*Chinese version published in 《Technology Innovation and Application/科技创新与应用》ISSN 2095-2945, 12th edition, page 69-70 *, *April 28 2020*

**Abstract**

The objectives to study planetary gear set often are to find torque ratio and velocity ratio between input and output shaft. This paper aims to achieve these objectives by develop a new method which is kinetics method in compare to traditional kinematics or tabular method[1]. This new method is direct and easy method, developed to study single stage planetary gear set in steady and unsteady state by first establishes the principle of force transfer at contact point between a pair of rotating cylinders. In others words the established principle described the power flow from one rotating cylinder to another by elaborates the force transfer mechanism at contact point between the rotating cylinders. By relying on the established principle, the model of force transfer in single stage planetary gear set for various arrangements could be derived. From the force transfer model of selected planetary gear set arrangement, the force transferred to output shaft of planetary gear set is shown in the free body diagram, hence the output torque could be directly obtained. The power at input and output shaft is a constant, which is equivalent to product of torque and angular velocity. By knowing the torque at output shaft, the angular velocity of output shaft then could be found.

**Keywords:** Planetary gear set, Torque ratio, Velocity ratio, Friction, Power Transmission

**1. Introduction**

Conventional approach in study planetary gear set is using tabular method[1]. The objective of this paper is to develop a direct method to obtain torque ratio & velocity ratio of input & output shaft of single stage planetary gear set in steady and unsteady state. First the underlying principle of force transfer at contact point in between a pair of rotating cylinders is established, follow by building the model of force transfer for two different arrangements of the planetary gear set[2]. Then the method in finding the torque ratio for the respective arrangement of planetary gear set is described and explained in logically and easy to understand manner.

**2. Nomenclature**

w_{out }Output angular velocity

w_{in} Input angular velocity

F_{t} Tangential force

F_{tr} Transfer force

F_{f} Friction force

F_{re} Reaction force

N Normal force

µ Coefficient of friction

F_{pt} Planet gear tangential force

F_{ret} Tangential component of reaction force

T_{in} Input torque

T_{o} Output torque

r_{p} Radius of planet gear

r_{s} Radius of sun gear

r_{c} Radius of carrier shaft

F_{c} Carrier shaft tangential force

Ѳ Angle

3**. Underlying Principle**

The dynamics at contact point of a pair of rotating cylinders used to transfer force from a rotating cylinder to another is studied and a model is developed. It is the underlying principle to study planetary gear set in both steady and unsteady state.

*Fig. 1: Model of transfer force from a rotating cylinder to another*

Tangential force, F_{t} responsible to rotate the cylinder B. If the tangential force, F_{t} is larger than friction force, F_{f} , the slip condition occurred. Gear set is used in such condition to avoid slippage.

**3.1 Single Stage Planetary Gear Set Arrangement 1 **

Ring gear : Fixed

Carrier : Output

Sun gear : Input

The established model to show the force interaction of the above arrangement as in Fig. 2 below.

*Fig, 2: Model of force transfer of single stage planetary gear set arrangement 1*

The F_{tr }could be decomposed into X-Y direction components as illustrate in Fig. 3 below.

*Fig. 3: X-Y directions components of transfer force, F _{tr} *

The F_{pt }acted on fixed ring resulted a reaction force, F_{re }on planet gear. The reaction force, F_{re }could decompose into XY direction components as illustrate in Fig. 4 below.

*Fig. 4: X-Y directions components of reaction force, F _{re} *

Input torque calculated as Fig. 5 below.

*Fig. 5: Illustration of input torque *

** Input torque, 𝑇_{in}=𝐹_{tr} . 𝑟_{s } ** (1)

Output torque at carrier is calculated as Fig. 6 below.

*Fig. 6: Illustration of output torque *

** Output torque, T_{o}=F_{ret} . ( 2_{rp}+r_{s}) ** (2)

(*2r _{p}+r_{s}*) is radius of ring gear, therefore torque ratio is simply the ring gear size divide by sun gear size. This finding is matching with the illustration of a gear manufacturer “

*wittenstein*” in the video “

*Made simple：Design and operating principle of a low-backlash planetary gearbox*”

*(1: 21 ~1:26)*[3]

*.*

Angular velocity ratio simply is the reciprocal of torque ratio.

For the example in which the planet gear size is equal to sun gear,

**3.2 Single Stage Planetary Gear Set Arrangement 2**

Ring gear : Output

Carrier : Input

Sun gear : Fixed

The model established to show the force interaction of the above arrangement as in Fig. 7 below.

*Fig. 7: Model of force transfer of single stage planetary gear set arrangement 2*

When the carrier rotated, F_{tr} is distributed to sun and ring gear as shown in Fig. 7. F_{tr} acted on fixed sun gear resulted a reaction force, F_{re} , this reaction force shall acts on ring gear. This means transfer force, F_{tr} generated by carrier finally caused the ring gear to rotate. By neglecting the friction force

**4. Conclusion**

The torque ratio of planetary gear set could easily be found by establish a model of force transfer for the selected arrangement of the planetary gear set based on the underlying principle described in this paper.

**References**

[1] Norton, Design of Machinery (2004), pp.495-499, McGraw-Hill.