The elastic potential energy u formula of spur gear teeth is written as:

Where:

ux – bending potential energy;

un – compression potential energy;

us – Shear potential energy.

The elastic potential energy is a function of the geometric parameters of the involute tooth profile, as shown in the figure:

The expressions of bending potential energy UX, compression potential energy UN and shear potential energy us are as follows:

Where:

αc – load angle at contact point C (°);

E – bending elastic modulus of spur gear material (MPA);

G – shear elastic energy of spur gear material (MPA);

C – Shear potential energy correction factor, 1.2 is selected in this paper;

YP – lower integral limit, at the tooth root circle;

YC – upper limit of integral.

The load angle is a function of the position of the involute tooth profile, and its expression is:

Where ξ C is the tooth profile parameter of the contact point, which is affected by the meshing position of the instantaneous spur gear. The expression is shown in the formula:

γb is the center angle formed by the tooth profile base circle radius of a single tooth, as shown in the formula:

Where:

Z1 – number of driving gear teeth;

rc – tooth profile radius of contact point (m);

rb – radius of base circle (m);

χ – Displacement coefficient;

αN – normal pressure angle (°).