Experimental and Numerical Study on the Flow Ripple of Circular-arc Gear Pumps Considering the Center Distance Deviation

: Novel circular-arc gear pumps effectively solve the problems of oil trapping and flow pulsation experi-7 enced with traditional gear pumps. However, the center distance deviation associated with assembly and installation during gear pump processing has an important influence on the outlet pressure pulsation characteristics of circular-arc gear pumps. First, the circular-arc tooth profile equation, conjugate curve equation and meshing line equation were derived to design the circular-arc gear meshing and center distance deviation functions. Second, the circular-arc gear tooth profile was accurately obtained. Then, a pressure pulsation characteristic simulation model for the novel circular-12 arc gear pumps considering the center distance deviation was established. The results show that with the increase of center distance deviation, the outlet flow rate of the arc gear pump increases first and then decreases greatly. Moreover, the center distance deviation has little effect on the independent tooth cavity pressure. Finally, the proposed ﬂuid dy-15 namic model is used to simulate a commercial circular-arc gear pump, which was tested within this research for mod-16 eling validation purposes. The comparisons highlight the validity of the proposed simulation approach.


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Among the existing positive displacement pumps, external gear pumps are mainly used in systems such as fuel 20 injection systems, automotive lubrication and transmission systems, and high-pressure cleaning and fluid delivery sys-21 tems [1]. The main advantages of this type of pump are that the manufacturing cost is low, the packaging is compact, it 22 can operate at high pressures, it is suitable for fluids in the high-viscosity range, and it has high resistance to fluid 23 variations and cavitation. However, the traditional involute external gear pump is limited by uneven outlet flows due 24 to an oil trapping phenomenon, which causes the pump to produce noise and vibrations [2]. 25 Researchers have begun to study unconventional tooth profiles, including arc gears, to further reduce the pulsation 26 of the outlet flow. Morselli [3] first described an arc gear pump in a patent, and Chen and Yang [4] published the first 27 research results obtained with an arc tooth profile in an external gear pump. Zhou Yang et al. [5] discussed the tooth 28 profile design in detail and combined involute and arc tooth profiles. Manring, Kasaragadda [6], Huang, Lian [7], and 29 others proposed a basic method to link spur gear tooth profile parameters with motion flow pulsation. Vacca et al. [8] 30 applied this method to circular arc gear pumps and theoretically analyzed determined that the outlet flow pulsation for 31 circular arc gear pumps was zero; however, experimental research results indicated that circular arc gear pumps yielded 32 very small pressure and flow pulsations. R. Massimo [9] reviewed different flow simulation methods to simulate the 33 flow of gear drives and external gear pumps before 2017. For the lumped parameter model, the flow simulation of 34 different pumps based on the control volume was analyzed, the control equation was given, and the latest research 35 results considering the effect of the interaction of fluid and mechanical components on leakage were introduced. Dipen 36 R et al. [10] simulated the internal fluid motion state of an involute external gear pump through FLUENT software, and 37 compared them through experiments, and obtained the good results. 38 The center distance deviation in the assembly of the circular-arc gear pump changes the contact position and contact 39 area of the gear tooth surface, thus affecting outlet pressure pulsation and resulting in vibration and noise. Therefore, 40 it is necessary to study how the center distance deviation affects the dynamic characteristics of the circular-arc gear 41 pump, which will help to guide the gear design and machining of the circular-arc gear pump, so as to lay a foundation 42 for giving full play to its performance advantages. In this paper, according to the principle of spatial meshing, we de-43 rived the end face tooth profile equation, conjugate tooth profile equation and meshing line equation for circular-arc 44 gears. On this basis, the effects of the tooth profile and center distance deviation on flow pulsation and pressure pulsa-45 tion are studied for a circular-arc gear pump, which provided the theoretical guidance for the circular-arc gear design. 46 47 2 of 13 Figure 1 shows the change in the meshing position with and without center distance deviation when the arc gears 49 are meshed. The driving gear produces an offset position∆ , and the offset position of the driven gear is zero. Figure 50 1(a) shows that the theoretical center distance is L, and the driving gear and the driven gear mesh at point M0 when 51 there is no center distance deviation. The driving gear and the driven gear mesh at point M1 when there is a center 52 distance deviation L  (which is positive here). Figure 1(b) is an enlarged view of the meshing position. 53 Because the driving and driven gears may have offset positions in reality and the operation process, CNC machin-54 ing technology and installation process are characterized by inherent errors, the center distance deviations in this article 55 are 0, 0.01 mm, and 0.02 mm.  The arc gear adopts a tooth profile based on an arc curve-involute-arc curve design; the top and root of the gear 62 have an arc curve design, and the side of the gear has an involute design to form a set of conjugate curves. 63 Because the tooth profile of the involute gear is symmetrical, we only assess half (N/2) of the tooth profile. The 64 curve ABCD from the highest point A on the gear tooth to the lowest point D in the adjacent tooth valley is selected as 65 the research focus. This curve segment is located within the angle∠AO1D, where∠AO1D =π/N and N is the number of 66 teeth. Taking the base circle center point O1 as the coordinate origin, the line connecting any point M on the base circle, 67 and circle center point O1 as the horizontal axis, we establish a plane rectangular coordinate system xOy, as shown in 68 Figure 2(a). This part of the curve is composed of the involute curve BC and two arc curves AB and CD, where the two 69 arc curves are tangential to the involute curve at points B and C. Due to the symmetry of the tooth profile [5], the arc 70 curve AB is perpendicular to the straight line O1A at point A, and the arc curve CD is perpendicular to the straight line 71 CD at point D. We obtain the tangents to the base circle separately through points B and C and the tangents at points B' 72 and C'. Since the two arc curves are tangential to the involute curve at points B and C, the intersection point A' of the 73 straight line BB' and the straight line O1A is the center of the arc curve AB, and the intersection point D' of the straight 74 line CC' and the straight line O1D is the center of the arc curve CD. 75 Assuming that the radius of the base circle is r and the angles ∠MO1A, ∠MO1C' and∠MO1B' are given by θA, θC 76 and θB, respectively, the involute tooth profile portion of the arc g∠ear pump can be represented by the above 4 param-77 eters. These four parameters are used to establish the involute equation, the coordinates of the centers of A1' and D1', 78 and the coordinates of the radii r1 and r2 to determine the expression of the addendum radius ra and the root radius rf. 79 The current point Q is established on the involute line, and the tangent to the base circle passing through Q is the 80 involute generation line. The tangent point is then Q', and the angle ∠Q'O1M is α.

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where 2 is a parameter of the arc angle of the CD segment. Then, the addendum radius is: The root radius is Since the tooth profile is formed by the smooth connection between the arcs A1B1 and C1D1 and the involute curve B1C1 91 and the connecting points of the arc and the involute are B1 and C1, the points B1 and C1 are both on the arc and the 92 involute curve; thus, the three curves are connected. Therefore, where 94 = tan + 2 = tan − 2 } (7) The above formula sets indicate that the tooth profile of the end face of the arc gear pump is influenced by the 95 base radius r; the opening angles θA, θC and θB; and the number of teeth. 96

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After determining the curve equation for a section of the driving gear, the conjugate curve equation of the driven 98 gear can be obtained by the principle of conjugate gear meshing [9]. The specific solution process is as follows. 99 As shown in Figure 2(a), points O1 and O2 are the centers of the gears, and point P is the pitch point of the gear 100 pair. The pitch circle radius is r1, and the center distance is a (O1O2=2r1). We take O1 as the origin of the coordinate system 101 S1 (O1-x1, y1), which is fixed, connected to gear 1 and rotates with gear 1. O2 is the origin of the coordinate system S2 (O2-102 x2, y2), which is fixed, connected to gear 2 and spins with gear 2. We use point P as the origin to fix the coordinate system 103 SP (P-x, y) on the meshed two-gear transmission plane. At the starting position, the coordinate axes y1 and y2 coincide 104 with y, and the coordinate axes x1 and x2 are parallel to x. 105 The known tooth profile ABCD is fixed and connected with the coordinate system S1 (O1-x1, y1). Gear 1 and the tooth 106 profile ABCD rotate counterclockwise together. The rotation angle of gear 1 is positive in the counterclockwise direction. 107 We can assume that the angle between the tangent at a point N (x1, y1) on the tooth profile ABCD and the axis x1 is γ. 108 The normal line intersects the pitch circle of gear 1 at point N', and the angle between the straight line O1 N' and the 109 tangent at point N is . Each point on the tooth profile ABCD is a contact point. By transforming coordinates of arc AB, 110 the involute curve BC, and the arc CD into coordinate system S2 (O2-x2, y2), which is fixed and connected to gear 2, we 111 can obtain the conjugate gear tooth profile equation: where is equal to 1. Within each tooth space, the following pressure build-up equation is solved for the pressure dynamics [11]: where is the bulk modulus, is the instantaneous tooth space volume, ̇ is the change rate of tooth space volume, 124 while Q is the flow in the tooth space. Each tooth space solves its own pressure build-up equation, i.e. Eq. (12). 125 The turbulent orifice connections of each tooth space are considered in this paper, similarly to the approach of [12]. 126 For this turbulent orifice connection, the flowrate is given by: Where the is the discharge coefficient, A is the orifice opening area, ∆ is the pressure difference between displace-128 ment chambers at both ends, and is the density taken at the average pressure between displacement chambers at 129 both ends. To take the influence of the Reynolds number on the discharge into account, the discharge coefficient is 130 modeled as [13].
where the is the user defined empirical maximum flow coefficient, and the typical value used is 0.7. The hyper-132 bolic tangent function is generally used to fit the case of low Reynolds number. The k is the predictive quantity of 133 Reynolds number, which is estimated as: The typical value of the critical Reynolds number for orifice plate is 1000 [13]. When the Reynolds number is low, 135 the hyperbolic tangent function makes the flow rate change linearly with the pressure difference and returns to the state 136 of flow field. The ℎ is the hydraulic diameter of orifice opening, and the velocity at orifice is estimated as 137 = √ 2∆ (16)

Implementation
138 The overall numerical model of an arc gear pump is established, as shown in Figure 3. The 1D model of the arc 139 gear pump is automatically generated so that the arc tooth profile can be accurately obtained. The center distance devi-140 ation was considered in advance. 141 Figure 3. The overall numerical model of an arc gear pump. 143 We established a three-dimensional model of the arc gear pump and then created a 1D model of the arc gear pump 144 according to the CAD import function. It should be noted that the center of the coordinate system of the arc gear pump 145 must be located at the 1D center of the driving gear, and the arc gear rotates counterclockwise. According to the 1D 146 model of the arc gear pump, the initial associated parameters can be obtained, as shown in Table 1. The initial param-147 eters of the overall simulation model were set as shown in Table 2 When the simulation model of the arc gear pump was established, each tooth volume during the meshing of the 153 driving gear and the driven gear was determined, as shown in  Figure 5 shows the influence of the center distance deviation on the pulsation of the pump outlet flow. Under the 167 light load condition (600rpm, 20bar), with the increase of center distance deviation, the outlet flow rate of the arc gear 168 pump increases first and then decreases greatly. When the center distance deviation is within 0.01mm, the outlet flow 169 rate of the arc gear pump increases gradually and has good dynamic characteristics. As the center distance deviation 170 increases to 0.02mm, the outlet flow rate of the arc gear pump decreases greatly, and the dynamic characteristics of this 171 pump become worse. Under the medium load condition (1480rpm, 80bar), with the increase of center distance devia-172 tion, the outlet flow rate of arc gear pump shows the same characteristics as the light load condition. In conclusion, the 173 arc gear pump is more sensitive to the center distance deviation. When the center distance deviation is controlled within 174 a certain range, the arc gear pump has better dynamic characteristics. Therefore, in order to give full play to the good 175 transmission performance of the arc gear pump and prevent excessive dynamic impact due to the center distance devi-176 ation, the dynamic response caused by the center distance deviation can be adjusted by optimizing the machining ac-177 curacy and assembly accuracy of the arc gear.

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The pump speed was set to 1480 r/min, and the pressure changes in each tooth volume for the driving and driven 180 gears were observed with center distance deviation, shown in Figure 6. When the arc gear pump starts to run, the first 181 tooth volume of the driving gear is completely connected to the pump inlet, and the pressure associated with the tooth 182 volume is the same as the inlet pressure. Approximately 90% of the first tooth volume of the driven gear is connected 183 to the pump outlet. Thus, at initial moment, pressure in the volume is lower than the outlet pressure. The pressure in 184 the tooth volume then gradually decreases with the operation of the pump. As the center distance deviation gradually 185 increases, the pressure in the first volume of the driving gear does not change, and the pressure in the first volume of 186 the driven gear has micro oscillations. When the arc gear pump continue to run, 95% of the second tooth cavity of the 187 driving gear and driven gear are connected to the pump outlet. At this time, the pressure in the tooth cavity is equal to 188 the pump outlet pressure. As the center distance deviation gradually increases, the pressure in the second cavity of the 189 driving and driven gears has no change. The third tooth cavity and of the driving gear is completely connected to the 190 pump outlet. At this time, the pressure in the tooth cavity is the same as the pump outlet pressure. The third tooth cavity 191 of the driven gear is an independent tooth cavity that is not connected to any port of the pump. The pressure in the 192 tooth cavity gradually reaches the outlet pressure. As the center distance deviation gradually increases, the pressure in 193 the third cavity of the driving gear has no change. The 4th tooth cavity and of the driving gear and driven gear are the 194 independent tooth cavities that are not connected to any port of the pump. As the center distance deviation gradually 195 increases, the pressure in the third cavity of the driving gear has no change. The 7th tooth cavity of the driving and 196 driven gears are connected to the pump inlet. The pressure on the driving gear smoothly decreases. Experiments were conducted on a commercial circular-arc gear pump produced by Settima. The ISO schematic of 200 the experimental setup is shown in Figure 7. The model under test was a 7-tooth gear pump of 32 cm 3 /rev displacement. 201 The tests were performed at the Shandong Shijing Machinery Co.Ltd， shown in Figure 8. The information about the 202 sensors used in the setup is presented in the Table 3. The fluid used during the test is the ISO VG 46 hydraulic oil.  Tests were performed for different shaft speeds (600 rpm, 1000 rpm, and 1480 rpm) with various pressure differ-208 entials up to 80 bar. The pump parameters were also measured to reproduce the gear profile in simulation. The com-209 parisons between outlet pressure oscillations and outlet flow oscillations are reported in the following part of this sec-210 tion. 211 Two representative cases of the outlet pressure oscillations and outlet flow oscillations comparisons are shown in 212 Figure 9 and 10. Figure 9 shows a high-speed and high-pressure case, in which the outlet pressure and flow behavior 213 are exhibited: for different center distance deviations, the outlet pressure oscillations are match the experiment, how-214 ever, the outlet flow rate have great oscillations, and the flow behavior is same to the case of center distance deviation 215 0 to 0.01mm. While Figure 10 shows a low-speed and low-pressure case, in which the outlet pressure and flow behavior 216 are exhibited: for different center distance deviations, the outlet pressure of simulation have not any change, but the 217 outlet pressure of experiment has oscillations, as well as the outlet flow result of experiment is greater than the simula-218 tion.

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This paper presents an experiment and numerical approach of modeling circular-arc gear pumps considering the 225 center distance deviation. A review on the gear profiles design of circular-arc gear pumps was first provided, and the 226 numerical modeling of circular-arc gear pumps then described. This numerical approach for modeling the geometric 227 evolutions of the displacement chambers permitted to study the kinematic flow ripple of circular-arc gear pumps. The 228 numerical model of an arc gear pump was established, in which the arc tooth profile can be accurately obtained, and 229 the center distance deviation was considered in advance. In different center distances, the pressure ripple and flow 230 ripple have been carried out in the light load condition and in the medium load condition with outlet pressures of 20, 231 and 80 bar and the constant rotational speed of 600 rpm and 1480 rpm. The results show that with the increase of center 232 distance deviation, the outlet flow rate of the arc gear pump increases first and then decreases greatly. The center dis-233 tance deviation has little effect on the independent tooth cavity pressure. In the final section of the paper, the proposed 234 fluid dynamic model is used to simulate a commercial circular-arc gear pump, which was tested within this research 235 for modeling validation purposes. The simulated outlet pressure and outlet flow ripple are compared against the meas-236 ured values. The comparisons highlight the validity of the proposed simulation approach. Next, the model can be used 237 to gain further insights on the circular-arc gear pump operation, such as including localized cavitation, internal flow 238 leakages. In order to give full play to the good transmission performance of the arc gear pump and prevent excessive 239 dynamic impact due to the center distance deviation, the dynamic response caused by the center distance deviation can 240 be adjusted by optimizing the machining accuracy and assembly accuracy of the arc gear. 241