A gantry dynamics analysis method

Abstract

The application belongs to the field of gantry cranes, and discloses a gantry dynamics analysis method, which comprises the following steps: establishing a full-order finite element model of a gantry crane, and obtaining a mass matrix and a stiffness matrix of the full-order finite element model; based on the modular structure characteristics of the gantry crane, dividing the full-order finite element model into a plurality of substructures, and decomposing the degrees of freedom of each substructure into internal degrees of freedom and boundary degrees of freedom; constructing a fixed boundary principal modal matrix and a constraint modal matrix of each substructure, and constructing a reduced-order basis matrix; using the reduced-order basis matrix, performing modal condensation on the mass matrix and the stiffness matrix of the full-order finite element model, generating a reduced-order mass matrix and a reduced-order stiffness matrix, and constructing a reduced-order dynamics equation; solving the reduced-order dynamics equation to obtain a reduced-order modal coordinate, and combining the reduced-order basis matrix and the reduced-order modal coordinate to inversely restore a displacement response vector of the full-order finite element model, which maximally retains the key dynamics information of the gantry crane through reduction.

Description

Technical Field

[0001] This invention relates to the field of gantry cranes, and more particularly to a method for dynamic analysis of gantry cranes. Background Technology

[0002] Gantry cranes are the core load-bearing and motion structures of equipment such as machine tools, lifting and transportation systems, and automated production lines. Their vibration response, stiffness, natural frequency, and coupled vibration characteristics directly determine the equipment's operational accuracy, stability, and service life, and are crucial for avoiding problems such as resonance, insufficient stiffness, and structural fatigue. During the research and development and optimization phases, dynamic simulation is an important basis for adjusting structural parameters and improving performance, and has become a core component of high-end gantry crane design.

[0003] Currently, finite element full-order models are commonly used for dynamic analysis in engineering. While this provides high computational accuracy, the large span and complex structure of gantry cranes, with tens to hundreds of thousands of degrees of freedom, mean that a single simulation can take hours to tens of hours. Even with software automation, it is still difficult to meet the efficiency requirements of rapid iteration in R&D, significantly extending the design cycle. Model reduction methods can significantly reduce degrees of freedom and improve computational efficiency, and have been validated in some complex structural fields.

[0004] However, existing polynomial interpolation-based order reduction methods are mainly applicable to beam structures, which are incompatible with the modular and multi-component collaborative structural characteristics of gantry frames. They lose the boundary coupling effects between beams, columns, and workbenches, failing to accurately reflect the collaborative vibration laws and making it difficult to meet the accuracy requirements of high-end equipment. Furthermore, these methods are typically limited to free vibration analysis, significantly reducing their applicability to highly nonlinear dynamic conditions such as high-speed motion, variable loads, and instantaneous impacts. Additionally, parameter adjustments require remodeling, resulting in poor reusability.

[0005] As equipment develops towards high speed, heavy load, and precision, the nonlinear dynamic analysis of gantry cranes urgently requires the support of efficient and high-precision order reduction models. Existing full-order models are inefficient, and general order reduction methods lack sufficient accuracy, leading to a contradiction between accuracy and efficiency in design, which restricts engineering applications.

[0006] Therefore, existing technologies still need improvement and development. Summary of the Invention

[0007] The purpose of this invention is to provide a dynamic analysis method for gantry cranes, which aims to solve the technical problem that existing gantry crane simulations using full-order models are time-consuming.

[0008] To achieve the above objectives, the solution provided by the present invention is as follows:

[0009] A method for dynamic analysis of a gantry crane includes: establishing a full-order finite element model of the gantry crane and obtaining the mass matrix and stiffness matrix of the full-order finite element model; based on the modular structural characteristics of the gantry crane, dividing the full-order finite element model into several substructures, and decomposing the degrees of freedom of each substructure into internal degrees of freedom and boundary degrees of freedom; based on the mass matrix, the stiffness matrix, and the division results of internal and boundary degrees of freedom, constructing fixed boundary principal mode matrices and constraint mode matrices for each substructure, and constructing reduced-order basis matrices based on the fixed boundary principal mode matrices and constraint mode matrices; using the reduced-order basis matrices, performing modal condensation on the mass matrix and stiffness matrix of the full-order finite element model to generate reduced-order mass matrices and reduced-order stiffness matrices, and constructing reduced-order dynamic equations based on the reduced-order mass matrix and the reduced-order stiffness matrix; solving the reduced-order dynamic equations to obtain reduced-order modal coordinates, and combining the reduced-order basis matrices and the reduced-order modal coordinates to invert and restore the displacement response vector of the full-order finite element model.

[0010] Preferably, based on the modular structural features of the gantry frame, when dividing the full-order finite element model into several substructures and decomposing the degrees of freedom of each substructure into internal degrees of freedom and boundary degrees of freedom, with the goal of minimizing the number of boundary degrees of freedom, based on the modular structural features of the gantry frame, the crossbeam, left column, right column and workbench of the gantry frame are divided into independent substructures, and the core degrees of freedom on the connection surface of each substructure are taken as boundary degrees of freedom, and the dispersed degrees of freedom inside each substructure are taken as internal degrees of freedom.

[0011] Preferably, the core degrees of freedom include translational degrees of freedom in three directions and rotational degrees of freedom in three directions.

[0012] Preferably, the decomposition of the degrees of freedom of each substructure is achieved by dividing the mass matrix and stiffness matrix of each substructure into blocks according to internal degrees of freedom and boundary degrees of freedom, and the block form is expressed as follows:

[0013]

[0014]

[0015] In the formula, Represents the mass matrix, Represents the stiffness matrix. and These are the mass and stiffness submatrices corresponding to the internal degrees of freedom. and These are the mass and stiffness submatrices corresponding to the boundary degree-of-freedom submatrices. , , and For cross-coupling submatrices, Indicates internal degrees of freedom. Indicates the boundary degrees of freedom.

[0016] Preferably, the step of constructing fixed boundary principal mode matrices and constraint mode matrices for each substructure based on the mass matrix, stiffness matrix, and the partitioning results of internal and boundary degrees of freedom, and constructing reduced-order basis matrices based on the fixed boundary principal mode matrices and constraint mode matrices, includes: fixing the boundary degrees of freedom of each substructure, solving the characteristic equations of the mass submatrix and stiffness submatrix corresponding to the internal degrees of freedom to obtain multiple internal degree of freedom eigenvectors, sorting the multiple internal degree of freedom eigenvectors in ascending order of natural frequencies, and selecting the first k sorted internal degree of freedom eigenvectors to construct fixed boundary principal mode matrices, where k is a positive integer; setting the inertial force of the substructure to zero, solving the static equilibrium equations based on the block form of the stiffness matrix to obtain the static mapping relationship of internal degrees of freedom with respect to boundary degrees of freedom, and constructing constraint mode matrices based on the static mapping relationship of internal degrees of freedom with respect to boundary degrees of freedom; and concatenating the selected fixed boundary principal mode matrices and constraint mode matrices to form reduced-order basis matrices.

[0017] Preferably, the characteristic equations of the mass submatrix and stiffness submatrix based on the internal degrees of freedom are expressed as:

[0018]

[0019] In the formula, Represents the internal free eigenvectors. This represents the generalized eigenvalue.

[0020] Preferably, the constraint mode matrix is ​​represented as:

[0021]

[0022] In the formula, Represents the constraint mode matrix. This is a cross-coupled submatrix.

[0023] Preferably, the reduced-order basis matrix is ​​expressed as:

[0024]

[0025] In the formula, Describes a reduced-order basis matrix. The identity matrix representing the boundary degrees of freedom dimension. The zero matrix for dimension matching, This represents the principal mode matrix with fixed boundaries.

[0026] Preferably, the reduced-order dynamic equation is expressed as:

[0027]

[0028]

[0029]

[0030] In the formula, Represents the reduced-order mass matrix. Represents the reduced stiffness matrix. For reduced-order load vectors, , For all-order external load vectors, Represents the reduced-order modal coordinates. This represents the second derivative of the reduced-order modal coordinate vector with respect to time. This represents the transpose of the reduced-order basis matrix.

[0031] Preferably, the displacement response vector of the full-order finite element model is expressed as:

[0032]

[0033] In the formula, This represents the displacement response vector of the full-order finite element model.

[0034] The gantry dynamics analysis method provided by this invention is based on a full-order finite element model, accurately constructing the mass and stiffness matrices to fully preserve the overall mechanical properties of the gantry. Simultaneously, considering the modular structure characteristics of the gantry, the full-order finite element model is divided into several substructures, and the internal and boundary degrees of freedom are decomposed. This not only conforms to the logic of actual engineering design but also simplifies complex structures, overcoming the bottleneck of low computational efficiency in traditional full-order analysis and improving the flexibility and operability of the analysis process. Furthermore, by constructing the fixed boundary principal modal matrix and constraint modal moments of each substructure... By using a reduced-order basis matrix, accurate modal reduction of the full-order mass and stiffness matrices is achieved. This significantly reduces the number of degrees of freedom and improves computational efficiency while preserving key structural dynamic information to the maximum extent, achieving a balance between computational efficiency and analytical accuracy, and avoiding the accuracy loss that is common in traditional reduction methods. Furthermore, by solving the reduced-order dynamic equations to obtain the reduced-order modal coordinates, and combining this with the inversion of the reduced-order basis matrix to obtain the full-order displacement response vector, the solution efficiency and result integrity are balanced. This provides accurate and reliable data support for the structural strength verification, vibration control, and optimization design of gantry cranes. Attached Figure Description

[0035] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0036] Figure 1 This is a flowchart of the gantry dynamics analysis method provided in the embodiments of the present invention;

[0037] Figure 2 This is a schematic diagram showing the boundary and internal degrees of freedom of the gantry provided in an embodiment of the present invention. Detailed Implementation

[0038] The terms "first," "second," "third," "fourth," etc. (if present) in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" or "having" and any variations thereof are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0039] For ease of understanding, the specific process of the embodiments of the present invention is described below. Please refer to [link / reference]. Figure 1 In this embodiment of the invention,

[0040] S101. Establish the full-order finite element model of the gantry crane and obtain the mass matrix and stiffness matrix of the full-order finite element model.

[0041] S102. Based on the modular structural characteristics of the gantry, the full-order finite element model is divided into several substructures, and the degrees of freedom of each substructure are decomposed into internal degrees of freedom and boundary degrees of freedom.

[0042] S103. Based on the mass matrix, stiffness matrix, and the division results of internal and boundary degrees of freedom, construct the fixed boundary principal mode matrix and constraint mode matrix of each substructure, and construct the reduced-order basis matrix based on the fixed boundary principal mode matrix and constraint mode matrix.

[0043] S104. Using the reduced-order basis matrix, modal reduction is performed on the mass matrix and stiffness matrix of the full-order finite element model to generate the reduced-order mass matrix and reduced-order stiffness matrix, and the reduced-order dynamic equation is constructed based on the reduced-order mass matrix and reduced-order stiffness matrix.

[0044] S105. Solve the reduced-order dynamic equations to obtain the reduced-order modal coordinates, and combine the reduced-order basis matrix and the reduced-order modal coordinates to invert and restore the displacement response vector of the full-order finite element model.

[0045] In this embodiment, based on the full-order finite element model, the mass matrix and stiffness matrix are accurately constructed, which can completely preserve the overall mechanical properties of the gantry. Simultaneously, considering the modular structure characteristics of the gantry, the full-order finite element model is divided into several substructures, and the internal and boundary degrees of freedom are decomposed. This not only conforms to the logic of actual engineering design but also simplifies complex structures, overcoming the bottleneck of low computational efficiency in traditional full-order analysis and improving the flexibility and operability of the analysis process. Furthermore, by constructing fixed boundary principal mode matrices, constraint mode matrices, and reduced-order basis matrices for each substructure, accurate modal reduction of the full-order mass and stiffness matrices is achieved. This significantly reduces the number of degrees of freedom and improves computational efficiency while maximizing the preservation of key structural dynamic information, achieving a balance between computational efficiency and analytical accuracy, and avoiding the accuracy loss that easily occurs in traditional reduced-order methods. In addition, by solving the reduced-order dynamic equations to obtain the reduced-order modal coordinates and combining them with the reduced-order basis matrix inversion to obtain the full-order displacement response vector, both solution efficiency and result integrity are considered, providing accurate and reliable data support for the structural strength verification, vibration control, and optimization design of the gantry.

[0046] In this embodiment, in step S101, a three-dimensional solid model of the gantry crane is constructed using finite element preprocessing software (such as ANSYS, Abaqus, etc.), i.e., a full-order finite element model. The full-order finite element model covers the core components of the gantry crane, including the crossbeams, left column, right column, worktable, and various connecting parts, accurately reproducing the actual structure's dimensions, assembly relationships, and material distribution. As a typical complex structure, the gantry crane's crossbeams are usually box-shaped or I-shaped cross-section beams, the columns are supporting structures, and the worktable is a moving part. The components are assembled through bolt connections or guide rails and sliders. These structural features must be accurately represented in the model to ensure the accuracy of subsequent dynamic analysis.

[0047] In this embodiment, after completing the geometric modeling, the material properties, mesh properties, and element properties of the full-order finite element model are defined. Specifically, material properties, including density, elastic modulus, and Poisson's ratio, are entered according to the design scheme. These parameters directly affect the values ​​of the mass matrix and stiffness matrix. For mesh properties, a second-order tetrahedral mesh type is preferred to balance computational accuracy and efficiency. Second-order elements can better describe the stress distribution and vibration modes of complex geometries. Element properties are defined, including the solid element type and the corresponding integration scheme. For dynamic analysis, full integration or reduced integration schemes are typically used, and the choice must be made based on the specific problem to avoid shear locking or hourglass modes.

[0048] In this embodiment, boundary conditions and load conditions are set according to the actual working state of the gantry, including applying fixed constraints at the bottom of the column to simulate the connection with the ground and constrain all degrees of freedom; setting motion constraints at the contact surface between the worktable and the crossbeam to simulate the sliding pair, allowing the worktable to translate along the guide rail direction while constraining other directions; and applying load conditions, including cutting load (simulating the cutting force during the machining process), self-weight load (considering the influence of gravity), and inertial load (considering the acceleration effect during the motion process).

[0049] Based on the above settings, the mass matrix is ​​obtained by solving using finite element software. and stiffness matrix The full-order dynamic equations can be expressed as:

[0050]

[0051] in, For the quality matrix, Here is the stiffness matrix. For displacement vectors, For acceleration vectors, For external load vector, This represents the total degrees of freedom of the full-order model.

[0052] In this embodiment, through precise modeling, reasonable mesh division, and realistic constraint and load settings, the full-order model can be ensured to truly reflect the dynamic characteristics of the gantry, avoiding excessive deviation between the simulation results after order reduction and the actual results due to model distortion.

[0053] In this embodiment, in step S102, based on the modular structural characteristics of the gantry, the full-order finite element model is divided into several substructures, and the degrees of freedom of each substructure are decomposed into internal degrees of freedom and boundary degrees of freedom. With the goal of minimizing the number of boundary degrees of freedom, based on the modular structural characteristics of the gantry, the crossbeam, left column, right column and workbench of the gantry are divided into independent substructures, and the core degrees of freedom on the connection surface of each substructure are taken as boundary degrees of freedom, and the dispersed degrees of freedom inside each substructure are taken as internal degrees of freedom.

[0054] In this embodiment, based on the modular assembly characteristics of the gantry frame, the full-scale model is divided into a beam substructure, a left column structure, a right column structure, a workbench structure, etc., and each substructure is denoted as... Each substructure is coupled through the boundary degrees of freedom of the connecting surfaces. The division method corresponds completely to the physical assembly relationship of the gantry frame. The beams and columns transfer loads through the connecting surfaces, and the worktable and beams are connected through guide rails and sliders. The dynamic characteristics of each substructure are relatively independent but also coupled with each other.

[0055] In this embodiment, minimizing the number of boundary degrees of freedom is the goal, aiming to retain as few boundary degrees of freedom as possible during the subsequent order reduction process, while still fully transmitting the coupling forces between substructures. This is because the boundary degrees of freedom are still retained in the form of physical coordinates after order reduction, and their number directly affects the final size of the reduced model.

[0056] The core degrees of freedom include three translational degrees of freedom and three rotational degrees of freedom, totaling six degrees of freedom. The three translational degrees of freedom correspond to displacements in the X, Y, and Z directions at the connection points, while the three rotational degrees of freedom correspond to rotations about the X, Y, and Z axes. These six degrees of freedom are chosen as the boundary degrees of freedom because, in three-dimensional space, any rigid body motion can be decomposed into three translational and three rotational components. Only by simultaneously transmitting these six degrees of freedom can the force and moment transmission between substructures be fully described. If only the translational degrees of freedom are selected and the rotational degrees of freedom are ignored, the bending moment transmission at the connection surfaces cannot be accurately simulated, resulting in the loss of boundary coupling effects.

[0057] Please see Figure 2 The red areas in the diagram represent the connection surfaces between substructures. The degrees of freedom on these connection surfaces are defined as boundary degrees of freedom; the large quantum structure degrees of freedom within the substructure are defined as internal degrees of freedom. Taking the connection between a beam and a column as an example, all nodes on the connection surface are located in the contact area, and the six degrees of freedom of these nodes are defined as boundary degrees of freedom; while the degrees of freedom of nodes far from the connection surface, such as the mid-span of the beam and the bottom of the column, are defined as internal degrees of freedom.

[0058] In this embodiment, the total degrees of freedom of each substructure are divided into internal degrees of freedom. and boundary degrees of freedom ,satisfy and , The total degrees of freedom of the full-order finite element model. This represents the total number of degrees of freedom within a single substructure. This represents the total number of degrees of freedom at the boundary of a single substructure. The number of internal degrees of freedom is much greater than the number of boundary degrees of freedom, which provides a prerequisite for significantly reducing the computational scale through modal condensation.

[0059] In this embodiment, the decomposition of the degrees of freedom of each substructure is achieved by dividing the mass matrix and stiffness matrix of each substructure into blocks according to internal degrees of freedom and boundary degrees of freedom. The block form is expressed as follows:

[0060]

[0061]

[0062] In the formula, Represents the mass matrix, Represents the stiffness matrix. Indicates internal degrees of freedom. Indicates the boundary degrees of freedom. and These are the mass and stiffness submatrices corresponding to the internal degrees of freedom, with dimensions of . , and These are the mass and stiffness submatrices corresponding to the boundary degree-of-freedom submatrices, with dimensions of . , , , and This is a cross-coupling submatrix with dimension . Since both the mass matrix and the stiffness matrix are symmetric matrices, they satisfy... , , , .

[0063] Substituting the partitioned matrix into the full-order dynamic equations, we can decompose them into two sub-equations related to the internal and boundary degrees of freedom, laying the mathematical foundation for solving the subsequent subsystems. The transformed full-order dynamic equations are in the following form:

[0064]

[0065] In the formula, , , These are the acceleration vector, displacement vector, and load vector for the internal degrees of freedom, respectively. , , These are the acceleration vector, displacement vector, and load vector of the boundary degrees of freedom, respectively.

[0066] In this embodiment, the complex structure is decomposed into simple substructures through modular partitioning, and the core boundary degrees of freedom and redundant internal degrees of freedom are separated by degree of freedom classification. This not only ensures that the coupling characteristics between substructures are not lost, but also provides clear optimization targets for subsequent order reduction processing.

[0067] In this embodiment, in step S103, based on the mass matrix, stiffness matrix, and the partitioning results of internal and boundary degrees of freedom, a fixed boundary principal mode matrix and constraint mode matrix for each substructure are constructed. A reduced-order basis matrix is ​​then constructed based on the fixed boundary principal mode matrix and constraint mode matrix. This includes: after fixing the boundary degrees of freedom of each substructure, solving the characteristic equations based on the mass submatrix and stiffness submatrix corresponding to the internal degrees of freedom to obtain multiple internal degree-of-freedom eigenvectors, and selecting the first from the multiple internal degree-of-freedom eigenvectors. First, construct the fixed-boundary principal mode matrix using the first-order mode; set the inertial force of the substructure to zero, and solve the static equilibrium equations based on the block form of the stiffness matrix to obtain the static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom, and construct the constraint mode matrix based on the static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom; then, concatenate the selected fixed-boundary principal mode matrix and the constraint mode matrix to form a reduced-order basis matrix.

[0068] In this embodiment, the characteristic equations based on the mass submatrix and stiffness submatrix corresponding to the internal degrees of freedom are expressed as follows:

[0069]

[0070] In the formula, Represents the eigenvectors of internal degrees of freedom. Represents generalized eigenvalues. and These are the mass submatrices and stiffness submatrices corresponding to the internal degrees of freedom.

[0071] Specifically, based on the principle of superposition of linear systems, the equations transformed from the full-order dynamic equations can be divided into two independent subsystems: homogeneous subsystems and non-homogeneous subsystems.

[0072] Homogeneous subsystems are represented as:

[0073]

[0074] The nonhomogeneous subsystem is represented as:

[0075]

[0076] satisfy as well as .

[0077] In the formula, This represents the acceleration vector of the homogeneous subsystem. Let represent the displacement vector of the homogeneous subsystem. This represents the displacement vector of the internal degrees of freedom of a substructure in a homogeneous subsystem. This represents the boundary degree of freedom displacement vector of a substructure in a homogeneous subsystem. This represents the acceleration vector of a non-homogeneous subsystem. Let represent the displacement vector of a non-homogeneous subsystem. This represents the displacement vector of the internal degrees of freedom of a substructure in a homogeneous subsystem. Let represent the boundary degree of freedom displacement vector of the substructure in a homogeneous subsystem.

[0078] Completely fix the boundary degrees of freedom of the substructure, that is, let At this point, the substructure retains only the vibration characteristics of its internal degrees of freedom, and the full-order dynamic equations degenerate into the characteristic equations of free vibration of the internal degrees of freedom, i.e.

[0079]

[0080] In the formula, This represents the internal degree-of-freedom acceleration vector of a substructure within a homogeneous subsystem. This equation describes only the free vibration law of the substructure itself and is not affected by the coupling of other substructures.

[0081] Using simple harmonic motion, assume that... , and ,therefore Furthermore, the generalized characteristic equation is derived, that is, the characteristic equation based on the mass submatrix and stiffness submatrix corresponding to the internal degrees of freedom is expressed as:

[0082]

[0083] The characteristic equations based on the mass and stiffness submatrices corresponding to the internal degrees of freedom are solved using numerical methods (such as the QR algorithm and the Jacobi algorithm). linearly independent eigenvectors of internal degrees of freedom (i.e., mode shape) and corresponding generalized eigenvalues (i.e., natural frequency), the natural frequency reflects the vibration characteristics of the substructure, and the mode shape reflects the displacement distribution law of the substructure at that frequency.

[0084] Since the lower-order modes of the substructure play a dominant role in the dynamic response, while the higher-order modes have minimal influence, the eigenvectors of multiple internal degrees of freedom are sorted in ascending order of natural frequencies, and the first k sorted eigenvectors are selected to construct the fixed-boundary principal mode matrix. .

[0085] in k is usually Less than 10%. For example, if the degrees of freedom within the substructure... =10000, then select the first 500 modes to construct the fixed boundary principal mode matrix. This ensures that vibration characteristics are not lost while significantly reducing the amount of data.

[0086] The selected k-th order modes are orthogonalized to ensure that the following orthogonal normalization condition is met, namely...

[0087]

[0088] In the formula, It is the transpose of the principal mode matrix with fixed boundaries. It is a k×k identity matrix. This is a diagonal matrix, where each diagonal element is the square of its corresponding natural frequency. Orthogonalization eliminates intermodal correlations, ensuring the accuracy of subsequent modal condensation.

[0089] Furthermore, yes A set of bases, therefore, exists and , making Therefore, the solution for the second subsystem is:

[0090]

[0091] In the formula, This represents the modal coordinate vector.

[0092] In this embodiment, the constraint mode matrix is ​​represented as:

[0093]

[0094] In the formula, Represents the constraint mode matrix. , The stiffness submatrix corresponds to the internal degrees of freedom. This is a cross-coupled submatrix.

[0095] Specifically, the inertial force of the substructure is set to zero, that is... At this point, the substructure is only considered in static equilibrium, and the dynamic equations degenerate into static equilibrium equations: This equation primarily describes the static response of the substructure under the action of boundary degrees of freedom, reflecting the coupling relationship between internal degrees of freedom and boundary degrees of freedom.

[0096] Substituting the partitioned matrix into the static equilibrium equations and expanding, we obtain:

[0097]

[0098] Since inertial forces are neglected and the equations only consider static equilibrium, the relationship between the internal and boundary degrees of freedom can be directly solved using matrix operations. Therefore, the expanded equations are rearranged to eliminate... and We obtain the static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom:

[0099]

[0100] The static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom reflects the influence of changes in the boundary degrees of freedom on the internal degrees of freedom, and is a core matrix characterizing the coupling properties of the substructure. Therefore, based on the static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom, the following is extracted: As a constraint mode matrix, this matrix fully describes the coupling law of the internal degrees of freedom moving with the boundary degrees of freedom, ensuring that the mechanical coupling characteristics between substructures are not lost after order reduction.

[0101] In this embodiment, the reduced-order basis matrix is ​​represented as:

[0102]

[0103] In the formula, Describes a reduced-order basis matrix. The identity matrix representing the boundary degrees of freedom dimension. for The identity matrix. Used to preserve complete information about the boundary degrees of freedom. A zero matrix with dimension matching ensures decoupling between internal and boundary degrees of freedom. Represents the constraint mode matrix. This represents the principal mode matrix with fixed boundaries.

[0104] The dimension of the reduced basis matrix A is ,because The reduced-order basis matrix has significantly fewer columns than the full-order model's degrees of freedom, providing a foundation for subsequent condensation.

[0105] In this embodiment, in step S104, the reduced-order dynamic equation is expressed as:

[0106]

[0107]

[0108]

[0109] In the formula, Represents the reduced-order mass matrix. Represents the reduced stiffness matrix. Represents the mass matrix, Represents the stiffness matrix. For reduced-order load vectors, , For all-order external load vectors, Represents the reduced-order modal coordinates. This represents the second derivative of the reduced-order modal coordinate vector with respect to time. Describes a reduced-order basis matrix. This represents the transpose of the reduced-order basis matrix.

[0110] Specifically, using reduced-order basis matrices Establish the mapping relationship between the displacement response vector of the full-order finite element model and the coordinates of the reduced-order modes. The displacement response vector of the full-order finite element model is expressed as:

[0111]

[0112] In the formula, This represents the displacement response vector of the full-order finite element model. Describes a reduced-order basis matrix. Represents the reduced-order modal coordinates. .

[0113] Next, the mapping relationship between the displacement response vector and the reduced-order modal coordinates of the full-order finite element model is substituted into the full-order dynamic equation, and both sides are multiplied on the left. By using the properties of matrix operations to eliminate redundant degrees of freedom, the initial reduced-order dynamic equations are obtained.

[0114]

[0115] In the formula, the reduced-order mass matrix Reduced stiffness matrix and reduced-order load vector They are respectively:

[0116]

[0117]

[0118]

[0119] Reduce the quality matrix Reduced stiffness matrix and reduced-order load vector Substituting the initial reduced-order dynamic equations, we obtain the final reduced-order dynamic equations, which are expressed as follows:

[0120]

[0121] The dimensions of both the reduced-order mass matrix and the reduced-order stiffness matrix are 1. Compared to a full-order matrix, the dimension is significantly reduced, resulting in a significant improvement in subsequent solution efficiency.

[0122] In this embodiment, in step S105, the reduced-order modal coordinates are obtained by solving the reduced-order dynamic equations, and the displacement response of the full-order model is inverted and restored using the coordinate mapping relationship. Finally, the simulation results that can reflect the actual dynamic characteristics of the gantry are output.

[0123] Specifically, in the process of solving the reduced-order equations, for linear steady-state conditions (such as vibration response under constant load), the modal superposition method is used to solve the problem, and the orthogonality of the reduced-order modes is used to simplify the calculation; for nonlinear conditions (such as high-speed impact and variable load conditions), the step-by-step integration method (such as the Newmark-β method and Wilson-θ method) can be used to solve the problem to ensure the stability and accuracy of the solution.

[0124] In this embodiment, taking the Newmark-β method as an example, the integration step size is set. The period is set to 1 / 20 to 1 / 10 of the highest natural frequency to ensure the capture of high-frequency vibration components; the convergence criterion is set as displacement residual ≤ 10. -6 or force residual ≤10 -3 To avoid numerical divergence, for example, if the highest natural frequency of the reduced-order model is 100Hz, corresponding to a period of 0.01s, then the integration step size is set to 0.001s.

[0125] In this embodiment, the reduced-order modal coordinates and their second derivatives are obtained through numerical solution. The reduced-order modal coordinates include all the dynamic response information of the reduced-order model, and the dimension is only a fraction of that of the full-order model. The solution time is greatly reduced from several hours to tens of hours to several minutes to tens of minutes.

[0126] In this embodiment, the reduced-order modal coordinates are substituted into the reduced-order basis matrix, and the displacement response vector of the full-order finite element model is obtained by inversion calculation. This vector includes the displacement information of all nodes of the full-order model, and the accuracy is consistent with the result obtained by directly solving the full-order equation.

[0127] Based on the displacement response vector of the full-order finite element model obtained by inversion calculation, the key dynamic characteristics of the gantry, such as vibration response, natural frequencies, mode shapes, and stress distribution, are further calculated. For example, the displacement response is converted into a frequency domain response through Fourier transform, and the natural frequencies and mode shapes of each order are extracted; the stress distribution of the core components is obtained through the stress calculation module to determine whether there are stress concentrations or regions exceeding the allowable stress of the material.

[0128] Optionally, the simulation results of the reduced-order model can be compared with those of the full-order model to verify the accuracy of the reduction. Key verification metrics include: low-order modal frequency error (≤1%), key node displacement response error (≤5%), and core component stress error (≤8%). If the error exceeds the allowable range, the steps described above must be repeated to adjust the number of modes selected (increasing the k value) or optimize the logic for constructing the reduced-order basis matrix until the accuracy requirements are met.

[0129] If there are areas with large local errors (such as near the connection surface), they can be corrected by locally refining the modes or adjusting the weight coefficients of the constraint mode matrix; if the overall error is large, it is necessary to check the modeling accuracy of the full-order model or the rationality of the substructure division, and repeat the preprocessing process.

[0130] In this embodiment, by efficiently solving the reduced-order equations and accurately restoring the full-order response, the simulation efficiency is greatly improved while ensuring the accuracy of the results. This solves the problems of long simulation time and low iteration efficiency of traditional full-order models, and provides a fast and reliable analysis method for gantry structure optimization.

[0131] The above description is merely a preferred embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A method for dynamic analysis of a gantry crane, characterized in that, include: Establish a full-order finite element model of the gantry crane and obtain the mass matrix and stiffness matrix of the full-order finite element model; Based on the modular structural characteristics of the gantry, the full-order finite element model is divided into several substructures, and the degrees of freedom of each substructure are decomposed into internal degrees of freedom and boundary degrees of freedom. Based on the mass matrix, the stiffness matrix, and the division results of internal and boundary degrees of freedom, the fixed boundary principal mode matrix and constraint mode matrix of each substructure are constructed, and the reduced-order basis matrix is ​​constructed based on the fixed boundary principal mode matrix and constraint mode matrix. Using the reduced-order basis matrix, modal condensation is performed on the mass matrix and stiffness matrix of the full-order finite element model to generate a reduced-order mass matrix and a reduced-order stiffness matrix, and a reduced-order dynamic equation is constructed based on the reduced-order mass matrix and the reduced-order stiffness matrix. Solve the reduced-order dynamic equations to obtain the reduced-order modal coordinates, and combine the reduced-order basis matrix and the reduced-order modal coordinates to invert and restore the displacement response vector of the full-order finite element model.

2. The gantry dynamics analysis method as described in claim 1, characterized in that, Based on the modular structural features of the gantry frame, the full-order finite element model is divided into several substructures, and the degrees of freedom of each substructure are decomposed into internal degrees of freedom and boundary degrees of freedom. With the goal of minimizing the number of boundary degrees of freedom, based on the modular structural features of the gantry frame, the crossbeam, left column, right column and workbench of the gantry frame are divided into independent substructures, and the core degrees of freedom on the connection surface of each substructure are taken as boundary degrees of freedom, and the dispersed degrees of freedom inside each substructure are taken as internal degrees of freedom.

3. The gantry dynamics analysis method as described in claim 2, characterized in that, The core degrees of freedom include translational degrees of freedom in three directions and rotational degrees of freedom in three directions.

4. The gantry dynamics analysis method as described in claim 2, characterized in that, The decomposition of the degrees of freedom of each substructure is achieved by dividing the mass matrix and stiffness matrix of each substructure into blocks according to internal degrees of freedom and boundary degrees of freedom. The block form is expressed as follows: In the formula, Represents the mass matrix, Represents the stiffness matrix. and These are the mass and stiffness submatrices corresponding to the internal degrees of freedom. and These are the mass and stiffness submatrices corresponding to the boundary degree-of-freedom submatrices. , , and For cross-coupling submatrices, Indicates internal degrees of freedom. Indicates the boundary degrees of freedom.

5. The gantry dynamics analysis method as described in claim 4, characterized in that, Based on the mass matrix, the stiffness matrix, and the division results of internal and boundary degrees of freedom, the fixed boundary principal mode matrix and constraint mode matrix of each substructure are constructed, and a reduced-order basis matrix is ​​constructed based on the fixed boundary principal mode matrix and constraint mode matrix, including: After fixing the boundary degrees of freedom of each substructure, solve the characteristic equations based on the mass submatrix and stiffness submatrix corresponding to the internal degrees of freedom to obtain multiple internal degree of freedom eigenvectors. Sort the multiple internal degree of freedom eigenvectors in ascending order of natural frequency, and select the first k sorted internal degree of freedom eigenvectors to construct the fixed boundary principal mode matrix, where k is a positive integer. By setting the inertial force of the substructure to zero, the static equilibrium equations are solved based on the block form of the stiffness matrix to obtain the static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom, and the constraint mode matrix is ​​constructed based on the static mapping relationship between the internal degrees of freedom and the boundary degrees of freedom. The selected fixed boundary principal mode matrix and the constraint mode matrix are concatenated to form a reduced-order basis matrix.

6. The gantry dynamics analysis method as described in claim 5, characterized in that, The characteristic equations for the mass submatrix and stiffness submatrix based on the internal degrees of freedom are expressed as follows: In the formula, Represents the internal free eigenvectors. This represents the generalized eigenvalue.

7. The gantry dynamics analysis method as described in claim 6, characterized in that, The constraint mode matrix is ​​represented as follows: In the formula, Represents the constraint mode matrix. This is a cross-coupled submatrix.

8. The gantry dynamics analysis method as described in claim 7, characterized in that, The reduced-order basis matrix is ​​represented as follows: In the formula, Describes a reduced-order basis matrix. The identity matrix representing the boundary degrees of freedom dimension. The zero matrix for dimension matching, This represents the principal mode matrix with fixed boundaries.

9. The gantry dynamics analysis method as described in claim 8, characterized in that, The reduced-order dynamic equation is expressed as: In the formula, Represents the reduced-order mass matrix. Represents the reduced-order stiffness matrix. For reduced-order load vectors, , For all-order external load vectors, Represents the reduced-order modal coordinates. This represents the second derivative of the reduced-order modal coordinate vector with respect to time. This represents the transpose of the reduced-order basis matrix.

10. The gantry dynamics analysis method as described in claim 9, characterized in that, The displacement response vector of the full-order finite element model is expressed as: In the formula, This represents the displacement response vector of the full-order finite element model.

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