Example 2: Gear Shift

• Basic system:
  • manual transmission in a car
  • allows to change the ratio of rotational speeds from motor and drive shaft
  • usually provides a few fixed ratios (gears)
  • complex internal structure
    • 
• Model in Modelica:
  • basic component GearWithExternalRatio
    • two connectors (flanges) and external input ratio r
    • equations
    • 
  • gear shift model
    • 
    • table to compute ratio from gear
    • additional inertias
  • complete model
    • 
  • run in Dymola with changing gear →
    •  Failed to reduce the DAE index.
• Mathematical problem:
  • event at gear shift r₁ → r₂
  • rotational speeds before event
    • 
  • rotational speeds after event
    • 
  • start values of ω_i, ω_o after event undefined
  • simple solution
    • fix one ω and compute the other
    • → often unrealistic behaviour
• Reality check:
  • problem appears in reality as well!
  • modern transmissions use synchronizer rings
    • adapt rotational speeds using friction
    • basically simple clutches
    • ω's adapt dynamically (according to connected inertias)
• Extended Model:
  • new gear shift model
    • 
    • one clutch per gear
  • shift logic component
    • detects gear change and opens/closes corresponding clutches
    • clutch of current gear is closed
    • other clutches are open
  • result
    • 
  • one new parameter for shifting time