# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## Abbreviations \begin{tabular}{l l l l} C & generating surface & R & base cone distance \\ O & origin of the coordinate system & \(A^{\prime}A\) & tangent line \\ \(OA^{\prime}\) & radius of the spherical radius & \(r_{f}\) & Gear tooth equation \\ \(n\) & normal vector of the tooth surface & \(\varphi\) & angle rolled by the occurring surface \\ \(\delta_{b}\) & base cone angle & \(\psi\) & the angle between OK and the instantaneous axis \\ \(\mathbf{M_{i}}\) & Transformation matrix of the gear projection & \(\mathbf{M_{\sum}}\) & Gear Assembly Variation Matrix \\  & to the zx plane & \(\beta\) & the angle of rotation of the gear projection onto the zx plane \\ \(\mathbf{M_{\theta i}}\) & rotation angle matrices for the driving & \(r_{i}\) & contact points vector \\  & and driven gears & & \\ \(n_{i}\) & normal vectors of contact points vector & \(Z\) & Teeth number \\ \(m\) & Module & \(\alpha\) & Pressure angle \\ \(B\) & Face width & \(ha\) & Addendum coefficient \\ \(c\) & Tip clearance coefficient & \(K_{e}\) & total meshing stiffness of a single gear pair \\ \(k_{b}\) & bending stiffness & \(k_{s}\) & shear stiffness \\ \(k_{b}\) & Hertz contact stiffness & \(k_{a}\) & axial compression stiffness \\ \(\delta k_{b}\) & bending stiffness of slice & \(\delta k_{s}\) & shear stiffness of slice \\ \(\delta k_{a}\) & axial compression stiffness of slice & \(j\) & Number of slices \\ \(\delta U_{b}\) & bending energy of slice & \(\delta U_{s}\) & shear energy of slice \\ \(\delta U_{a}\) & axial compressive energy of slice & \(h\) & the distances from the meshing point to the gear teeth's line of symmetry \\ \(t\) & the distances from the meshing point & \(F\) & interaction force \\  & to the base circle & & \\ \(F_{a}\) & radial force & \(F_{b}\) & tangential force \\ \(E\) & elastic modulus & \(G\) & shear modulus \\ \(I_{p}\) & polar moment of inertia of the section & \(\rho\) & distance from the point on the cross-section to the center \\ \(K\) & time-varying meshing stiffness of straight bevel gears & \(l\) & the number of meshing pairs. \\ \(E_{x}\) & Modification of Lead crown relief & \(\delta\) & deformation \\ \(K_{p}\) & stiffness matrix of the pinion & \(\mathbf{s^{m}}\) & total deformation matrix \\ \(\delta^{p}\) & deformation matrix of the pinion & \(\mathbf{E_{x}}\) & total modification matrix \\ \(\mathbf{F_{d}}\) & total meshing force matrix & \(F_{m}\) & resultant force of the sliced meshing forces \\ \(F_{e}\) & meshing force error & \(\xi\) & cycle coefficient \\ \(E_{k}\) & Modification of tooth profile modification & \(Le\) & The distance from the modification position to the tooth tip. \\ \(xc\) & The distance from the start of the modification & \(\sigma\) & elastic deformations of tooth pairs within the double-tooth \\  & to the tip of the tooth. & & contact region under loading conditions. \\ \(K_{t}\) & meshing stiffness of the tooth profile modification & \(\lambda\) & directional rotation radius \\ \(L\) & Shaft end length & D & external diameter \\ \(I_{dn}\) & Diameter moment of inertia & \(I_{dm}\) & Polar moment of inertia \\ \(bl\) & Backlash & \(c_{m}\) & Meshing damping \\ \(\mathbf{K_{b}}\) & Bearing stiffness matrix & \(\mathbf{C_{b}}\) & Bearing Damping Matrix \\ \(\mathbf{V_{l}}\) & Construct vector & \(\mathbf{q_{m}}\) & the deflection of gear pair of action \\ \(\delta_{m}\) & dynamic transmission error & \(e_{s}\) & geometric transmission error \\ \(Ldr\) & load distribution ratio & \(\mathbf{M^{m}}\) & mass matrices of the gear pair \\ \(\mathbf{C^{m}}\) & stiffness matrix of the gear pair & \(\mathbf{K^{m}}\) & damping matrix of the gear pair \\ \(\mathbf{q^{e}}\) & the displacement vector & \(\mathbf{M^{e}}\) & mass matrix of the Timoshenko beam element \\  & of the Timoshenko beam element & \(\mathbf{K^{e}}\) & stiffness matrix of the Timoshenko beam element \\ \end{tabular}