedom of the support are as follows:

[0021]

[0022]

[0023] In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , ; For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator and These represent the rotation angles of the support around the x-axis and y-axis, respectively. and These represent the rotation angles of the outer shell around the x-axis and y-axis, respectively.

[0024] Furthermore, when the number of excitation devices in the secondary structural unit is m, the general dynamic equations for the three degrees of freedom of the support are as follows:

[0025] (2)

[0026]

[0027] (3)

[0028]

[0029] (4)

[0030] In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , ; For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator and These represent the rotation angles of the support around the x-axis and y-axis, respectively. and These are the rotation angles of the outer shell about the x-axis and y-axis, respectively; and These are the moments of inertia of the support about the x-axis and y-axis about its center of mass, respectively. , for and The second derivative of .

[0031] Furthermore, in step 3), the steps for deriving the general dynamic equations for the three degrees of freedom of the outer shell include:

[0032] Step 3.1) Derive the general dynamic equations for the shell with only one first-order dynamic structural element, including the vertical displacement of the shell's center of mass. The outer shell rotates about the y-axis by an angle The outer shell rotates about the y-axis by an angle ;

[0033] Step 3.2) Based on the general dynamic equations of the shell with three degrees of freedom when there is only one first-order dynamic structural unit, derive the general dynamic equations of the shell with three degrees of freedom when there are n first-order dynamic structural units in the general model.

[0034] Furthermore, the general dynamic equations for the three degrees of freedom of the shell in the case of only one dynamic first-order structural unit are as follows:

[0035]

[0036]

[0037] (5)

[0038] In the formula, variables To determine the upper bound of the summation depending on index q, This refers to the number of vibration isolators located beneath the outer casing.

[0039] Furthermore, the general dynamic equations for the three degrees of freedom of the shell in the case of n first-order dynamic structural units are as follows:

[0040]

[0041]

[0042] (6)

[0043]

[0044]

[0045] (7)

[0046]

[0047]

[0048] (8)

[0049] In the formula, variables To determine the upper bound of the summation depending on index q, This refers to the number of vibration isolators located beneath the outer casing.

[0050] Furthermore, the steps for deriving the general dynamic equations of the inactive excitation device itself include:

[0051] S1) When there is one non-functional excitation device in the secondary structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0052] (9)

[0053] S2) When there are k inactive excitation devices in the secondary structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0054] . . .

[0058] (10)

[0059] S3) When there is one non-functional excitation device in the primary structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0060] (11)

[0061] S3) When there are r non-functional excitation devices in the first-level structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0062] . . .

[0065] (12)

[0066] S4) When both the primary structural unit and the secondary structural unit have one inactive excitation device, construct the general dynamic equation for the inactive excitation device itself, i.e.:

[0067]

[0068] (13)

[0069] When constructing a first-level structural unit and a second-level structural unit with r and k inactive excitation devices respectively, the general dynamic equation of the inactive excitation device itself is:

[0070] . . .

[0074]

[0075] . . .

[0078] (14)

[0079] Furthermore, the origin dynamic stiffness of the components connecting the two ends of the vibration isolator in a multi-degree-of-freedom system. As shown below:

[0080] (15)

[0081] In the formula, For frequency; For acceleration admittance;

[0082] Stiffness parameters in the general dynamic equation As shown below:

[0083] (16)

[0084] In the formula, k is the stiffness of the vibration isolator. The origin dynamic stiffness of the upper connecting parts. The origin dynamic stiffness of the lower connecting parts.

[0085] Furthermore, the general dynamic equations of a multi-degree-of-freedom system are characterized by the mass matrix M, damping matrix C, stiffness matrix K, and excitation vector F.

[0086] The mass matrix M is shown below:

[0087] (17)

[0088] The stiffness matrix K is shown below:

[0089]

[0090] (18)

[0091] (19)

[0092] (20)

[0093] (twenty one)

[0094] (twenty two)

[0095] (twenty three)

[0096] (twenty four)

[0097] (25)

[0098] The activation vector F is shown below:

[0099] (26)

[0100] In the formula, variable c is the upper limit of the summation of dependent variable q, f is the number of all first-level structural units of work, variable a is the upper limit of the summation of dependent variable p, and e is the number of excitation devices for all work in the second-level structural unit.

[0101] This invention, based on a modular design concept and comprehensively considering the origin dynamic stiffness parameter, proposes a Modified Dynamic Equation (MDE) method that integrates dynamic stiffness correction. This method aims to achieve efficient modeling and accurate prediction of vibration response results for complex assembled systems. It decomposes the complex system into independent functional modules, incorporates the origin dynamic stiffness into the dynamic equations, predefines and parameterizes the local dynamic characteristics of each module, and, combined with the connection rules between modules, automatically assembles the global system equations. This not only effectively avoids the redundancy caused by repetitive derivations in traditional modeling but also significantly improves the prediction accuracy of the system's vibration response, providing a solid theoretical framework for rapid response to structural changes in the system. Attached Figure Description

[0102] Figure 1 It is a two-degree-of-freedom spring oscillator system;

[0103] Figure 2 As the basic structural unit;

[0104] Figure 3 A schematic diagram of a general dynamic model;

[0105] Figure 4 A schematic diagram of the six degrees of freedom of the general model;

[0106] Figure 5 This is a schematic diagram of the vertical displacement at the connection point of the vibration isolator;

[0107] Figure 6 This is a schematic diagram of the displacement variable of the vibration isolator.

[0108] Figure 7 This is a schematic diagram showing the change in the number of vibration isolators;

[0109] Figure 8 A schematic diagram showing the change in the number of excitation devices for a secondary structural unit;

[0110] Figure 9 A schematic diagram showing the change in the number of vibration isolators in a primary structural unit;

[0111] Figure 10 This is a schematic diagram illustrating the change in the number of primary structural units;

[0112] Figure 11 There is one non-functional excitation device in the secondary structural unit.

[0113] Figure 12 There are k inactive excitation devices in the secondary structural unit.

[0114] Figure 13 There is one non-functional excitation device in the primary structural unit.

[0115] Figure 14 There are r non-functional excitation devices in a first-level structural unit.

[0116] Figure 15 Each of the primary and secondary structural units has one non-functional excitation device.

[0117] Figure 16 There are r and k inactive excitation devices for the first-level structural unit and the second-level structural unit, respectively.

[0118] Figure 17 This is a diagram illustrating the acquisition of dynamic stiffness at the origin.

[0119] Figure 18 This is a graph showing the dynamic stiffness spectrum characteristics at the origin.

[0120] Figure 19 Schematic diagram of the stiffness of each component;

[0121] Figure 20 The parameters of the general model to be constructed. Detailed Implementation

[0122] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0123] Example 1:

[0124] See Figures 1 to 20 The modular dynamics modeling method for complex assembled systems, which combines origin dynamic stiffness correction, includes the following steps:

[0125] Step 1) Define the dynamic second-order structural unit and the dynamic first-order structural unit;

[0126] Step 2) Based on the dynamic second-level structural unit, derive the general dynamic equations of the support in three degrees of freedom when all excitation devices are working;

[0127] Step 3) Based on the first-order structural unit of dynamics, derive the general dynamic equations of the three degrees of freedom of the outer shell when all excitation devices are working;

[0128] Step 4) Derive the general dynamic equations of the non-working excitation devices when the first-level structural unit and the second-level structural unit have r and k non-working excitation devices respectively;

[0129] Step 5) Determine the multi-degree-of-freedom system to be modeled, and represent the multi-degree-of-freedom system as one second-order dynamic structural unit and n first-order dynamic structural units, and record the model parameters ( , , ,r, , , , ); m represents the number of working excitation devices in the secondary structural unit, n represents the number of working primary structural units, k represents the number of non-working excitation devices in the secondary structural unit, r represents the number of non-working primary structural units, a represents the number of vibration isolators below the excitation devices in the secondary structural unit, b represents the number of vibration isolators below the supports in the secondary structural unit, c represents the number of vibration isolators below the excitation devices in the primary structural unit, and s represents the number of vibration isolators below the outer shell.

[0130] Step 6) Calculate the origin dynamic stiffness of the components connected at both ends of the vibration isolator in the multi-degree-of-freedom system;

[0131] Step 7) Input the number of first-level structural units, the number of excitation motors and vibration isolators from Step 5), and the stiffness of the vibration isolators from Step 6) into the general dynamic equations of the three degrees of freedom of the support, the general dynamic equations of the three degrees of freedom of the shell, and the general dynamic equations of the non-working excitation device itself to obtain the general dynamic equations of the multi-degree-of-freedom system.

[0132] Example 2:

[0133] The modular dynamic modeling method for complex assembly systems, which combines origin dynamic stiffness correction, is the same as in Example 1. Further, the dynamic secondary structural unit includes a support, several excitation devices located above the support, and several vibration isolators arranged in the upper and lower layers of the support; the vibration isolators arranged in the upper layer of the support are located between the support and the excitation devices; the vibration isolators arranged in the lower layer of the support are located between the support and the outer shell.

[0134] The model contains several primary structural units, each of which includes an excitation device, and each excitation device is equipped with several vibration isolators.

[0135] The excitation device of the first-level structural unit of dynamics is fixed to the outer shell through vibration isolators.

[0136] Example 3:

[0137] The modular dynamic modeling method for complex assembly systems, combined with origin-point dynamic stiffness correction, has the same technical content as any one of Examples 1-2. Furthermore, the steps for deriving the general dynamic equations for the three degrees of freedom of the support include:

[0138] Step 2.1) Construct the general dynamic equations for the three degrees of freedom of the support when the secondary structural unit has only one excitation device;

[0139] Step 2.2) Based on the general dynamic equations of the support with three degrees of freedom when the secondary structural unit has only one excitation device, construct the general dynamic equations of the support with three degrees of freedom when the secondary structural unit has m excitation devices.

[0140] Example 4:

[0141] The modular dynamic modeling method for complex assembly systems, which combines origin dynamic stiffness correction, is technically the same as any one of Examples 1-3. Furthermore, when the number of excitation devices in the secondary structural unit is 1, the general dynamic equations for the three degrees of freedom of the support are as follows:

[0142]

[0143]

[0144] In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , ; For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator and These represent the rotation angles of the support around the x-axis and y-axis, respectively. and These represent the rotation angles of the outer shell around the x-axis and y-axis, respectively.

[0145] Example 5:

[0146] The modular dynamic modeling method for complex assembly systems, which combines origin dynamic stiffness correction, is technically the same as any one of Examples 1-4. Furthermore, when the number of excitation devices in the secondary structural unit is m, the general dynamic equations for the three degrees of freedom of the support are as follows:

[0147] (2)

[0148]

[0149] (3)

[0150]

[0151] (4)

[0152] In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , ; For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator and These represent the rotation angles of the support around the x-axis and y-axis, respectively. and These are the rotation angles of the outer shell about the x-axis and y-axis, respectively; and These are the moments of inertia of the support about the x-axis and y-axis about its center of mass, respectively. , for and The second derivative of .

[0153] Example 6:

[0154] The modular dynamic modeling method for complex assembled systems, combined with origin dynamic stiffness correction, has the same technical content as any one of Examples 1-5. Further, in step 3), the steps for deriving the general dynamic equations for the three degrees of freedom of the outer shell include:

[0155] Step 3.1) Derive the general dynamic equations for the shell with only one first-order dynamic structural element, including the vertical displacement of the shell's center of mass. The outer shell rotates about the y-axis by an angle The outer shell rotates about the y-axis by an angle ;

[0156] Step 3.2) Based on the general dynamic equations of the shell with three degrees of freedom when there is only one first-order dynamic structural unit, derive the general dynamic equations of the shell with three degrees of freedom when there are n first-order dynamic structural units in the general model.

[0157] Example 7:

[0158] The modular dynamic modeling method for complex assembled systems, which combines origin dynamic stiffness correction, is technically the same as any one of Examples 1-6. Furthermore, the general dynamic equations for the three degrees of freedom of the shell in the case of only one dynamic first-level structural unit are as follows:

[0159]

[0160]

[0161] (5)

[0162] In the formula, variables To determine the upper bound of the summation depending on index q, This refers to the number of vibration isolators located beneath the outer casing.

[0163] Example 8:

[0164] The modular dynamic modeling method for complex assembly systems, which combines origin dynamic stiffness correction, is technically the same as any one of Examples 1-7. Furthermore, the general dynamic equations for the three degrees of freedom of the shell in the case of n first-level dynamic structural units are as follows:

[0165]

[0166]

[0167] (6)

[0168]

[0169]

[0170] (7)

[0171]

[0172]

[0173] (8)

[0174] In the formula, variables To determine the upper bound of the summation depending on index q, This refers to the number of vibration isolators located beneath the outer casing.

[0175] Example 9:

[0176] The modular dynamic modeling method for complex assembled systems, which combines origin dynamic stiffness correction, has the same technical content as any one of Examples 1-8. Furthermore, the steps for deriving the general dynamic equations of the non-working excitation device itself include:

[0177] S1) When there is one non-functional excitation device in the secondary structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0178] (9)

[0179] S2) When there are k inactive excitation devices in the secondary structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0180] . . .

[0184] (10)

[0185] S3) When there is one non-functional excitation device in the primary structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0186] (11)

[0187] S3) When there are r non-functional excitation devices in the first-level structural unit, construct the general dynamic equation of the excitation device itself, that is:

[0188] . . .

[0191] (12)

[0192] S4) When both the primary structural unit and the secondary structural unit have one inactive excitation device, construct the general dynamic equation for the inactive excitation device itself, i.e.:

[0193]

[0194] (13)

[0195] When constructing a first-level structural unit and a second-level structural unit with r and k inactive excitation devices respectively, the general dynamic equation of the inactive excitation device itself is:

[0196] . . .

[0200]

[0201] . . .

[0204] (14)

[0205] Example 10:

[0206] The modular dynamic modeling method for complex assembly systems, which combines origin-point dynamic stiffness correction, has the same technical content as any one of Examples 1-9. Further, it addresses the origin-point dynamic stiffness of the components connected at both ends of the vibration isolator in a multi-degree-of-freedom system. As shown below:

[0207] (15)

[0208] In the formula, For frequency; For acceleration admittance;

[0209] Stiffness parameters in the general dynamic equation As shown below:

[0210] (16)

[0211] In the formula, k is the stiffness of the vibration isolator. The origin dynamic stiffness of the upper connecting parts. The origin dynamic stiffness of the lower connecting parts.

[0212] Example 11:

[0213] The modular dynamic modeling method for complex assembled systems, which combines origin dynamic stiffness correction, has the same technical content as any one of Examples 1-10. Furthermore, the multi-degree-of-freedom system includes the ship's power system and related vibration reduction and isolation devices.

[0214] Example 12:

[0215] The modular dynamic modeling method for complex assembled systems, which combines the origin dynamic stiffness correction, has the same technical content as any one of Examples 1-11. Furthermore, the general dynamic equation of a multi-degree-of-freedom system is characterized by the mass matrix M, damping matrix C, stiffness matrix K, and excitation vector F.

[0216] The mass matrix M is shown below:

[0217] (17)

[0218] The stiffness matrix K is shown below:

[0219]

[0220] (18)

[0221] (19)

[0222] (20)

[0223] (twenty one)

[0224] (twenty two)

[0225] (twenty three)

[0226] (twenty four)

[0227] (25)

[0228] The activation vector F is shown below:

[0229] (26)

[0230] In the formula, variable c is the upper limit of the summation of dependent variable q, f is the number of all first-level structural units of work, variable a is the upper limit of the summation of dependent variable p, and e is the number of excitation devices for all work in the second-level structural unit.

[0231] Example 13:

[0232] The modular dynamics modeling method for complex assembled systems, which incorporates origin-based dynamic stiffness correction, includes the following steps:

[0233] Basic Theory of Dynamics

[0234] In the field of engineering technology, dynamic characteristics are a crucial performance indicator, essentially representing the relationship between force and displacement. In model dynamics analysis, displacement-related mechanical factors not only include inertial forces but also require consideration of two other important forces: damping forces, which are positively correlated with velocity, and elastic forces, whose magnitude maintains a linear relationship with displacement changes. The coupled effects of these three forces constitute the core elements of the dynamic equations of a mechanical vibration system. In practical engineering problems, complex mechanical structures typically contain continuous masses and elastic elements. To facilitate analysis and calculation, discretization methods are often used to simplify the actual physical system into a multi-degree-of-freedom system composed of lumped masses, springs, and dampers.

[0235] For linear systems, the vibration differential equations obtained by the Newton-Euler method or the Lagrange method can be finally simplified to the following matrix differential equation form.

[0236] (1)

[0237] In the formula, For the system's degrees of freedom, For the quality matrix, Here is the stiffness matrix. This is the damping matrix. For example... Figure 1 Consider a simple two-degree-of-freedom spring oscillator system containing inertial force, damping force, and elastic force. Its expression is as follows:

[0238] (2)

[0239]

[0240]

[0241]

[0242] According to the vibration differential equation, using the "time-domain differential theorem", by performing Fourier transforms on both sides of equation (1), we can obtain:

[0243] (3)

[0244] After sorting, the relationship between system input and output can be obtained:

[0245] (4)

[0246] General model construction

[0247] In the dynamic analysis of complex mechanical systems, based on specific application areas and analysis objects, their structural characteristics can be summarized and generalized to construct a general model for that specific application object. This is a key prerequisite for achieving efficient system prediction and index allocation. This paper takes the ship's propulsion system and its supporting vibration reduction and isolation devices as the research object. Ships typically have several excitation devices that provide various energy and power to the entire ship, including steam engines, turbines, diesel engines, and electric motors, to ensure normal navigation. At the same time, these excitation devices are also significant sources of strong vibration and noise. The vibrations they generate are transmitted from connections such as engine mounts to various parts of the ship, affecting other structures.

[0248] To effectively cut off the aforementioned vibration transmission path, single-stage or two-stage vibration isolation devices are commonly used in engineering for vibration reduction and isolation. Based on the general structure of the aforementioned ship propulsion system, this paper abstractly constructs the following... Figure 3The modular general dynamic model shown is mainly composed of two types of basic modules: one is... Figure 2 (a) shows a secondary structural unit, which includes a support and a variable number of excitation devices above the support. The second is... Figure 2 (b) shows a primary structural unit, which contains a single excitation device. The number of vibration isolators in both basic units is variable. Therefore, as shown... Figure 3 The general dynamic model shown can be adjusted according to the model structure or actual needs.

[0249] Will as Figure 3 The general model shown is simplified to a mass-spring-damping system, i.e. - - The system. Because under experimental conditions, the obtained information is often the acceleration information of the excitation device, the degrees of freedom of the excitation device are no longer considered when deriving the dynamic equations of the entire system. Instead, the motion information of the excitation device is directly used as the input condition, and it is assumed that all vibration isolators connected to the same excitation device have the same motion state at the connection ends, i.e., the acceleration information generated by the excitation device itself. Figure 3 When performing dynamic analysis on the entire system, the outer shell is subjected to the action of the supports in the secondary structural units and the excitation device in the primary structural units. Therefore, the degrees of freedom of the entire equation will change with the changes in the model structure. The six most critical degrees of freedom are as follows: Figure 4 As shown: Vertical displacement of the center of mass of the outer shell The outer shell rotates about the x-axis by an angle The outer shell rotates about the y-axis by an angle Vertical displacement of the support's center of mass The rotation angle of the support around the x-axis Rotation angle about the y-axis When there are non-functional excitation devices in the model, the degrees of freedom of the entire equation will increase. The effects of changes in various structural parameters on the dynamic equations will be discussed below.

[0250] Derivation of Extendable Dynamic Equations

[0251] To refine the multi-degree-of-freedom coupling equations of the system, the rotation angles of the support and the outer shell about the x-axis and y-axis are taken into account. Assume the connection point between the vibration isolator and a rigid body is P, and its position vector in the rigid body's local coordinate system is... The absolute translational displacement of the rigid body's center of mass is .

[0252] Based on the micro-vibration assumption, due to the rotation angle and Minimal, its corresponding coordinate rotation matrix and It can be approximated by equations (5) and (6):

[0253] (5)

[0254] (6)

[0255] After undergoing a combination of spatial rotation and translation, the connecting point P is mapped to a new coordinate system in the absolute inertial coordinate system. for:

[0256] (7)

[0257] Expand the matrix and discard higher-order insignificant quantities (such as...) Only the vertical displacement of point P along the Z-axis is extracted. Simplifying, we get:

[0258] (8)

[0259] In the formula, Let be the vertical displacement of the rigid body's center of mass. Let x be the angle of rotation of the rigid body about the x-axis. Let be the rotation angle of the rigid body about the y-axis, and x and y be the x and y coordinates of the vibration isolator connection point relative to the horizontal distribution of the rigid body's center of mass, as shown below. Figure 5 As shown.

[0260] To facilitate understanding of the meaning of each variable in the subsequent derivation equations, the vertical displacement at the connection point between the vibration isolator and the support and the outer shell is considered. Variable subscripts are named according to the following rules, such as... Figure 6 As shown in the figure. The meanings of the variables in the figure are as follows: Indicates the outer shell Vertical displacement of the vibration isolator connected below. Indicates the relationship between the secondary structural unit and the outer shell. Vertical displacement of the vibration isolator connected above. Indicating a secondary structural unit, related to the support Vertical displacement of the vibration isolator connected below. In a second-order structural unit, the first Vibration isolators and supports below the excitation device Vertical displacement at the upper connection point; Indicates the first In each primary structural unit, with the outer shell Vertical displacement at the upper connection point This indicates the first non-functional excitation device in the secondary structural unit. Represents the second-order structural unit. A non-functional actuator This indicates the first non-functional first-level structural unit. Indicates the first A non-functional primary structural unit This indicates the vibration isolator and support below the first inactive excitation device in the secondary structural unit. Vertical displacement at the upper connection point In a second-order structural unit, the first Vibration isolator and support below the non-functional excitation device Vertical displacement at the upper connection point This indicates the first non-functional first-level structural unit. Indicates the first A non-functional primary structural unit This indicates that in the first non-functional primary structural unit, it is related to the outer shell. Vertical displacement at the upper connection point Indicates the first In a non-functional primary structural unit, with the outer shell Vertical displacement at the upper connection point. The third letter in all variable subscripts. All represent the first unit in the same group of vibration isolation units. There are one vibration isolator. Since the excitation device is relatively small, we no longer consider its rotational degree of freedom; the motion information of each excitation device is directly used as the input at the end point of the vibration isolator below it. , , , , , .

[0261] When all excitation devices in the model are working

[0262] The Influence of Second-Level Structural Unit Variation on the Dynamic Equation

[0263] The secondary structural unit of the general dynamic model of this research object consists of a support frame, several excitation devices, and several vibration isolators under each excitation device. Changes in the number of excitation devices and vibration isolators affect the stress on the support frame, thus affecting its three degrees of freedom of motion. Taking the vertical degree of freedom of the support frame as an example... lateral tilt degree of freedom and pitch freedom Taking the example of the changes in the secondary structural units, we can illustrate the impact of these changes on the overall dynamic equation of the research object.

[0264] (1) Changes in the number of vibration isolators

[0265] like Figure 7As shown, when only considering the change in the number of vibration isolators, assuming the number of excitation devices is 1, the number of vibration isolators in the upper and lower layers of the secondary structural unit support increases from 1 to [missing information]. Individual and With a given number of vibration isolators, the dynamic equation can be obtained as shown in equation (9). It can be seen from the equation that when the number of vibration isolators changes, it is only necessary to add the corresponding damping and stiffness terms to the equation.

[0266]

[0267] (9)

[0268] In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , The same applies to the following text. For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator are specified; other variables are named according to the previously mentioned naming rules, such as 6 and... Figure 7 As shown.

[0269] (2) Changes in the number of excitation devices

[0270] In a secondary structural unit, changes in the number of excitation devices also affect the dynamic equations. For example... Figure 8 As shown, when considering the number of excitation devices increasing from 1 to... When the number of vibration isolators of the excitation device remains unchanged, the equation for the vertical displacement degree of freedom of the support is given by equation (10). It can be seen that as long as the corresponding number of excitation terms is controlled, models with different numbers of excitation devices can be flexibly calculated and analyzed.

[0271]

[0272] (10)

[0273] In the formula, This represents the number of excitation devices in the secondary structural unit; the meanings of the other variables are the same as above.

[0274] Following the same analytical approach, the rotation angle of the support around the x-axis can be directly derived. and the angle of rotation around the y-axis The dynamic equations are in the following form:

[0275] The support tilt motion degree of freedom The equation is:

[0276]

[0277] (11)

[0278] The bracket has a degree of freedom of pitch motion. The equation is:

[0279]

[0280] (12)

[0281] The Influence of Changes in First-Order Structural Elements on the Dynamic Equations

[0282] In the general dynamic model of this research object, the first-level structural unit consists of a single excitation device and several parallel vibration isolators. Changes in this unit and the number of its internal vibration isolators will alter the dynamic response and motion characteristics of the outer shell. Since all first-level structural units are rigidly fixed to the outer shell base, they only affect the motion state of the outer shell's three degrees of freedom, specifically the vertical degree of freedom. lateral tilt degree of freedom and pitch freedom Taking this as an example, we will analyze in detail the impact of changes in the first-level structural unit on the equation.

[0283] (1) The effect of changes in the number of vibration isolators

[0284] like Figure 9 As shown, while keeping other structural and parameter parameters of the system constant, when considering only the expansion effect of the number of vibration isolators within a single-level structural unit, it is assumed that the total number of vibration isolators within the unit gradually increases from the initial single-unit (1) to multiple units (...). The extended comprehensive dynamic equation can be derived as shown in equation (13). From the equation, it can be seen that the increase or decrease in the number of physical vibration isolators is intuitively mapped to the linear superposition of each parallel transmission path in the dynamic differential equation. That is, when the number of vibration isolators changes, there is no need to reconstruct the overall framework at the equation construction level. It is only necessary to add or subtract the corresponding local stiffness coefficient term and damping coefficient term at the force end of the equation.

[0285]

[0286]

[0287] (13)

[0288] In the formula, variables To determine the upper bound of the summation depending on index q, This represents the number of vibration isolators located beneath the outer shell. From this derivation, it's easy to see that this dynamic model possesses strong versatility and extensibility. In subsequent numerical solutions, it's only necessary to introduce a summation operator and flexibly control the number of accumulated terms (i.e., parameters). This allows for efficient and accurate calculations and characteristic analysis of complex derivative models equipped with varying numbers of vibration isolators.

[0289] (2) The impact of changes in the number of primary structural units

[0290] Furthermore, changes in the total number of first-order structural units in the system's macroscopic topology also significantly affect the overall dynamic equations. For example... Figure 10 As shown, when the total number of primary structural units under consideration is expanded from the initial 1 by translation to... When there are 1 unit (and the number of vibration isolators inside each unit is arbitrarily given), the vertical translational degree of freedom of the outer shell can be derived. The dynamic equations on the equation are shown in equation (14). As can be seen from this equation, the addition of physical modules strictly follows the superposition principle in mathematical modeling. It is only necessary to expand the dimension of the summation term in the mechanical terms of the equation to achieve rapid construction of complex models.

[0291]

[0292]

[0293] (14)

[0294] In the formula, This represents the number of primary structural units. Following the modular overlay analysis logic described above, and using the spatial coordinates of each connection node as the lever arm for torque conversion, the rotation angle of the outer shell around the x-axis can be directly derived. and the angle of rotation around the y-axis The dynamic equations are in the following form:

[0295] Degrees of freedom of lateral tilting motion of the outer shell The equation is:

[0296]

[0297]

[0298] (15)

[0299] shell pitch motion degrees of freedom The equation is:

[0300]

[0301]

[0302] When the model contains a non-functional excitation device

[0303] When a model contains a non-functional excitation device, this device, lacking an input stimulus, has its degrees of freedom coupled with the entire system, thus affecting the overall system response. Consequently, the model's dynamic equations will change due to the influence of the non-functional excitation device.

[0304] There are non-functional excitation devices in the secondary structural unit.

[0305] (1) When there is one non-functional excitation device in the secondary structural unit

[0306] The system of equations constructed at this point is 7-dimensional, that is... , , , , , , There are a total of 7 degrees of freedom. The first 6 degrees of freedom ( , , , , , The equation for the 7th degree of freedom is the same as that derived earlier. The equations for the degrees of freedom of the non-working excitation device are as follows.

[0307] (17)

[0308] In the formula, The number of vibration isolators below the excitation device. For the mass of the excitation device, This provides the acceleration motion information for the excitation device. The first one below the excitation device The velocity motion information of the end where the vibration isolator is connected to the bracket, i.e. , Similarly, that is The same applies below.

[0309] (2) When there are k inactive excitation devices in the secondary structural unit

[0310] The system of equations constructed at this point is (6 + k) dimensional, i.e. , , , , , , … There are a total of (6+k) degrees of freedom. The first 6 degrees of freedom ( , , , , , The equation for ) is the same as that derived earlier, from the 7th to the (6 + k)th degree of freedom ( … The equations for the degrees of freedom of the non-working excitation device are as follows.

[0311] . . .

[0315]

[0316] There are non-functional excitation devices in the primary structural unit.

[0317] (1) When there is one non-functional excitation device in the primary structural unit

[0318] The system of equations constructed at this point is 7-dimensional, that is... , , , , , , There are a total of 7 degrees of freedom. The first 6 degrees of freedom ( , , , , , The equation for the 7th degree of freedom is the same as that derived earlier. The equations for the degrees of freedom of the non-working excitation device are as follows.

[0319] (19)

[0320] (2) When there are r non-functional excitation devices in the primary structural unit

[0321] The system of equations constructed at this point is (6 + r) dimensional, i.e. , , , , , , … There are a total of (6 + r) degrees of freedom. The first 6 degrees of freedom ( , , , , , The equation for ) is the same as that derived earlier, from the 7th to the (6 + r)th degree of freedom ( … The equations for the degrees of freedom of the non-working excitation device are as follows.

[0322] . . .

[0325] (20)

[0326] When both primary and secondary structural units have non-functional excitation devices

[0327] (1) When each of the primary structural unit and the secondary structural unit has one non-operating excitation device

[0328] The system of equations constructed at this point is 8-dimensional, that is... , , , , , , , There are a total of (8) degrees of freedom. The first 6 degrees of freedom ( , , , , , The equation for the 7th to 8th degrees of freedom is the same as the one derived earlier. , The equations for the degrees of freedom of the non-working excitation device are as follows.

[0329]

[0330]

[0331] (1) When the first-level structural unit and the second-level structural unit have r and k non-operating excitation devices respectively.

[0332] The system of equations constructed at this point is (6 + k + r) dimensional, i.e. , , , , , , … , … There are a total of (6 + k + r) degrees of freedom. The first 6 degrees of freedom ( , , , , , The equation is the same as the one derived earlier, from the 7th to the (6 + k + r)th degree of freedom ( … , … The equations for the degrees of freedom of the non-working excitation device are as follows.

[0333] . . .

[0337]

[0338] . . .

[0341]

[0342] In summary, changes in the model structure will cause changes in the matrix dimensions and the expressions of matrix elements in the vibration differential equations. Based on the above-mentioned laws governing the influence of model structure changes on the dynamic equations, the overall dynamic equations of the model studied in this paper can be obtained. These equations can adapt to changes in multiple parameters of second-order and first-order structural units, which greatly improves the efficiency of deriving the dynamic equations of different structural models.

[0343] Correction of the dynamic equations based on the origin dynamic stiffness

[0344] As the complexity and precision requirements of industrial systems increase, the limitations of classical dynamic equations become increasingly apparent. Their assumptions of linearity, static boundary conditions, and simplified handling of local dynamic parameters often lead to significant discrepancies between theoretical predictions and finite element simulations or physical experiments. This is because the dynamic characteristics of the interface are not fully characterized. In particular, the "origin dynamic stiffness" parameter of components directly coupled to the vibration isolator reflects the frequency-varying local stiffness characteristics of the system under dynamic loads, and its amplitude exhibits significant nonlinearity with the excitation frequency. Traditional modeling methods implicitly assume that the isolated components (such as equipment bases or mounting platforms) are ideal rigid bodies, neglecting their local deformation characteristics under dynamic load excitation. Using only static stiffness parameters fails to accurately characterize energy dissipation and inertial effects under excitation, ultimately causing prediction errors in response amplitude and spectral characteristics. Especially near the resonant frequency, the nonlinear decay of the origin dynamic stiffness may cause impedance mismatch in the system, leading to excessive prediction errors in vibration amplitude by traditional models. To address this issue, the dynamic equations are modified by introducing a frequency-varying function of the origin dynamic stiffness, thereby establishing a multi-degree-of-freedom coupled dynamic model to improve the calculation accuracy of the dynamic equations. Incorporating the origin dynamic stiffness into the dynamic equations not only allows for a more accurate quantification of the coupling effect between the excitation source, structure, and vibration isolator, but also provides crucial physical basis for optimizing vibration and noise suppression strategies.

[0345] Calculation of dynamic stiffness parameters at the origin

[0346] The dynamic stiffness at the origin of a component is a key indicator characterizing the dynamic deformation resistance of a structure. It is defined as the stiffness characteristics exhibited by the structure in a specific frequency domain when the excitation point and response point originate from the same source in the free state of the model component. The calculation of dynamic stiffness is based on dynamic equations, taking into account the mass, stiffness, damping, and frequency characteristics of the external excitation force of the component. Under dynamic conditions, the relationship between force and displacement is usually expressed as a complex quantity, and dynamic stiffness can be described as the ratio of force to displacement.

[0347] The dynamic equation of a single-degree-of-freedom vibration system is given by equation (1). In the frequency domain, through Fourier transform, equation (1) can be transformed into equations (3) and (4).

[0348] Dynamic stiffness is defined as the ratio of force to displacement in the frequency domain. Equation (4) is further transformed into:

[0349] (twenty three)

[0350] Because measuring acceleration signals is more convenient than measuring displacement or velocity signals, acceleration signals are typically used when acquiring vibration signals. Furthermore, when the excitation force input is a unit force, the expression for dynamic stiffness in the frequency domain can be obtained as follows:

[0351] (twenty four)

[0352] That is, the dynamic stiffness at the origin can be obtained by applying a unit force excitation and picking up the acceleration response at the same point and calculating it using equation (24).

[0353] The dynamic stiffness parameters of the upper reference point at the midpoint of the long side of a certain bracket component were obtained using the finite element method simulation, such as... Figure 18 Using HyperMesh software, in a free state, apply a unit force perpendicular to the support's normal direction (Z-axis) at the location shown in the component drawing. Simultaneously, the acceleration response at that point is captured. Substituting the obtained acceleration response data into Equation 24 yields the dynamic stiffness parameters at the origin, whose amplitude and phase characteristic curves are shown below. Figure 18 As shown.

[0354] Corrected dynamic equations

[0355] After measuring the origin dynamic stiffness of the components connecting the two ends of the vibration isolator, it is incorporated into the overall dynamic equation. For example... Figure 19 As shown, at this point, the stiffness in the equation... This is the static stiffness of the vibration isolator. The origin dynamic stiffness of the upper connecting parts The origin dynamic stiffness of the connecting parts at the lower end The sum of the three is shown in the following formula.

[0356] (25)

[0357] The total dynamic stiffness model of the system in the frequency domain was constructed using equation (25). The model uses the static stiffness of the vibration isolator. Using the principle of linear superposition as a baseline, the origin dynamic stiffness of the connecting components between the upper and lower ends is introduced. and This approach transforms the idealized connection, originally described only by static parameters, into a frequency-varying function that incorporates the dynamic characteristics of the boundary structure. By... and As a frequency-dependent correction term directly coupled into the stiffness expression, it not only endows the equation with the ability to describe the local flexibility effect, but also realizes the mapping from a single static stiffness to a complex dynamic stiffness across the entire frequency band.

[0358] Overall dynamic equations

[0359] Based on the preceding discussion of the evolution of the number of excitation devices and vibration isolators in secondary and primary structural units, and by fully considering the effect of the coupling of the origin dynamic stiffness parameter on the correction of the system boundary conditions, this paper successfully constructs a general overall dynamic equation framework that can adapt to changes in the model structure. Figure 20The general parametric model architecture shown here, in order to achieve a unified definition for various complex assembly conditions, introduces the following core configuration parameters: the secondary structural unit contains... Each device has an excitation device, and each device is configured with an excitation device below it. There is a vibration isolator, and there is one under the bracket. One vibration isolator. Meanwhile, the system has several vibration isolators connected in parallel. Each unit has a primary structural unit, and each unit has an excitation device below it. There are [number] vibration isolators, and the number of vibration isolators below the outer casing is [number]. Each secondary structural unit contains A non-functional excitation device, with several components connected in parallel in the system. A non-functional primary structural unit.

[0360] Based on the modular parameter definition logic described above, when dealing with complex target systems with different equipment sizes and support layouts, only the actual model structure parameters need to be defined. , , ,r, , , , By substituting the equations derived above, the global dynamic differential equations of the target system can be directly assembled without the need for re-deriving the mathematical equations. This method significantly reduces the time cost of preliminary theoretical modeling while truly achieving efficient and adaptive modeling of complex multi-layer vibration isolation systems.

[0361] The coefficient matrix of the entire dynamic equation has a dimension of (6 + k + r). The M, C, K, and F matrices of the general dynamic equation are summarized as follows:

[0362] The mass matrix M is as follows:

[0363] (26)

[0364] The stiffness matrix K is as follows:

[0365] The stiffness matrix is ​​a 6th-order block matrix in the upper left corner. as follows:

[0366]

[0367] The stiffness matrix K is a block matrix consisting of rows 1 to 6 and columns 7 to (6 + r + k) from the top right corner. as follows:

[0368]

[0369]

[0370] The stiffness matrix K is a block matrix consisting of rows 7 to (6 + k + r) from the bottom right corner and columns 7 to (6 + k + r). as follows:

[0371]

[0372] In the formula, Let be the number of vibration isolators below the k-th inactive excitation device in the secondary structural unit. Let K be the number of vibration isolators in the r-th non-functional first-order structural unit. The stiffness matrix K is divided into blocks from row 7 to row (6 + k + r) and columns 1 to 6. With the top right corner block matrix The transpose is the same, as shown below.

[0373]

[0374] In summary, the overall stiffness matrix K is as follows.

[0375]

[0376] The construction method of the damping matrix C follows the same theoretical framework as the stiffness matrix, and its specific expression can be obtained by analogy with the derivation process of the stiffness matrix (replacing K with C).

[0377] The activation vector F is as follows:

[0378]

[0379] In the formula, variable c is the upper limit of the summation of dependent variable q, f is the number of all first-level structural units of work, variable a is the upper limit of the summation of dependent variable p, and e is the number of excitation devices for all work in the second-level structural unit.

[0380] By substituting the corresponding feature parameters into the constructed general scalable dynamic equations for different models, the differential equations of complex models can be quickly established.

Claims

1. A modular dynamics modeling method for complex assembled systems combining origin dynamic stiffness correction, characterized in that, Includes the following steps: Step 1) Define the dynamic second-order structural unit and the dynamic first-order structural unit; Step 2) Based on the dynamic second-level structural unit, derive the general dynamic equations of the support in three degrees of freedom when all excitation devices are working; Step 3) Based on the first-order structural unit of dynamics, derive the general dynamic equations of the three degrees of freedom of the outer shell when all excitation devices are working; Step 4) Derive the general dynamic equations of the non-working excitation devices when the first-level structural unit and the second-level structural unit have r and k non-working excitation devices respectively; Step 5) Determine the multi-degree-of-freedom system to be modeled, and represent the multi-degree-of-freedom system as one second-order dynamic structural unit and n first-order dynamic structural units, and record the model parameters ( , , ,r, , , , m represents the number of working excitation devices in the secondary structural unit, n represents the number of working primary structural units, k represents the number of non-working excitation devices in the secondary structural unit, r represents the number of non-working primary structural units, a represents the number of vibration isolators below the excitation devices in the secondary structural unit, b represents the number of vibration isolators below the supports in the secondary structural unit, c represents the number of vibration isolators below the excitation devices in the primary structural unit, and s represents the number of vibration isolators below the outer shell. Step 6) Calculate the origin dynamic stiffness of the components connected at both ends of the vibration isolator in the multi-degree-of-freedom system; Step 7) Input the model parameters from Step 5) and the stiffness of the vibration isolator from Step 6) into the general dynamic equations of the three degrees of freedom of the support, the three degrees of freedom of the shell, and the general dynamic equations of the non-working excitation device itself to obtain the general dynamic equations of the multi-degree-of-freedom system.

2. The modular dynamic modeling method for complex assembled systems combined with origin dynamic stiffness correction as described in claim 1, characterized in that, The dynamic secondary structural unit includes a support frame, several excitation devices located above the support frame, and several vibration isolators arranged in the upper and lower layers of the support frame; the vibration isolators arranged in the upper layer of the support frame are located between the support frame and the excitation devices; the vibration isolators arranged in the lower layer of the support frame are located between the support frame and the outer shell. The model contains several primary structural units, each of which includes an excitation device, and each excitation device is equipped with several vibration isolators. The excitation device of the first-level structural unit of dynamics is fixed to the outer shell through vibration isolators.

3. The modular dynamic modeling method for complex assembly systems combined with origin dynamic stiffness correction as described in claim 2, characterized in that, The steps to derive the general dynamic equations for the three degrees of freedom of the support include: Step 2.1) Construct the general dynamic equations for the three degrees of freedom of the support when the secondary structural unit has only one excitation device; Step 2.2) Based on the general dynamic equations of the support with three degrees of freedom when the secondary structural unit has only one excitation device, construct the general dynamic equations of the support with three degrees of freedom when the secondary structural unit has m excitation devices.

4. The modular dynamic modeling method for complex assembled systems combined with origin dynamic stiffness correction as described in claim 3, characterized in that, When the number of excitation devices in the secondary structural unit is 1, the general dynamic equations for the three degrees of freedom of the support are as follows: In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , ; For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator and These represent the rotation angles of the support around the x-axis and y-axis, respectively. and These represent the rotation angles of the outer shell around the x-axis and y-axis, respectively.

5. The modular dynamic modeling method for complex assembled systems combined with origin dynamic stiffness correction according to claim 3, characterized in that, When the number of excitation devices in the secondary structural unit is m, the general dynamic equations for the three degrees of freedom of the support are as follows: （2） （3） （4） In the formula, variables For dependency indexes The upper limit of the summation, This refers to the number of vibration isolators under the support frame. , , ; For stent Vertical acceleration motion information, and The first Below the first excitation device The stiffness and damping parameters of each vibration isolator and Each is a stent The first one below The stiffness and damping parameters of each vibration isolator and These represent the rotation angles of the support around the x-axis and y-axis, respectively. and These are the rotation angles of the outer shell about the x-axis and y-axis, respectively; and These are the moments of inertia of the support about the x-axis and y-axis, respectively, around its center of mass; , for and The second derivative of .

6. The modular dynamics modeling method for complex assembled systems combined with origin dynamic stiffness correction according to claim 1, characterized in that, Step 3) involves deriving the general dynamic equations for the three degrees of freedom of the outer shell, including: Step 3.1) Derive the general dynamic equations for the shell with only one first-order dynamic structural element, including the vertical displacement of the shell's center of mass. The outer shell rotates about the y-axis by an angle The outer shell rotates about the y-axis by an angle ; Step 3.2) Based on the general dynamic equations of the shell with three degrees of freedom when there is only one first-order dynamic structural unit, derive the general dynamic equations of the shell with three degrees of freedom when there are n first-order dynamic structural units in the general model.

7. The modular dynamics modeling method for complex assembled systems combined with origin dynamic stiffness correction as described in claim 6, characterized in that, The general dynamic equations for the three degrees of freedom of the shell in the case of only one first-order dynamic structural unit are as follows: （5） In the formula, variables To determine the upper bound of the summation depending on index q, This refers to the number of vibration isolators located beneath the outer casing. The general dynamic equations for the three degrees of freedom of the shell in the case of n first-order dynamic structural elements are as follows: （6） （7） （8） In the formula, variables To determine the upper bound of the summation depending on index q, This refers to the number of vibration isolators located beneath the outer casing.

8. The modular dynamics modeling method for complex assembled systems combined with origin dynamic stiffness correction as described in claim 1, characterized in that, The steps to derive the general dynamic equations for the inactive excitation device include: S1) When there is one non-functional excitation device in the secondary structural unit, construct the general dynamic equation of the excitation device itself, that is: （9） S2) When there are k inactive excitation devices in the secondary structural unit, construct the general dynamic equation of the excitation device itself, that is: . . . （10） S3) When there is one non-functional excitation device in the primary structural unit, construct the general dynamic equation of the excitation device itself, that is: （11） S3) When there are r non-functional excitation devices in the first-level structural unit, construct the general dynamic equation of the excitation device itself, that is: . . . （12） S4) When both the primary structural unit and the secondary structural unit have one inactive excitation device, construct the general dynamic equation for the inactive excitation device itself, i.e.: （13） When constructing a first-level structural unit and a second-level structural unit with r and k inactive excitation devices respectively, the general dynamic equation of the inactive excitation device itself is: . . . . . . （14） In the formula, k represents stiffness.

9. The modular dynamics modeling method for complex assembled systems combined with origin dynamic stiffness correction according to claim 1, characterized in that, The origin dynamic stiffness of the components connected at both ends of a multi-degree-of-freedom system vibration isolator As shown below: （15） In the formula, For frequency; For acceleration admittance; Stiffness parameters in the general dynamic equation As shown below: （16） In the formula, k is the stiffness of the vibration isolator. The origin dynamic stiffness of the upper connecting parts. The origin dynamic stiffness of the lower connecting parts.

10. The modular dynamic modeling method for complex assembled systems combined with origin dynamic stiffness correction according to claim 1, characterized in that, The general dynamic equations of a multi-degree-of-freedom system are characterized by the mass matrix M, damping matrix C, stiffness matrix K, and excitation vector F. The mass matrix M is shown below: （17） The stiffness matrix K is shown below: （18） （19） （20） （21） （22） （23） （24） （25） The activation vector F is shown below: （26） In the formula, variable c is the upper limit of the summation of dependent variable q, f is the number of all first-level structural units of work; variable a is the upper limit of the summation of dependent variable p, and e is the number of excitation devices of all work in the second-level structural unit.

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