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 (°).