# Study on Vibration Reduction Performance of Gear Pairs Made by a High-Strength Fe-Mn Damping Alloy
## 1 Introduction
[16] proposed a micromechanical formula based on two macroscopic scalar (integral martensitic volume fraction) and tensor (average strain) internal variables to simulate the constitutive equation of shape memory alloy, which was calculated by embedding the UMAT form into ABAQUS, and the obtained results were compared with the experimental results. [PERSON] [17] proposed a three-dimensional finite element material law for an elastoplastic analysis of unidirectional metal matrix composites, implemented in ABAQUS by the user subroutine UMAT. [PERSON] [18; 19; 20] built a nonlinear damping model, developed the ABAQUS user subroutine, and obtained the finite element method using the modified Marshall principle to simulate the nonlinear high-damping characteristics of martensite shape memory alloy. The hysteresis curve of the shape memory alloy was compared with the experimental hysteresis curve to verify the correctness of the founded nonlinear damping model. [PERSON] et al. [21] established the isotropic nonlinear viscoelastic constitutive model of solid propellant considering damage evolution, and the simulation calculation was realized using the user subroutine UMAT of ABAQUS. The stress tests of uniaxial and biaxial specimens were carried out under different loading conditions, different strain rates and temperatures. The experimental data were in good agreement with the simulation results. Based on the thermodynamic model of martensitic transformation, [PERSON] et al. [22] embedded the constitutive model describing the thermodynamic behavior of SMA in the ABAQUS material library, which could automatically update the relevant operators of the model in the calculation process. [PERSON] et al. [23] proposed a viscoelastic nonlinear damping model to simulate the mechanical behavior of martensitic shape memory alloy according to the modified Masin rule [24], developed the ABAQUS subroutine, and obtained the hysteretic curve of a Cu Al Mn metal plate under a traction-compression cyclic load, which was consistent with the actual test results. [PERSON] et al. [25] used the implicit user-defined material subroutine of ABAQUS to calculate the mechanical parameters related to strain rate and Poisson's ratio, which was applied to the study of mixed-modefractures of symmetrical double cantilever beams. Based on the creep experiments of polymer composites under different stresses, [PERSON] et al. [26] established a three-dimensional nonlinear viscoelastic constitutive model combined with the Schapery equation and used the standard finite element program UMAT to predict the creep deformation of composite guide plates after 15 years of storage. [PERSON] et al. [27] established a numerical integration algorithm of the coupled constitutive model based on the continuous damage mechanics theory in the finite element software ABAQUS employing the user subroutine UMAT. By comparing the numerical prediction results of the model and the experimental data under different loading conditions, it was confirmed that the model has the ability to simulate the nonlinear viscoelasticity and damage behavior of viscoelastic polymers. To further clarify the main damping source of Fe-17.5 Mn-0.022C alloy under low strain amplitude, the effects of different thermomechanical treatments on defect structure and damping in cold drawing were studied to improve the damping capacity of high-strength Fe-Mn based alloy [28]. [PERSON] et al. [29] investigated the effects of quenching temperature and ageing on variations in vacancy concentration and segregation of solute atoms and their relationship with damping capacity in a Fe-Mn alloy. Some of the damping capacity can be remarkably improved by lowering the vacancy concentration, but it can also be deteriorated by the segregation of carbon atoms. A higher damping capacity can be obtained by furnace cooling or quenching and then ageing in an Fe-Mn alloy with lower carbon content or the addition of Co or Nb.