### 3.1 Structure Representation

The power flow analysis of multistage PGTs usually requires the calculation of speed and torque. However, the number of teeth was not determined in the conceptual design stage. Therefore, a structural representation that can easily visualize the essential functions of the transmission without addressing the complexities of planetary gear kinematics is helpful. In this study, a geometric model was used as the analysis tool. It provides two functions: qualitative kinematics for power flow analysis and a structural representation for power flow visualization. Qualitative kinematics and structural representations are also physical meanings of the geometric model.

Figure 3 shows the functional diagram of a basic planetary gear set. Two fundamental circuits, 2-34 and 2-14, can be identified, according to Buchsbaum et al. [24].

The basic planetary gear set shown in Figure 3(a) can be represented as a geometric model in which the vertices represent the center link, including the sun, ring, and carrier, and the horizontal dashed line represents the planet, as shown in Figure 3(b). Each solid line in a geometric graph connects two vertices, one representing the central gear, and the other representing the carrier. This solid line is referred to as a circuit line because its vertices belong to the same fundamental circuit.

The fundamental circuit, represented by the circuit line, is the building block of the geometric model. According to the definition of a fundamental circuit, planet 4 is shared by two fundamental circuits. Therefore, the horizontal dashed line 4 “connects” vertices 1 and 3 belonging to different fundamental circuits. Another link shared in Figure 3(a) is carrier 2, represented by the intersection point of the two circuit lines, as shown in Figure 3(b).

When a link is used as the central link in different fundamental circuits, we use a vertical dashed line to indicate its sharing. This type of sharing includes cases where it is used as a carrier and center gear or as different center gears in different fundamental circuits. An example of a more complex PGT represented by this sharing among the fundamental circuits is shown in Figure 4. In this train, link 2 is shared by fundamental circuits 2-35, 2-45, and 1-26. The intersection point of circuit lines 2-3 and 2-4 indicates that shared link 2 acts as a carrier in circuits 2-35 and 2-45. This intersection point is connected to the vertex of another circuit line 1-2 by a vertical dashed line. This vertical dashed line indicates that the function of link 2 in the other fundamental circuit, namely 1-26, is the central gear. Finally, because link 4 acts only as a common sun for fundamental circuits 2-45 and 1-46, two circuit lines, 1-4 and 2-4, are connected by a vertical dashed line on the vertices representing central gear 4.

As shown in Figures 3(b) and 4, the vertexes representing the ring gear are always on the circuit line with a positive slope, while the vertexes representing the sun gear are always on the circuit line with a negative slope. Therefore, there is a correspondence between the type of gear and slope of the circuit line. This is explained in Section 3.2 because of its kinematic meaning. In Figure 4, the dashed line between vertexes 1 and 2 is an auxiliary line [21] in which all the vertex representing carriers should be located.

### 3.2 Kinematics

In contrast to other representations, a geometric model was developed to directly represent the kinematics of PGTs based on their structural representation. Each circuit line strictly corresponds to the Willis equation for the associated fundamental circuit. The slope of the circuit line is equal to the ratio of teeth in the Willis equation. The relationship between the positions of different circuit lines corresponds to the structural connections of the train. The lengths of the various circuit lines are determined by both the kinematic constraint of the train (isokinetic constraint) and structural connection. When equipped with Cartesian coordinates, the value of the horizontal projection of the dashed line representing the planet onto the vertical axis is the angular velocity of the planet, as shown by *ω*_{4} in Figure 3(c). Similarly, the value of the vertical projection of the vertex representing the central link on the horizontal axis is the angular velocity of the central link, as shown by *ω*_{3}, *ω*_{2}, and *ω*_{1} in Figure 3(b).

The representation of kinematics comes from the definite physical meaning of the slope of the circuit line: the tooth ratio of the center gear to the planet. Therefore, the circuit line represents the type of fundamental circuit, that is, the internal gear mesh or external gear mesh. Because the tooth ratios of the internal and external gear meshing pairs have different symbols, the slope of the circuit line is a signed number.

Let *λ* represent the slope of the circuit line, *ω* the angular velocity of the link, and *Z* the number of teeth in the gear. The kinematic equations [25] of a fundamental circuit can then be reformulated as

$$\left\{ \begin{gathered} \lambda_{sp} = \frac{{\omega_{p} - \omega_{c} }}{{\omega_{s} - \omega_{c} }} = - \frac{{Z_{s} }}{{Z_{p} }}, \hfill \\ \lambda_{rp} = \frac{{\omega_{p} - \omega_{c} }}{{\omega_{r} - \omega_{c} }} = + \frac{{Z_{r} }}{{Z_{p} }}, \hfill \\ \end{gathered} \right.$$

(1)

where the subscripts *s*, *r*, *p*, and *c* denote the sun, ring gear, planet, and carrier, respectively. Because the dimensions of the ring gear are constant greater than those of the planet in the internal gear mesh, we obtain

$$\lambda \left\{ {\begin{array}{*{20}c} {\,\, = \lambda_{sp} < 0\quad {\text{external gear mesh,}}} \\ { = \lambda_{rp} > 1\quad {\text{internal gear mesh}}{.}} \\ \end{array} } \right.$$

(2)

The tooth ratio of the ring gear to the sun within a planetary gear set can be expressed as follows through the geometric meaning of the slope of a straight line:

$$- \frac{{Z_{r} }}{{Z_{s} }} = \frac{x}{y} = \frac{{\lambda_{rp} }}{{\lambda_{sp} }},$$

(3)

where *x* and *y* are the lengths of the vertical projection of the circuit line on the horizontal axis, as shown in Figure 3(c).

When only qualitative kinematics are desired, the slope of each circuit line is arbitrary under the constraint of the structure, and Eq. (2). Nevertheless, it does not affect the correctness of the qualitative kinematics. Therefore, one can easily draw a geometric graph for a PGT. We used Figure 4 as an example to illustrate the steps of the geometric model drawing. Without loss of generality, the drawing begins with the first stage from the left.

Consider an arbitrary point in the plane as link 2. It is generally convenient to choose a carrier as the starting point for drawing. It is also permissible to begin with any other link.

Starting from link 2, draw a straight line to the lower left with an arbitrary slope greater than one. The length of the straight line was arbitrary and its endpoint was link 3. This step will provide fundamental circuit 2-35.

Starting from link 2, draw a straight line down to the right at any slope less than zero, until the endpoint has the same horizontal height as link 3. The end of the straight line is part of link 4, which belongs to the first planetary stage, that is, the sun of the first planetary stage. This step yields fundamental circuit 2-45.

Connecting 3 and 4 with a dashed line gives the planet 5.

Starting from 2, we draw a vertical dashed line of an arbitrary length. The endpoint of the vertical dashed line is still link 2. However, it indicates the part of link 2 that belongs to the second planetary stage, that is, the ring of the second planetary stage.

Starting from the ring of the second planetary stage, draw a straight line to the upper right with an arbitrary slope greater than 1 and intersect the (imaginary) auxiliary line with slope 1. The intersection point is the other carrier (link 1). It should be noted that if an imaginary auxiliary line is drawn, it should be drawn as a dashed line. This step gives fundamental circuit 1-26.

Starting from 1, draw a straight line to the lower right, with the endpoint being the intersection of the horizontal dashed line passing ring 2 and the vertical dashed line passing sun 4. This is the sun of the second planetary stage. The straight line gives fundamental circuit 1-46. The horizontal dashed line between the newly obtained point and ring 2 corresponds to planet 6. It should be noted that the slope of straight line 1-4 is not arbitrary. This is naturally constrained by the previous steps.

The qualitative kinematics of the train are determined from the geometric model: the angular velocity of all central links increases from left to right, and the angular velocity of all planets increases from bottom to top. When any central link is fixed, the rotation of the other central links is reversed on both sides.

This study does not deal with the exact kinematic solution. When the exact solution needs to be obtained, the geometric model must be placed in a Cartesian coordinate system, and the slope of each circuit line must be given. From the above process of drawing the geometric model, it is clear that the geometric graph is unique when the slopes of all the circuit lines are given. For a detailed explanation of the geometric method and exact solution of the kinematics, refer to Ref. [21].

### 3.3 Torque Analysis

Eliminating the planet speed from Eq. (1) returns the Willis’ ratio. Together with the power balance and torque balance equation [15] of the fundamental circuit, Willis’ ratio leads to a well-known result for a fundamental circuit: the torque ratios of the three shafts are functions of Willis’ ratio. Several studies have documented these results.

In this study, we considered only PGTs of the ring-sun type because of the parallel-connected constraint. For the basic planetary gear set shown in Figure 3, the above result leads to a definite relationship of the signs among the torques of the fundamental links: the torques of the sun and ring have the same sign, and they are opposed to that of the carrier. This relationship can be expressed in the context of the geometric method using Eq. (3):

$$\frac{{T_{s} }}{{T_{r} }} = \frac{y}{x},$$

(4)

$$T_{c} = - \left( {\frac{x + y}{x}} \right)T_{r},$$

(5)

and

$$T_{c} = - \left( {\frac{x + y}{y}} \right)T_{s}.$$

(6)

Considering the torque balance on the planet, Eq. (4) can be graphically expressed as shown in Figure 5.