STATIC ANALYSIS of a SINGLE STAGE HYDRAULIC CYLINDER

V.GómezRodríguez,Juan José CabelloEras,Hernán HernándezHerrera,Rafael GoytisoloEspinosa. Bolivariano Technological Institute of Technology. Guayaquil. Ecuador. 1 vgomez@bolivariano.edu.ec Faculty of Engineering, Universidad de la Costa.Barranquilla, Colombia. jcabel1o2@cuc.edu.co Faculty of Engineering, Universidad de la Costa.Barranquilla, Colombia. hhernand16@cuc.edu.co #4 Faculty of Engineering, Universidad de Cienfuegos. Cienfuegos, Cuba. ragoyti@ucf.edu.cu


B. Static analysis of hydraulic cylinders.
With the development of computer-aided methods, and theavailability of new mathematical approaches, attemptshave been made to overcome difficulties posed byequations governing domains containing stepvariations inproperties [16], [17]. The model proposed by [15] considered the cylinder as a beam with step variationsin rigidity, subjected to perfect loading (no eccentricities),and lacking of initial curvature. In this approach thepossible influence of a loose fit in the joints is notconsidered, nor the effect of self-weight in the case whenthe cylinder adopts a position at an angle to the vertical. In paper [18] analyzed the influence of the extension length on loads distribution through sliding contacts along the boom. In another work [19], a method is used which permitsdetermination of stability characteristics for cylinder of anynumber of stages. The results obtained in that work aresuperior to results obtained in other works. A deficiency inprevious methods stems from the fact that the self-weightof the cylinder is not considered, despite the fact that it hasa substantial influence on the deformation of the rod in anarticulated system. The influence is due to the bendingcaused by the self-weight, and to the moment causing asagging equal only to the deformation produced by theloose fit of the sliding joint in each stage. In other work [20] the bending is determined without considering that therod is subjected to a combination of transverse andlongitudinal flexure. In further studies empirical methodshave also been presented to determine the stability ofhydraulic cylinders, and the finite element method hasbeen used, in which the cylinder is modeled as a columnwith a cross-section that varies in the longitudinaldirection.

II. THEORY. C. Modeling of the system
In the scheme of analysis presented in this study thefollowing factors are taken into account, which generallyhave not been considered in previous studies: • Loose fit existing between the piston -body androd -and the axle box.
• Self-weight of the cylinder and the hydraulicfluid. It is considered that the two components of the model, CAand CB (Fig. 1b), have a rigidity equal to that of acylindrical tube (sleeve) R 1 and that of a rod, respectively R 2 . Thecomponents are subject to a uniformly distributed loadingequal to the combined sum of the various components andthe hydraulic fluid, divided by the length. The transitionpoint C is the point where the axis of the cylindrical tubeand the axis of the rod intersect, when the system deformsdue to the action of applied loads. The weight W of thesliding connection between the cylindrical body and therod is considered as a load concentrated at the transitionpoint C, as indicated in Fig. 1b. The weights of the axlebox and the piston head are included in the concentratedload W acting at the transition point C.
In Fig. 2   In this an cylinder. compatib stage hyd ' but w passes th deviation part, owi In Fig. 5 lines is d moment nalysis, the dif These differe bility to obtain draulic cylinde ithout any ax hrough these n at any point ng to the flexu 5 the portion o due to the slo of the system

III. EQUATIONS FOR THE CALCULATION SCHEME.
For an arbitrary section of the tube part, for example the section 1-1 of Fig. 5, the flexural moment that produces the curvature is obtained by taking the product of the flexural rigidity with the second derivative of the displacement for this part, namely: The flexural moment in this section produced by theexternal loads and the reactions is: (2) Where: y 1 -displacement at a distance z in the cylinder,measuredwith respect to the left support. For equilibrium the internal and external moments must beequal. Thus combining equations (1) and (2) one obtains.

(3)
Where: An analysis is next carried for the rod part of thestructural member (section 2-2, Fig. 5), for which one obtains: 2 (4) Where: The differential equations (3) and (4) describe thebehavior of the displacements in the tube and the rod parts of a hydraulic cylinder carrying an axial load.The boundary conditions, and the continuity conditions to be satisfied in solving these equations are: The solution for equations (3) and (4) Where: Where C 1, C 2, D 1 , and D 2 are constants to bedetermined from the conditions (5). Application of the equations (5) leads to the following values for the constants C 1 and C 2 .
sin 1 cos (10) With the application of the continuity conditions at the sliding joint, i.e. at z=l c and substituting the values of C 1 and C 2 from equations (10) and (11) Where: The equations for the slope are obtained as the first derivative of the displacements (6) and (7) The values of the constants C 1 , C 2 , D 1 and D 2 in the displacement equations (6) and (7) and the equations for the slope (14) and (15) are obtained from the relations (10), (11), (17) y (18), respectively. The slope β is evaluated using a procedure proposed by the authors. The other two unknown values, β c y β v which appear in the equations of displacement and slope and in the equations to determine C 1 , C 2 , D 1 and D 2 are found by enforcing the following conditions of compatibility. On combining the equations (10), (11), (14), (15), (17), (18), and (20)  Solving simultaneously the equations (21) y (22) oneobtains:
The equation for the flexural moment for the tube part of the body is found using equations (12), and similarly the flexural moment in the rod part of the body is found using the equation:

(27)
IV. CONCLUSIONS. An approach has been presented for the static analysis of a hydraulic cylinder of a single stage. The approach is comprehensive, and permits consideration of a number of factors, which have not been accounted for in previous studies. The equations obtained allow to improve the design of the hydraulic cylinders providing economical designs with appropriate safety factors.