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Since a revenue train is useless, unless it can be moved from origination to destination; starting it is all important. One of the most salient quantities in loco listings is the draw bar force at a dead start or starting tractive effort (Fs). This may be measured at a test plant, but more often it is computed from other parameters. For steam locos the basic equation is: Fs = K * P * C^2 * S / D. Where Fs = Force in pounds, P = Boiler pressure in pounds per square inch, C = Cylinder diameter in inches, S = Piston stroke length in inches, D = Driver diameter in inches and K is dependent on several factors, including cutoff setting of valve gear, pressure drop in steam delivery pipes from boiler to pistons, heat losses and all calculation constants. NOTE: On steam locomotives, an engine comprises all the cylinders, pistons, rods, valve gear and drivers linked together as one mechanically independent unit. While a locomotive consists of all the engines together. Expressed as a percentage, cutoff is the ratio of the steam admission time, through the intake valves, to the total time of a power stroke. At 100% the intake valve would be open throughout the entire power cycle with full pressure applied. While at 50% the intake valve is closed half way through the stroke and steam expansion develops the remaining diminishing pressure on the piston. Lowering cutoff reduces the effective piston force. K is typically around .75 at 50% cutoff, .80 at 78% and .85 at 90% for well designed engines. Apparently due to design restraints, .90 was the maximum usually achieved with no cutoff. Wherever possible, once moving, the cutoff was reduced to take advantage of steam expansion and reduce consumption by adjusting valve gear settings. The optimum utilization of this was considered a prime rating of an engineer's ability, since it conserved water. Since piston force Fp is the pressure P in pounds/square inch (psi) times the area, the cylinder diameter C is squared with the constants thrown into K. Fp = P * (¶ / 4) * C^2 = K * P * C^2 Fp is applied via the crosshead and main rod to the crank pin whose radius is 1/2 the stroke S. This develops a TORQUE = Td. Td = (1/2 S) * Fp in-lb Since torque is constant at any radius on a wheel, this is passed to the tires as: Td = (1/2 D) * Te.__or__ Fs = Td / (1/2 * D) Thus solving for Fs and cancelling out the (1/2)'s: Fs = Fp * (1/2 S) / (1/2 D) = K * P * C^2 * S / D lb. Note, since this is a simple leverage calculation, the ratio of disadvantage Rd is similar to gear ratios. Rd =Fs / Fp = S / D. For a single power stroke cycle, , Fp follows a half wave sine curve, due to the circular path of the crank pin. For each cylinder half the crank rotation is a push, while the other half is a pull. Each is zero at the ends and maximum at the middle. However since drivers are quartered, placing opposite crank pins at 90° apart and steam is admitted alternately in both piston directions; there are four overlapping power stokes per driver revolution. The peak at one piston coincides with the zero of the other and when added together produce a nearly flat line; which resembles a well filtered power supply voltage. Cutoff will influence the bumps in the line. This is very similar to a four cylinder, two cycle, gas engine output. An adjunct of quartering is that there is no dead spot regardless of initial starting position. As built the PRR I-1s had K = .75 @ 50% cutoff, P = 250 psi, C = 30.5", S= 32" and D = 62". The computed starting tractive effort was: Fs = .75 * 250 * 32^2 * 30.5 / 62 = 90,024 lb Later in the I-1sa modifications, the cutoff was raised to .78% for K= .80 and Te of 96,026 lb. Locos with multiple cylinders, engines or compound mallets require additional calculations, since some parameters may differ. Often at lower pressure, exhaust steam from a first pair of cylinders was used to supply additional, larger diameter cylinders, as with mallets. Using the same pressure on both, the PRR Q's had one engine with six drivers and large cylinder diameters, while the other had four drivers and smaller cylinders. Where the pressure is the same or known, each engine can be computed separately, and added together for the total. For two identical engines at the same pressure, K is doubled. Since the recommended practice for compound mallets was that Fp should be the same on all engines with the same number of drivers, this may be used for most older applications. Fs = 2 * K * P * C^2 * S / D. For compound locos using other cylinder ratios than the recommended, complications arise. Baldwin used an adjustment factor A, based on the square of the ratio of the high pressure cylinder diameter Ch to the low pressure cylinder diameter Cl. This was often referred to as the cylinder volume ratio. A = (Ch^2 / Cl^2) +1 = (Ch / Cl)^2 + 1. Yielding: Fs = 2 * K * P * Ch^2 * S / D / A. Or in the more usual awkward form. Fs = 2 * K * P * Ch^2 * S / D / [(Ch^2 / Cl^2) +1] . The N & W USRA variation Y-3 had K = .85, P = 270 psi, Ch = 25", Cl = 39", S = 32" and D = 56". A = (Ch^2 / Cl^2) +1 = (Ch / Cl)^2 + 1 = (25^2 / 39^2) + 1 = (625 / 1521) + 1 = 1.411. Fs = 2 * K * P * Ch^2 * S / D / A = 2 * .85 * 270 * 25^2 * 32 / 56 / 1.411 = 116154 lb. With geared locos, the gear ratios must be taken into account. For non-steam types of motive power, the starting tractive effort is determined by the stall TORQUE of the motor(s) and the transmission ratio. Often this is derived from the stall torque Ts of electric motor(s). At this point, maximum current is applied; limited only by winding resistance. This develops maximum torque on most motors; which in turn is transferred to the driver(s) to produce Fs. On some early electrics with above frame motors, the motor torque was applied through quartered crank pins on opposite shaft ends, through main rods to driver crank pins. These were similar to steam locos with the motor cranks replacing pistons and crossheads. Again the main rod forces are sinusoidal with both sides adding to a nearly straight line. Good examples were the PRR DD-1, L-5 and original FF-1. In this case the main rod force (Fm) is found from the motor torque Tm and the shaft center-crankpin distance (Sm). Fm = Tm / Sm. Through the main rod this force is applied to the driver crank pin at axle-crankpin distance Sd to create a torque Td. Td = Fm * Sd = Tm * Sd / Sm. Fs is derived from the axle torque and 1/2 driver diameter. Fs = Td / 1/2 D = Tm * Sd / Sm / 1/2 D = 2 * Tm * Sd / Sm / D. Diesels and most other electrics have motors directly geared to axles. The torque transfer from motor shaft to axle is determined by the overall ratio (G). It should be noted that the PRR and some others listed inverted gear ratios so that a 3:1 was written as 1:3. Td = G * Tm. Thus starting tractive effort is: Fs = 2 * Td / D = 2 * G * Tm / D. The method is the same as used in models. Unfortunately the necessary parameters are rarely listed by model makers. Factor of adhesion A is a closely related value, which is simply the weight on drivers (Wd) divided by the starting tractive effort (Fs). A = Wd / Fs Due to the surface molecular characteristics and reaction between the steel driver tires and the steel railhead, the static coefficient of friction (C) is about .25. This is found from the force (F/ required to initiate sliding of a given weight (W). If a 25 lb force is required to start sliding a 100 lb weight, the coefficient is .25. The ideal factor of adhesion (A) is the reciprocal of the coefficient of friction. Commonly values are around A = 4. Locos with lower values were slippery; while higher were considered sluggish. As tires and rails became polished the coefficient decreased, increasing the tendency to slip. Water and ice further agrevate conditions requiring the use of sand for a better bite. The PRR K-4 had about 209,300 lb on 3 driving axles and a tractive effort of 44,460 for a sluggish factor of adhesion of 4.71. In spite of this, very frequently, on hot summer evenings in Atlantic City; a familiar chug-chug-chug-chug-ratitat-tat-tat-tat-tat reverberated through the neighborhood; when a K-4 tried to start a long train, over filled with DFTD's ("down for the day") or SHOEBY's (people bringing shoe box lunches). The PRR GG-1 had tons of cast cement blocks mounted on the frame to increase factor of adhesion. Diesel starting tractive efforts were often rated at C = 25% or A = 4. In the model world, we are more likely to use ounces or grams than pounds; but since C and A are ratios, they are dimensionally independent. Due to the lower coefficient of friction between rails and drivers; more often they are approximately: C = 20% and A = 5. If possible, only by adding weight can they approach the prototype Te values. "Ideally" a loco with 15 oz on the drivers would have a 3 oz drawbar pull which translates to approximately, just over 40 cars on a level track in HO. Once moving, the operation becomes a new ball game. Except on steeper grades or higher speeds, factor of adhesion plays a minor role in pulling limitation . Here the prime limiter is the maximum horse power that can be generated by the prime mover. The output power is the drawbar pull times the speed. This also holds true for models. The motor output must provide sufficent mousepower to handle the train on grades; else the train will stall as with the prototype. |
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