The ZF 9 Speed Automatic - part 2

For fun and profit I've coded working simulations of automatic transmissions using in m, c, Simulink, and even Excel spreadsheets.  My latest effort is compiled C/C++ to make a GNU octave .oct file for the ZF nine speed transmission, with INI file initialization via a flex based lexer.  The fact that I'm posting this on Christmas Eve pretty much proves I have no clue how to take a real vacation.  On the other hand I'm sitting on the couch with my kid, and in guy terms that counts as interacting.

The basics are  working at a very basic level - open converter, locked in a gear - and I thought it'd be a good idea to keep some notes.  In particular the gear matrix was more tedious than I'd like... partly because I eschewed my beloved lever analogs in favor of free body analyses... but looking at the body analytic result I am left wondering if the lever analog was obscuring the symmetric gear matrix.  I'll present the result first, and then work up to it from first principles:

• J is a diagonal matrix representing gear inertiae for the acceleration terms
• N and f  are gear contact gains and forces 
• S is gear speed constraints (e.g. tooth count ratios)
• C is clutch torque application 
• X is external torque application
• H represents a "holding" clutch, D represents a "dragging" clutch

The ZF nine speed has four simple planetary gear sets, with a novel concentrically nested arrangement of the first two.  Gear ratio is determined by a combination of six clutches.  Any simple planetary has four "nodes:" a ring gear, sun gear, pinion gears, and the carrier for the pinion gears.  Generic torque to force balance equations can be expressed as:

In matrix form these equations can be expressed as:

The transmission has four gear sets with four nodes each, which would imply a total of sixteen torque:acceleration equations.  The total is reduced by rigid connections between gear components:

• A1 and A2 and R3 are connected
• A3 and R4 are connected
• S3 and S4 are connected

If the converter turbine shaft is included in gear train analysis (which is probably a good idea) there are twelve independent torque:acceleration equations.  Clutches are named by which gears they control.

These equations can be expressed more compactly in matrix form, but to help understanding  I think it's better to take it one step at a time.

Here's the explicit inertia torque terms:

Vehicle inertia adds to output carrier inertia.  Vehicle inertia reflected to the output carrier is:

Here's the explicit gear:gear contact forces:

It may not be immediately obvious, but the speed constraints could be take as simply the transpose of the contact force matrix.

The clutch torque application matrix is:

In the case of the clutch torque (if you're keeping track of the meaning of the matrix rows), it is fairly obvious the speed constraints associated with clutch torque actions are simply the transpose of the torque matrix.

The actions of holding clutches should be moved to the acceleration/gear contact force matrix.  Dragging clutches (or controlled slip clutches) remain with the externally applied torques.  

Last, the externally applied torque matrix, which is fairly boring:

That's all you need for gear analyses.  For input torque I added an open converter using simple K factor and torque ratio lookups, and for output torque I used estimated road load torque (reflected to the output carrier through the axle).  Add water, compile with mkoctfile, debug, debug some more and eventually I had a simple torque converter/transaxle simulation that runs 100 faster than real time!

- nzvyyx