A DSP text gives an example of an ideal mass striking an ideal string here:


This is drawn as an equivalent electrical circuit as follows, where each R represents one of the two string segments the mass interacts with (ie. the string segment to the left of the mass and the string segment to the right):

enter image description here

A piano hammer is imagined slightly differently as a mass driving a damped spring against the string here:

enter image description here

They state: "The impedance of this plucking system, as seen by the string, is the parallel combination of the mass impedance $ ms$ and the damped spring impedance $ \mu+k/s$. The damper $ \mu $ and spring $ k/s$ are formally in series."

I am wondering if anyone would be able to draw the equivalent electrical circuit for this interaction. I am trying to figure out how to write equations for the forces and velocities of a hammer modeled in this way but I think I need the diagram first.

This is my best guess. I put the damper and spring in series, and their combination in parallel to the mass which has the force exerted to it directly. I am uncertain of the + or - directions for the damper and spring:

enter image description here

Furthermore, in the first case of the mass striking a string, because all the elements were in series, I believe the sum of forces for the elements had to equal zero ($Fm(t) + Fr(t) + Fr(t) + Fext(t) = 0$). What would that equation of forces look like for this circuit?

I'm guessing it would be something like an equation system based on the parallel sections:

$Fm(t) + Fr(t) + Fr(t) + Fext(t) = 0$

$Fk(t) + Fr(t) + Fr(t) + Fext(t) = 0$

$Fm(t) = Fk(t)$

Is any of that right?


  • 1
    $\begingroup$ This might be a question for the people over at physics SE... $\endgroup$ – MBaz Dec 8 '19 at 14:57
  • $\begingroup$ Yeah fair enough. I was thinking that myself. I'll try to put together my best attempt at what I think it should look like and post there. What I'm working on (physical instrument modeling) overlaps DSP and physics I suppose. $\endgroup$ – mike Dec 8 '19 at 17:20
  • $\begingroup$ I design circuits, and I would not confuse myself by making an electrical analog. Why not just write the differential equations for the mass-spring-damper system and extract the impedance? $\endgroup$ – TimWescott Dec 9 '19 at 21:45
  • $\begingroup$ I don't know how to do that except as directed by that textbook which requires you to understand the analogous circuit. $\endgroup$ – mike Dec 14 '19 at 20:59

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