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Criterion of stability of hydraulic cylinder and method of increasing its reliability under longitudinal-transverse loads

Currently, in most cases, the study of the operability of hydraulic cylinders in terms of their bearing capacity is carried out by means of assessing the longitudinal stability of a compressed rod with the variable cross section. In this case, the inclination of the rod in space is not taken into account. However, during operation, as a rule, a hydraulic cylinder inclined in space due to longitudinal-transverse loading is deformed in the vertical longitudinal plane with the appearance of a full-scale deflection. So, the limit value of the longitudinal compressive force is less than the limit compressive force of a stable rod, which implies that a rod having an initial curvature is more susceptible to bending, appearance of residual deformations in its elements and has less reliability than a straight, vertically arranged one. As the guide elements of the hydraulic cylinder wear out, its total deformation increases, which leads to an increase in longitudinal-transverse loads. With regard to long-stroke hydraulic cylinders, a special danger is their immediate stop in space when the working equipment of the machine meets an insurmountable obstacle. The purpose of the study is a scientifically based description of the areas of functioning of the hydraulic cylinder in the conditions of longitudinal-transverse bending or stability, as well as the boundaries separating them


Introduction
Multilink equipment of modern hydroficated road and construction machines (RCM) is driven by double-acting hydraulic cylinders with a one-sided rod, the operation of which is accompanied by their complex plane-parallel movement in space. At present, double-acting hydraulic cylinders with a one-sided rod are widely used in modern road and construction machines (RCM). Functionally, every hydraulic cylinder is exposed to a whole complex of operational longitudinal-transverse loads due to which it is deformed in the vertical longitudinal plane and a complete deflection appears, which sharply increases the acting bending loads and reactions in movable pressurized couplings. Currently, in most cases, the study of the operability of hydraulic cylinders according to the parameters of their (load) bearing ability is carried out by means of assessing the longitudinal stability of a compressed rod with variable cross section. Moreover, according to its flexibility, a conclusion is drawn on the applicability of either Euler formula or Yasinsky formula, which, unfortunately, in no way take into account the inclination of the rod in space. However, during operation, as a rule, a hydraulic cylinder inclined in space due to longitudinal-transverse loading is deformed in the vertical longitudinal plane with the appearance of a complete deflection. In this case, the limit value of the longitudinal compressive force is less than the limit compressive force of a stable rod, which implies that a rod having an initial curvature is more susceptible to bending and the appearance of residual, plastic deformations in the sections of its elements and has less reliability than a straight, vertically arranged one. As the friction surfaces of the elements of the hydraulic cylinder wear out, which again leads to an increase in its total deformation and, respectively, to an increase in the acting longitudinal-transverse loads, the operating conditions of the hydraulic cylinder deteriorate with greater intensity, as a result of which its reliability and service life are reduced both in terms of its bearing and sealing abilities.
With regard to long-stroke hydraulic cylinders, their immediate stop in space is a particular danger when a multi-link equipment of a machine meets an insurmountable obstacle. All of the listed disadvantages of the traditional design of the hydraulic cylinder are eliminated by bringing the hydraulic cylinder from the state of longitudinal-transverse bending to the state of stability or close to that through the support of the body (liner) of the hydraulic cylinder with an intermediate support.
Hence, the aim of the work is a scientifically based description of the areas of functioning of the hydraulic cylinder under conditions of longitudinal-transverse bending or stability, the boundaries separating them, as well as developing recommendations for improving the design of the hydraulic cylinder. The object of the study is a hydraulic cylinder. The subject of the study is methods for assessing its bearing capacity in conditions of longitudinal-transverse bending and stability [1][2][3][4][5][6][7][8].

Materials and Methods
The methodology of the article includes the provisions of the mechanics of a deformed body, hydraulics, and technical operation.
Data processing was carried out using MathCAD and MathLab programs.

Results
The hydraulic cylinder during operation is subject to longitudinal-transverse loading, leading to its longitudinal-transverse bending in the vertical longitudinal plane [1].
The longitudinal-transverse bending of an element should mean its deformation under the action of the longitudinal force. It happens due to the three main reasons: either there is an initial irregularity due to which the longitudinal force creates a bending moment relative to the center of gravity of the cross sections; or there is an eccentricity of the load application, i.e., the line of action of the longitudinal force does not pass through the center of gravity for solid sections or the center of bending for hollow ones; or there is a transverse load causing a deflection, which, in turn, creates a non-zero bending moment from the longitudinal force PS.
In the case with RCM cylinders, especially long-stroke ones, all the three cases take place, as a result of which calculation is possible only according to the deformed scheme, that is, with full account for the total deflection of the hydraulic cylinder.
However, in a number of works, the calculation of compressed-curved rods during their longitudinal-transverse loading is recommended not to be carried out according to permissible stresses, but according to permissible loads. So, in [2], the limit value of the longitudinal compressive force E S P for the hydraulic cylinder and its allowable value are calculated by Euler formula.
from the condition of loss of stability by a hydraulic cylinder or another similar element. Where: E is Young's modulus of the rod material, I is the minimum moment of inertia of its section, and l is the length of the rod.
It is clear that the rod with greater flexibility, when other parameters are unchanged, has lower compressive and bending compressive strength, due to which the depletion of the bearing strength for stability occurs, as a rule, before the material's safety factor has been exhausted.
However, when calculating longitudinal-transverse bending, it makes no sense to talk about the loss of stability in the plane of action of the transverse load, since, firstly, at any value of the axial force PS, the rod experiences bending, and when it increases, there is no qualitative change in the nature of the deformation, as is the case with loss of stability. Secondly, Euler formula is applicable only under the condition where: λ is the flexibility of the considered rod, depending on its reduced length μl, dimensions and shape of the cross section, that is, on the main central radius of inertia i, which is usually minimal min i ; l is the length of the rod; lim λ is limit or boundary flexibility for the material of the rod, depending only on the physico-mechanical properties: Young's modulus E and the limit of proportionality pr σ of the material of the rod; μ is the effective length factor. Hence, the task of describing the boundary conditions and areas of functioning of the hydraulic cylinder, in which it may lose its working capacity due to loss of bearing capacity either because of limit longitudinal-transverse deformations, or because of the loss of stability, is of interest.
Currently, mechanical engineering aims to create hydraulic cylinders with parameters for the main and additional series in units of measurement according to standards In relation to RCM hydraulic cylinders, these parameters lie in the ranges: p = (6 ... 25) The speed of movement of the rod in these documents is not specified, but lies in the range of dz/dt = (0.1 ... 1.0) m/s, and, as applied to RCM, does not exceed 0.5 m / s. This paper considers one of the most common hydraulic cylinders with the following characteristics: p = 25 MPa; z = 1400 mm; D1,3 = 140 mm; D2,4 = 90 mm.
The hydraulic cylinders of multi-link RCM during the operation of the machine make significant spatial plane-parallel movements, which are described by the possible and working ranges of their movements [3,4].
In accordance with the design schemes of the equipment of some multi-link RCMs proposed in [3,4], as well as their geometric characteristics and parameters of the hydraulic cylinders used in them [1,3], possible ranges of the spatial arrangement of the hydraulic cylinders as applied to the boom, stick and bucket hydraulic cylinders, for example, of single-bucket excavators EO-3322A and EO-4121A with a backhoe, are 88°, 103°, 214° and 90°, 105°, 191°, respectively [3][4][5]. Moreover, the ranges of the spatial arrangement of these hydraulic cylinders only due to the movement of their own rods make up 88°, 9°, 7° and 90°, 6°, 9°, respectively [5].
Taking into account the provisions of [1], the compressive stresses σcomp(xσ)arising in a dangerous cross section xσ of the hydraulic cylinder rod under longitudinal-transverse loading without considering the kinematic features of the working equipment of a specific RCM are analytically described by the equation: Where: the first summand gives the value of the normal stress from the action of the longitudinal compressive force РS, the second provides the value of the greatest compression stresses caused by the transverse load МQ(xσ) from the weight of the hydraulic cylinder, the third and fourth ones are the same, caused by additional bending in the presence of eccentricity е(хσ) in its supports and total deflection yT(xσ), which are the shoulder of the application of longitudinal compressive force РS; F(xσ) and W(xσ) are the area and moment of resistance of the rod section, respectively; xσ is the coordinate of its dangerous cross section.
The total deflection yT(x) of the hydraulic cylinder is equal to the sum of its components At the initial moment of operation, the deflection yT(x) can be represented by the sum ).
In equations (3) and (4) ) (  Replacing the stresses σsrt(xσ) with the limit σls and, transforming the equation (3) with respect to the longitudinal compressive force, we obtain the equation for finding its maximum value hor B S P at which a horizontally or inclined hydraulic cylinder loses its working capacity under longitudinal-transverse loading as a result of limit deformation of its rod, in this case, a circular solid section (Fig. 1 Since record (6) is valid for a horizontal and inclined hydraulic cylinders, then for a vertical one it transforms into (7) due to the fact that the characteristics (6) Where, Θ k is the angle of inclination of the hydraulic cylinder to the surface of gravity. In this case, record (7) does not change regardless of the spatial location of the hydraulic cylinder.
The maximum allowable value of the longitudinal compressive force stab S P , provided that the hydraulic cylinder loses its working capacity due to its stability loss according to recommendation [6], is found from equation (  However, Euler formula is applicable only under condition (2) [2], which imposes a restriction on equations (11) and (12).
Considering that the length coefficient μ, which determines the nature of the fastening of supports, for most common in RCM double-acting hydraulic cylinders with a single-   (13), is not applicable, and any rod, due to its low flexibility because of its relatively small length and significant diameter, must be assumed to be stable. Figure 3 shows the dependences of the flexibility λ of the hydraulic cylinder on its full length lhz for various values of the internal diameter of the liner D3 according to GOST 6540-68 "Hydraulic cylinders and pneumatic cylinders. Main sizes", as applied to RCM, taking into account their limit flexibility lim λ .
Given this information (Figs. 2 and 3), special attention from the standpoint of the hydraulic cylinder losing operability due to its loss of stability should be given to the rods of significant lengths with small diametrical dimensions.       In this case, it is mandatory to check the operability of the designed hydraulic cylinder using the stability criterion proposed above (12), the result of which can be the following  (6) and (7),.
Here, the first inequality (16) corresponds to the condition of disruption of the hydraulic cylinder's operability due to the possible loss of its bearing capacity [5] as a result of loss of stability, while the second (17) and third (18) entries indicate the same, but because of the extreme longitudinal-transverse deformation of the long bearing elements of the hydraulic cylinder, mainly its rod (Fig. 1).
It should be noted that hydraulic cylinders with great flexibility must first be tested for stability. For this, it is recommended to use the proposed stability criterion (12). Moreover, in all cases, one should take into account the parameters of their greatest operational loading and the characteristics of the spatial arrangement, as applied to a specific RCM. P are present, the hydraulic cylinder operability should be assessed by the characteristics of the stress-strain state of the long elements of the hydraulic cylinder and the reactions in its movable, sealed "pistonliner" and "rodguide sleeve" couplings.
The main drawback of the existing design of a double-acting reciprocating hydraulic cylinder with a one-sided rod is that before the application of the operational longitudinal compressive force, it has full deflection, defined as the sum of the deflection due to the misalignment of its main load-bearing elements (rod and liner), because of the presence of gaps in its "piston -liner" and "rodguide sleeve" couplings, deflection as a result of the presence of a possible initial (technological) curvature of the long elements (rod and body), regulated by the technological tolerance of misalignment of manufacturing long products, as well as the deflection from the action of shear forces -the weights of these elements (Figure 9 and 10.). After the application of the operational longitudinal compressive force (Fig. 11), that is, when liquid is supplied under pressure to the piston cavity of the hydraulic cylinder, the total deformation of the hydraulic cylinder (Fig. 12) increases [1] and, being the shoulder of the application of this force, leads to an increase in the total bending moment, which can cause critical stresses and, accordingly, the appearance of plastic deformations at the hydraulic cylinder rod and the subsequent loss of operability by the   As the friction surfaces of the elements of the hydraulic cylinder wear out, again leading to an increase in its total deformation (Fig. 13), respectively, to an increase in the acting longitudinal-transverse loads, stresses (Fig. 13) and reactions (Fig. 14) in movable pressurized couplings, the operating conditions of the hydraulic cylinder deteriorate with greater intensity [5], which leads to a decrease in its reliability and service life, both in its bearing and in sealing capacities [6].

Discussion
The listed disadvantages of the traditional hydraulic cylinder design can be eliminated by bringing the hydraulic cylinder from the state of longitudinal-transverse bending to the state of stability or close to that through the support of the hydraulic cylinder body (liner) with an intermediate sensor support (Fig. 15). Bringing the hydraulic cylinder from the state of longitudinal-transverse bending to the state of stability or close to that through supporting the body (liner) of the hydraulic cylinder by an intermediate sensor backup support suggests several possible support options depending on the amount of supporting force when the following conditions are met, namely: the absence of reaction in the movable sealed "rod -guide sleeve" coupling, that is, R1= 0 (Fig. 16); the absence of reaction in the movable sealed "piston -liner" coupling, that is, R2= 0 (Fig. 17) and the absence of a complete deflection of the hydraulic cylinder at the point of connection of the hydraulic cylinder body with the backup support, that is, yT=0 (Fig. 18), which is practically accompanied by lack of full deflection of the hydraulic cylinder rod.    In addition, as follows from the analysis of the dependences shown in Figures 19-21, this support option is characterized by the minimum value of R2 reaction in the "pistonliner" coupling (Fig. 22). Moreover, judging by the direction of the reactions (Fig. 23 and 24), with the support of the hydraulic cylinder, according to the last two options, the contact points of the long supporting elements of the hydraulic cylinder change to the opposite. Regarding the total deflection yT, it should be noted that the second support option, in relation to the first one, is somewhat preferable (Figs. 23 and 24).  However, which is noteworthy, both values are much less than the total deflection of the hydraulic cylinder of the traditional design (Fig. 13), which confirms the relevance and feasibility of the research conducted in this direction.

Conclusions
1. The most preferred option for supporting the hydraulic cylinder should be considered the one with an intermediate sensor backup support, which realizes the support force ) 0 ( 1  R F , additionally relieving the least reliable "rod -guide sleeve" coupling of the hydraulic cylinder, bordering on the environment saturated with abrasive dust.
2. Reducing the reactions R1 to zero and R2 by more than seven times acting in movable sealed "rod -guide sleeve" and "piston -liner" couplings, which, in turn, contributes to a significant reduction in the wear rate of their constituent elements, which increases the durability of the hydraulic cylinder.
3. A decrease of the total deflection yγ more than four times is accompanied by a decrease in stresses σCOMP by (13 ... 15) %. 4. In addition, the support of the hydraulic cylinder makes it possible to reduce the stress σCOMP due to the total decrease in the second and third summands of equation (1)  5. When creating an intermediate sensor backup support, it is necessary to take into account the nonlinear nature of the support force F. 6. The design parameters of the auxiliary hydraulic cylinder of the intermediate sensor support should be considered for a specific hydroficated RCM taking into account its working process and loading mode. So, for example, the working process of the hydraulic cylinder of the handle of the bucket excavator of size IV with the backhoe is limited by the characteristics z and Θ, respectively (0.0 m ... 0.8 m) by (17° ... 18.5°), which greatly simplifies the design of the supports. 7. All of the above indicates the relevance of research in this direction, the results of which contribute to increasing the efficiency of the use of hydroficated RCM in general.