Trapezoidal & Quadrilateral Speed Time Curve of Traction

In order to find out the performance of an electric traction service at different schedule speeds, speed-time curves are replaced by trapezoidal and quadrilateral curves. From these simplified curves (trapezoidal and quadrilateral curves), the relationship between acceleration, retardation, average speed, and distance can be obtained.


The use of these simplified curves saves the labor of repeating the calculations a large number of times and also while performing calculations there is no need for much information related to motor characteristics.


While constructing trapezoidal and quadrilateral speed time curves, the following points need to keep in mind,
  • The area enclosed by the curve, acceleration, and braking retardation values must be similar to that of the original speed-time curve.
  • The running and coasting periods are replaced by a constant speed period for a long-distance run as shown by the trapezoidal speed time below.
  • The initial acceleration and coasting periods are extended for the short distance of run as shown by the quadrilateral speed-time curve.
  • The area under the trapezoidal and quadrilateral curves and the actual speed time curve should be same.

In the two simplified curves, the trapezoidal speed-time curve corresponds to the speed-time curve of mainline service and the quadrilateral speed-time curve corresponds to the speed-time curve of urban and suburban services where short distances are involved. Let us see the construction and calculations of trapezoidal and quadrilateral speed-time curves.

Trapezoidal Speed Time Curve :

The trapezoidal speed-time curve is a close approximation to the speed-time curve of mainline service where the free-running period of the locomotive is long. The below shows the trapezoidal speed-time curve OPQR where speed is taken on the y-axis and time on the x-axis.

Trapezoidal Speed Time Curve

Let,
  • S = Distance between two stations or stops in meters.
  • t = Actual run time between two stops in sec.
  • α = Acceleration during starting point in m/s2.
  • β = Retardation during braking in m/s2.
  • Vp = Peak value of speed in m/sec.
  • Vavg = Average value of speed in m/sec.
  • ta = Time of acceleration in sec.
  • tb = Time of free-running period in sec.
  • tc = Time of braking in sec.
The total distance between stops is given by the area of trapezium OPQR.
S = Area of OPQR
S = Area of OPT + Area of PQST + Area of QRS
S = A1 + A2 + A3 ...(1)
We know that acceleration is defined as the rate of change in velocity. From the figure, we have,
α = Vp/ta and β = Vp/tc ...(2)
In the above curve, the area of triangle OPT is,
A1 = 1/2 × Base × Height = 1/2 × ta × Vp ...(3)
Area of PQST,
A2 = length × Breadth = Vp × tb ...(4)
Area of triangle OPT,
A3 = 1/2 × Base × Height = 1/2 × tc × Vp ...(5)
Therefore, substituting equations 3, 4, 5 in equation 1, we get,
S = 1/2 × ta × Vp + Vp × tb + 1/2 × tc × Vp ...(6)
But from figure, we have,
tb = [t - (ta + tc)]
Substituting above tb value in equation 6 we get,
Trapezoidal Speed Time Curve
From equation 2 we have,
α = Vp/ta ⇉ ta = Vp/α ...(8)
β = Vp/tc ⇉ tc = Vp/β ...(9)
On adding equation 8 and 9 we get,
ta + tc = Vp × 1/α + 1/β
Substituting the above ta + tc value in equation 7, we get,
Trapezoidal Speed Time Curve
In the above equation, the +ve sign cannot be adopted, as the value of Vp obtained by using the +ve sign will be much higher than what is possible in practice. Hence -ve sign will be used and therefore, we have,
Trapezoidal Speed Time Curve
From the above equation, an unknown quantity can be determined by substituting the values of known quantities.

Quadrilateral Speed Time Curve :

The quadrilateral speed-time curve is a close approximation to the speed-time curve of urban or suburban services where the distance between stops is small. The below shows the quadrilateral speed-time curve OSTU where speed is taken on the y-axis and time on the x-axis.

Quadrilateral Speed Time Curve of Traction

Let,
  • S = Distance between two stations or stops in meters.
  • VS = Speed at the end of the acceleration period in m/sec.
  • VT = Speed at the end of coasting period in m/sec.
  • α = Acceleration during starting point in m/s2.
  • βc = Retardation during coasting period in m/s2.
  • β = Retardation during braking in m/s2.
  • t = Actucal run time between two stops in sec.
  • ta = Time of acceleration in sec.
  • tb = Time of coasting peirod in sec.
  • tc = Time of braking in sec.
Total distance between stops is given by,
S = Area of OSTU
S = Area of OSW + Area of STVW + Area of TUV
S = A1 + A2 + A3 ...(1)
We know that acceleration is defined as the rate of change of velocity. From the figure, we have,
α = VS/ta, β = VT/tc, and βc = (VS - VT)/tb ...(2)
In the above curve, area of traingle OSW,
A1 = 1/2 × Base × Height = 1/2 × ta × VS ...(3)
Area of STVW,
A2 = (VS + VT/2) tb ...(4)
Area of traingle TUV,
A3 = 1/2 × Base × Height = 1/2 × tc × VT ...(5)
Therefore, substituting equations 3, 4, 5 in equation 1, we get,
S = VS/2 × (ta + tb) + VT/2 × (tb + tc) ...(6)
But from figure, we have,
t = ta + tb + tc
t - tc = ta + tb
t - ta = tb + tc
Substituting above (t - tc) and (t - ta) values in equation 6 we get,
Quadrilateral Speed Time Curve of Traction
From equation 2 we have,
α = VS/ta ⇉ ta = VS/α ...(8)
β = VT/tc ⇉ tc = VT/β ...(9)
Substituitng equation 8 and 9 in equation 7 we get,
Quadrilateral Speed Time Curve of Traction
From the curve, the retardation during coasting period is,
βc = (VS - VT)/tb
VT = VS - βc tb ...(11)
But from figure, we have,
t = ta + tb + tc
tb = t - ta - tb ...(12)
Substituting equation 12 in 11 we get,
VT = VS - βc (t - ta - tc)
Substituting ta, tc values from equations 8 and 9 in the above equation we get,
Quadrilateral Speed Time Curve of Traction
Thus from equations 10 and 13, the distance between two stops and speed at the end of the coasting period can be calculated by knowing the values of α, βc, and β.


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