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Horsepower versus Speed

In January, 1998, I posted a couple of responses to a question asked by one of the Alpine listers. The question was "How much Horsepower do I need to go 130 mph?" (or something similar..). I have updated the analyses to revise the drag coefficient, coefficient of friction between the road and tire, and to add rolling resistance due to tire flex. I ask your forgiveness if the formatting turns out crappy...

Remember...

HP = (Drag)*(Velocity)/550 + Wheel Bearing Loss (HP)

Wheel Bearing Loss (HP) = Mt * N /63000 where: Mt = F * f * D * N F = Bearing Radial Load (lbs) = 600 lbs f = Friction Factor = 0.0018 D = Bearing Bore (inches) = 1.125 inches N = RPM = (Velocity - ft/sec)*(60 - sec/min) / (pi * tire dia (ft)) = 9.55 * Velocity

Drag is the sum of all forces acting in opposition to forward motion. In the previous analyses, I had considered aerodynamic drag and tire side slip (from toe in/out). Now I have included tire rolling resistance due to pressure and road surface (25 psig and smooth concrete). Here goes...

Drag = Aero + Tireslip + Rolling Resistance

Aero = (Rho)(V*V)(Cd)(S)/2
     where: Rho = Std Air Densities (sea level = 0.002377 slugs/ cu. ft)
                                       2000 ft = 0.002241   "     "
                                       4000 ft = 0.002109   "     "
              V = Velocity in ft/sec
             Cd = Drag Coefficient = 0.45 ( was 0.50)
              S = Alpine Frontal Area (sq ft) = 18 sq ft (est.)
Tireslip = (Mu) * P * sin(B) * 2 
     where: Mu = Coefficient of friction, rubber - road = 0.62 (was 0.50) 
             P = Weight on One Front Tire (lbs) = 600 lbs 
             B = ARCTAN(Toe in/out / contact patch length) (B is slip angle) 
               = ARCTAN(0.125/6) = 1.2 degrees 
Rolling Resistance = fr * W 
     where: fr = resistance coefficient 
               = fo + 3.24 *fs* (Sp/100)^2.5 
            Sp = Speed (mph) 
            fo = Basic Coefficient = 0.012 
            fs = Speed Effect Coefficient = 0.0076 
             W = Vehicle Weight = 2400 lbs 

Putting it all together and plugging in the numbers to get...

HP = (600)*(0.0018)*(1.125)*(V)*(9.55)/63000            (Wheel Bearings) 
     + (0.002377)*(V*V*V)*(0.45)*(18)/1100              (Aerodynamics) 
     + (0.62)*(600)*(sin(1.2))*(2)*(V)/550              (Tire Slip) 
     + ((0.012 + 3.24 *(0.0076)*((V)*1.467/100)^2.5) 
        *2400)*V/550                                    (Rolling Res) 

Which reduces to

HP = 0.0001842*V + 0.0000175*V^3 + 0.028*V +  
        + (0.052 + 0.011*(.007*V)^2.5)*V 

Plugging that into Excel 4.0 and varying the second term (density) and find the data in Table 1 and plotted in Figure 1!

Table 1
SpeedVel.HPHP HP
(mph)(ft/sec)(s/l)(2000')(4000')
57.340.600.590.59
1014.671.231.231.23
1522.01 1.951.941.93
2029.34 2.802.782.75
2536.68 3.823.773.72
3044.01 5.054.964.88
3551.35 6.536.396.26
4058.68 8.318.117.91
4566.0210.4310.159.87
5073.3512.9412.5512.16
5580.6915.8715.3514.84
6088.0218.6017.9417.94
6595.36 23.2022.3321.49
70102.6927.6826.6025.55
75110.0332.7631.4330.14
80117.3638.4936.8735.30
85124.7044.9142.9741.09
90132.0352.0649.7547.52
95139.3759.9957.2854.65
100146.7068.7465.5862.52
105154.0478.3574.7071.16
110161.3788.8884.6880.61
115168.71100.3795.5790.91
120176.04112.85107.40102.10
125183.38126.38120.21114.23
130190.71140.99134.06127.33
135198.05156.74148.98141.44
140205.38173.67165.01156.61
145212.72191.82182.20172.86
150220.05211.24200.58190.25
155227.39231.97220.21208.81
160234.72254.06241.13228.58
170249.39302.48286.97271.93
175256.73328.91311.99295.58
180264.06356.88338.47320.61
185271.40386.43366.44347.05
190278.73417.61395.95374.95
195286.07450.46427.05404.34
200293.40485.03459.77435.28

Copyright (C) 1998 - 2004, all dates inclusive, L.E. Mayfield - All Rights Reserved
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