# What is the electrical equivalent circuit for this interaction?

• mikejm
In summary, a textbook provides an example of an ideal mass striking an ideal string and both cases are modeled as equivalent circuits with different damping.
mikejm
A textbook 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 (where the damping is roughly 0):

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 how 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.

Here is my best guess. I put the damper (which is u=0) and spring in series, and their combination in parallel to the mass. I am uncertain of the + or - directions for the damper and spring:

In the first case of the mass striking a string, because all the elements were in series, I believe the sum of forces (voltages) 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 that correct?

In the first case, I should have thought that the moving mass is equivalent to a lossless inductor with a short circuit in which a direct current is flowing. When it strikes the string, a switch, having the two resistors across it in parallel, is opened. It becomes an L-R ciruit having a time constant of 2L/R.
In the second case, it looks as if the initial mass (inductance) has a damping resistance in parallel to it. With DC circulating current, there is no voltage across the damping resistor, but when the switch opens, the forward EMF is developed across the damping resistor, so dissipating energy.
In a practical case, I suspect that the damping is chosen to minimise the noise of the hammer; in other words, at higher frequencies than the fundamental of the string.
These cases look rather like a spark transmitter, where a capacitor is charged by DC and then connected to an inductor and a resistive load, so that damped oscillations occur.

Trying to draw such equivalences is asking for trouble. It is far better to work the original problem as given than to try to trade it for another problem.

anorlunda and phinds

## 1. What is an electrical equivalent circuit?

An electrical equivalent circuit is a simplified representation of a complex electrical system or interaction. It uses a combination of electrical components, such as resistors, capacitors, and inductors, to model the behavior of the original system. This allows for easier analysis and understanding of the system's properties.

## 2. How is an electrical equivalent circuit created?

An electrical equivalent circuit is created by analyzing the behavior of the original system and determining which electrical components and their values best represent that behavior. This is often done through experimentation or through mathematical modeling and simulation.

## 3. What is the purpose of an electrical equivalent circuit?

The purpose of an electrical equivalent circuit is to simplify the analysis of a complex electrical system or interaction. It allows for easier understanding of the system's behavior and can also be used to design and optimize the system for specific purposes.

## 4. Can an electrical equivalent circuit accurately represent all systems?

No, an electrical equivalent circuit is a simplified representation and may not accurately capture all aspects of a system. It is typically used for linear systems and may not work well for non-linear systems or those with complex behaviors.

## 5. How is an electrical equivalent circuit useful in practical applications?

An electrical equivalent circuit can be useful in practical applications by providing a simplified model for designing and analyzing electrical systems. It can also aid in troubleshooting and predicting the behavior of a system, making it a valuable tool for engineers and scientists.

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