Before analyzing the ratchet tooth root bending stress, it is necessary to introduce the incision effects theory (Verkhvsky et al., 1967). Take the example of a steel plate with an incision, which is shown in Fig. 1.

First, we assume that the incision is fairly shallow. Thus, the depth of the incision crack under a collapsing force will not affect the entire width of the steel plate, as in the following: 12tρ<a, where ρ is the curvature of the incision. t is the depth of the incision. a is the width of the steel plate.

For this kind of plate, the effects of a shallow incision would be limited to a certain depth, as follows: 2a0=2tρ, where a0 is the maximum depth of the incision crack under a collapsing force.

According to the analysis of the gear, the maximum stress should be generated at the tooth root fillet when the ratchet is being loaded. To ensure that the designed ratchet meets the strength requirement, the folded section hypothesis is introduced to analyze this maximum stress.

In the folded section hypothesis, the equivalent critical cross section includes two parts, where the first one can be determined by the theory of the incision depth effects. For most ratchets, it satisfies the requirement that is 2tρ<a, so the length of the first one is as follows: 3lAB=2trρ, where tr is the distance between the addendum circle and dedendum circle of the ratchet.

Along the length lAB, we make the first polyline. Then, we draw a radial from the tooth root fillet center and go through point A, which is the mid-point of this arc. We take point A as the center and the length lAB as the radius to draw another arc; this arc intersects the radial at a point which is named point B. The position of points A and B is shown in Fig. 2.

The second part of the equivalent critical cross section is determined by its geometric structure. We connect points B and O so that the line BO intersects the axle hole at point C. The polyline ABC is the projection of the equivalent critical cross section on the front (it is also named the component hazard section).

As can be seen from Fig. 2, ratchet teeth are asymmetrical structures. If the ratchet is cut along the radial direction and the circular contour is straightened, then a steel plate with asymmetrical incision is shown. For the asymmetrical structure, its neutral layer can be determined by the area of the stress diagram as follows (Verkhvsky et al., 1967) 4(ρ+ya)-a0-(ρ+ya)ln⁡ρρ+a0+(ya-a0)22=yb22ya+yb=h, where a0 is the projection of lAB in the radial direction, ya is the distance between the dedendum circle and neutral layer, and yb is the distance between the neutral layer and the axle hole.

According to the neutral plane, when the ratchet is applied at force F, the torque M is as follows: 5M=F×L, where L is the length of the arm of the force.

For the bending deformation, tension and compression occur simultaneously. The ratchet is no exception. When the ratchet tooth is loaded, the polyline AB and BD will be tense. To analyze its deformation, we draw another polyline A1B1C1 in the same way as the polyline ABC, but below it, take a microelement KF between AB and A1B1 and take another microelement K1F1 between BD and B1D1. When the ratchet tooth deforms, the line BD is pivoted by an angle Δγ around the point D to B2D and the line AB around to A2B2. This is shown in Fig. 3. The microelement KN is stretched to KF, and K1N1 is stretched to K1F1. Now assume that there is no lateral pressure between microelement. According the Hook's law, the stress of microelement KN and K1N1 is as follows: 6σ1=ε1E=FNKFE=[(ya-a0)cos⁡α+u1]ΔγE(ρ+lAB-u1)Δα7σ2=ε2E=F1N1K1F1E=u2ΔγE(ρ+lAB)Δα⋅cos⁡α. In Eq. (6), KN is as follows: 8KN=(ya-a0)Δγ⋅cos⁡α+u1Δγ, where ε1 and ε2 are the relative elongation of the microelement, E is the elasticity modulus, u1 is the distance between the point B and the microelement KF, and α is the angle of the line AB and the line O1O.

The compression occurs in the BD of the second polyline BC. It is also pivoted by an angle Δγ around the point D to DC2. Now take a microelement K2N2 between the line DC and the line D1C1. When DC is pivoted, then the microelement K2N2 is compressed into K2F2. Then, assume that there is no lateral pressure between microelement. According the Hook's law, the stress of microelement K2N2 is as follows: 9σ3=ε3E=F2N2K2F2Eu3ΔγEρ+lABΔα⋅cos⁡α, where u3 is the distance between the point D and the microelement K2F2.

In the following, according to the moment equilibrium condition: 10M=∫0lABσ1bya-a0cos⁡α+u1du1+∫0ya-a0σ2bu2du2+∫-yb0σ3bu3du3, where b is the width of the ratchet.

Substitute Eqs. (6), (7) and (9) into Eq. (10) and take the terms that do not change with u1 and u2 and u3 outside the integral symbol as follows: 11ΔγEΔα=Mb[∫0lAB[(ya-a0)cos⁡α+u1]2du1ρ+lAB-u1+∫0ya-a0u22du3(ρ+lAB)cos⁡α+∫-yb0u32du4(ρ+lAB)cos⁡α]. For convenience, let the integral in the square bracket equal N and solve it as follows: 12N=∫0lAB(ya-a0)cos⁡α+u12du1ρ+lAB-u1+∫0ya-a0u22du3(ρ+lAB)cos⁡α+∫-yb0u32du4(ρ+lAB)cos⁡α=(ya-a0)2cos⁡2αln⁡ρ+lABρ+2(ya-a0)cos⁡α(ρ+lAB)ln⁡ρ+lABρ-lAB-(ρ+lAB)lAB-lAB22+(ρ+lAB)2ln⁡ρ+lABρ+(ya-a0)33(ρ+lAB)cos⁡α+yb33(ρ+lAB)cos⁡α. Here, N is the section factor which represents the geometric properties of the ratchet.

Substitute Eqs. (11) and (12) into Eqs. (6), (7) and (9). The stress at any point on the polyline ABC can be obtained by the following equation: 13σ1=M[(ya-a0)cos⁡α+u1]bN(ρ+lAB-u1)14σ2=Mu2bN(ρ+lAB)cos⁡α15σ2=Mu3bN(ρ+lAB)cos⁡α. To obtain the maximum stress of the critical cross section of the equivalent at the point A, let u1=lAB. 16σ1=M[(ya-a0)cos⁡α+lAB]bNρ, where σ1 is the maximum ratchet tooth root bending stress. Compare σ1 with the bending fatigue limit stress σFlim⁡. If σ1<σFlim⁡, then the design ratchet meets the strength requirement.

To acquire the maximum stress, the strain gauges were pasted on the tooth root fillet, which is shown in Fig. 11.

The structure of the ratchet test bed is shown in Fig. 12. It consists of a baseboard, shaft, pawl, lock nut, ratchet and support plate. The baseboard was bolted on the working table of an electronic universal testing machine. The pawl was fixed on the loading end of electronic universal testing machine. The shaft support is the L-shaped plate, which was bolted to the test bed bottom. The shaft and the ratchet were connected by the buttress thread, which only transmits power in one direction. The lock nut was soldered to the shaft to prevent the rotation of the ratchet under load.

The experiment was carried out as follows: we operated the electronic universal testing machine to lower the pawl to the level of the working surface of the ratchet. Then, we loosened the shaft support bolts and pushed the test bed to the position where the pawl coincides with the ratchet-tooth-loaded surface. Thereafter, we adjusted the ratchet and loading pawl until full contact was realized.

Then, we applied force to the working surface of ratchet and recorded the readings on strain gauge when the force reached the set value.

In the experiment process, each ratchet tooth was pasted into two sets of strain. We averaged the two sets of data and multiplied the elastic modulus E to obtain its stress. The jaw vise was used to assemble and disassemble the ratchet, and two teeth were deformed in this process. So, only four teeth could be tested. Finally, the stress data of three teeth were collected successfully. These experimental data are shown in Fig. 13, and the mathematical model results are also shown in the figures to make a contrast. Its deviations under different forces are shown in the right-hand side of Fig. 13.

As can be seen from Fig. 13, the experimental data fit well with the model analysis data and are also larger. The average deviation between the mathematical model and the experiment of tooth 1 is approximately 6.9 %, tooth 2 is approximately 6.8 %, and tooth 3 is approximately 9 %.

It also should be noted that the average deviation between the mathematical model and experiment is more than 6 %, and the experimental data are closer to the simulation data. This further verifies that the non-standard ratchet, which has a big shaft hole, has a better torsion resistance.

Apart from that, the deviation between the experimental data and the simulation data and the model analysis data is not constant and shows a decreasing trend with the loading force. The reason for that has two aspects. First, the low machining precision of the ratchet test bed causes a tiny gap between the lock nut and the ratchet, and it could not be eliminated completely by preloading. So, the ratchet will rotate on a tiny angle, with the loading force and results in a variable deviation.

Second, the ratchet tooth will have a tiny deformation as the loading force increases in the loading process. This deformation changes the contact condition and results in a variable deviation.