Gear Dynamic Analysis Using Reduced 3D Models
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Solution Overview
Problem
Current methods for obtaining dynamic parameters of meshing gears either rely on quick but imprecise simple models or computationally intensive 3D finite element models, requiring significant time and resources.
Innovation Solution
A computer-implemented method using reduced order 3D models with iterative integration algorithms, such as the Newmark algorithm, to determine dynamic parameters like dynamic transmission error and stresses, significantly reducing computational time while maintaining precision by leveraging angular step solutions and component mode synthesis transformations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If simple models are used for dynamic parameter analysis, then computational time is reduced, but precision of results deteriorates
Solution Approach 1:
The gear model is segmented into a reduced set of critical degrees of freedom (DOFs) that capture the essential dynamic behavior. By identifying and retaining only the most influential DOFs related to tooth contact and gear deformation, the model achieves computational efficiency while preserving accuracy for critical dynamic parameters.
Solution Approach 2:
The full 3D gear model is replicated and rotated by 360°/N to generate multiple rotated full 3D models. This copying approach allows the reduced order model to be applied iteratively across different angular positions, maintaining precision through comprehensive coverage while avoiding the computational burden of analyzing the complete full 3D model at every position.
2Measurement precision
If full 3D finite element models are used, then precision of dynamic parameters is improved, but computational time and resources increase significantly
Solution Approach 1:
The method extracts only the essential dynamic characteristics from the full 3D model by identifying critical DOFs and creating a reduced order model. This extraction process removes unnecessary computational complexity while retaining the key information needed for accurate dynamic parameter analysis, achieving precision without the full computational burden.
Solution Approach 2:
The model undergoes parameter changes by transforming from a full 3D representation to a reduced order representation with fewer DOFs. This parameter transformation, based on component mode synthesis, changes the dimensional parameters of the model while preserving the essential dynamic behavior, enabling faster computation with maintained precision.
3Measurement precision
If full 3D models are used for large gears, then analysis precision is maintained, but computational resources and time required increase to weeks or months
Solution Approach 1:
For large gears, the model is segmented to identify and retain only the critical DOFs that dominate the dynamic response. This segmentation approach is particularly effective for large gears where the full 3D model would contain an extremely large number of nodes, enabling the analysis to focus on the most influential regions and parameters.
Solution Approach 2:
The method utilizes the periodic nature of gear meshing by rotating the reduced order model through discrete angular positions (360°/N) and applying periodic boundary conditions. This periodic approach captures the cyclic dynamic behavior of large gears efficiently, avoiding the need to model the entire continuous rotation with full 3D detail.
Data Source
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AI summary
A computer implemented method comprising the steps of receiving a gear tooth load condition comprising a set of tooth loads and representing a meshing load over a mesh cycle; defining or receiving a reduced order 3D model of a gear comprising at least one tooth base, a tooth flank and tooth tip and including information of a full 3D model such as a total number N of gear teeth; determining z sets of reduced order model mass and stiffness parameters, wherein the z-th set refers to a z-th angular position of the model about an axis defining at least a portion of said tooth flank as an involute flank; and wherein the z sets comprise a Z1 set representing a first model angular position identifying the beginning of the mesh cycle and a Z set representing a second model angular position identifying the end of the mesh cycle; determining reduced order displacements, including tooth contact loss, and/or speeds and/or accelerations of model nodes by executing an iterative integration algorithm applied to said mass and stiffness parameters; wherein an expansion matrix is applied to said reduced order displacements and/or speeds and/or accelerations to obtain all DOF of said full 3D model; said full 3D model is rotated by 360°/N to obtain a rotated full 3D model and a reduction matrix is applied to the rotated full 3D model nodes to obtain a rotated reduced 3D model when a time iteration of the integration algorithm corresponds to said Z1 set at the beginning of the mesh cycle.