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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