Analysis of the reliability of the plates of complex shapes for the cams of automatic half-hose machines by the criterion of fatigue strength

The purpose of the work is to develop the algorithm for calculation of the probability of failure-free operation of flexible plates of complex shapes for the cams, using the criterion of fatigue strength, to make decisions on minimization of their sizes with the subsequent evaluation of the implemented measures. The studies are performed on the example of a plate for the cam, the configuration of which provides both beam and double console deformations at the same time. The comprehensive approach is presented, which includes dynamic analysis of the needle impact on the cam surface, computer simulation of geometric parameters of the plate by the finite element method (FEM) according to the condition of strength, and classical principles of mathematical statistics and Theory of Probability. The finite-element model with a volumetric 8-node element is used, which allows, with ease of calculations, to obtain the results with a high accuracy when determining the stress-deformed state of the study object. The computer simulation is performed using Code_Aster software with free access. Also, in the calculation of reliability, the random nature of the multifactorial influence on the value of the plate fatigue limit and the loading variations of the cam are considered. When determining the equivalent forces, taking into account the cyclical operation of the machine, the forces that affect the cams during the production of one product are taken as a loading. The numbers of cycles corresponding to these loadings are calculated by the formulas, which are made as a result of the analysis of trajectories of needles motion concerning the cams when knitting different sections of the product. When calculating equivalent stresses, only stresses that influence the accumulation of fatigue failures are used, that is, the values greater than the half of the detail fatigue limit. The proposed computational approach can be used to generate the cams with flexible work plates of different configurations, including under the equal stress condition.


Introduction
In the automatic half-hose machines, technological trajectories of the needles are set by the movements of their heels after the cams, accompanied by impact loadings and accumulation of fatigue damages. It is obvious that the intensification of the speed modes of the automatic half-hose machines and the enhancement of technological capabilities lead to their further growth. A promising solution that can increase the reliability of the automatic half-hose machines is to reduce the impact force in a "needle-cam" pair, primarily by increasing the flexibility of the cam.

Literature review
The bibliography that concerns this way of improvement is presented in [1,2,3], where the design features of the cams with flexible work plates (FWPs) are mainly considered. Some works [4,5] are devoted to the validation of FWPs by the criterion of static strength. However, given that FWPs are classified as the elements that are cyclically loaded and limited in size, we consider it appropriate to analyze them for fatigue strength in probabilistic figuration.

Research methodology
The purpose of the work is to propose the algorithm for calculation of the probability of failure-free operation of flexible plates of complex shapes for the cams, using the criterion of fatigue strength, to make decisions on minimization of their sizes with the subsequent evaluation of the implemented measures. The studies are performed on the example of a plate for the cam, the configuration of which provides both beam and double console deformations at the same time and the dimensions of which are obtained in [4] as the result of the computer simulation.
Classically, according to the criterion of fatigue strength, the probability of failure-free operation of the part is determined depending on the quantile [6]:   according to the medial values of fatigue is an important step in the calculation of fatigue. The traditional provisions, given in [7], are used: where Кthe total coefficient of multifactorial impact, which is determined by the formula: ( The coefficients k  , (3), respectively characterize the influence of concentration of normal stresses, absolute dimensions (scale factor), quality of processed surface, the anisotropy of material and surface strengthening on 1D   . The values of coefficients are determined according to the tables and graphs, which, in turn, are made according to the results of the experiments, or calculated in the absence thereof.
The dependence [8] is used in the calculation of k  : where qthe coefficient of metal sensitivity to the concentration of normal stresses;  theoretical coefficient of concentration.
Since for the structural bearing steel SHKH  According to (4), it is possible to take k= 1. Considering the dimensions of the dangerous section of the FWP console, it is also obvious that there is no influence of the scale factor, i.e. kd=1.
In terms of quantity, the influence of the quality of the detail surface on its fatigue limit is determined by the formulas: where RZthe roughness of the surface of the plate.
For the FWP of the cam, the average height of fine irregularities is RZ  3.2 mcm [10]. Taking (5, a), we have: The value of the anisotropy coefficient is determined according to the table [7] at B > 1200 MPa: Considering the instructions, provided in [7,8], the following values of the coefficient kv are recommended: 1.0without surface strain hardening; 1.15with surface strain hardening. In accordance with the technology of FWPs manufacturing, the value kv = 1.15 should be selected.
Applying the calculated and accepted values of coefficients to the (3), we have K = 1.24. As a rule, in the calculations, we have K = 1.5…3. The deviation occurs, first of all, because of the small dimensions of the dangerous section of the plate compared to most real parts. The reduction K is also influenced by the technology of the FWP production. Then, according to (2), we have  Table 1.
The forces are calculated by the formula from [1], the reliability of which is verified experimentally: where Vxthe horizontal component of the needle heel speed, which is equal to the circular speed of the points on the needle cylinder surface; the angle of inclination of the working surface of the cam to the horizontal line; mred, Credthe reduced mass and rigidity in the "needle-surface cam" pair; KCcoefficient that takes into account the deformation of the bend of the needle rod at the moment of impact; Frthe resistance force of the needle movement in the race, which is created specifically to eliminate the willful lowering of the needles; hthe damping coefficient.
In the calculations, the reduced mass mred is considered equivalent to the mass of the needle and Cred is calculated as in case of the series connection of rigidity of the needle during the interaction with a non-deformable cam (which is determined in the manner similar to h, considering the frequency characteristics of the oscillograph chart [1] and the rigidity of FWP of the cam, calculated in [4]).
The results of the calculations of needle loadings on FWPs of the needle-lifting and stitch cams by the formula (6) at different speed modes are presented in Table 1. Analytical determination of stresses in the FWP of the cam as a statically indeterminate spatial construction of complex shapes with two rigid structures is voluminous and uninformative in the end result. Therefore, the computer simulation is used, which also makes it possible to eliminate most of the assumptions made in the analytical  [11]. A finite-element model of the FWP is developed (Figure 1), where a volumetric 8-node finite element is used, which is simple for calculations and allows obtaining results with a high accuracy when determining the stress-deformed state of the study object. The simulation is performed using Code_Aster software with free access [12,13,14]. The maximum stresses at the FWPs points for all six loading modes are presented in Table 1, and sample screenshotsin Figure 2.  Considering the cyclicity of the automatic half-hose machines work, it is advisable to take forces acting on the cams during the production of one product (sock) as the loading. The number of cycles of loading i z is set for typical socks with jacquard weaving on the ankle and foot areas, with classic heel and toe, which are knitted during the direct and reverse rotations of the needle cylinder and with additional technological rows. The selection of needles by the cam with an incomplete closing of the third system during the knitting of spandex of the product, by the ornamented gates of three systems during the formation of the pattern on the ankle areas (the average number of included gates is 2,23 gates [15]), by the malfunction of gates on the foot area in the second and third systems, the devices for turning on and off the needles when knitting heel and toe tabs, as well as the number of needles and knitting systems, are considered. The calculation of i z takes into account the results of the analysis of trajectories of the movement of the needles in relation to the cams when knitting different sections of the products. The formulas for determination of i z in accordance with the modes of loading and the corresponding calculated numbers are presented in Table 2. Table 1. Summary information about loading of the cams.
The real loading with an obvious regularity of alternation of different levels during the cycle of knitting of one typical product is replaced by that one, which is equivalent in the degree of accumulation of the fatigue failure, by the formula [16]: where i i i р z z  the relative occurrence of the loading i F (the calculated values are given in Table 1).  Note. In Table 2, the following notations are used: 1. m = 2the number of simultaneously selected needles for one reverse movement of the cylinder; r the number of cams, which make the looped rows at one rotation of the cylinder; 2. Applying the values, presented in Table 1, to the (7)

Results
According to (1) If we set a goal that both needle-lifting cam and stitch cam have the same probability of failure-free operation, for example, () pt = 0.981, then, concerning the needle-lifting cam, we should solve the inverse task on determination of geometric parameters of its FWP, which would provide the corresponding equivalent stress lif  = 398.0 MPa. The adjustment is made by the width of the console beam В , where we have the dangerous section. The screenshot at such a stress at В = 3.17 mm is presented in Figure 4. Using the results of similar calculations, the graphical dependence of () pt on В ( Figure 5) is constructed, which is convenient to use when designing a plate for the needle-lifting cam at any given probability of failure-free operation.

Conclusions
The comprehensive approach to estimation of the probability of failure-free operation of plate of complex shape for the cam of automatic half-hose machine, using the criterion of fatigue strength, is proposed, which includes dynamic analysis, computer simulation by the finite element method (FEM), and principles of mathematical statistics and Theory of Probability. On the example of the plate of the cam as the detail, the sizes of which are limited, the feasibility of using the strength analysis in probabilistic figuration is shown, which in comparison with the traditional calculations (that consider the normalized coefficients of safety) provides a given level of reliability together with minimization of sizes. To describe the stress state of the plate as a statically indeterminate spatial construction of IOP Publishing doi:10.1088/1757-899X/1164/1/012012 9 complex shape with two rigid structures, the finite element method is used in Code_Aster software. The proposed computational approach can be used to generate the cams with flexible work plates of different configurations, including under the equal stress condition.