The effect of braid angle on hydraulic hose geometry

Citation: Dýrr, F.; Bureˇcek, A.; Hružík, L.; Polášek, T.; Ledvoˇn, M.; Dvoˇrák, L. The Effect of Braid Angle on Hydraulic Hose Geometry. Processes 2024,12, 152. https:// doi.org/10.3390/pr12010152 Academic Editor: Qingbang Meng Received: 12 December 2023 Revised: 2 January 2024 Accepted: 5 January 2024 Published: 8 January 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). processes Article The Effect of Braid Angle on Hydraulic Hose Geometry Filip Dýrr * , Adam Bureˇcek , Lumír Hružík, Tomáš Polášek, Marian Ledvoˇn and Lukáš Dvoˇrák Department of Hydromechanics and Hydraulic Equipment, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic; [email protected] (A.B.); lumir[email protected] (L.H.); [email protected] (T.P.); [email protected] (M.L.); [email protected] (L.D.) *Correspondence: [email protected] Abstract: Hydraulic hoses are part of most hydraulic systems, from industrial hydraulics with open loop hydraulic systems to mobile hydraulics with closed loop hydraulic systems. The design parameters of hydraulic hoses may influence the duty cycle dynamics of these systems. One of the factors that influence the behavior of a hydraulic hose under pressure loading is the steel braid angle with respect to the hydraulic hose axis. This work aims to determine the effect of the hydraulic hose braid angle on the change in its geometry. The next objective is to determine the forces that occur at the hose ends under pressure loading. The stresses occur when fluid pressure is applied to the inner wall of the hydraulic hose. Consequently, these stresses are transferred to the hose ends through the steel braid or spiral. The phenomenon of the neutral braid angle provides a balance between the stresses generated inside the hydraulic hose. Therefore, hydraulic hose manufacturers try to produce hydraulic hoses with a neutral braid angle, because the lifetime of the hydraulic hose is also related to this. As part of this research work, an experimental device was constructed in order to measuring the properties of hydraulic hoses. When the hose was loaded with fluid pressure, the change in hose geometry was measured and the angle of the hose braid was measured simultaneously. Upon the measurements, the effect of the braid angle on the hose behavior under pressure loading was determined. Keywords: braid angle; hose geometry; hydraulic hose; tensile force 1. Introduction The parts of hydraulic systems are connected to each other by hydraulic lines, which can be formed from hydraulic hoses. There are advantages and disadvantages to using hydraulic hoses as hydraulic lines. The main advantage is the relative movement of the connected parts with respect to each other, Another advantage can be the reduction in pressure peaks during hydraulic shock due to the hydraulic capacity of the hydraulic hoses [ 1 , 2 ]. However, the mentioned influence of the hydraulic capacity of the hose also has the opposite effect; that is, the decrease in the hydraulic system’s stiffness. Wang described the problem of double-acting cylinder position control, where the hydraulic hose influences the system dynamics. Wang compared the PID controller with the ADRC (Active Disturbance Rejection Control), where the ADCR variant of the controller performs better and can compensate for the influence of the hydraulic hoses [ 3 ]. Previously, the influence of different parameters on the performance of hoses has been investigated [ 4 ]. The hydraulic capacity, bulk modulus [ 5 , 6 ] and viscoelastic properties [ 7 , 8 ] of hydraulic hoses are also related to the design. A hydraulic hose consists of a rubber inner tube to ensure tightness. The next design component is the braid or spiral, which determines the maximum pressure loading. Hydraulic hoses are available with one or more braids (spirals) depending on the working pressure [ 9 ]. With more braids (spirals), the flexibility of the hose is reduced. The working pressure of these hoses is up to 400 bar in common high-pressure hydraulic applications. A hydraulic hose must be able to withstand the maximum working pressure and must not be Processes 2024,12, 152. https://doi.org/10.3390/pr12010152 https://www.mdpi.com/journal/processes Processes 2024,12, 152 2 of 15 damaged even by short-term overloading. For this reason, pressure tests of hydraulic hoses are carried out [ 10 ]. For high-pressure hydraulic hoses, the braid or spiral is made from steel wire [ 11 , 12 ]. For some applications, the hose braid or spiral can be made from aramid fiber or polyvinyl acetate [ 13 , 14 ]. When a hydraulic hose is loaded with fluid pressure, stresses are generated and transferred through the braid to the hose ends. Tensile forces are applied to the hose fitting if the hose is not installed correctly. This can result in damage to the hose and failure of the entire hydraulic system. Therefore, the importance of the right hose mounting is paramount. The angle of the hydraulic hose braid has a major influence on the hydraulic hose deformation and the forces acting on the hose ends [ 15 – 18 ]. When the hose is loaded by fluid pressure, the hose steel braid tends to deform so that the opposing strands of the braid are at a neutral braid angle to each other at which the axial and hoop stress components are in balance. This deformation causes relative friction between the steel braid and the rubber tube or rubber interlayer of the hose. The degree of deformation of a hydraulic hose influences its service life [ 19 ]. The solution to this problem is the method of braiding the steel wires at a neutral braid angle ϕN = 54.7356 ◦ to the hose axis. In this case, the balance between axial and hoop stresses is ensured. When manufacturing hydraulic hoses, the aim is to maintain the required braid angle. This is ensured by the right combination of the braiding rate and feed rate of the manufacturing machine [20]. The aim of this work is to determine the influence of the braid angle on the deformation of the hydraulic hose and the forces acting on the hose ends. Within the framework of this research, an experimental device was created in order to test hydraulic hoses. The result is the determination of the dependence of the change in hose length and tensile force on the braid angle for hydraulic hoses with different internal diameters and different braid designs. This work serves as a summary of the measured data, which can be used for further research in the field of mathematical 3D modelling, for example in the field of finite element analysis of the deformation stress of hydraulic hoses. 2. Theoretical Background The definition of the neutral braid angle, at which balance is reached between the stresses generated, is based on the theory of a closed cylinder of radius rand wall thickness s. The working pressure pof the fluid acting on the hose inner wall generates hoop stress σOand axial stress σA(see Figure 1) in the hose wall [21]. Processes 2024, 12, x FOR PEER REVIEW 2 of 16 A hydraulic hose must be able to withstand the maximum working pressure and must not be damaged even by short-term overloading. For this reason, pressure tests of hydraulic hoses are carried out [10]. For high-pressure hydraulic hoses, the braid or spiral is made from steel wire [11,12]. For some applications, the hose braid or spiral can be made from aramid fiber or polyvinyl acetate [13,14]. When a hydraulic hose is loaded with fluid pressure, stresses are generated and transferred through the braid to the hose ends. Tensile forces are applied to the hose fitting if the hose is not installed correctly. This can result in damage to the hose and failure of the entire hydraulic system. Therefore, the importance of the right hose mounting is paramount. The angle of the hydraulic hose braid has a major influence on the hydraulic hose deformation and the forces acting on the hose ends [15–18]. When the hose is loaded by fluid pressure, the hose steel braid tends to deform so that the opposing strands of the braid are at a neutral braid angle to each other at which the axial and hoop stress components are in balance. This deformation causes relative friction between the steel braid and the rubber tube or rubber interlayer of the hose. The degree of deformation of a hydraulic hose influences its service life [19]. The solution to this problem is the method of braiding the steel wires at a neutral braid angle φN = 54.7356° to the hose axis. In this case, the balance between axial and hoop stresses is ensured. When manufacturing hydraulic hoses, the aim is to maintain the required braid angle. This is ensured by the right combination of the braiding rate and feed rate of the manufacturing machine [20]. The aim of this work is to determine the influence of the braid angle on the deformation of the hydraulic hose and the forces acting on the hose ends. Within the framework of this research, an experimental device was created in order to test hydraulic hoses. The result is the determination of the dependence of the change in hose length and tensile force on the braid angle for hydraulic hoses with different internal diameters and different braid designs. This work serves as a summary of the measured data, which can be used for further research in the field of mathematical 3D modelling, for example in the field of finite element analysis of the deformation stress of hydraulic hoses. 2. Theoretical Background The definition of the neutral braid angle, at which balance is reached between the stresses generated, is based on the theory of a closed cylinder of radius r and wall thickness s. The working pressure p of the fluid acting on the hose inner wall generates hoop stress σO and axial stress σA (see Figure 1) in the hose wall [21]. Figure 1. Hoop and axial stress components in a closed hydraulic hose [21]. The axial force FA acting on the closed end of the hose is given below [21]: FA = π × r2 × p , (1) where p is the fluid pressure and r is the inner radius of the hose. The axial stress σA generated in the wall of the hydraulic hose is given by the axial force FA acting in the cross-sectional area of the hydraulic hose under the condition that r >> s [21]: Figure 1. Hoop and axial stress components in a closed hydraulic hose [21]. The axial force FAacting on the closed end of the hose is given below [21]: FA=π×r2×p, (1) where pis the fluid pressure and ris the inner radius of the hose. The axial stress σA generated in the wall of the hydraulic hose is given by the axial force F A acting in the cross-sectional area of the hydraulic hose under the condition that r>> s[21]: σA=FA 2×π×r×s, (2) Processes 2024,12, 152 3 of 15 where sis the wall thickness of the hose. We obtain the expression for axial stress σA by modifying Equations (1) and (2), as given below [21]: σA=p×r 2×s. (3) The hoop force F O acting on the unit length of the hydraulic hose as given below [ 21 ]: FO=1×2×r×p, (4) where 1 is the unit length. The hoop stress σO expressed per the unit length of the hydraulic hose is defined by Equation (5) [21]: σO=FO 1×2×s. (5) We obtain the expression for hoop stress σO by modifying Equations (4) and (5), as given below [21]: σO=p×r s. (6) Comparing Equations (3) and (6), it can be seen that the axial stress σA is half of the hoop stress σO . Figure 2shows a section of hydraulic hose where the wire tensions acting on the braid wires are indicated. A braid angle ϕ is given for the braid wire and the longitudinal axis of the hydraulic hose, as shown in Figure 2on the left and right [21]. Processes 2024, 12, x FOR PEER REVIEW 3 of 16 σA = FA 2 × π × r × s , (2) where s is the wall thickness of the hose. We obtain the expression for axial stress σA by modifying Equations (1) and (2), as given below [21]: σA = p × r 2× s. (3) The hoop force FO acting on the unit length of the hydraulic hose as given below [21]: FO = 1× 2 × r × p , (4) where 1 is the unit length. The hoop stress σO expressed per the unit length of the hydraulic hose is defined by Equation (5) [21]: σO = FO 1× 2 × s. (5) We obtain the expression for hoop stress σO by modifying Equations (4) and (5), as given below [21]: σO = p × r s. (6) Comparing Equations (3) and (6), it can be seen that the axial stress σA is half of the hoop stress σO. Figure 2 shows a section of hydraulic hose where the wire tensions acting on the braid wires are indicated. A braid angle φ is given for the braid wire and the longitudinal axis of the hydraulic hose, as shown in Figure 2 on the left and right [21]. Figure 2. The wire tensions acting on the braid wires [21]. When an element of unit length is released, the height of the element is equal to tanφ (see Figure 2). For the balance between the hoop and axial stresses when the hose is subject to internal fluid pressure, Equation (7) for the hoop tension TO and Equation (8) for the axial tension TA must satisfy: T O = T ⋅ sin φ = p × r s × 1 × s, (7) T A = T ⋅ cos φ = p × r 2 × s tan φ × s , (8) where 1∙s is the area of the unit length element on which the hoop tension TO acts and tanφ∙s is the area of the unit length element on which the axial tension TA acts. The neutral braid angle φN can be defined using the trigonometric [21]: tan 𝜑  = T O T A . (9) Substituting Equations (7) and (8) into Equation (9), the expression is as follows [21]: Figure 2. The wire tensions acting on the braid wires [21]. When an element of unit length is released, the height of the element is equal to tan ϕ (see Figure 2). For the balance between the hoop and axial stresses when the hose is subject to internal fluid pressure, Equation (7) for the hoop tension T O and Equation (8) for the axial tension TAmust satisfy: TO=T·sinϕ=p×r s×1×s, (7) TA=T·cosϕ=p×r 2×stanϕ×s, (8) where 1 · sis the area of the unit length element on which the hoop tension T O acts and tanϕ·sis the area of the unit length element on which the axial tension TAacts. The neutral braid angle ϕNcan be defined using the trigonometric [21]: tanϕN=TO TA . (9) Substituting Equations (7) and (8) into Equation (9), the expression is as follows [21]: tanϕN=2 tanϕ. (10) Processes 2024,12, 152 4 of 15 We obtain the expression for the neutral braid angle by modifying Equation (10) [ 21 ]: ϕN=tan−1√2=54.7356 ◦. (11) Table 1shows the changes in the geometry of the hydraulic hose caused by an internal fluid pressure increase. When the initial braid angle ϕ becomes greater than the neutral braid angle ϕN , the length of the hydraulic hose increases, and the hose diameter decreases. When the initial braid angle ϕ is less than the neutral braid angle ϕN , the length of the hydraulic hose decreases, and the hose diameter increases. In both cases, the volume of the hose increases as the internal fluid pressure increases. With a neutral braid angle, there is no change in geometry due to the change in braid angle [22]. Table 1. Geometric changes of the hydraulic hose when the internal fluid pressure increases [22]. Hose Geometry Changes ϕ<ϕNϕ>ϕN Length Decreases Increases Diameter Increases Decreases Volume Increases Increases The change in the braid angle due to the internal fluid pressure increase in the hose causes the axial force to be applied on the hose ends. In the case of hose shortening, a tensile force F T should be generated which acts on the hose fitting. In the case of hose extension, on the other hand, a pushing force FTH should be generated. Manufacturers of high-pressure hydraulic hoses perform many tests on their products, which may include pressure or temperature tests or chemical resistance tests. One of the tests that are performed on high-pressure hoses is a test where the shortening or elongation of the hose is measured [ 23 ]. This test is also part of the SAE J343 standard, which specifies the procedures for performing tests on high-pressure hydraulic hoses. This standard gives detailed instructions for performing the test and specifies allowable values for hose elongation or shortening depending on the hose design and size. The percentage of hydraulic hose elongation or shortening may vary depending on the standard and the hose. For example, Fitch stated limits that allow a hose to be shortened by 6% of the original length and elongated by 2% of the original length in his publication [ 23 ]. The testing of high-pressure hoses for possible elongation or shortening under working pressure is a common practice carried out by manufacturers, but these data are not widely available. The added value in this research will be the simultaneous sensing of the braid angle due to the removal of the rubber cover and the subsequent determination of its effect on hose shortening or elongation. 3. Experiment For this study, experimental equipment was constructed in order to test hydraulic hoses with different inner diameters and design types (see Figure 3). The left part of the figure shows the equipment design. The middle part of the figure shows the experimental equipment photo, and the right side shows a detail of the hose under test. To measure the braid angle under the fluid pressure, the hydraulic hose cover was removed from the hydraulic hose. The hydraulic hose cover did not affect the pressure capability, but only protected the hose from external influences. In this way, the angle of the hose outer braid could be measured visually. With two and more layers of braids, only the angle of the outer braid could be visually read without damaging the hose. For this reason, only hydraulic hoses with one braid were evaluated. Figure 4shows on the left side the tested hydraulic hoses. Figure 4shows on the right side a detail of the hose braid. Table 2shows the technical data and geometric dimensions of the tested hydraulic hoses. The table shows two types of braid for the tested hoses, namely SC and SN. Both types of hose braids are suitable for high pressure hydraulics. However, the SC designation defines the possibility Processes 2024,12, 152 5 of 15 of a tighter bend radius for these hoses, which is suitable for installations where space is at a minimum. Processes 2024, 12, x FOR PEER REVIEW 5 of 16 shows two types of braid for the tested hoses, namely SC and SN. Both types of hose braids are suitable for high pressure hydraulics. However, the SC designation defines the possibility of a tighter bend radius for these hoses, which is suitable for installations where space is at a minimum. Figure 3. Experimental equipment for testing hydraulic hoses. Figure 4. Measured hydraulic hoses on the left side and detail of the braid on the right side. Table 2. Technical data of tested hydraulic hoses. Hydraulic Hose Inner Diameter din Outer Diameter dout Wall Thickness s Length l Type of Braid Maximal Working Pressure pmax (mm) (mm) (mm) (mm) (-) (bar) DN12_A 13 19 3 1355 1SC 160 DN12_B 13 19.5 3.25 1345 1SN 160 DN12_C 13 20 3.5 1371 1SN 160 DN16_A 16 22 3 1350 1SC 130 DN16_B 16 23 3.5 1345 1SN 130 DN16_C 16 24 4 1350 1SN 130 Figure 3. Experimental equipment for testing hydraulic hoses. Processes 2024, 12, x FOR PEER REVIEW 5 of 16 shows two types of braid for the tested hoses, namely SC and SN. Both types of hose braids are suitable for high pressure hydraulics. However, the SC designation defines the possibility of a tighter bend radius for these hoses, which is suitable for installations where space is at a minimum. Figure 3. Experimental equipment for testing hydraulic hoses. Figure 4. Measured hydraulic hoses on the left side and detail of the braid on the right side. Table 2. Technical data of tested hydraulic hoses. Hydraulic Hose Inner Diameter din Outer Diameter dout Wall Thickness s Length l Type of Braid Maximal Working Pressure pmax (mm) (mm) (mm) (mm) (-) (bar) DN12_A 13 19 3 1355 1SC 160 DN12_B 13 19.5 3.25 1345 1SN 160 DN12_C 13 20 3.5 1371 1SN 160 DN16_A 16 22 3 1350 1SC 130 DN16_B 16 23 3.5 1345 1SN 130 DN16_C 16 24 4 1350 1SN 130 Figure 4. Measured hydraulic hoses on the left side and detail of the braid on the right side. Table 2. Technical data of tested hydraulic hoses. Hydraulic Hose Inner Diameter din Outer Diameter dout Wall Thickness s Length lType of Braid Maximal Working Pressure pmax (mm) (mm) (mm) (mm) (-) (bar) DN12_A 13 19 3 1355 1SC 160 DN12_B 13 19.5 3.25 1345 1SN 160 DN12_C 13 20 3.5 1371 1SN 160 DN16_A 16 22 3 1350 1SC 130 DN16_B 16 23 3.5 1345 1SN 130 DN16_C 16 24 4 1350 1SN 130 DN19_A 19 26 3.5 1345 1SC 105 DN19_B 19 27 4 1345 1SN 105 DN19_C 19 27 4 1345 1SN 105 Processes 2024,12, 152 6 of 15 Figure 5shows a simplified scheme of the experimental equipment. The source of the pressurized fluid was a hydraulic power unit which supplies fluid to the channel P. The hydraulic hose H was connected to the pressure line via a ball valve BV. The ball valve was only used to close the pressure line when changing the tested hose H. The top end of the hydraulic hose H was screwed to a top steel plate, which was connected to the aluminum frame. The required pressure value pwas set by the pressure proportional relief valve PRV. Working fluid passed through PRV to channel T. The fluid pressure value pwas measured by the pressure sensor PS. To measure the tensile force F T of the hose in the longitudinal axis, the bottom end of the hose was attached to the force sensor FS. This tensile force F T of the hose increased with increasing the working pressure p. The force sensor FS was attached to a bottom steel plate, which was fixed into the frame structure. To measure the change in length of the hose, the bottom end of the hose was connected to a bracket which was fitted in the linear guides on the sides. This variant allowed one degree of freedom in the longitudinal axis of the hydraulic hose H. As the working pressure pincreased, the change in the hydraulic hose length ∆ loccurred. The length change ∆ lof the hose was determined by the laser distance sensor LS1. Simultaneously, the diameter dof the hydraulic hose braid was measured by the optical micrometer LS2. A photo of the hose braid with the removed cover was taken with the camera CAM. The photos of the braids were taken in the working pressure range p= (0 ÷ 140) bar. The pressure sensor PS and the force sensor FS were connected to measurement instrument, MS5070, from Hydrotechnik. The working fluid was mineral oil. The used parts are listed in Table 3. Processes 2024, 12, x FOR PEER REVIEW 6 of 16 DN19_A 19 26 3.5 1345 1SC 105 DN19_B 19 27 4 1345 1SN 105 DN19_C 19 27 4 1345 1SN 105 Figure 5 shows a simplified scheme of the experimental equipment. The source of the pressurized fluid was a hydraulic power unit which supplies fluid to the channel P. The hydraulic hose H was connected to the pressure line via a ball valve BV. The ball valve was only used to close the pressure line when changing the tested hose H. The top end of the hydraulic hose H was screwed to a top steel plate, which was connected to the aluminum frame. The required pressure value p was set by the pressure proportional relief valve PRV. Working fluid passed through PRV to channel T. The fluid pressure value p was measured by the pressure sensor PS. To measure the tensile force FT of the hose in the longitudinal axis, the bottom end of the hose was attached to the force sensor FS. This tensile force FT of the hose increased with increasing the working pressure p. The force sensor FS was attached to a bottom steel plate, which was fixed into the frame structure. To measure the change in length of the hose, the bottom end of the hose was connected to a bracket which was fitted in the linear guides on the sides. This variant allowed one degree of freedom in the longitudinal axis of the hydraulic hose H. As the working pressure p increased, the change in the hydraulic hose length Δl occurred. The length change Δl of the hose was determined by the laser distance sensor LS1. Simultaneously, the diameter d of the hydraulic hose braid was measured by the optical micrometer LS2. A photo of the hose braid with the removed cover was taken with the camera CAM. The photos of the braids were taken in the working pressure range p = (0 ÷ 140) bar. The pressure sensor PS and the force sensor FS were connected to measurement instrument, MS5070, from Hydrotechnik. The working fluid was mineral oil. The used parts are listed in Table 3. Figure 5. Simplified scheme of the experimental equipment. Table 3. List of used parts. Symbol Name Type (Producer) Measuring Range Measuring Accuracy PRV proportional relief valve DBEBE 6X (Rexroth , Hong Kong, China) - - PS pressure sensor PR400 (Hydrotechnik, Obergünzburg, Germany) (0–250) bar ±0.25% of full scale FS force sensor FO 200 (Hydrotechnik) (0–5) kN ±0.5% of full scale Figure 5. Simplified scheme of the experimental equipment. Table 3. List of used parts. Symbol Name Type (Producer) Measuring Range Measuring Accuracy PRV proportional relief valve DBEBE 6X (Rexroth, Hong Kong, China) - - PS pressure sensor PR400 (Hydrotechnik, Obergünzburg, Germany) (0–250) bar ±0.25% of full scale FS force sensor FO 200 (Hydrotechnik) (0–5) kN ±0.5% of full scale LS1 laser distance sensor optoNCDT (Micro epsilon, Hong Kong, China) (0.5–200) mm ±0.08% of full scale LS2 optical micrometer LS-7070 (Keyence, Walnut Creek, CA, USA) (0.5–65) mm ±3µm CAM camera FASTCAM MINI UX (Photron, Tokyo, Japan) - - Processes 2024,12, 152 7 of 15 4. Results and Discussion The static properties of nine hydraulic hoses were measured and evaluated. For comparison, hydraulic hoses with one steel braid and different inner diameters (din = 13, 16 and 19 mm) were selected. Two experiments were performed for each hydraulic hose. In the first experiment, the dependence hose length strain εl with respect to the working pressure pwas determined. In this experiment, the top end of the hose under test was tightly threaded to the frame and the bottom end of the hose was attached to a linear guide that allowed movement in the longitudinal axis of the hose. In this experiment, the working fluid pressure pacting on the inner wall of the hydraulic hose caused the hose length to shorten. In the second experiment, the dependence of the hose tensile force F T with respect to the working pressure pwas determined. The fitting of the bottom end of the hose was performed through the force sensor FS into the bottom fitted steel plate, which was rigidly connected to the frame of the equipment. In this case, there was no shortening of the hydraulic hose as in the first experiment. The working pressure pcaused an increase in the tensile force F T , which was transferred by the braid steel wires to the hydraulic hose ends. In both experiments, the braid angle of the tested hydraulic hose was simultaneously evaluated for working pressure p min = 0 bar a p max = 140 bar. Figure 6shows the method for the evaluation of the hydraulic hose braid angle. The evaluation of the braid angle was performed using Photron FASTCAM Viewer 4 (PFV4) software. Due to the optical distortion of the angle, the braid wires that crossed relative to each other in the center of the hydraulic hose were evaluated. This was the point where the least optical distortion occurred, which is due to the curvature of the hydraulic hose. Subsequently, the angle αi was evaluated by using the function “angle 2” with two plotted lines parallel to the braid of the hose. Subsequently, the angle of the braid ϕi with respect to the longitudinal axis of the hydraulic hose was determined using Equation (12): ϕi=180 −αi 2. (12) Processes 2024, 12, x FOR PEER REVIEW 8 of 16 Figure 6. Evaluation of the hose braid angle. Figure 7 shows the details of the evaluation of the single angles αi for one measurement. The plotted line follows the selected braid wire along the length at which minimal curvature occurs. To refine the results for this measurement, five angles α1 to α5 were evaluated for one image taken. Three measurements of the dependence of length strain εl on the working pressure p and three measurements of the dependence of tensile force FT on the working pressure p were performed for one hydraulic hose. For the measured angle values for a specific working pressure and hose, the arithmetic mean was determined, given by (13): φ  = 1 n  φi n i= 1 , (13) where n is the number of angle measurements. The measured angle values αi are included in Table 4. The arithmetic mean of the braid angle φ is supplemented by the measurement uncertainty type A, which is equal to the sample standard deviation of the arithmetic mean and is given by Equation (14): uAφ =Sφ  =  ∑  φi − φ   2 n i=1 n ( n−1 ) . (14) Figure 7. Details of the angle αi evaluation. The measured values of the angle αi and the calculated values of the braid angle φi for the hydraulic hose DN19_C are included in Table 4. From the calculated values φ1 to φn, the arithmetic mean of the braid angle with type A measurement uncertainty was determined according to Equations (13) and (14). In this way, the initial braid angle φin corresponding to the hydraulic hose without working pressure load p was determined. The initial braid angle φin was the same for both types of measurements, due to the same i  Figure 6. Evaluation of the hose braid angle. Figure 7shows the details of the evaluation of the single angles αi for one measurement. The plotted line follows the selected braid wire along the length at which minimal curvature occurs. To refine the results for this measurement, five angles α1 to α5 were evaluated for one image taken. Three measurements of the dependence of length strain εl on the working pressure pand three measurements of the dependence of tensile force F T on the working Processes 2024,12, 152 8 of 15 pressure pwere performed for one hydraulic hose. For the measured angle values for a specific working pressure and hose, the arithmetic mean was determined, given by (13): ϕ=1 n n ∑ i=1 ϕi, (13) where nis the number of angle measurements. The measured angle values αi are included in Table 4. The arithmetic mean of the braid angle ϕ is supplemented by the measurement uncertainty type A, which is equal to the sample standard deviation of the arithmetic mean and is given by Equation (14): uAϕ=Sϕ= v u u u t n ∑ i=1(ϕi−ϕ)2 n(n−1). (14) Processes 2024, 12, x FOR PEER REVIEW 8 of 16 Figure 6. Evaluation of the hose braid angle. Figure 7 shows the details of the evaluation of the single angles αi for one measurement. The plotted line follows the selected braid wire along the length at which minimal curvature occurs. To refine the results for this measurement, five angles α1 to α5 were evaluated for one image taken. Three measurements of the dependence of length strain εl on the working pressure p and three measurements of the dependence of tensile force FT on the working pressure p were performed for one hydraulic hose. For the measured angle values for a specific working pressure and hose, the arithmetic mean was determined, given by (13): φ  = 1 n  φi n i= 1 , (13) where n is the number of angle measurements. The measured angle values αi are included in Table 4. The arithmetic mean of the braid angle φ is supplemented by the measurement uncertainty type A, which is equal to the sample standard deviation of the arithmetic mean and is given by Equation (14): uAφ =Sφ  =  ∑  φi − φ   2 n i=1 n ( n−1 ) . (14) Figure 7. Details of the angle αi evaluation. The measured values of the angle αi and the calculated values of the braid angle φi for the hydraulic hose DN19_C are included in Table 4. From the calculated values φ1 to φn, the arithmetic mean of the braid angle with type A measurement uncertainty was determined according to Equations (13) and (14). In this way, the initial braid angle φin corresponding to the hydraulic hose without working pressure load p was determined. The initial braid angle φin was the same for both types of measurements, due to the same i  Figure 7. Details of the angle αievaluation. The measured values of the angle αi and the calculated values of the braid angle ϕi for the hydraulic hose DN19_C are included in Table 4. From the calculated values ϕ1 to ϕn , the arithmetic mean of the braid angle with type A measurement uncertainty was determined according to Equations (13) and (14). In this way, the initial braid angle ϕin corresponding to the hydraulic hose without working pressure load pwas determined. The initial braid angle ϕin was the same for both types of measurements, due to the same initial conditions. The end braid angle ϕen corresponded to the maximum working pressure pmax = 140 bar . The determination of the end braid angle ϕen was performed separately for each type of measurement due to the different conditions (loose/fit hose end). Table 4. Measured braid angle values for hydraulic hose DN19_C. DN19_C Evaluating the Initial Braid Angle ϕin for Pressure pmin = 0 bar Evaluating the End Braid Angle ϕen for Pressure pmax = 140 bar α1÷αn [◦] ϕ1÷ϕn [◦] ϕin [◦] α1÷αn [◦] ϕ1÷ϕn [◦] ϕen [◦] εl=f(p) 75.393 52.30 52.26 ±0.03 72.7458 53.27 53.44 ±0.04 Measurement 1 75.4747 52.26 72.7458 53.63 75.8255 52.09 72.9397 53.53 75.8728 52.06 73.4766 53.26 75.3752 52.31 73.0621 53.47 Measurement 2 75.8248 52.09 73.1139 53.44 75.2628 52.37 72.9507 53.52 76.0046 52.00 72.7508 53.62 76.0819 51.96 73.3767 53.31 75.9478 52.03 72.7527 53.62 Measurement 3 75.6965 52.15 72.5419 53.73 75.7654 52.12 73.3178 53.34 74.9611 52.52 73.2721 53.36 75.5103 52.24 73.5703 53.21 75.4931 52.25 73.4296 53.29 Processes 2024,12, 152 9 of 15 Table 4. Cont. DN19_C Evaluating the Initial Braid Angle ϕin for Pressure pmin = 0 bar Evaluating the End Braid Angle ϕen for Pressure pmax = 140 bar α1÷αn [◦] ϕ1÷ϕn [◦] ϕin [◦] α1÷αn [◦] ϕ1÷ϕn [◦] ϕen [◦] FT=f(p) 75.7271 52.14 52.26 ±0.03 73.704 53.15 53.25 ±0.07 Measurement 1 75.0143 52.49 72.7037 53.65 75.7063 52.15 73.7333 53.13 75.5178 52.24 73.6536 53.17 75.9032 52.05 73.9443 53.03 Measurement 2 75.1601 52.42 73.3606 53.32 75.1492 52.43 72.8422 53.58 75.089 52.46 73.2932 53.35 75.7168 52.14 73.0096 53.50 75.3231 52.34 73.6646 53.17 Measurement 3 74.9483 52.53 74.3733 52.81 75.2163 52.39 73.752 53.12 75.1989 52.40 72.9056 53.55 74.9691 52.52 73.3729 53.31 75.3156 52.34 74.2968 52.85 The braid angles ϕin and ϕen were evaluated in the same way for all hydraulic hoses tested. The change in the braid angle ∆ϕ , given by Equation (15), is the important factor in the hose length change ∆ lor in the hose tensile force F T when the fluid pressure pis applied: ∆ϕ=ϕen −ϕin. (15) 4.1. Evaluation Hose Length Strain with Respect to the Working Pressure Table 5provides an overview of all initial braid angles ϕin , end braid angles ϕen and braid angle changes ∆ϕ achieved by each hydraulic hose when measuring the dependence of the hydraulic hose length strain εlon the working pressure p. Table 5. Measured values of initial braid angles ϕin , end braid angles ϕen and braid angle changes ∆ϕof the tested hydraulic hoses for the experiment εl=f(p). Hydraulic Hose Inner Diameter din Outer Diameter dout Wall Thickness s Initial Braid Angle ϕin End Braid Angle εl=f(p) ϕen Change in Braid Angle ∆ϕ (mm) (mm) (mm) (◦) (◦) (◦) DN12_A 13 19 3 51.01 ±0.08 52.36 ±0.06 1.35 DN12_B 13 19.5 3.25 53.05 ±0.10 53.86 ±0.10 0.81 DN12_C 13 20 3.5 53.80 ±0.05 54.12 ±0.05 0.32 DN16_A 16 22 3 52.51 ±0.02 53.69 ±0.06 1.18 DN16_B 16 23 3.5 52.31 ±0.05 52.85 ±0.10 0.54 DN16_C 16 24 4 52.77 ±0.08 53.74 ±0.02 0.97 DN19_A 19 26 3.5 53.31 ±0.03 53.96 ±0.03 0.65 DN19_B 19 27 4 53.03 ±0.05 53.90 ±0.07 0.87 DN19_C 19 27 4 52.26 ±0.03 53.44 ±0.04 1.18 A summary graph of all hydraulic hoses for the experiment εl =f(p) can be seen in Figure 8. For evaluating the results, it is important to consider several factors that may affect the individual dependencies. Both the initial braid angle ϕin and the actual change in the braid angle ∆ϕ must be considered. Based on the theory presented in Section 2, the greater the difference between the initial ϕin braid angle and the neutral angle, the greater the potential for geometric change in the hose. While increasing the working pressure p, the braid angle changes from ϕin to ϕen . The greater the change in the braid angle ∆ϕ during the increase in working pressure p, the greater the length strain εl will be. The initial braid