# A Novel Ring Spring Vibration Isolator for Metro Superstructure
## 2 Theoretical Derivation and Numerical Simulation of Ring Spring  
### Analytical Analysis  
#### 2.1.1 Stress  
When the ring spring receives the axial force \(F\), the inner ring generates compressive stress whereas the outer ring generates tensile stress. The contact surfaces of the inner rings and the outer rings both have compressive stress, thus producing tensile stress on the contact surface in the circumferential direction [15]. For the inner ring, the tensile stress offsets some of the inner ring's compressive stress, so the inner ring's maximum compressive stress is not on the contact surface, but on the cross-section. It will be more favorable for stress verification to not consider the compressive stress on the inner ring's contact surface, but directly calculate the compressive stress caused by the circumferential component force [16], while in terms of the outer ring, the maximum tensile stress is generated on the inner surface of the outer ring after the tensile stress is combined with the compressive stress in the outer ring section. As for the strength ability, the maximum stress of both inner rings and outer rings should be smaller than the allowable stress. The maximum compressive stress of the inner ring [8]:  
\[\sigma_{1}=\frac{F}{\pi A_{1}\tan(\beta+\rho)}\leq\sigma_{1p} \tag{1}\]  
The maximum compressive stress of the outer ring [8]:  
\[\sigma_{2}=\frac{F}{\pi A_{2}\tan(\beta+\rho)}\bigg{[}1+\frac{2A_{2}}{\
u D(h- \delta_{0})(1-\mu\tan\beta)}\bigg{]}\leq\sigma_{2p} \tag{2}\]  
where \(D\) is the diameter of the spring; \(h\) is the ring's height; \(\sigma_{1p}\) is the inner ring's allowable stress; \(\sigma_{2p}\) represents the outer ring's allowable stress; \(v\) is the Poisson ratio which is 0.3; \(\delta_{0}\) refers to the distance between two inner or outer rings under no load; \(A_{1}\) and \(A_{2}\) refer to the area of the inner ring and the outer ring's cross-section, respectively; \(\beta\) is the cone angle between the inner rings and the outer rings; \(\rho\) is the friction angle of the conical surface; and \(\mu\) is the friction coefficient. The calculation formula (Equation (2)) is based on the assumption that the wall thickness of the ring spring is smaller, which is 1/20-1/30 of the diameter, and the ring spring is simplified as a thin-walled member [8]. Due to the lateral deformation, the tensile stress \(\sigma_{2}^{\prime}=\frac{1}{v}\sigma_{c}\) in the circumferential direction of the inner surface is caused by the compressive stress \(\sigma_{c}\) of the outer ring on the conical contact surface. Equation (2) is only applicable to a thin-walled ring spring. The standard aberration will be larger if Equation (2) is used to compute the maximum compressive stress of a thick-walled spring. A spring with wall thickness that is 1/10-1/20 of the diameter is considered as a thick-walled spring, which produces shortening deformation in the direction perpendicular to the load. Its material also produces elongation deformation along the load direction. Thus, the compressive stress caused on the contact surface of the cone results in inner surface tensile stress \(\sigma_{2}^{\prime\prime}=v\sigma_{c}\) in the circumferential direction. Currently, ring springs are thin-walled members, and they are mostly used in mechanical fields. If the ring spring is considered for use in construction fields, the wall thickness should be increased to 1/10-1/20 of the diameter. This thickness is more appropriate to be used in construction. As the wall thickness is distinct, we derive a new formula. After deriving a new formula, we revised and supplemented the formula of the ring spring's maximum stress (Equation (2)), and proposed the new formula that calculates the maximum tensile stress of the outer ring of the thick-walled ring spring as Equation (3). In the third section of the numerical simulation, the finite element software ABAQUS is also used to explore and verify the proposed correction formula (Equation (3)) in detail.