## Abbreviations  
### Nomenclature  
\(e\)static transmission error  
\(E_{j}\)manufacturing eccentric error of the component (\(j=s\), \(r1\), \(r2\), \(c\), \(p1\), \(\dots\), \(pN\))  
\(E_{\text{spi}}\)tooth profile error in the \(i\)th sun-planet mesh (\(i=1\), \(2\), \(\dots\), \(N\))  
\(\text{I}_{j}\)moment of inertial of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\), \(p1\), \(\dots\), \(pN\))  
\(\text{F}(t)\)exciting force induced by the transmission error and time-varying mesh stiffness  
\(\text{G}\)gyroscopic matrix due to the rotation of the carrier  
\(\text{k}_{\text{spi}}^{\text{R}}(t)\)time-varying mesh stiffness of the right-side \(i\)th sun-planet mesh (\(i=1\), \(2\), \(\dots\), \(N\))  
\(\text{k}_{\text{spi}}^{\text{L}}(t)\)time-varying mesh stiffness of the left-side \(i\)th sun-planet mesh (\(i=1\), \(2\), \(\dots\), \(N\))  
\(\text{k}_{\text{r1 pi}}^{\text{L}}(t)\)time-varying mesh stiffness of the right-side \(i\)th ring-planet mesh (\(i=1\), \(2\), \(\dots\), \(N\))  
\(\text{k}_{\text{2 pi}}^{\text{L}}(t)\)time-varying mesh stiffness of the left-side \(i\)th ring-planet mesh (\(i=1\), \(2\), \(\dots\), \(N\))  
\(\text{k}_{j}\)bearing supporting stiffness of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\), \(p1\), \(\dots\), \(pN\))  
\(\text{k}_{\text{fi}}\)tangential support stiffness of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\))  
\(\text{K}_{\text{m}}\), \(\text{K}_{\text{b}}\)stiffness matrix related to gear mesh and bearing supporting  
\(\text{K}_{\text{av}}\)stiffness matrix related to the centrifugal effect of planets  
\(m_{j}\)mass of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\), \(p\))  
\(\text{M}\)mass matrix of the whole system  
\(N\)number of planets  
\(\tau_{j}\)base circle radius of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(p\))  
\(\tau_{c}\)radius of the circle passing through the planet centers \begin{tabular}{l l} \(t\) & time \\ \(T_{j}\) & external torque acting on the component \(j\) (\(j=s\), \(c\)) \\ \(T\) & external torque vector applied to the system \\ \(u_{j}\) & torsional linear displacements of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\), \(1\), \(\ldots\), \(N\)) \\ \(\omega_{c}\) & rational angular speed of the carrier \\ \(x_{i}\), \(y_{i}\) & lateral translational displacements of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\), \(1\), \(\ldots\), \(N\)) \\ U & system displacement vector \\ \(z_{i}\) & axial motions of the component \(j\) (\(j=s\), \(r1\), \(r2\), \(c\), \(1\), \(\ldots\), \(N\)) \\ \(\alpha_{t}\) & transverse pressure angle \\ \(\beta_{b}\) & base helix angle \\ \(\varphi_{i}\) & planet assembly position angle \\ \(\varphi_{\rm si}\) & \(\varphi_{i}-\alpha_{\rm t}\) \\ \(\varphi_{\rm ri}\) & \(\varphi_{\rm i}+\alpha_{\rm t}\) \\ \(\Theta^{R}_{spi^{\prime}}\Theta^{L}_{spi}\) & relative mesh deflection of elastic elements used in the right-side and left-side \\  & sun-planet-gear mesh \\ \(\Theta^{R}_{r1 pi^{\prime}}\Theta^{L}_{r2 pi^{\prime}}\) & relative mesh deflection used in the \(i\)th ring-planet mesh spring for right and left ring \\  & gear \\ \(\Theta_{xxi}\), \(\Theta_{yxi}\), \(\Theta_{zxi}\) & relative deflections between the \(i\)th planet and the carrier in \(x\)-, \(y\)-, and \(z\)-direction of \\ \end{tabular} \begin{tabular}{l l} \(\Theta_{xxi}\), \(\Theta_{yxi}\), \(\Theta_{zxi}\) & bearing spring \\ \end{tabular} 
* (12) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Influence of manufacturing errors on dynamic floating characteristics for herringbone planetary gears. _Nonlinear Dyn._**2018**, _93_, 361-372. [CrossRef]
* (13) [PERSON]; [PERSON]; [PERSON] Load sharing performances of herringbone planetary gears considering manufacturing errors. _J. Cent. South Univ. (Sci. Technol. )_**2016**, _47_, 474-481. * (14) [PERSON]; [PERSON]; [PERSON] Effect of sun gear manufacturing eccentric errors on dynamic performances of herringbone planetary gear train. _J. Northeast Univ. (Nat. Sci. Ed. )_**2015**, _36_, 709-714.