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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, its complex geometry is ignored, and it is treated as a rigid body with its weight preserved. The inner diameter of this rigid ring corresponds to the diameter of the large ring gear and the connection point with the spring plate, while the outer boundary aligns with the pitch circle of the gear. Two primary approaches are used to analyze the spring plate: the analytical method and the finite element method, both of which are discussed in detail below. Unless otherwise noted, all force values are expressed in kilograms (kg), and all lengths are in millimeters (mm).
In the analytical approach, the relative motion between the ring gear and the kiln body is described using three parameters: a linear displacement at a certain angle and a rigid-body angular displacement. The deformation of the spring plate is directly influenced by the movement of the ring gear, which allows us to establish a relationship between the internal forces in the spring plate and the displacements of the ring gear. These internal forces arise from the meshing forces between the gears and the gravitational load. As a two-dimensional problem, we can derive three equilibrium equations involving the three unknown displacements of the ring gear. Solving these equations mathematically will ultimately determine the behavior of the system.
Once the spring plate is simplified into an equivalent beam with uniform tensile and bending stiffness, it is divided into beam elements. There are a total of several beam units involved. At the connection point between the spring plate and the large ring gear, there are six degrees of freedom. When modeling the large ring gear, the unit nodes are coordinated with the beam elements, and the large ring gear is represented using shell elements. However, since the finite element 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, the displacements along the Z-axis and rotations about the X and Y axes are zero. The connection between the spring plate and the large ring gear is a hinge, so the translational displacements of the hinge points on each spring plate are constrained to match those of the corresponding main node on the large ring gear. However, the angular displacements around the Z-axis remain independent for each pair of points.
This structure is subjected to two types of external loads. The idea of reducing the thickness of the spring plate to buffer impact loads is not practical. The radial stiffness of the spring plate assembly is very high, and even with reduced thickness, it cannot effectively absorb the additional forces caused by the bending of the kiln body, especially the forces generated by top teeth contact.
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