When studying the traveling wave vibration of bevel gear, in order to calculate the resonant speed of bevel gear, we must know the natural vibration frequency and vibration mode of bevel gear. The natural vibration frequency of bevel gear is divided into static frequency and dynamic frequency. The natural frequency of the bevel gear seen from the static coordinate is the static frequency (f), and the natural vibration frequency of the bevel gear seen from the dynamic coordinate when the bevel gear rotates is the dynamic frequency (fd). The relationship between the two is:
Where: B – dynamic frequency coefficient, which is related to the shape, size, vibration mode and vibration order of bevel gear.
Because the static frequency of bevel gear is generally very high, and the difference between dynamic frequency and static frequency is very small, the static frequency of bevel gear is usually used to replace the dynamic frequency in engineering, which will be verified in calculation and analysis.
At present, there are two methods to determine the natural frequency of bevel gear: calculation method and test method;
Where:
da, df – diameter of addendum circle and dedendum circle of bevel gear;
E – longitudinal elastic modulus of bevel gear material;
ρ M – density of bevel gear material;
μ— Poisson’s ratio of bevel gear material;
R2 – inner diameter of bevel gear rim;
U – transmission ratio;
For bevel gears with complex shape, the finite element method is usually used to accurately calculate their natural vibration frequency. The solution method is to discretize the bevel gear structure according to the finite element method, add all external loads varying with time to the corresponding nodes, and establish the finite element dynamic equation:
Where:
[M] – quality matrix;
[C] – damping matrix;
[K] – stiffness matrix;
{q” (t)}, {q’ (t)}, {q (t)} – acceleration, velocity and displacement matrices respectively;
{f (t)}, – load matrix.