Simplified Climbing- Design Theory and -Test Results T. S. HUFF, Engineer of Road Design, and F. H. SCRIVNER, Supervising Research Engineer, Texas Department A simplified theory of the motion of heavy vehicles on grades is presented. A set of speed-distance curves computed from the theory, based on values of maximum sustained speeds observed in Arizona, is given as the current basis for design of climbing in Texas. Speed-distance curves representing the observed performance of a test vehicle on 11 grades are compared with the corresponding curves developed from the theory. Fair agreement was found, and it was concluded that the simplified theory is accurate enough for use in the design of climbing lanes.

• CONSIDER a vehicle (Figure 1) of gross that although the net driving force must weight, W, travelling at a variable vel• satisfy Equation 1 involving the acceleration ocity, V, on a inclined at an angle, and the grade angle, it may also be ex• e, with the horizontal, the value of e pressed independently as some function of being taken as positive if the vehicle is as• velocity only, since each of its components cending and negative if it is descending. If is a function of velocity only. g represents the acceleration of gravity For example, if the truck operates at a and t the time, then, neglecting that part known maximum sustained velocity on any of the driving force required to impart grade, the numerical value of P/W corres• angular acceleration to rotating parts, we ponding to that velocity may be immediately may write the force equation, calculated from Equation 1, which in this case reduces to P/W = sin 9, and that Wdv P - W sin e, where P, magnitude of P/W will always exist at that g Ht velocity, at least approximately, regard• a variable, may be termed the net driving less of the value of the acceleration. In force acting on the vehicle. The above Figure 2 we have plotted values of P/W equation may be rewritten in the form computed in this way against correspond• ing values of the velocity, v, from basic P 1 dv sin e data supplied mainly by Willey (1.) in 1950, W=gc[f ^ (1) and applying to an average heavy vehicle The net driving force is the total traction operating on mountain grades in Arizona. exerted by the driving wheels against the The points plotted in Figure 2 are con• , less wind resistance and road nected by straight lines to form a con• surface resistance. Again neglecting in- tinuous graph of P/W versus v. Each ertial resistance to angular acceleration, straight line segment extending from, say, it follows that if the truck is always operated Vn to v^ may be represented by an at the highest possible speed and always equation bf^the form. within the range of engine speed recom• P /W = av + b mended by the manufacturer, then the total (2) driving force must be expressible, at least where varies within the interval, Vjj approximately, as a single-valued function to n + 1> and a and b are constant of the velocity only. Air resistance in within the same interval. still air is usually considered to be a func• From Equations 1 and 2 we may form a tion of the velocity only, and we shall as• third equation, not containing P/W ex• sume that no wind exists. We shall also plicitly, which becomes the general motion assume that the type and roughness of the equation for the vehicle, as follows: pavement do not change and, therefore, that the road surface resistance may be dv - gav + g (sin 6 b) = 0 (3) taken as constant, or at most as a function of velocity only. We therefore conclude where v is restricted to the velocity in- terval, to v^+j, and a and b are interval to v^+i- Thus, during the constant in the same interval. time t, the velocity changes from VQ to V, the vehicle travels a distance x, and the ratio, net driving force to gross weight, changes m value from (avQ + b) to (av + b). (The logarithm is taken to the base, e).

Pa net driving forci {|b>) grosi Mfight (Ibt) In using Equations 4 for calculating the vihielB It OBCtntfing distance traveled or time consumed by a vehicle while it changes velocity over an Mass X Acctlaralion > Forct, interval greater than that for which a and "fSt-p- b are constant, it is necessary to compute WMrB w = vtlocilr {lt/s»),and the increments of distance and time cor• t ' limm (Mc) responding to each subinterval of the type, (Tha additionol driving force raquirtd lo oeeelarota rotating pans II noglocttd ) Vn to v'n_ ' 'n to ^n+l. and Vn+i, to v, and to add these increments in order to Figure 1. obtain total distance and total time.

MAXIMUM SUSTAINED SPEEDS NUMERICAL VALUES USED IN PLJOTTING GRAPH FOR USE IN Psrcent Sine = v EQUATIONS (91 ,( Max 6rad« P/w (fWllc) le II9I5 Ellimottd - 0096049 11919 T 06983 - 0067991 12926 6 05990 10 27 - 0093993 094811 5 04994 19 20 - 0022699 0796S0 4 03997 17 60 - 00r5594 067419 5 02999 24 00 -0010267 094631 02000 J3 73 - 00090909 037039 00000 73 33

NOTE TU .olun of 0 oiul D opply batnan Iht nlociliti indicated Far tROmplt, b. 079690 in tha valeeity ronga liam 13 20 la 1760 ft / neluiKa

• William E Willa>;uiailll Trach Spaada: a ,JAN 1990,P92

vdt /sec)

Figure 2. Graph of Equation 2. We now denote the position of the vehicle TEXAS DESIGN METHOD at any instant by its coordinate, x, meas• ured in the direction of motion from a sta• Figure 3 shows speed-distance curves tionary point on the grade behind the truck. computed from Equations 4, the value of We also stipulate that x = t = 0 when v = the constants, a and b, having been taken V(j, and that the grade angle, 6, is con• from Figure 2. By interpolating between stant. Then the solution of Equation 3 may these curves, one may determine the ap• be written in the following form suitable proximate speed, in the range from 0 to for the construction of speed-distance and 47 miles per hour, of Willey's (j.) aver• time-distance curves: age heavy vehicle at any point on any series of successive grades ranging between X = (sin e - b) t minus 7 percent and plus 7 percent pro• (4) vided the speed at one point on the series of grades is known. The upper limit of wherein t = 1_ log/av + b - sin e 47 mph. was selected because that figure ag \avo + b sin e J was the average speed of trucks on ap• proximately level grades in Texas. and both v and v, pre restricted to the EXAMPLE OF USE OF CURVES WARRANTS FOR CLIMBING LANES iDashed Imai on graph indicot* itapt taken in CLASS B HIGHWAYS Providt climbing lont ond parking shouldtr finding propdr location for climbing lont ihawn CLASS C HIGHWAYS — OlSirable treotmsnt lama os for CLASS 8 HIGHWAYS i&=2sa; on ikoten I Minimum treotmcnl convarl to climbing lone _30M PH ond SPEED-DISTANCE CURVES CLASS D HIGHWAYS Moki studies to determine feosibility of converting occeleroting -Traniitions. stioulder too climbing lone.tohing into oecount ESTIMATED FOR (I) construction costs and, (21 volume of heavy trucks CLIMBING LANE A TYPICAL HEAVY TRUCK CLASS E HIGHWAYS — Climbing lonee not considered necessary

decelirating dvetltroting DECELERATION ACRFl FRATinM on grades indicoted 0 n 9 rades indicated h 32 -5% 1 % K M' f • 1 —t— V U ft / li / / ? f J / r i f ^ k W 'A '1% Vill

• \ 3% • 1 1 / 1 1 • 1 '/ • i • • 3 4 5 6 3 4 S S DISTANCE (Thoutandt of f«et) DISTANCE (Thousands of fsal)

Figure 3. In using the chart for design purposes, DATA PERTAINING TO TEST VEHICLE AND vertical curves are generally ignored and CONDITIONS OF OPERATION speeds are usually taken from Figure 3 on 1. Vehicle identification; International R-195 Tractor. 2. Vehicle overall maximum dimensions: (a) height, 7, 75 the assumption that the vehicle travels in a feet; (b) width, 7. 75 feet. straight line from one point of grade inter• 3. Total gross weight: 57,180 lb. 4. Manufacturer's maximum gross vehicle weight rating: section to the next. Vertical curves can 50, 000 lb. be broken up into straight-line segments, 5. Gear ratios: (a) transmission, 6. 98, 3. 57, 1. 89, 1. 00, 0. 825 (overdrive); (b) aux. trans., none; (c) axle, of course, if the additional accuracy is 6.5, 8.86; (d) total gear reductions, 61. 84, 45. 37, considered worthwhile. 31. 63, 23. 21, 16. 75, 12. 28, 8. 86, 6. 50, 7. 31, and 5. 36. 6. Tire size: 10. 00 by 20. 7. Net engine power at sea level: 146 hp. at 2, 600 rpm. * ROAD TEST OF A HEAVY VEHICLE 8. Altitude: 950 feet. 9. Road service type and condition: bituminous, good. In December 1953 a road test was con• •Brake horsepower, 162 at 2, 800 rpm. ducted by the Planning Survey of the Texas Highway Department (2) on a section of The curves of Figure 3 have been used ranch-to-market Road"93 in Travis and in the design of climbing lanes in Texas Burnet counties west of Austin in an effort since 1952. An example of the design to provide data from which the theory being procedure is given in the figure. Briefly, used in design of climbing lanes could be it consists in finding the point on an as• checked or corrected, if necessary. The cending grade where the speed drops to vehicle used was an International Harvester 30 mph. , and the next subsequent point R-195, two-axle truck tractor (146 net where the speed has increased to 30 mph. horsepower at sea level) and a 33-foot, and the truck is accelerating. These two Hobbs tandem-axle, flat-bed trailer, both points form the limits of the tangent sec• loaned to the department free of charge by tion of the climbing lane. Reversed curves, the respective manufacturers. Table 1 525 feet in length, are added to each end of gives essential data pertaining to the trac• the tangent. Thus the design vehicle is tor. The trailer was loaded with steel removed from the general stream piling, the gross weight of tractor and at a speed somewhat greater than 30 mph. , trailer being 57, 180 lb. (see Figure 4). and likewise is returned at a speed ex• In running the tests, pneumatic tubes, or ceeding 30 mph. detectors, which actuated electric switches

Figure 4. Vehicle used in road test. Tractor is over a pneumatic detector. when run over, were first stretched trans• (Figure 6) recorded the time each of the verse to the highway at 100-foot intervals four axles passed over the pneumatic tubes on a selected grade. Two instruments, an during the test. Esterline-Angus 20-pen graphic recorder Eleven grades ranging from 700 to 1,500 with about "^lo-second accuracy (Figure 5) feet in length, and from 0. 16 percent to and an oscillograph and camera with tuning- 7. 62 percent in inclination, were used in fork timer accurate to about yiooo second the test. In all test runs, the driver, an employee of the department, attempted to maintain the highest-possible speed while remaining within the range of engine speed recom• mended by the manufacturer and marked on the speedometer. The test procedure was as follows:

Up-Grade Acceleration Runs

The driver approached the grade at the bottom at a very-low speed (1 or 2 mph,). When within 3 or 4 feet of the first detector, he accelerated as rapidly as possible and continued to accelerate until he had passed over the last detector at the top of the grade. If he had not reached maximum sustained speed at that time, he returned to the bottom of the grade and repeated the run, except that he approached the first detector at approximately the speed and in the gear he had previously passed the last detector. This procedure was followed Figrure 5. Recorder measured time required until maximum sustained speed was at• for truck to travel distance between de• tained. tectors.

Figure 6. Oscillograph and camera in portable darkroc recorded time each axle passed over a detector. Up-Grade Deceleration Runs types described above. (Figure 7). Both recording instruments performed well, but The driver attempted to approach the it was impractical to operate the oscillo• grade at the bottom at a speed equal to or graph continuously during most test runs greater than 47 mph. and attempted to because the fast-moving recording paper reach the top at the highest-possible vel• would be exhausted before the truck had ocity. K, after passing the last detector, finished the run. his speed was still greater than maximum sustained speed, he returned to the bottom of the grade and repeated the run, except ANALYSIS OF ROAD-TEST RESULTS that he approached the first detector at Approximate values of velocities for approximately the speed and in the gear he use in plotting speed-distance curves rep• had previously passed the last detector. resenting the test runs were computed as The process was continued until the vel• follows from the basic data: (1) From 20- ocity on the grade was reduced to maximum pen recorder data, the velocity at the in• sustained speed. stant when the front axle of the vehicle was midway between two successive detectors Down-Grade Acceleration Runs was taken equal to the distance between detectors (100 feet) divided by the corre• At the top of the grade the driver ap• sponding time interval. (2) From oscillo• proached the first detector at 1 or 2 mph., graph data, the velocity when the mid-point then accelerated as rapidly as possible. between the second and third axles was

UP-GRADE UP-GRADE DOWN-GRADE Decalararion Runt Accalaration Rum Acceleration Rune Groda 610% Grade 0 16% Grade -0 77%

Run 4

Run 9 Run 3

Run 6 Run 2 Run 5

Run 4 ^^^^

Run Z a 20

Run 3

DISTANCE (IPOO')

Figure 7. A few of the 118 speed-distance curves developed from the several runs made on each grade. Runs are numbered in chrono• logical order.

If, on passing the last detector, he had not over a detector was taken equal to the dis• attained a speed of 47 mph., he returned tance between those axles (18. 62 feet) di• to the top of the grade, making his ap• vided by the corresponding time interval. proach on the second down-grade run at Accelerations for use m Equation 1 the speed and in the gear he had previously were computed from oscillograph data only, passed the last detector, and again ac• since such computations require more- celerated as rapidly as possible. The accurate data than velocity determinations. process was repeated until he attained a At low velocities, approximately simultan• speed of at least 47 mph. on the grade. eous values of acceleration and velocity All told, there were 118 test runs of the were computed from the time intervals between the passage of three successive points represent periods during which the axles over one detector. In order to con• truck was apparently traveling at maxi• vert the time data to accelerations, use mum sustained speed, that is, when the was made of finite difference forms of the acceleration was equal to zero. Ignoring derivatives, dVdt^ and dx/dt. At higher areas of the graph where the scatterii^ of velocities, time intervals between the pas• points was too wide to indicate any con• sage of one axle over three successive sistency in the data, an average line was detectors were used in the difference drawn through the remaining points. This equations. line was taken to represent the graphical

Moi wilainad oelocity computed b, SAE method, veriut P/W = tin 8

Max susfoined velocity obMrved during rood leet, vflriue P/W • tin 8

Velocity vertut P/W computed from occelerolion obterved during rood tett (See equotion 3) Gropb of P/W uied in computing tpeed-dittonce oiryet

Figure 8. Graph of P/W versus v from 1953 road-test data (solid line) and from SAE "Truck Ability Prediction Procedure" (dashed line). The approximate acceleration and the form of P/W expressed as a function of grade angle being known for a given instant, velocity only. these values were substituted in Equation 1 On the same graph values of sin 6 were for dv/dt and fl, respectively, and the nu• plotted against the corresponding maximum merical value of P/W was computed for sustained speeds computed by a method that instant. Each computed value of P/W proposed by the Society of Automotive Eng• was then plotted against the corresponding ineers (3). These values, plotted as points velocity in Figure 8, where the solid points enclosed in triangles, are based entirely represent instants when the acceleration on the factors pertaining to the truck and was different from zero, and the circled test environment given in Table 1.

Figure 9. Figure 10. 8

(The reason for the wide scattering of points on Figure 8 is not known, but it might have been due in part to unavoidable variations in wind direction, wind velocity and driver behavior. Some of the scatter• ing might also have resulted from the in• herent inaccuracies encountered in the sub-

t s 0i9T«HCE IIPOO)

Figure 13. parent tracing cloth, was placed over a j„i corresponding test curve plotted to the same scale, and the velocity lines (hori• zontal lines) on the two graphs were i : ; ; matched. The computed curve was then »•"•••"«»• moved horizontally, keeping the velocity Figure 11. lines matched, until it appeared to pass stitution of difference equations for differ- through the midpoint of the test curve, ential equations. And some scattering could be expected because of variations in the force required to change the angular ve• locity of rotating parts while the truck accelerated at varying rates). I Next, from the graph of average values | of P/W versus velocity (Figure 8), and by • use of Equations 4, a set of three speed- distance curves (up-grade deceleration, up-grade acceleration, and down-grade acceleration) was plotted for each of the eleven test grades. D)ITI.tt 1,0001

Figure 14. Then the test curve was traced on the cloth with the computed curve. If the curve so transferred coincided with the computed curve, then it could be concluded that, within the range of velocities covered by the test curve, the computed curve repre-

—iWi-

OISTANCE lipOOl

Figure 12. Finally, the 118 speed-distance curves (Figure 7) previously plotted directly from the observed data, (referred to hereafter as "test curves") were compared with the corresponding speed-distance curves com• puted by use of the graph of average values of P/W (referred to hereafter as "computed curves") in the following mannen The computed curve, drawn on trans- Figure 15 9

ing force acting at any sustained velocity was, on the average, greater than the net driving force acting at the same velocity when the vehicle was accelerating or de• celerating. This apparent anomoly in vehicular performance might be explained by the fact that the driver, while rapidly accelerating or decelerating, had little time for searching out the best gear, whereas his sustained speed on any grade occurred only after he had had ample time to find the proper gear for that grade. In this connection it was also noted that the maximum sustained speeds computed for the test vehicle by the method recommended by SAE (see points enclosed in traingles in Figure 8) agreed rather well with the

Diitonct djOOO') observed values (the circled points in Fig• ure 8) except in the velocity range of about Figure 16. 14 ft. per sec. to 30 ft. per sec. (9. 5 mph. to 20. 5 mph.). In this range the values sented the test data well. On the other computed by the SAE method were some• hand, the contrary was true if the test what greater than the observed values. curve departed substantially from the com• In spite of the exceptions noted above, puted curve. Figures 9 through 17 show the speed-distance curves computed from these comparisons. the graph of average values of the ratio, net driving force to gross weight (Figure COMPARISON OF TEST RESULTS WITH 8), appeared to represent the average per• THEORY formance of the test vehicle fairly well. Therefore, Figure 18, which was made up Although fair agreement frequently -ex• isted between the shapes of the speed- by use of Figure 8 and Equations 4, for in• distance curves plotted directly from the tegral values of grade percentages, may data, (the solid lines of Figures 9 through be taken as a general summary of the av• 17) and the curves computed from the graph erage performance of the test vehicle. If of average values of P/W (the dashed lines detailed comparisons of Figure 18 are of Figures 9 through 17), two major ex• made with Figure 3, it will be seen that ceptions are noteworthy: the test vehicle was generally slower than 1. On many runs, the test curves in• the design truck in current use in Texas. dicated some irregularity in the motion of the vehicle, apparently caused in part by gear shifting. This irregularity was es• pecially noticeable on some of the up• grade deceleration runs at velocities ap• proaching maximum sustained speed, when the vehicle frequently first slowed to 2 or 3 mph. below maximum sustained speed and then accelerated. 2. The observed maximum sustained speed was frequently from 1 to 3 mph. greater than the speed shown on the com• puted curves. The reason for this dis• crepancy may be found by reference to Figure 17. Figure 8, where it can be seen that most CONCLUSIONS of the circled points (which represent net driving force to gross weight at maximum 1. Inspection of the test curves of Fig• speed) lie above the line representing the ures 9 through 17 indicate that, even under average of all points. Thus, the net driv• controlled conditions, the relation between EXAMPLE OF USE OF CURVES

(Ooihad linaa on graph indicofa staps tahan in finding propar location for climbing lono ahown J*=299L. on iketch 1 3aM PH ond SPEED-DISTANCE CURVES a

on grades Indicated on grodes indicated

3 4 5 3 * S 6 DISTANCE (Thousonds of f66t) DISTANCE (Thousonds of r66t)

Figure 18. 11

the speed and the distance travelled by the full-scale tests. average heavy vehicle handled by a driver of probably better than average skill may ACKNOWLEDGEMENTS not always be consistent. 2. The speed-distance curves com• The following department employees puted on the assumption of a net driving were especially helpful in conducting the force which varies only with velocity agreed road test; Reed Baker, K. G. Crawford, fairly well with the corresponding curves W. R. Welty, Hubert Henry, S. W. Adair, plotted directly from test data, at least in and Gene Voltz. Miss Leah Moncure did those cases where the vehicular perform• the numerical work in connection with the ance was consistent. Therefore it appears reduction of the data and the preparation of that the simplified theory (Equations 4) is the graphs. sufficiently accurate for use in design of For their generous cooperation thanks climbing lanes. are also due Johnny Wright, of Hobbs 3. The Society of Automotive Engineers Trailer Company; H. T. Rosell and J. H. has provided a method for computing max• Gillum, of International Harvester; P. J. imum sustained speeds for any gross Rudolph, of the Petty Geophysical Engi• ' weight to horsepower ratio (3). Values so neering Company; and C. M. Ogle, of the computed, if used in conjunction with the Texas Motor Transportation Association. simplified theory of truck motion present• Fred B. Lautzenhiser, Chairman of the ed herein, should make it possible to pre• SAE Subcommittee on Classification and dict, at least approximately, the behavior Evaluation of Transportation Engineering on grades of vehicles of any gross-weight- Formulas, furnished formulas used in to-horsepower ratio without resorting to computing maximum sustained speeds (3),

References

1. "Uphill Truck Speeds", by William by Willey. E. WiUey, Roads and Streets, January 2. "Texas Highways", Texas Highway 1950, p. 52. See also, "Uphill Speeds Department, Austin, Texas, January, of Trucks", Highway Research Board 1954, p. 14. Proceedings, 1949, p. 304; and "Sur• 3. "Truck Ability Prediction Pro• vey of Downhill Speeds of Trucks on cedure", Second Edition. SP-82, Society Mountain Grades", Highway Research of Automotive Engineers, Inc., 29 West Board Proceedings, 1950, p. 322, both 39th , New York 18, New York,