Sunday, March 26, 2017

The development of the slider crank equation

Something for the torsional vibes folks who discover my blog... I've been meaning to put this up for some time!  I used to teach a grad course at MTU on the design of internal combustion engines.  Part of the background needed to understand torsional vibration in slider crank mechanisms is the complex description of the motion of the piston with crank angle (time if you will).  So here you go!

For offset pins Lichty is a good reference.

So cool! So here we start with the angle the connecting rod makes with the crank arm...

CFT is Charles Frederick Taylor and his book published from the MIT Press on Design of Engines

Generally the error of a truncated expansion is easily accepted when you consider cylinder to cylinder pressure variation due to porting, standing waves, blow down, volumetric efficiency etc... Ultimately what we want from these equations is gas torque.  The turning power created by the engine.
My O-Speed note was for an example of an overspeed consideration not discussed further in these lecture notes

The differentiation is straight forward...
So here you have the kinematic relationships.  Next - what good are they - really cool if you want to figure out the gas load forces.

To make these equations yield meaningful side forces, and piston scuff calcs you will need a gas pressure v crank angle vector to apply with proper phase angle.

This is the heart of the torsional gas load.  Turning effort.  This complex wave can be reduced to frequency (by crank order) and phase.  It always acts in the opposite direction of the inertia torque so they form the complex turning effort. That then becomes the forcing function for an ordal superposition forced torsional response. YEE-HAAW!

The radial force can be used to calculate bearing fluid film thickness - but that's another story...

Hope the math folks liked the post!


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