Analysis of self-locking factors of single wedge two-way backstop

: In this study, the structural composition and working principle of the single wedge two-way backstop are analysed, and the factors affecting the self-locking of the single wedge two-way backstop are explored. Two wedge shapes are established, the contact stress between the wedge and the outer casing is calculated by ANSYS and compared with the stress value calculated by Hertz theory, to verify the applicability of the finite element method.


Introduction
The overrunning clutch and backstop are the basic components of the mechanical transmission. The difference between overrunning clutch and backstop is that the outer casing of the overrunning clutch is generally not fixed and acts as an input or output shaft, while the outer casing of the backstop is stationary. There are two axes for input and output. The overrunning clutch can be divided into contact type and non-contact type according to the working principle, and the contact types are usually roller overrunning clutch, slider overrunning clutch and link-type overrunning clutch. Non-contact type is generally wedge clutch. The backstops are generally divided into single and bidirectional. At present, in the large-scale mechanical equipment, such as coal mines, cement machinery, roller-type overrunning clutch or one-way backstop is used, but for some small machines, such as subway or new energy vehicle, the two-way backstop has been favoured a lot.
At present, domestic scholars have made certain achievements in the research of two-way backstop. Wu et al. of Tongji University analysed the reverse locking and forward transferring functions of the single wedge two-way backstop and verified the feasibility of it [1]. Gao et al. of Tianjin University of Science and Technology applied the single wedge two-way backstop to the elevator. Besides, the calculation and analysis of the working principle of the wedge were carried out [2]. Wang and Sai verified the application of the single wedge two-way backstop in the transmission system of the logging instrument and the reliability of it [3]. Liao and Hou calculated and analysed the key parameters of the wedge on the two-way backstop, and the application of the backstop in the transmission system of the micro-resistance scanning imaging tool was introduced in combination with the wear-resistant design of the backstop [4]. Nowadays, the research on the two-way backstop is limited to the theoretical calculation. In this paper, the finite element method and the theory are combined to analyse the two-way backstop and verify the correctness of the suppose.

Principle implementing the reverse lock function of the single wedge two-way backstop
As shown in Fig. 1, in the sectional view of the single wedge twoway backstop, the wedge is installed between input and output shafts, the input shaft and the output shaft are opposite to each other.
The single wedge two-way backstop can only drive the driven shaft when the input shaft rotates. When the output shaft moves, the backstop will be in a stuck state. This paper studies the factors that affect the self-locking of the two-way backstop. The contact stress between the wedge and the outer casing is the focal point of the analysis [5]. Since the input shaft and output shaft do not make contact when the backstop in the stuck state, the force existing at the point c of the output shaft is studied. The force diagram is shown in Fig. 2.
As shown is the force analysis diagram of the single wedge two-way backstop. The points a and b are the tangent points of the wedge and the outer casing, the spring force on the wedge is ignored [6,7]. The force of the driven shaft on the wedge is F c (the frictional force exists at the point c), the force of the outer casing and the wedge are F 1 and F 2 , for the convenience of calculation, the point a is the coordinate system origin, and regard F f 1 is the X-axis, F N1 is the Y-axis. A Cartesian coordinate system is established, and the self-locking equation of the backstop is obtained

Research on self-locking factors
The wedge angles are θ 1 , θ 2 , θ 4 and the supporting forces of the outer casing are F N1 , F N2 . The horizontal distance between the long side of the wedge and the centre of the outer casing is Δ 1 . Similarly, referring to the lower margin, the horizontal distance between the short side of the wedge and the centre of the outer casing is Δ 2 . The coefficient of friction is μ, the vertical distance between the output shaft and the wedge action point c to the centre line oo 1 is d, by formulae (1)-(5), it can be seen that the factors affecting the F 1 and F 2 mainly have Δ 1 , Δ 2 , μ, d, R, so the influence of these factors on F 1 , F 2 and F c is discussed.

Hertz theoretical calculation
At the same time, in order to verify the correctness of the suppose, the contact stress between the wedge and the outer casing at the contact point is obtained according to the Hertz contact theory and the rectangular contact between the wedge and the outer casing becomes a width of 2a and a length of b by line contact: The contact radius of the inner surface of the outer casing is R 1 , the radius of the contact surface of the wedge is R 2 , E 1 , E 2 , ν 1 , ν 2 are, respectively, the elastic modulus and Poisson's ratio of the two contact materials. Take

Analysis of the influence of wedge shape on the contact stress
Two different wedge shapes are established in Figs. 8 and 9. The two sides of the wedge 1 are tangent to the inner radius of the outer casing, and the wedge 2 is not tangent to the inner radius of the outer casing, and the sharp edges of the wedge 1 and the wedge 2 are, respectively, inverted. Rounded corner R0.5.

Finite element simulation results and analysis
The contact stress of wedge 1, wedge 2 and outer casing was analysed by ANSYS. We can be seen in Figs. 10 and 11, the maximum stress of the outer casing is 82.186 MPa. The maximum stress of the wedge is 271.71 MPa (Figs. 12-14). At the same time, the contact state, contact pressure and slip distance of the contact area between the two different types of wedges and the outer casing are further compared. The results are shown in Tables 2 and 3. In summary, sliding distance and contact pressure of the wedge 2 are small, under the same conditions, the stress level of the wedge 1 is higher, so that the function of the wedge 1 is better.  Table 1 Effects of five factors on F 1 , F 2 and F c Value In this paper, briefly described the difference between the overrunning clutch and the backstop, and introduces the development of the single wedge double-way backstop and the structure, working principle and self-locking condition of the single wedge double stop. Five factors, including Δ 1 , Δ 2 , μ, d, R, the factors affecting the self-locking of the wedge are analysed. The increase of the Δ 1 , Δ 2 , μ will increase the contact forces F 1 and F 2 , while the F c will remain unchanged. F 1 and F 2 increase with the increase of R, and the increase of R does not affect F c . F 1 , F 2 and F c will decrease with the increase of d. The appropriate parameters are selected, Δ 1 = 3.5, Δ 2 = 3, d = 13, R = 17.5, μ = 0.15 obtain F 1 = 3704 N, F 2 = 3218 N, F c = 1499 N. Combine the Hertz contact theory to calculate the wedge stress, and obtain the average stress value of 48 MPa, which is not much different from the simulation value of ANSYS.
The wedges of two different shapes are designed and simulated by ANSYS software. By comparing the contact state, contact pressure and sliding distance, the contact pressure of the wedge 1 is smaller, so the shape of the wedge is better, which can be used for improving the lives of the wedge.