## 2 Modulation Effects and Dynamics Symptom of Planet Gear Fault
### Dynamic Model of Planetary Gear  
Dynamic model of planetary gear is shown in Figure 4a, some important parameters are given as well. For different dynamic components, dynamic parameters are distinguished according to the initial letter of their names. Specifically, c presents carrier, r presents ring gear, s presents sun gear, and p presents planet gear. When it is necessary to refer to a planet gear, subscript like \(n\) = 1, 2,..., _NP_ will be used, and _NP_ is planet gear number. For convenience to describe motion of each dynamic gear components, several coordinates are used, which is shown in Figure 4a. Coordinate _OXY_ is fixed to the earth, it is static all the time. Figure 4: Planetary gearbox: (**a**) dynamic model and (**b**) relative displacement. Figure 3: Hanning function-based signal transfer path: (**a**) Hanning function based model of signal transfer path; (**b**) Modulation effect of a four-planet-gear planetary gearbox. Coordinate \(Oxy\) is fixed to the carrier, in other words, \(Oxy\) rotates with the carrier, its rotating speed is exactly same with the carrier, and the \(x\) axis of \(Oxy\) overlaps with first planet gear center. Coordinate \(O_{n}x_{n}y_{n}\) is also rotating coordinate, its origin is located at the center of planet gear \(P_{n}\). Physical meaning of dynamic parameters in Figure 4 is explained as follows:  
\(x_{i}\), \(y_{i}\) (\(i\) = s, r, c, 1, 2,..., \(NP\)). Translational displacement of gear components, which are evaluated in \(Oxy\). \(u_{i}\) (\(i\) = s, r, c, 1, 2,..., \(NP\)). Rotational displacement of gear component, also evaluated in \(Oxy\). \(u_{i}\) = \(r_{i}\theta_{i}\), where \(r_{i}\) is radius of base circle for gear components or center circle of carrier, \(\theta_{i}\) is angular displacement. \(\varphi_{n}\) (\(n\) = 1, 2,..., \(NP\)). Phase difference between \(OO_{n}\) and \(x\) axis of \(Oxy\), \(\varphi_{n}\) = 2\(\pi\) (\(n\) - 1)/\(NP\). \(k_{P}\), \(c_{\mathrm{P}}\). Planet gear bearing stiffness and damping. \(k_{i}\), \(c_{i}\) (\(i\) = c, r, s). Radial stiffness and damping of carrier, ring gear, and sun gear. \(k_{it}\), \(c_{it}\) (\(i\) = c, r, s). Tangential stiffness and damping of carrier, ring gear, and sun gear. Relative displacement analysis of planetary gearbox is shown in Figure 4b, and following is the detailed analysis of relative displacement between sun and planet gear. Relative displacement of other meshing pair can be derived in the same way. Defining the positive direction as pointing from sun gear to planet gear along the meshing line, by projecting sun gear displacement and planet gear displacement to the meshing line, the relative displacement between sun gear and planet gear can be derived as the following:  
\(x_{\mathrm{s}}\) and \(y_{\mathrm{s}}\) are translational displacements of sun gear in \(x\) direction and \(y\) direction, their projections are -\(x_{\mathrm{s}}\)sin\(\varphi_{\mathrm{s}nt}\) and \(y_{\mathrm{s}}\)cos\(\varphi_{\mathrm{s}nt}\), where \(\varphi_{\mathrm{s}nt}\) = \(\varphi_{n}\) - \(\alpha_{\mathrm{s}}\); \(x_{n}\) and \(y_{n}\) are translational displacements of planet gear in \(x\) direction and \(y\) direction, respectively, and their projections are -\(x_{n}\)sin\(\varphi_{\mathrm{s}nt}\) and \(y_{n}\)cos\(\varphi_{\mathrm{s}n}\); \(u_{\mathrm{s}}\) and \(u_{n}\) are rotational displacements of sun gear and planet gear, respectively, and their projections are \(u_{\mathrm{s}}\) and \(u_{n}\), respectively.