Furthermore, a helical edge curve with constant helix angle on the cylinder and the ball-head, the exact solution of the helical groove cross-section equation, and the mathematical model of the grinding wheel cross-section, are also presented in this study.
In this procedure, a recently developed 2.5D finite element method, which can consider wheel rotation but only requires a 2D mesh over a cross section containing the wheel axis, is used to calculate the vibration response of the wheel.
According to the 2.5D FEM and 2.5D BEM, only the cross section of the wheel has to be meshed.
Equation (9) shows that the radial vibration of the cross section boundary of the wheel is harmonic at the same frequency as the wheel rail force, and its vibration displacement amplitude is periodic function of (alpha).
It can be shown that, observed from OXYZ, the radial displacement vector of a cross section boundary of the wheel is given by (see Fig. 2.
The nodal normal vibration velocity of the cross section boundary of the wheel can be obtained by using the 2.5D FEM, as shown in Sect. 2.
Now by submitting the nodal normal vibration velocity vector, (overline{varvec{upsilon}}_{nm}) of the cross section boundary of the wheel into Eq. (24), the nodal sound pressure vector (overline{varvec{p}}_{m}) can be obtained.
Integrals in Eq. (25) are evaluated using the boundary element method combined with the nodal normal vibration velocity vector (overline{varvec{upsilon}}_{nm}) and nodal pressure vector (overline{varvec{p}}_{m}) of the cross section boundary of the wheel.
The normal of the cross section boundary is directed into the wheel.
Then, the vibration response of the wheel is taken as acoustic boundary condition and the 2.5D acoustic boundary element method, which only requires a 1D mesh over the boundary of the above cross section, is utilised to calculate the sound radiation of the wheel.
