# Dimensionless of dynamic equation of bevel gear

The dynamic equation of bevel gear is a nonlinear second-order differential equation with variable parameters. If it is solved directly according to the formula, because the international system of units is used for all physical quantities in the equation, and the order of magnitude of each physical quantity varies greatly. For example, the order of magnitude of the comprehensive meshing stiffness of the bevel gear is between, the torsional stiffness of the transmission shaft is between, the order of magnitude of the meshing damping of the bevel gear is between, and the order of magnitude of the torsional damping of the transmission shaft is between, The order of magnitude of the response displacement of the system is, which makes the order of magnitude of each variable coefficient in the dynamic equation very different, which will bring great difficulties to the solution of the equation. If the numerical analysis method is used to solve the equation, because the order of magnitude of the parameters in the equation is very different, it is difficult to select the appropriate step size, and the calculation error is difficult to control, resulting in the failure of the solution. Therefore, in the actual engineering calculation and analysis, this kind of equation should be normalized and dimensionless before solving, and the physical equation should be transformed into a specific mathematical equation.

The normalized dimensionless treatment method for the dynamic equation of central bevel gear is as follows. First, let:

Where:

Rin – radius of input transmission shaft;

Rout – radius of central transmission rod;

RZ – radius of driving bevel gear base circle;

RC – radius of driven bevel gear base circle;

θ In – input angular displacement of transmission shaft;

θ Out – output angular displacement of central transmission rod;

θ Z – angular displacement of driving bevel gear;

θ C – angular displacement of driven bevel gear;

Substitute the formula into the formula, and subtract formula 2 from Formula 1, formula 2 from Formula 3, and formula 3 from formula 4 to obtain:

Define the time t when the quantity is normalized, and define B as the characteristic size. Through T and B, we can get:

Bring the formula into:

Of which:

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