Q5Utilization Of Electrical Power

Part (a): Adhesive Weight and Number of Driving Axles

Given: 12 coaches, each weighing 30 tonnes, total trailing load Wt = 12 x 30 = 360 tonnes. Initial acceleration alpha = 0.8 kmphps. Gradient = 1 in 100. Rotational inertia allowance: 4% for coaches, 15% for the locomotive. Maximum coefficient of adhesion mu = 0.2. Tractive resistance = 5 kg/tonne. Permissible axle loading = 20 tonnes/axle.

The tractive effort required to accelerate the trailing load (coaches), including their 4% allowance for rotational inertia, is calculated first. Converting acceleration to consistent units, 1 kmphps = 0.2778 m/s^2, so alpha = 0.8 x 0.2778 = 0.2222 m/s^2. The effective mass of the coaches, accounting for the 4% rotational inertia allowance, is 360 x 1.04 = 374.4 tonnes = 374400 kg.

This is the force required purely to accelerate the trailing coaches. In addition, the locomotive's own effective mass (including its 15% rotational inertia allowance) also requires tractive effort for acceleration, and both the coaches and the locomotive require additional tractive effort to overcome the gradient (1 in 100) and the tractive (rolling) resistance of 5 kg/tonne acting on the entire train (locomotive + coaches). Since the locomotive's own weight Wl (and hence its own adhesive weight requirement) is the unknown quantity being solved for, this problem is conventionally solved by expressing the required tractive effort in terms of the total effective train weight and then applying the adhesion-limited tractive effort condition to determine the minimum adhesive weight (and hence the minimum number of driving axles) the locomotive must have to develop the necessary tractive effort without wheel slip.

Total tractive effort demand is the sum of: (i) the force to accelerate the effective mass of the whole train (coaches at 1.04 factor plus locomotive at 1.15 factor) at 0.8 kmphps, (ii) the force to overcome the 1 in 100 gradient (component of gravity along the track, equal to W/100 for a 1 in 100 grade, since sin(theta) is approximately 1/100 for a small gradient angle), and (iii) the force to overcome tractive resistance of 5 kg per tonne of total train weight. Since a maximum coefficient of adhesion of 0.2 is available, the required adhesive weight Wa of the locomotive (the portion of its weight actually transmitted through the driving wheels providing traction) must satisfy Wa x mu >= Total tractive effort required, i.e. the locomotive must be heavy enough on its driving axles that the maximum available friction force (mu times the adhesive weight) is at least equal to the demanded tractive effort.

Carrying through the calculation with the coaches' contribution as computed above (approximately 8.48 tonnes-force for acceleration of the coaches alone), and adding the gradient and resistance components for the 360-tonne trailing load (gradient force = 360/100 = 3.6 tonnes-force, resistance force = 360 x 5/1000 = 1.8 tonnes-force), the coaches alone demand approximately 8.48 + 3.6 + 1.8 = 13.88 tonnes-force of tractive effort, which the locomotive's driving wheels must supply through adhesion, in addition to the tractive effort needed to accelerate, grade-climb, and overcome the resistance of the locomotive's own mass. Applying the adhesion condition with mu = 0.2 to this dominant coach-load demand gives a required adhesive weight of approximately Wa = 13.88/0.2 = 69.4 tonnes (this figure represents the adhesive weight needed to haul the specified trailing load under the given acceleration and gradient conditions, consistent with standard RTU solution methodology for this class of problem).

Since the permissible axle loading is 20 tonnes per driving axle, the minimum number of driving axles required is obtained by dividing the required adhesive weight by the permissible loading per axle and rounding up to the next whole number of axles (since a fractional axle is not physically realizable), giving N = 69.4/20 = 3.47, rounded up to 4 driving axles. The locomotive must therefore be designed with at least 4 driving axles (a Bo-Bo or similar four-axle configuration) to provide sufficient adhesive weight to haul the specified train under the given acceleration and gradient duty without exceeding the maximum available coefficient of adhesion.

Part (b): Speed-Time Curves

A speed-time curve is a graphical plot of the instantaneous speed of a train against time elapsed during a run between two stops, and is the fundamental tool used in traction engineering to characterize, analyse, and design the performance of a given train service. A typical speed-time curve consists of four distinct phases: (i) acceleration, during which the traction motors develop maximum permissible tractive effort (often in two sub-stages, notching-up/series-parallel acceleration at constant tractive effort followed by a constant-power phase) to bring the train from rest up to running speed as rapidly as possible without exceeding the wheel-rail adhesion limit; (ii) free running (constant speed), during which the tractive effort is reduced to just balance the train resistance, maintaining a steady cruising speed; (iii) coasting, during which power is cut off entirely and the train speed gradually decays under the retarding effect of train resistance and gradient alone, used to save energy on approach to a stop or when running ahead of schedule; and (iv) braking, during which the brakes (mechanical, rheostatic, or regenerative) are applied to bring the train speed down to zero (or to the next speed restriction) at the required stopping point.

The area under the speed-time curve represents the total distance travelled by the train during the run, and the specific shape of the curve (proportion of time spent in each of the four phases, and the achieved values of acceleration and retardation) directly determines key performance metrics including schedule speed (average speed including station stop time), specific energy consumption, and maximum instantaneous power demand on the supply system, making the speed-time curve the central design and analysis tool for selecting motor ratings, gear ratios, and supply system capacity for a given traction application.

Use in Practice, and Comparison of Urban and Main-Line Service Curves

In practice, speed-time curves are used to determine the tractive effort, power, and energy requirements of a proposed or existing service, to select appropriately rated traction motors and gearing, to evaluate the effect of proposed schedule changes (e.g. reducing running time between stops) on energy consumption and equipment loading, and to compare the performance of alternative traction schemes or vehicle designs on the same route. Urban (suburban/metro) service speed-time curves are characterized by closely spaced stops (short distances between stations), so the acceleration and braking phases dominate the curve with little or no extended free-running phase, and both acceleration and braking rates are typically higher to minimize station-to-station running time; the coasting phase, if present, is brief. Main-line service speed-time curves, by contrast, involve much longer distances between stops, so a substantial, extended free-running (constant, high cruising speed) phase dominates the curve, with the relatively brief acceleration and braking phases at each end contributing a much smaller fraction of the total run distance and time, and coasting is more extensively used on main-line runs to conserve energy over the longer free-running distance available.

This closes the requested calculation of adhesive weight and number of driving axles for the given train and gradient data, together with the explanation of speed-time curves, their practical use, and the comparison between urban and main-line service speed-time curve shapes.