# Strength check of spiral bevel gear planetary reducer

The contact analysis of meshing gears of planetary gear train is carried out by using transient dynamic model. To simplify the model and improve the efficiency. The main process is as follows:

1) Contact settings. Gear 3 tooth surface is contact surface, gear 4 tooth surface is target surface. The meshing gear contact is set to have friction, and the friction coefficient is 0.05.

2) Meshing. Mesh generation should be as fine as possible to ensure the simulation accuracy.

3) Set the load step. Apply an angular velocity load to gear 3, the value of which is:

ω3= ω2= ω1·z1/z2= 6 000×21/45= 2 800 r / min;

The torque applied to gear 4 under rated condition is as follows:

T4 = T3 · Z4 / Z3 = 11.9 × 32 / 24 = 15.9 nm, and the torque under emergency stop condition is 47.7 nm.

As shown in the figure, the maximum von Mises equivalent stress of gear 3 under rated condition is 313.56 MPa, and that of gear 3 tooth root under emergency stop condition is 675.18 MPa. The yield limit of gear material is 850 MPa, which meets the strength requirements in both working conditions.

The traditional theoretical formula based on material mechanics is different from von Mises equivalent stress in finite element method based on elastic mechanics. Their basic theories are different, and their simplifications and assumptions are also different. The traditional method treats the tooth as a cantilever beam and simplifies the model; the von Mises equivalent stress of the finite element method is calculated by three principal stresses according to the fourth strength theory. The fourth strength theory is also called distortion energy density theory. According to the theory, plastic yield will occur as long as the maximum distortion energy density reaches the limit distortion energy density of material under uniaxial tension no matter what stress state the material is in. According to this theory, the ultimate strength conditions are as follows:

The stress ratio calculated by finite element method and traditional theory under rated working condition is calculated

Under the condition of emergency stop, the ratio is as follows:

From the calculation results, the ratio of the two is too large, and the difference between the ratio results under different working conditions is too large.

According to the gear design and experimental verification of the reducer, the results of the finite element method are closer to the analytical solution and test results with the support of reasonable model and algorithm. 