Railway & Locomotive Traction Engineering, AREMA Train Resistance, Ruling Grade & Power
Railroad Train Resistance, Ruling Grade & Drawbar Power
Calculate Specific Rolling + Aerodynamic Resistance (N/t), Ruling Grade + Curve Resistance (N/t), Total Coupler Drawbar Tractive Effort (kN), and Rail Drawbar Power (kW).
On steel-on-steel freight rail, a modest 1.15% mountain grade adds 5x MORE drag than rolling friction and aerodynamic drag combined!
Tapered steel wheels on hardened `136RE` rail roll with extraordinarily low journal + flange resistance (`r_{run} = 6.0 + (125 / w_{axle}) + 0.085 * V + 0.0011 * V^2` N/tonne at Speed `V` km/h and Axle Load `w_{axle} = M_{train} / N_{axles}` tonnes/axle per metric AREMA freight correlations)—just `~23` N/tonne (`0.23%` friction coefficient!) at `72` km/h! That extreme rolling efficiency makes freight consists hyper-sensitive to **Gravity Ruling Grade `G%`** (`1.0%` rise = `10` m per `1,000` m = **`98.0665` N/tonne** exact potential-energy component!) plus **Uncompensated Curve Flange Drag** (`0.04%` equivalent grade per AREMA degree of curvature `D_c` = **`3.92266 * D_c` N/tonne**)! Summing **`r_{tot} = r_{run} + r_{gc}`** over Train Gross Mass `M` (`tonnes`) sizes **Total Coupler Tractive Effort `F_{TE} = (M * r_{tot}) / 1,000`** (`kN`) and **Drawbar Rail Power `P_{rail} = F_{TE} * (V / 3.6)`** (`kW`)!
Verify on a `4,800`-Tonne (`48`-car, `192`-axle -> `25.0` t/axle) Intermodal/Grain Consist—Gross_Train_Mass `M` = 4,800 t (B2), Total_Axles `N_{ax}` = 192 (B3), Track_Speed `V` = 72.0 km/h (`20` m/s, B4), Ruling_Grade `G` = 1.15% (B5), and Track_Curvature `D_c` = 2.50° (B6): Level Running Resistance equals `22.82` N/t (E2) vs Grade+Curve Resistance `122.58` N/t (E3), demanding `697.9` kN Tractive Effort (E4) and **`13,958.9` kW** (`18,719` hp -> `5` six-axle `4,400`-hp diesel-electric locomotives at `85%` transmission efficiency! E5)!
| AREMA Freight Haulage KPI (4,800 t, 192 Axles: Fast 72 km/h @ 1.15% vs Mountain 32 km/h @ 2.20%) | Formula | Verified Output |
|---|---|---|
| Level Running Resistance (N/t) | 6.0 + 125/(M/Nax) + 0.085*V + 0.0011*V^2 | 22.82 N/t (14.85 N/t @ 32 km/h) |
| Grade + Curve Resistance (N/t) | 98.0665 * Grade_Pct + 3.92266 * Curve_deg | 122.58 N/t (231.44 N/t @ 2.20%, 4.0° curve!) |
| Total Drawbar Tractive Effort (kN) | (Train_Mass_t * (r_run + r_gc)) / 1000 | 697.9 kN (1,182.2 kN coupler limit check!) |
| Required Drawbar Power (kW) | Tractive_Effort_kN * (Speed_km_h / 3.6) | 13,958.9 kW (10,508.1 kW @ 32 km/h) |
AREMA Modified Running Resistance, Ruling Grade + Curvature & Rail Power Formula
Validate `Train_Mass M >= 50` t, `Axle_Count >= 4`, realistic axle load `M / N_{ax}` in `[5, 45]` t/axle, `Speed V` in `[5, 200]` km/h, `Grade G` in `[-3.0%, 5.0%]`, and `Curvature D_c` in `[0°, 18°]`!
=LET(mt_t, B2, na_x, B3, vs_k, B4, gp_c, B5, cd_g, B6, wa_t, mt_t / na_x, rr_n, 6 + (125 / wa_t) + 0.085 * vs_k + 0.0011 * (vs_k ^ 2), rg_n, 98.0665 * gp_c + 3.92266 * cd_g, ft_k, (mt_t * (rr_n + rg_n)) / 1000, IF(OR(mt_t < 50, na_x < 4, wa_t < 5, wa_t > 45, vs_k < 5, vs_k > 200, gp_c < -3, gp_c > 5, cd_g < 0, cd_g > 18), "Invalid Consist Input", ROUND(ft_k * (vs_k / 3.6), 1)))2.20% Continental-Divide Mountain Crawl (`32` km/h, `4.0°` Curve) Coupler Knuckle Limit Check
Climb a `2.20%` mountain pass with `4.0°` reverse curves at `32` km/h (`B4 = 32`, `B5 = 2.2`, `B6 = 4.0`): Power E5 actually drops to `10,508.1` kW, YET Tractive Effort E4 surges to **`1,182.2` kN** (`265,800` lbf)—approaching the `~1,735` kN AAR Grade E head-end coupler knuckle fatigue limit and requiring mid-train **Distributed Power (`DP`)** radio helpers!
=IF(ISTEXT(E5), E5, LET(wa_t, B2 / B3, rr_n, 6 + (125 / wa_t) + 0.085 * B4 + 0.0011 * (B4 ^ 2), rg_n, 98.0665 * B5 + 3.92266 * B6, ROUND((B2 * (rr_n + rg_n)) / 1000, 1)))Try the example
Use the original values or change them to test your understanding.