An analytical modelling approach for overall torsional stiffness of rotate vector reducer

Abstract

Analytical modelling for overall torsional stiffness of rotate vector (RV) reducer has been a meaningful research priority owing to its over-constrained and multi-crank actuation structural characteristics. This paper presents an analytical modelling approach for overall torsional stiffness of RV reducer. The approach starts from analysing stiffness topological relations among all components, in order to convert the stiffness modelling of entire machine into the stiffness calculation of several subsystems. Then, time-varying meshing information including contact points, contact pins, contact deformation, and multi-tooth mesh stiffness, are precisely identified with consideration of profile modification and loads. Subsequently, the nonlinear stiffness of turning-arm bearings and support bearings is calculated through the force analysis of crankshafts. On this basis, the overall torsional stiffness model of RV reducer is developed, which concerns various stiffness parameters of high-speed stage and low-speed stage. Furthermore, the sensitivities and influences of stiffness parameters on the overall torsional stiffness of RV reducer are systematically analysed. Finally, the effectiveness and accuracy of the proposed model are validated by the physical prototype test. The outcomes of this paper are beneficial to the stiffness prediction and design of RV reducer according to its performance requirement.

Appendices

Appendix 1

1.1 Calculation method of ${\text{ }}k_{{{\text{sp}}}}^{e}$

The sun gear and planetary gears in the high-speed stage are involute gears, and the mesh stiffness between a single gear pair is calculated by [35]

$$\frac{1}{{k_{{{\text{sp}}}}^{{{\text{single}}}} }} = 0.04723 + \frac{0.15551}{{z_{{\text{s}}} }} + \frac{0.25791}{{z_{{\text{p}}} }} - 0.00635x_{{\text{s}}} - 0.11654\frac{{x_{{\text{s}}} }}{{z_{{\text{s}}} }} - 0.00193x_{{\text{p}}} - 0.24188\frac{{x_{{\text{p}}} }}{{z_{{\text{p}}} }} + 0.0529x_{{\text{s}}}^{{2}} + 0.00182x_{{\text{p}}}^{2}$$

(35)

The total mesh stiffness is expressed as

$$k_{{{\text{sp}}}} = \left( {0.75\varepsilon + 0.25} \right)k_{{{\text{sp}}}}^{{{\text{single}}}}$$

(36)

Thus, the equivalent torsional stiffness of the mesh stiffness between the sun gear and planetary gear is expressed as

$$k_{{{\text{sp}}}}^{e} = k_{{{\text{sp}}}} r_{{\text{s}}}^{{2}} i_{{\text{q}}}^{{2}}$$

(37)

where x_s and x_p represent the modification coefficient of sun gear and planetary gear, respectively; ε denotes the gear contact ratio; r_s represents the radius of the sun gear base circle.

1.2 Calculation methods of ${\text{ }}k_{{{\text{s}}}}^{e}$ and ${\text{ }}k_{{{\text{a}}}}^{e}$

The input shaft and crankshaft are divided into multiple shaft segments, so their torsional stiffness can be calculated by

$$\frac{1}{{k_{{\text{s}}} }} = \sum\limits_{i = 1}^{{n_{{\text{s}}} }} {\frac{1}{{k_{{{\text{s}}i}} }}}$$

(38)

$$\frac{1}{{k_{{\text{a}}} }} = \sum\limits_{i = 1}^{{n_{{\text{a}}} }} {\frac{1}{{k_{{{\text{a}}i}} }}}$$

(39)

where n_s and n_a represent the numbers of the shaft segments of the input shaft and crankshaft, respectively.

The equivalent torsional stiffness of the both can be further expressed as

$$k_{{\text{s}}}^{e} = k_{{\text{s}}} i_{{\text{q}}}^{2}$$

(40)

$$k_{{\text{a}}}^{e} = k_{{\text{a}}} z_{{\text{c}}}^{{2}}$$

(41)

1.3 Calculation method of ${\text{ }}k_{{{\text{b}}}}^{e}$

Referring to Fig. 5, the bending deformation of the crankshaft is expressed as

$$\delta_{{\text{b}}} = \delta_{{\text{q}}} - \delta_{{\text{H}}}$$

(42)

where δ_q and δ_H represent the bending deformation along the rotation direction caused by F_q and F_H, respectively.

Then the equivalent torsional stiffness of the bending stiffness of the crankshaft can be expressed as

$$k_{{\text{b}}}^{e} = k_{{\text{b}}} r_{{\text{H}}}^{{2}}$$

(43)

Appendix 2

See Table 

Table 4 Test data

4.