Posted on Sat 16th May 2015 12.22PM in category Qtrs1thru3_2015
I apologize for the delay in writing, as I think I said something about “next week” in my first post! I’ve been very busy with exciting improvements to our Key Force 1 technology and business models, and also with writing another article for the PTG journal. These first two posts should serve as a teaser for the more detailed analysis of a piano action’s ratios (note the plural), given in the upcoming article.
In the last post, I showed how the static/action ratio differs significantly, depending upon where on the hammerhead one focuses. The two most common points of focus are the tip/crown and the approximate center of mass (c.m.). I modeled a typical grand action in CAD, and calculated both of these static ratios (SR’s) across the pre-let-off stroke. I also calculated the Gear Ratio (GR) at many points along the stroke, graphing it alongside the static ratios. To determine the SR’s, I used vertical movement (displacement) of the appropriate hammerhead point (along with the movement at the AP of course). In this post, I will calculate the continuously-varying SR values using both displacement and force components, again with the help of a typical CAD action model. I’ll call the latter method a force-based method for determining SR. I will also provide a mathematical proof, showing that the displacement-based and force-based SR values should be exactly the same. Finally, I will introduce a “levered gear” model, which helps not only in differentiating between SR and GR, but also in mathematically connecting the two parameters!
Figure 5 shows a CAD model for a typical action, as it appears about two mm into its keystroke. It is very similar to the 17 mm-knuckle model from the first post. In this case, however, the horizontal lever arm components – to both the tip and the c.m. – are also shown. This allows us to calculate the force-based SR, right along with the displacement-based SR. This is done by simply taking advantage of the instantaneous GR, which is the exact multiplier for transferring induced torque at the hammer to a resulting torque at the key. Of course, GR (actually, its square) has the additional mathematical duty of reflecting the local inertia of the hammer assembly to an effective inertia at the key! This latter duty is not the topic of this post, however.
Figure 6 is a more generic view of a CAD model, accompanied by pertinent dimensional variables and equations. The theoretical calculation of the force-based static ratio is made possible by the gear ratio being easily extracted as the model is swept through the pre-let-off range. For this model, the GR and displacement-based SR are graphed across the stroke in Figure 7. When the force-based SR values (at the c.m.) were calculated, they were found to coincide almost exactly with the displacement-based values based on the c.m.. In fact, their average values are in agreement to within 1/20th of one percent! When the focus was moved to the tip/crown of the hammerhead, the values also converged (to within 3/10ths of one percent!). Thus, the force-based static ratio is essentially identical to the displacement-based static ratio, as long as one focuses on the same hammerhead point.
This static ratio convergence can also be shown mathematically. Figure 8 shows the action model again, but with extra focus on the hammerhead and resulting displacement vectors. The proof works equally well, no matter where the focal point is for the hammerhead. In Figure 8, the focal point was chosen to be the approximate center of mass. When the moving line “Line 1” is in the region below horizontal, the angle must be set up a bit differently (using a 90 - qhmr), but the result is exactly the same.
As a precursor to the upcoming article in the Piano Technicians Journal, I will show a “levered gear” model (Figures 9 and 10) at two points in the keystroke. This model will ultimately help illustrate the huge differences between the static ratio and the gear ratio. It will also be used to connect the two mathematically. The main reason for the SR varying through the stroke is because the GR varies through the stroke. For a given GR, changes in either shank orientation or the relative lever arm lengths (Lhmr vs. LAP) govern the SR. The upcoming article will cover that extensively. Note the visual definition of GR in Figure 10, as the change in angular rotation of the hammer assembly, divided by that of the keystick. An equivalent definition is the resulting torque produced at the keystick, divided by an induced torque at the hammer assembly. The static ratio is all about “how much does this point here move when that point over there moves a given amount?”. Or equivalently, “how much force is generated at that point over there (the AP usually) for a given induced downward force at the hammerhead?”. The SR could almost be referred to as a “point ratio”.