## 3 Dynamic Model and Equations of Motion  
### Dynamic Model  
Figure 2 shows the developed transverse-axial-torsional coupling dynamic model of the herringbone planetary gear set. All the gears (i.e., the sun, two rings, and planets) and the planet-carrier are considered to be rigid bodies. Bearing components are modeled by linear springs between their housings and bodies. Linear springs applied along the action line denote gear mesh interactions. The damping, gravity and friction impact are neglected, and the tilting motion is ignored. Each member possesses four degrees of freedom (DOFs), including one axial translation, two lateral translations, and one rotation. As demonstrated in Figure 2, there are three kinds of reference frames established, namely (1) the static reference frame \(OXYZ\), (2) the dynamic reference frame \(Oxyz\) rotating around the coordinate origin \(O\) together with the carrier, and (3) the dynamic reference frame \(O_{i}x_{i}y_{i}z_{i}\) rotating with the carrier, whose origin \(O_{i}\) is at the \(i\)th planet-gear's center, and \(x_{i^{\prime}}\), \(y_{i}\)-axis are, respectively, in the radial and tangential direction.  
Translational coordinates \(x_{i}\), \(y_{i}\), \(z_{i}\) (\(i=c\), \(r1\), \(r2\), \(s\)) are respectively assigned to the carrier, right ring gear, left ring gear, and sun gear. Rotational coordinates are given by \(u_{i}=r_{i}\theta_{i}\) (\(i=c\), \(r1\), \(r2\), \(s,1\), \(2\),..., \(N\)), where \(\theta_{i}\) is the rotation; \(r_{i}\) (\(i=r1\), \(r2\), \(s\), \(1\), \(2\),..., \(N\)) are, respectively, the base radius for the right ring, left ring, sun and planets, and \(r_{c}\) is the circle radius from the center of each planet-gear to that of the carrier. Translational coordinates \(x_{j}\), \(y_{j}\), \(z_{j}\) (\(j=1\), s..., \(N\)) are assigned to the \(j\)th planet-gear, denoting the absolute radial, tangential, and axial deflections of the \(j\)th planet-gear. The vector [\(x_{s}\), \(y_{s}\), \(z_{s}\), \(u_{s}\), \(x_{r1}\), \(y_{r1}\), \(z_{r1}\), \(u_{r1}\), \(x_{r2}\), \(y_{r2}\), \(z_{r2}\), \(u_{r2}\), \(x_{c}\), \(y_{c}\), \(z_{c}\), \(u_{c}\), \(x_{1}\), \(y_{1}\), \(z_{1}\), \(u_{1}\),..., \(x_{N}\), \(y_{N}\), \(z_{N}\), \(u_{N}\)]\({}^{T}\) is selected as the generalized coordinates to create the lumped-parameter dynamic analytical model for the HPGT system. \(\alpha_{t}\) denotes the transverse pressure angle. \(e\) represents the static transmission error. \(k\) denotes the stiffness. \(\varphi_{i}=2\pi(i-1)/N\) represents the installation position angle of the planet \(i\), and \(\varphi_{i}\) is assumed to be positive when anticlockwise. \(\beta_{b}\) is the base helix angle. Symbols \(L\) and \(R\) respectively represent the left side and right side of the herringbone gears in the HPGT system. \(spi\) denotes the \(i\)th external meshing pair. \(r1 pi\) and \(r2 pi\) stand for the \(i\)th right- and left-side internal meshing pairs, respectively.  
Figure 2: Bending-torsional-axial coupling dynamic model of herringbone planetary gears taking both sides of each gear into consideration. For simplicity, only one planet–gear is exhibited and the carrier is not displayed in the figure.