- Original Article
- Open Access

# Response Sensitivity to Design Parameters of RV Reducer

- Yu-Hu Yang
^{1}Email author, - Chuan Chen
^{1}and - Shi-Yu Wang
^{1}

**31**:49

https://doi.org/10.1186/s10033-018-0249-y

© The Author(s) 2018

**Received: **29 May 2016

**Accepted: **8 June 2018

**Published: **20 June 2018

## Abstract

Dynamic characteristic significantly affects performance of RV reducer. The current researches mainly pay attention to free vibration properties of RV reducer. In order to satisfy the increasing demand on high performance, response sensitivity is analytically studied on the basis of cyclic symmetry structure. Based on the structure characteristics, a dynamic model is developed by taking into account the influence of bearing stiffness, crankshaft bending stiffness and mesh stiffness within planetary and cycloidal stages. For the model, governing equation of motion is derived and solved by Fourier series method. The solution revealed that forced vibrations at primary frequency are well defined structural. There exist three typical forced vibration modes: rotational, translational and planetary component modes. Response sensitivity to basic design parameters is obtained as closed-form expressions by differential method. With the typical vibration modes, response sensitivity is simplified and classified into three types. Calculation of sensitivity implies that vibrations of the output wheel are sensitive to eccentricity. As eccentricity increases, sensitivity of translation decreases first and then increases, but sensitivity of rotation always increases. The proposed method for analyzing response sensitivity provides some principles for selecting parameters for RV reducer from the point of view of forced vibration.

## Keywords

- RV reducer
- Mesh phasing
- Vibration modes
- Sensitivity

## 1 Introduction

RV reducer is widely used in industrial robots due to the advantages such as compact structure, multi-teeth meshing and high carrying capacity, large transmission ratio and high transmission efficiency. However, vibration significantly affects the positioning and repeatability, especially for high performance applications, such as semiconductor manufacturing and space industry.

RV reducer is a newly developed reducer on the basis of cycloid gear. Although the vibration of spur gear has studied by many researchers, the vibration of cycloid gear has received little attention. Refs. [1, 2] develop dynamic model of a single stage cycloid drive. Zhang et al. [3, 4] formulated a dynamic model of RV reducer which considered mesh and bearing stiffness. Thube et al. [5] established finite element model of cycloid speed reducer and analyzed the load and stress distributions. Hsieh [6] developed mathematical model for the dynamic analysis of cycloid speed reducer. These studies have laid a foundation for the construction of dynamic model of RV reducer.

Known from literatures above, scholars mainly pay attention to free vibration properties. Although wide study has been carried out in the analysis of mechanical transmission [7], research about dynamic response and parameter selection of RV reducer is scare. Parameter sensitivity analysis provides an effective method for dynamic analysis and parameter selection. Because of this, researchers addressed the parameter sensitivity to modal behaviors of the planetary gears [8, 9]. Zhu et al. [10] researched the effects of pin stiffness on natural frequencies. Zhang et al. [3, 4] numerically studied the sensitivity of the crankshaft bending stiffness and bearings support stiffness to natural frequencies of RV reducer. The above studies normally employ numerical calculations. To achieve more finds, Refs. [11–16] analytically examine the modal sensitivity based on typical free vibration modes of planetary gears. These studies provide valuable insights into sensitivity calculation of free vibration.

While sensitivity of free vibration can provide information for parameters selection, sensitivity of forced vibration can aid the selection in a more straight way. However, there is little literature on this respect, especially for RV reducer. Related research was generally carried out by using numerical method based on spur planetary gears [17–21]. Refs. [22, 23] investigate dynamic response of gear systems with Runge–Kutta method. Although numerical methods are sufficiently accurate, they are time-consuming and general results are difficult to obtain. However, studies using analytical method are rarely found. Zhang et al. [24] derived general formulations of dynamic response sensitivity with ideal model based on differential matrix theory. Liu et al. [25, 26] investigated response sensitivity of compound planetary gears. Yang [16] proposed a method based on the Fourier series and calculated sensitivity of forced vibration to mass and stiffness of helical planetary gears. This method is mathematically effective, though the relationships between sensitivity calculation and structured vibrations need further research.

As typical structured vibration, mesh phasing has been reported by the previous researchers [27, 28]. Parker et al. [29–31] studied the effects of mesh phasing in planetary gears and proved that it is based solely on system symmetry. Because of cyclic symmetry structure, such behaviors should still be found for RV reducer, where the phasing describes the link between parameters of high-speed stage, including planet gear number and sun gear tooth number, and those of low-speed stage, including the cycloid gear number and pin number, and forced vibration. However, whether it exists or not and to what degree basic parameters affects the structured vibration need to be studied.

The objective of this paper is to analytically study response sensitivity to parameters of RV reducer on the basis of cyclic symmetry structure. Therefore, a lumped parameter model was built up by incorporating various stiffness factors and the governing equation was derived and analytically solved. On the basis of analytical solution, response sensitivity to basic parameters was derived as closed-form expressions. The research is helpful for the analysis of response sensitivity and parameter selection of RV reducer.

## 2 Governing Equation of Motion

*M*planets and crankshafts,

*N*cycloid gears and output wheel are treated as rigid bodies. Component flexibility, bearings and gear meshes are represented by linear springs. The supports of the components are modeled as two perpendicular springs with equal stiffness. The transverse stiffness of the sun, planets, crankshafts, cycloid gears, and output wheel are designated as

*k*

_{s},

*k*

_{a},

*k*

_{Hb},

*k*

_{cb}and

*k*

_{o}. The torsional stiffness of the sun, planet and output wheel are represented as

*k*

_{st},

*k*

_{H}and

*k*

_{ot}. The gear meshes are modeled by springs acting along the line of action. The sun-planet and cycloid-pin mesh stiffness are

*k*

_{si}and

*k*

_{bj}. The stiffness of the pinwheel is not taken into account in the modeling since it is much larger. The external torques are

*T*

_{s}and

*T*

_{o}which are respectively applied to the sun and output wheel.

The coordinates are also illustrated in Figure 2. Each component has two translational and one rotational motion. Thus the system has 6*M *+ 3*N *+ 6 degrees of freedoms. Using a fixed basis, translational coordinates (*x*_{s}, *y*_{s}) and (*x*_{o}, *y*_{o}) are assigned to the sun and output wheel, as shown in Figure 2(a). Coordinates (*x*_{pi}, *y*_{pi}), (*x*_{Hi}, *y*_{Hi},) and (*x*_{cj}, *y*_{cj}) are respectively assigned to planet *i*, crankshaft *i*, and cycloid gear *j*, where *i *= 1, 2, …, *M* and *j *= 1, 2, …, *N*, as shown in Figure 2(b). Circumferential planet and cycloid gear locations are specified by the angles *ψ*_{
i
} and *ψ*_{
j
}. The *x*_{pi} and *x*_{Hi} coordinates are chosen to be positive from the geometric center of the sun to that of the arbitrarily chosen first planet. The *x*_{cj} coordinate is chosen to be positive from the geometric center of the pinwheel to that of the arbitrarily chosen first cycloid gear. All rotational coordinates are chosen to be *θ* which are positive in counterclockwise. This is illustrated in Figure 2, where *θ*_{s}, *θ*_{pi}, *θ*_{Hi}, *θ*_{ci}, and *θ*_{o} are respectively assigned to the sun, planets, crankshafts, cycloid gears and output wheel.

_{b}and K

_{m}are the inertia, support stiffness, and mesh stiffness matrices. F is the applied external torque. q is the generalized coordinate vector. They are given in the Appendix. The associated free vibration equation is

## 3 Primary Frequency Vibration From Mesh Stiffness

Response induced by mesh frequency excitation has structured behaviors due to the symmetry structure and mesh stiffness variation.

### 3.1 Primary Frequency Response

_{1l}and A

_{2l}are the cosine and sine components of \({\tilde{\varvec{F}}}\); B

_{1l}and B

_{2l}are those of \(\Delta {\varvec{q}}\), and

*ω*

_{ l }is the

*l*th frequency.

_{ l }and A

_{ l }are given in the Appendix.

*l*th amplitude of the response is

*τ*th freedom is

Please note that Eq. (1) describes a parametric vibration, but the derivation above ignores a so-called second order term \(\Delta {\varvec{K}}\Delta {\varvec{q}}\), which transfers the original parametric vibration to a forced one. Parametric vibration includes basic frequency and side frequency. The derivation here retains basic frequency and removes side frequency which has less influence on vibration than basic frequency [34, 35]. So this paper only studies primary frequency of parametric vibration of RV reducer.

### 3.2 Typical Vibration Modes at Primary Frequency

Ref. [30] shows by the vector addition of forces and moments that planetary gears exist typical vibration modes: rotational, translation and planet modes. The conclusion is based solely on the mesh force periodicity and system symmetry. Due to cyclic symmetry structure, RV reducer should have similar typical vibration modes as planetary gears, which is demonstrated by simulation in Section 3.3. Therefore, for RV reducer with *M* planets and crankshafts and *N* cycloid gears, the forced vibration has a well-defined structure. There are three typical vibration modes: rotational, translational and planetary component modes. The properties of each mode are summarized as below.

#### 3.2.1 Rotational Modes

#### 3.2.2 Translational Modes

#### 3.2.3 Planetary Component Modes

### 3.3 Numerical Results on Vibration Modes

Parameters of example RV reducer

Component | Mass | Moment of inertia | Base diameter |
---|---|---|---|

Sun | 1.30 | 4.44 × 10 | 10.57 |

Planet | 0.88 | 1.01 × 10 | 48.63 |

Crankshaft | 0.40 | 7.56 × 10 | 2.20 |

Cycloid gear | 2.76 | 2.09 × 10 | 85.80 |

Carrier | 15.33 | 1.06 × 10 | 63.50 |

Stiffness of example RV reducer

Component | Stiffness | Torsional stiffness |
---|---|---|

Sun | 4.19 × 10 | 1.16 × 10 |

Crankshaft | 2.33 × 10 | 6.99 × 10 |

Support bearing | 9.84 × 10 | ‒ |

Turning arm bearing | 9.76 × 10 | ‒ |

Planet gear | 2.68 × 10 | ‒ |

Cycloid gear | 8.35 × 10 | ‒ |

*ω*

_{ l }excited at the

*l*th harmonic should occur at the mesh frequency of

*ω*

_{ l }/

*l*. However, the resonant peaks corresponding to different natural frequency are not excited at all harmonics. In Figure 4(a), frequencies at ①, ②, and ③ associated with translational modes show resonant peaks. Peaks at ④ and ⑤ are corresponding to ② and ③ at the third and fifth harmonics. The other peaks at the two harmonics do exist. They are small but present. Refined increments of mesh frequency would make them more apparent. However, the natural frequencies which are associated with rotational and planetary component modes do not show resonant peaks at all harmonics.

Similar situation occurs in Figure 4(b). The frequencies associated with rotational modes show resonant peaks, but the frequencies associated to the other two modes do not show peaks. In Figures 4(c) and (d), the frequencies corresponding to all three modes show resonant peaks. The peaks at ① and ② in Figure 4(c) are associated with planetary component modes.

Obviously, the result in Figure 4 shows that RV reducer has typical forced vibration modes: translational, rotational and planetary component modes. However, RV reducer scarcely has more than three planetary components. The calculated results verify conclusions in Section 3.2 due to cyclic symmetry of RV reducer. The typical vibration modes can be used to simply the formula of response sensitivity.

## 4 Response Sensitivity to Parameters

*l*th harmonic response sensitivity \({{\partial B_{1l}^{\tau } } \mathord{\left/ {\vphantom {{\partial B_{1l}^{\tau } } {\partial \xi }}} \right. \kern-0pt} {\partial \xi }}\) and \({{\partial B_{2l}^{\tau } } \mathord{\left/ {\vphantom {{\partial B_{2l}^{\tau } } {\partial \xi }}} \right. \kern-0pt} {\partial \xi }}\).According to Eq. (9), response sensitivity

*S*is

With the obtained three typical vibration modes, one can have corresponding response sensitivity as follows.

### 4.1 Sensitivity of Rotational Modes

To rotational modes, when \(\xi\) denotes transverse support stiffness of central components, only translational position of central components of G has terms. Besides, B_{lR} has terms only corresponding to rotational position of central components. It is a zero vector with B_{lR} premultiply by G which means the forced vibration of rotational modes is independent of transverse support stiffness of the central components. Thus, there is no need to calculate sensitivity of the forced vibration of rotational modes to transverse support stiffness. It can reduce the calculation time and provide some rules to suppress harmful vibration.

### 4.2 Sensitivity of Translational Modes

Similar to rotational modes, sensitivity of the forced vibration of translational modes to torsional stiffness of central components is zero which means the forced vibration of translational modes is independent of torsional stiffness of the central components.

### 4.3 Sensitivity of Planetary Component Modes

Sensitivity of the vibration of planetary component modes to both translational and torsional stiffness of central components is zero, implying these modes are insensitive to stiffness of the central components.

## 5 Numerical Results on Response Sensitivity to Parameters

### 5.1 Sensitivity to Stiffness

*k*

_{s}and

*k*

_{st}for a range of sun speeds. The mesh frequency related to the sun speed ranges from 0 Hz to 2500 Hz. In Figure 5(a), the peak curves corresponding to resonant response occur when mesh frequency coincides with the first and second natural frequencies associated with translational modes. The resonant curves arise slowly as the support stiffness increases, because the natural frequencies arise with an increase in the stiffness. Likewise, similar behaviors can be found in Figure 5(b). The results can help choose working condition from the perspective of sensitivity.

*k*

_{s}and

*k*

_{st}at 1000 r/min and 2000 r/min. In Figure 6(a), sensitivity decreases nonlinearly as support stiffness increases and flattens out when stiffness is high. The speed has more influence when support stiffness is low. Similar phenomenon happens to torsional stiffness in Figure 6(b).

### 5.2 Sensitivity to Mass and Moment of Inertia

*m*

_{s}and moment of inertia

*J*

_{s}of the sun for a range of sun speeds. In Figure 7(a), the peak curves corresponding to resonant response occur when mesh frequency coincides with natural frequencies associated with translational vibration modes. The resonant curves decrease as mass and moment of inertia increase. It is because that the natural frequencies decrease as mass and moment of inertia increase. Likewise, similar situation appears in Figure 7(b). Moreover, compared Figure 7(a) with Figure 5(a), it suggests that translation of the output wheel is more sensitive to mass of the sun than the support stiffness. Compared Figure 7(b) with Figure 5(b), it implies that rotation of the output wheel is more sensitive to the torsional stiffness of the sun than the moment of inertia.

*m*

_{s}and

*J*

_{s}at 1000 r/min and 1250 r/min. In Figure 8(a), sensitivity increases as mass of the sun increases and gradually leads to steep when mass is large. The speed has more influence when the mass of the sun is large. Similar phenomenon occurs to moment of inertia in Figure 8(b).

### 5.3 Sensitivity to Eccentricity

The above results imply that sensitivity of some typical forced vibration occur peaks when mesh frequency coincides with natural frequency. To stiffness parameters, the vibration is more sensitive as stiffness is low. However, to mass and inertia parameters, the vibration is more sensitive as they are large.

## 6 Experiments

Natural frequencies from experiment and analytical model

Mode | Experiment | Analytical model | Error |
---|---|---|---|

1 | 148.95 | 154.56 | 3.77 |

2 | 387.88 | 396.09 | 2.12 |

3 | 681.47 | 697.16 | 2.30 |

Table 3 shows the first three natural frequencies obtained from dynamic testing and analytical calculation. Results of two methods are in good agreement and the maximum amplitude difference is within 4%, which verifies the effectiveness of the analytical dynamic model.

## 7 Conclusions

- (1)
RV reducer is a two-stage planetary gear train with cyclic symmetry structure. A dynamic model of RV reducer is established by incorporating bearing stiffness, crankshaft stiffness and mesh stiffness. The central components and an arbitrary number of planetary components are considered.

- (2)
Typical forced vibration modes are verified and the properties are identified. There are three typical vibration modes: rotational, translational and planetary component modes. For each mode, vibration in certain harmonics of mesh frequency is suppressed.

- (3)
Response sensitivity of three typical vibration modes to parameters are derived and expressed in closed-form formula with the typical modes. The vibration is more sensitive to stiffness as it is low. On the contrary, the vibration is more sensitive to mass and inertia as they are large.

- (4)
Eccentricity has influence on both translational and rotational vibration of the output wheel. Sensitivity to translation decreases first and then increases as eccentricity increases, but to rotation it always increases. Vibration is more sensitive to eccentricity for higher speeds.

## Declarations

### Authors’ Contributions

Y-HY was in charge of the whole trial; Y-HY, CC and S-YW wrote the manuscript; CC assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

Yu-Hu Yang, born in 1962, is currently a professor at *School of Mechanical Engineering, Tianjin University, China*. His research interests include theory of mechanism and machinery dynamics. Tel: +86-13502108719; E-mail: Yangyuhu@tju.edu.cn.

Chuan Chen, born in 1985, is currently a PhD candidate at *School of Mechanical Engineering, Tianjin University, China*. He received his bachelor and master degree from *Northeastern University, China*. His research interests include gear dynamics. E-mail: chenchuan1985728@126.com.

Shi-Yu Wang, born in 1974, is currently an associate professor at *School of Mechanical Engineering, Tianjin University, China*. His research interests include machinery dynamics. E-mail: wangshiyu@tju.edu.cn.

### Competing Interests

The authors declare no competing financial interests.

### Funding

Supported by National Hi-tech Research and Development Program of China (863 Program, Grant No. 2011AA04A102), National Nature Science Foundation of China (Grant No. 51175370), Application of Basic Research and Frontier Technology Research Key Projects of Tianjin (Grant No. 13JCZDJC34300).

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## Authors’ Affiliations

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