21.1 SECTION 21 TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL Constructing Mathematical Models for Analyzing Hydrostatic Transmissions 21.1 Selecting a Clutch for a Given Load 21.12 Clutch Selection for Shaft Drive 21.13 Sizing Planetary Linear-Actuator Roller Screws 21.16 Designing a Rolling-Contact Translation Screw 21.21 Selection of a Rigid Flange-Type Shaft Coupling 21.30 Selection of Flexible-Coupling for a Shaft 21.32 Selection of a Shaft Coupling for Torque and Thrust Loads 21.34 High-Speed Power-Coupling Characteristics 21.35 Selection of Roller and Inverted-Tooth (Silent) Chain Drives 21.38 Cam Clutch Selection and Analysis 21.42 Timing-Belt Drive Selection and Analysis 21.43 Geared Speed Reducer Selection and Application 21.47 Power Transmission for a Variable- Speed Drive 21.48 CONSTRUCTING MATHEMATICAL MODELS FOR ANALYZING HYDROSTATIC TRANSMISSIONS Construct a mathematical model of vehicle performance for a construction vehicle powered by a hydrostatic transmission when the vehicle is driven by a 45-hp (33.6- kW) engine at 2400 rpm, with a high idle-speed of 2600 rpm. The vehicle has a supercharge pump rated at 2 hp (1.5 kW). Other vehicle data are: loaded radius, r L ϭ 14.5 in (36.8 cm); gross vehicle weight, W g ϭ 8500 lb (3825 kg); weight on drive wheels, W w ϭ 5150 lb (2338 kg); final drive ratio, R fd ϭ 40:1; coefficient of slip, C s ϭ 0.8; and coefficient of rolling resistance, C r ϭ 60 lb/1000 lb (27.2 kg/ 454 kg) of gross vehicle weight. The vehicle is powered by a hydrostatic trans- mission with a 2.5-in 3 /rev (41-mL/rev) displacement pump, rated at 5000 lb / in 2 (34.5-MPa). Compare the performance produced by using a 2.5-in 3 /rev (41-mL / rev) displacement fixed-displacement motor and a 2.5-in 3 /rev (41-mL / rev) dis- placement variable-displacement motor with an 11-degree displacement stop. Other pump and motor data are given on performance curves available from the pump manufacturer. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Source: HANDBOOK OF MECHANICAL ENGINEERING CALCULATIONS 21.2 DESIGN ENGINEERING Calculation Procedure: 1. Determine the vehicle speed at maximum tractive effort The theoretical pump displacement required to absorb the input horsepower from the engine, using the nomenclature at the end of this procedure, is: 396,000 H p D ϭ pt PN pp Substituting, 396,000 (45 Ϫ 2) D ϭ pt 5,000 (2,400) 3 ϭ 1.42 in /rev (23.3 mL / rev) Next, find the horsepower-limited displacement from D ϭ DE ppttp Substituting, D ϭ 1.42 (0.92) p 3 ϭ 1.31 in /rev (21.5 mL / rev) Now we must find the pump flow from DNE pp v p Q ϭ p 231 Substituting, 1.31 (2,400) (0.88) Q ϭ p 231 ϭ 12 gal / min (0.76 L / s) Using the motor torque curve from the manufacturer for the pump being con- sidered, similar to Fig. 1, these data give a motor torque, T m ϭ 1800 lb/in (203.3 Nm) at a motor speed of 960 rpm. Maximum tractive effort is given by TREn m ƒd ƒdm T ϭ emax r L Substituting, 1,800 (40) (0.9) T ϭ emax 14.5 ϭ 4,469 lb (2029 kg) The vehicle speed at this tractive effort is given by Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL TRANSMISSIONS, CLUTCHES, ETC. 21.3 100 99 98 97 96 95 94 80 70 60 50 40 30 20 10 0 40 35 30 25 20 15 10 5 0 100 90 80 70 60 50 Volumetric efficiency, E rv (%) Outlet flow, Q p (gpm) Input horsepower, H p (hp) Pump speed, N p (rpm) (a) Overall efficiency, E pod (%) FIGURE 1 (a) Typical pump performance curves relate input horsepower, fluid outlet flow rate, speed, and pressure to volumetric and overall efficiency. (b) Motor performance curves relate output horsepower, fluid inlet flow rate, speed, and pressure to volumetric and overall efficiency. (Machine Design.) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL 1000 500 0 60 50 40 30 20 10 0 30 25 20 15 10 5 0 100 80 60 40 20 0 Output torque, T m (lb-in.) Output horsepower (hp) Inlet flow, Q p (gpm) Overall efficiency, E moa (%) Motor speed, N m (rpm) (b) FIGURE 1 Continued. SI values for Fig. 1a and 1b: gpm L / sec lb-in. Nm 00 0 0 5 0.32 500 56.4 10 0.63 1000 112.9 15 0.95 20 1.26 psi MPa 25 1.58 200 1.4 30 1.89 250 1.72 35 2.2 500 3.4 40 2.5 1000 6.89 2000 13.8 3000 20.7 4000 27.6 5000 34.5 1.5 cu in / rev (24.6 mL / revf) 2.5 cu in / rev (41 mL / rev) hp kW 00 10 7.46 20 14.9 30 22.4 40 29.8 50 37.3 60 44.8 70 52.2 80 59.7 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL TRANSMISSIONS, CLUTCHES, ETC. 21.5 FIGURE 2 Performance curve for vehicle analyzed in calculation procedure. (Ma- chine Design.) Fig. 2 SI lb kg mph m/ sec 00 1000 454 2 0.89 2000 908 4 1.78 3000 1362 6 2.68 4000 1816 8 3.58 5000 2270 10 4.47 Nr mL N ϭ v 168 RE ƒd ƒd Substituting, 960 (14.5) N ϭ v 168 (40) ϭ 2.1 mi / h (0.939 m / s) Plot these computed values as point A on the tractive-effort vs. vehicle-speed curve, Fig. 2. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL 21.6 DESIGN ENGINEERING 2. Determine the tractive effort at the maximum vehicle speed From the pump performance curve obtained from the manufacturer, the maximum pump flow is 25.2 gal/min (1.59 L / s) at 2700 lb / in 2 18.6 (MPa) and 2.5 in 3 /rev (41 mL / rev). From these data, the motor torque curves for the fixed-displacement motor give N m ϭ 2240 rpm and T m ϭ 1000 lb / in (112.9 Nm). The maximum vehicle speed produced by the fixed-displacement motor is Nr mL N ϭ v 168 RE ƒd ƒd Substituting, 2,240 (14.5) N ϭ v 168 (40) ϭ 4.8 mi / h (2.15 m/s) The tractive effort at this speed is TREn m ƒd ƒdm T ϭ emax r L Substituting, 1,000 (40) (0.9) T ϭ e 14.5 ϭ 2,482 lb (1127 kg) Plot these values as point B on Fig. 2. From the curves for the variable-displacement motor, N m ϭ 3580 rpm and T m ϭ 560 lb/in (63.2 Nm). Therefore, as before, maximum vehicle speed produced by the variable-displacement motor is 3,580 (14.5) N ϭ v 168 (40) ϭ 7.7 mi / h (3.44 m / s) And the tractive effort, as before, is: 560 (40) (0.9) T ϭ e 14.5 ϭ 1,390 lb (631 kg) Plot these values as point C on Fig. 2. 3. Find intermediate points on the tractive-effort vs. vehicle-speed curve To plot an intermediate point on the curve, Fig. 2, a pump flow of 21 gal/min (1.325 L/s) is chosen arbitrarily. For the fixed-displacement motor, this flow gives N m ϭ 1800 rpm and T m ϭ 1200 lb/in (135.5 Nm). Therefore, vehicle speed and tractive effort are Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL TRANSMISSIONS, CLUTCHES, ETC. 21.7 1,800 (14.5) N ϭ v 168 (40) ϭ 3.8 mi / h (1.698 m / s) 800 (40) (0.9) T ϭ e 14.5 ϭ 2,979 lb (1352 kg) which are plotted as point D on Fig. 2. For the variable-speed motor, N m ϭ 2600 rpm and T m ϭ 800 lb/in (90.3 Nm). Therefore, the vehicle speed and tractive effort are: 2,600 (14.5) N ϭ v 168 (40) ϭ 5.6 mi / h (2.5 m / s) 800 (40) (0.9) T ϭ e 14.5 ϭ 1,986 lb (901.6 kg) Plot these values as point E on Fig. 2. 4. Find the maximum theoretical speed for each motor type The final point needed to construct the performance curve is the maximum theo- retical speed of the vehicle for each type of motor. For the fixed-displacement motor, maximum motor speed is given by NDEE pmax p v p v m N ϭ mmax nD mm Substituting, 2,600 (2.5) (0.95) (0.95) N ϭ mmax 2.5 ϭ 2,346 rpm The maximum theoretical vehicle speed is found from Nr mmax L N ϭ v max 168 R jd Substituting, 2,346 (14.5) N ϭ v max 168 (40) ϭ 5.1 mi / h (2.5 m / s) which is plotted as point F on Fig. 2. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL 21.8 DESIGN ENGINEERING For the variable-displacement motor, maximum motor speed is 2,600 (2.5) (0.95) (0.95) N ϭ mmax 1.5 ϭ 3,911 rpm and the maximum theoretical vehicle speed is 3,911 (14.5) N ϭ v max 169 (40) ϭ 8.4 mi / h (3.75 m / s) which is plotted as point G on Fig. 2. 5. Refine the curve with rolling resistance and tractive effort at wheel slip To refine the curve, rolling resistance, tractive effort at wheel slip, and gradability must be determined. Rolling resistance is found from R ϭ WC rg v r Substituting, 8,900 (60) R ϭ r 1,000 ϭ 510 lb (231.5 kg) Tractive effort at wheel slip is given by T ϭ WC ews Single-path T ϭ 0.6 WC ews Dual-path Substituting, T ϭ 5,150 (0.8) e ϭ 4,120 lb (1870 kg) The gradability at slip is given by T Ϫ R er Ϫ 1 G ϭ tan sin 100 ͫͩ ͪͬ W g v Substituting, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL TRANSMISSIONS, CLUTCHES, ETC. 21.9 4,120 Ϫ 510 Ϫ 1 G ϭ tan sin 100 ͫͩ ͪͬ 8,500 ϭ 47% for the fixed-displacement motor. The maximum gradability for the variable-displacement motor, limited by 5000 lb/in 2 (34.45 MPa), is 2,482 Ϫ 510 Ϫ 1 G ϭ tan sin 100 ͫͩ ͪͬ 8,500 ϭ 24% These data are also shown on the tractive-effort curve, Fig. 2, which now gives a complete picture of vehicle performance. Related Calculations. The analytical technique presented here allows the hy- drostatic transmission to be evaluated on paper, and necessary changes made before the unit is actually built. The procedure uses a series of calculations that gradually define transmission and vehicle data. With these data, a curve can be constructed, Fig. 2, so that vehicle performance can be predicted for the entire operating range. The first step in the analysis is to compare the ‘‘application values’’ of the vehicle and the available transmissions. This comparison provides a simple way to match vehicle requirements to transmission capabilities. The vehicle application value expresses vehicle requirements and depends on required vehicle speed and maximum tractive effort. For single-path applications only one transmission is used. For dual-path applications, where two transmissions are used, the transmission on each side of the vehicle must be treated as if it were a single-path system. In such a case, a normal assumption is that 60 percent of the total weight on the drive wheels transfers to one side of the vehicle when it ne- gotiates a slope or turn. The transmission application value expresses transmission capabilities and depends on motor torque and speed. If the transmission application value is greater than that for the vehicle, the proposed transmission is viable, and calculations to size properly the transmission can be made. If the vehicle application value is greater, consideration must be given to increasing pump speed, using variable-displacement motors, increasing pump displacement, lowering maximum vehicle speed, or accepting a lower vehicle trac- tive effort. If a comparison of application values indicates that a proposed transmission is adequate, a more refined procedure must be used to size the transmission. This ensures that the transmission is applied within its horsepower rating, and that it meets vehicle power requirements. The input power available to the transmission is the net engine flywheel horse- power (kW) at full-load governed speed, less the horsepower (kW) required for supercharge and auxiliary no-load losses. For a transmission to operate satisfacto- rily, input horsepower (kW) must not exceed its rated horsepower (kW). In dual- path applications, power is split between two pumps and 80 percent of the total input horsepower (kW) should be used in this calculation. Thus, up to 80 percent of the total input horsepower (kW), is assumed to be directed to one pump. Next, the transmission’s ability to produce the required tractive effort and pro- pelling speed must be checked. For these calculations, the pump and motor per- Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL 21.10 DESIGN ENGINEERING TABLE 1 Coefficients of Rolling Resistance Surface Rubber tire kg /454 kg Crawler kg/454 kg Concrete 10–20 4.5–9.1 40 18.2 Asphalt 12–22 5.5–9.9 40 18.2 Packed gravel 15–40 6.8–18.2 40 18.2 Soil 25–40 11.4–18.2 40 18.2 Mud 37–170 16.8–77.2 — — Loose sand 60–150 27.2–68.1 100 45.4 Snow 25–50 9.1–22.7 — — Units are lb /1,000 lb (kg/ 454 kg) gross vehicle weight. formance curve for the proposed transmission must be available, usually from the manufacturer. The maximum tractive effort is limited by the pump relief-valve setting or wheel slip. For multiple-path systems, maximum tractive effort must be divided by the number of motors and multiplied by 0.6 before the comparison is made. The final calculation required to determine whether a transmission meets vehicle requirements is to check maximum vehicle speed. If the transmission can produce the required tractive effort and speed, it is sized properly. However, if speed is too low and tractive effort acceptable, consideration should be given to increasing pump speed, using a variable-displacement motor, or decreasing the ratio of the final drive. If speed is acceptable but tractive effort too low, give consideration to increasing the final drive ratio. The resultant loss in maximum speed can be recovered by increasing pump speed or by using a variable-displacement motor. Once the transmission is sized to meet vehicle requirements, a mathematical model can be generated to predict system performance. The calculations necessary to produce the model take into account such factors as pump speed, pump and motor displacement, and pump and motor efficiency. The expected vehicle perform- ance is represented by a tractive-effort vs. speed curve. The first two steps in generating the math model are to define the upper and lower limits on the curve. The upper limit is the vehicle speed produced at the maximum tractive effort; the lower limit is the tractive effort produced at maximum vehicle speed. Typically, four intermediate points on the performance curve are sufficient to provide a rough approximation of vehicle performance. Six to eight points may be required for a complete analysis. These points are calculated as shown here, except that motor torque and speed are determined for pump displacements between max- imum displacement and displacement at maximum tractive effort. To complete the analysis, a number of factors must be calculated to determine how they affect vehicle performance. One factor that must be considered is rolling resistance, the portion of tractive effort required to overcome friction and move the vehicle. For the vehicle to move, available tractive effort must be greater than the rolling resistance. If the actual coefficient of rolling resistance is not known, the values in Table 1 can be used. Few vehicles operate only on level ground; so the slope or grade it can climb must be determined. This factor, called gradability, is calculated as shown above. Gradability can be determined for any point along the tractive-effort vs. speed curve, up to the slip-limited effort. Gradability at wheel slip is usually the upper limit. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. TRANSMISSIONS, CLUTCHES, ROLLER-SCREW ACTUATORS, COUPLINGS, AND SPEED CONTROL