Baker Valve Gear Simulator in Python

What started off as a compile-time project has finally yielded some results.

As some of you are aware, my interests in engineering exceed computing and from time to time take the front seat for brief periods. In this case, I've spent two weekends volunteering at the Southeastern Railway Museum shops with the hope of working on the preservation and restoration of the exquisite A&WP 290. The neat thing about steam locomotives is they're incredibly overt exhibitions of thermodynamics and mechanics. The boiler is a giant heat exchanger that sends steam to cylinders where work is performed. A mechanical linkage outside the frame transmits this power to the wheels and uses the wheel position to configure valves that admit and exhaust the working fluid.

One of the things I was interested in was a mathematical model of this linkage. This sort of modelling has applications in robotics and kinematics simulations, so it's not too bizarre of a topic for a computer engineer to dabble in. To simulate the valve gear, I constructed a set of polynomial equations modelling each rod as a constant length between two points capable of 2D motion in a plane. The collection of pin joints and rigid rods determines a finite set of possible static equilibria for the system. Enough points can be computed analytically so a numerical non-linear equation solver can compute the others.

I wrote a Python application [distributed as source] using SciPy's fsolve() to simulate the running gear. Plots generated with matplotlib provide visual output.

Here are some of the results. This wikipedia article explains the Baker valve gear and this animated gif shows it in operation [generated from an existing valve gear simulator which I've not yet used to validate mine]. Other valve gears are explained on this page if you're interested.

The following plot illustrates the state of the running gear for several wheel rotations. Not every part is a two-joint rod; there is a bellcrank and an eccentric crank mounted to the main crank pin. See the Wikipedia article for a better illustration.




The following plot illustrates the offsets of the crosshead and cylinder valves for two different cutoffs and two different laps. Cutoff is the angle between vertical and the valve gear's yoke. 0 degrees is effectively neutral, and +25 degrees is "well forward." Setting it to a negative angle reverses the locomotive. Lap is, in a manner of speaking, the amount of motion derived from the crosshead; the valves can move this much without actually admitting any steam. It's worth noting that lap is built into the valve gear, while cutoff is a setting that is adjusted during operation.




To achieve smooth operation, it's desirable to admit steam into the cylinder before the piston has reached front or back dead center. This "lead" is effectively a phase offset that depends on the direction of motion. The amount of lap affects the amount of lead by mixing the 90-degree offset motion of the valve gear with the crosshead via the combination lever. The following plot illustrates the displacement of the cylinder valves as zero-mean curves to enable a direct comparison of 0" and 3.5" of lap. In practice, it's easy to add a constant linear offset to the cylinder valve spindle, and this is one of the variables that must be tuned when the locomotive is overhauled.




The rod sizes and geometry used in this simulation were estimated from photographs of A&WP 290 as well as direct measurements. I didn't get every measurement I wanted, as I was taking them by hand with a tape measure that falls apart, and real locomotives are 3 dimensional and quite large. Nevertheless, this was an interesting project and I'm very gratified by the results. Maybe one day I'll build a 7.5" gauge live steamer and dust off this code for design purposes...