## 16/03/2021

The simplified speed – time curve for urban and sub-urban service for electrical traction is shown in the Figure.

# Quadrilateral Speed – Time Curve

The study of Quadrilateral speed time curve is given below.

Let

α = Acceleration in km per hour per second

βC = Coasting retardation in km per hour per second

β = Braking retardation in km per hour per second

V1 = Maximum speed at the end of acceleration ( km per hour )

V2 = Speed at the end of coasting period ( km per hour )

T = Total time of run ( second )

Acceleration time in second t1 = V1 / α  …. ( 1 )

Coasting time in second t2 = V1 – V2 / βC ….. ( 2 )

Braking time in second t3 = V2 / β …… ( 3 )

Total distance travelled in ( km )

S = Distance travelled during acceleration ( Area ABC )  +

Distance travelled during coasting ( Area BDEF ) +

Distance travelled during retardation ( DEF )

S = ( ½ V1t1 / 3600 ) + { ( V1 + V2 ) / 2 × ( t2 / 3600 ) } + ( ½ V2t3 / 3600 )

S = ( V1t1 / 7200 ) + { ( V1t2 / 7200 ) + V2t2 / 7200 ) } + ( V2t3 / 7200 )

S = { V1 ( t1 + t2 ) / 7200 } +  { V2 ( t2 + t3 ) / 7200 }

As t1 + t2 + t3 = T

S = { V1 ( T – t3 ) / 7200 } +  { V2 ( T – t1 ) / 7200 }

S = V1T / 7200 + V2T / 7200 – V1t3 / 7200 – V2t1 / 7200

Now t = V1 / α and t3 = V2 / β

S = { T ( V1 + V2 ) / 7200 } –  ( V1 ×  V2 ) / 7200 β –  ( V1 ×  V2 ) / 7200 α

S = { T ( V1 + V2 ) / 7200 } –  V1V2 / 7200 β –  V1V2 / 7200 α

7200S = T ( V1 + V2 ) – V1V2 ( 1 / α + 1 / β )

From equation ( 2 )

t2 βC = V1 – V2

V2 = V1 – t2 βC

V2 = V1 – βC ( T – t1 – t3 )

V2 = V1 – βC ( T – V1 / α  – V2 / β )

V2 = V1 – βC T + βC V1 / α +  βC V2 / β

V2 – βC V2 / β  = V1 – βC T + βC V1 / α

V2 ( 1 – βC / β )  = V1 – βC ( T + V1 / α )

V2 = V1 – βC ( T + V1 / α ) / ( 1 – βC / β )

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