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To simplify the analysis, the original spring plate is modeled as an equivalent beam that maintains the same tensile and bending stiffness as the actual structure. For the large ring gear, the complex internal geometry is ignored, and it is treated as a rigid body with its mass preserved. The inner diameter of the ring gear corresponds to the connection point with the spring plate, while the outer diameter matches the pitch circle. Two primary methods are used to analyze the spring plate: analytical methods and finite element analysis (FEA). These approaches are discussed in detail below. Unless otherwise stated, all forces are expressed in kilograms, and all lengths are given in millimeters.
In the analytical approach, the relative motion between the ring gear and the kiln body is described using three parameters: linear displacement at a specific angle and a rigid-body angular displacement. The deformation of the spring plate is directly influenced by the movement of the ring gear, which establishes a relationship between the internal forces in the spring plate and the displacements of the ring gear. These internal forces result from the meshing force between the ring gear and the kiln, as well as gravitational effects. As a two-dimensional problem, we can derive three equilibrium equations based on the three unknown displacements of the ring gear. The next step involves solving these equations mathematically.
After simplifying the spring plate into an equivalent beam with uniform tensile and bending stiffness, it is divided into beam elements. There are a total of several beam units used in the model. At the connection point between the spring plate and the large ring gear, there are six degrees of freedom. To ensure compatibility between the ring gear's model and the beam elements, the large ring gear is represented using shell elements. However, since the FEA software being used does not support shell elements, we replace them with a combination of membrane and plate elements, totaling 25 such units. Since the problem is planar, displacements along the Z-axis and rotations about the X and Y axes are considered zero. The connection between the spring plate and the large ring gear is a hinge, meaning that the translational displacements at the hinge points are equal to those of the corresponding main nodes on the ring gear. However, angular displacements around the Z-axis remain independent for each pair of points.
This system is subjected to two types of external loads. The idea of reducing the thickness of the spring plate to absorb impact loads is not practical. Additionally, the radial stiffness of the spring plate is high, so even with reduced thickness, it cannot effectively absorb the additional forces caused by the bending of the kiln body.
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