Trapezoidal Speed - time Curve

 

• The speed – time curve of the main line service is replaced by trapezoidal curve. 
• The running and coasting period in the trapezoidal curve is simply replaced by constant speed as shown in the figure. 
• The area of the speed – time curve represents total distance travelled by the train.

Let

α = Acceleration in km / second

β = Retardation in km / second

V_m = Crest speed in km / hour

T = Total time in second

Acceleration time t₁ = V_m / α …. ( 1 )

Retardation time t₃ = V_m / β ……( 2 )

Free running time t₂ = T – ( t₁ + t₃ ) ….. ( 3 )

                                 = T – ( V_m / α + V_m / β )

Total distance of run in km ( S )

= Distance travelled during acceleration +

   Distance travelled during free run +

   Distance travelled during braking

= Triangle Area OAB + Rectangle Area ABCD + Triangle Area CDE

= V_mt₁ / ( 2 × 3600 ) + V_mt₂ / ( 3600 ) + V_mt₃ / ( 2 × 3600 )  ….. ( 4 )

Substitute value of ( 1 ), ( 2 ) and ( 3 ) in the equation ( 4 )

S = V_m² / 7200 α + V_m { T – ( t1 + t3 ) } / 3600 + V_m² / 7200 β

S = V_m² / 7200 α + V_m / 3600 { T – ( t1 + t3 ) } + V_m² / 7200 β

S = V_m² / 7200 α + V_m / 3600 { T – ( V_m / α + V_m / β ) } + V_m² / 7200 β

S = V_m² / 7200 α + V_mT / 3600 – V_m² / 3600α – V_m² / 3600β ) } + V_m² / 7200 β

S = V_m² / 7200 α – V_m² / 3600α + V_mT / 3600 + V_m² / 7200 β – V_m² / 3600β

S = V_mT / 3600 – V_m² / 7200 α  – V_m² / 3600β

                                     OR

Vm² / 3600 ( 1 / 2α + 1 / 2β ) – V_mT / 3600 + S = 0

Let us consider that k = ( 1 / 2α + 1 / 2β )

kVm² / 3600 – V_mT / 3600 + S = 0

kVm² – V_mT + 3600S = 0

Now

a = k

b = T

c = 3600S

Solution of the equation

         V_m = b ± √ ( b² – 4ac ) / 2a

         V_m = T ± √ ( T²  – 4( k ) ( 3600S ) / { 2k }

        V_m = T ± √ ( T²  – 4( k ) ( 3600S  ) / { 2k }

        V_m = T ± √ ( T² / 4k²  – ( 3600S / k )

• If we use positive sign for the solution of the equation, the value of V_m should be much higher than the expected. 
• The negative sign is used in practice.

        V_m = T – √ ( T² / 4k²  – ( 3600S / k )

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