# What is the electrical equivalent circuit for this interaction? (Is this correct?)

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):

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

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?

Thanks.

• This might be a question for the people over at physics SE... – MBaz Dec 8 '19 at 14:57
• 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. – mike Dec 8 '19 at 17:20
• 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? – TimWescott Dec 9 '19 at 21:45
• I don't know how to do that except as directed by that textbook which requires you to understand the analogous circuit. – mike Dec 14 '19 at 20:59