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12/03/2021

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.


trapezoidal-speed-time-curve.png


Let

α = Acceleration in km / second

β = Retardation in km / second

Vm = Crest speed in km / hour

T = Total time in second

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

Retardation time t3 = Vm / β ……( 2 )

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

                                 = T – ( Vm / α + Vm / 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

= Vmt1 / ( 2 × 3600 ) + Vmt2 / ( 3600 ) + Vmt3 / ( 2 × 3600 )  ….. ( 4 )

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

S = Vm2 / 7200 α + Vm { T – ( t1 + t3 ) } / 3600 + Vm2 / 7200 β

S = Vm2 / 7200 α + Vm / 3600 { T – ( t1 + t3 ) } + Vm2 / 7200 β

S = Vm2 / 7200 α + Vm / 3600 { T – ( Vm / α + Vm / β ) } + Vm2 / 7200 β

S = Vm2 / 7200 α + VmT / 3600 – Vm2 / 3600α – Vm2 / 3600β ) } + Vm2 / 7200 β

S = Vm2 / 7200 α – Vm2 / 3600α + VmT / 3600 + Vm2 / 7200 β – Vm2 / 3600β

S = VmT / 3600Vm2 / 7200 α  – Vm2 / 3600β

                                     OR

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

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

kVm2 / 3600 VmT / 3600 + S = 0

kVm2 VmT + 3600S = 0

Now

a = k

b = T

c = 3600S

Solution of the equation

         Vm = b ± √ ( b2 – 4ac ) / 2a

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

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

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

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

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

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