Equivalent mechanical system for a simple filter

An LC filter is, ideally, lossless, so a mechanical model wouldn't include dashpots, which are dissipative.

One mechanical model is spring-mass-spring-mass... corresponding to L...C...L...C...

---- | | in-----/////----| |----////-----etc---out | | ----

which is a lowpass.

Collins makes mechanical filters sort of like this...

I think the discs may be machined out of a solid rod, which makes this a bandpass filter.

I've also seen a string of anchored rods, sort of like a row of flagpoles connected by coupling wires, which would also be a bandpass.

John

Thanks for the response! I included the dashpot because it was in the original schematic, and I think it's to limit the current going into the circuit. I don't know if it'd be accurate for me to call this a "filter", since it was suggested as a way to convert square to semi- decent sine waves.

So (excluding the resistor), are you suggesting that it's a spring- mass-spring system?

Thanks again! Dave

An electrical passive Butterworth is a string of Ls and Cs. It is usually designed to be terminated on one or both ends by a resistive source and/or a resistive load, which would have mechanical equivalents, perhaps source and load, perhaps dashpots if the mechanical impedances weren't right.

If your excitation frequency is constant, you could use a bandpass filter, coupled resonators like the Collins things or coupled rods/flagpoles. If not, you'd need a lowpass, like the spring-mass example.

John

It looks like I was wrong in calling it a Butterworth filter. After some more searching, it looks like it's a pi section filter / pi filter. Back to the original question, what would the mechanical equivalent of a pi filter be?

Thanks again, Dave

"Pi" expresses the topology of the filter. "Butterworth" expresses the transfer function. A pi filter could be a Butterworth, a Bessel, a Chebychev, or some other transfer function, depending on the values of the elements.

A pi filter would be pretty much the spring-mass thing I sketched. The basic pi might be mass-spring-mass, or spring-mass-spring. It could have more L-C (spring-mass) sections; the more sections, the higher-order the transfer function and potentially the sharper the frequency cutoff.

The simplest lowpass is an R-C filter, sloppy first-order transfer function, which would correspond to

dashpot ______ | in -----| ======-------/mass/-----------out |______

Mechanical filters are cool, since many sections can be machined out of a hunk of stuff, and Qs are usually higher than electronic parts can achieve, especially at low frequencies.

John

How do you transfer an electrical signal to a mechanical filter and then back again at the other end? Voice coils? I seem to recall those Rockwell mechanical filters have rather large insertion losses -- >10dB -- which I always figured was primarily due to the electricalmechanical transfers rather than losses inherently within the mechanical parts themselves.

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Wouldn't an R-C filter be a dashpot and a spring? I thought caps were modeled as springs (i.e. they store energy and their voltage is dependent on charge), and inductors were modeled as masses (i.e. they resist a change in current, like a mass resists acceleration).

Thanks, Dave

I disagree. I don't know anything higher than algebra with derivatives and stuff, but I visualize it differently from what it sounds like you guys are talking about.

I use the "water pipe" model. I see a resistance (which is, duh, resistance ;-)), a spring-loaded membrane (a cap; the membrane is parallel to the plates - actually, it's the gap)[1], a flywheel coupled to a positive- displacement pump/turbine (the inductor)[1], and another cap (see above ;-)).

From what I was told "dashpot" means, it sounds like what you'd put in parallel with the cap to represent its losses. With the inductor (flywheel-turbine/pump assy) you'e probably achieve the same thing with a Prony brake or some such.

And, of course, the load is whatever the water moves.

Notes:

1. A capacitor opposes a change in voltage; an inductor opposes a change in current.

Hope This Helps! Rich

one way is to build it from piezoelectric material

resonators, crystals, and SAW filters are mechanical filters.

at lower frequencies, yes.

l

Sorry, but your water pipe model is probably going to muddify rather that illuminate. Idealized dash-pots, springs and masses are *exact* mathematical analogs of their lumped electrical cousins - no fudging, hedging or hand waving is required.

However, it might be a good idea for Dave to bite the bullet and study electric circuits, for the sake of notational convenience if anything. There are lots of sources on the web. For example:

You can experiment with circuits using a simulator such as LTspice, which is free:

-- Joe

the water model works well down to DC, the springs and masses model doesn't seem to.

here's a boost converter implemented in the water model :)

bye

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Idealized springs, masses and dash pots work just as well as their lumped electrical analogs at DC (and all other frequencies), because the underlying differential equations are identical. Of course, they are all idealizations so their behaviours diverge from those of real devices, but that's another issue.

q (charge) x (position) q' (current) x' (velocity) q'' (rate of change of current) x'' (acceleration) V (voltage) F (force) R (resistance) b (viscosity)

1/C (inverse of capacitance) k (spring constant) L (inductance) m (mass)

V =3D Lq'' + Rq' + q/C resonance at SQRT(1/LC) F =3D mx'' + bx' + kx resonance at SQRT(k/m)

-- Joe

-- snip--

Idealized springs, masses and dash pots work just as well as their lumped electrical analogs at DC (and all other frequencies), because the underlying differential equations are identical. Of course, they are all idealizations so their behaviours diverge from those of real devices, but that's another issue.

q (charge) x (position) q' (current) x' (velocity) q'' (rate of change of current) x'' (acceleration) V (voltage) F (force) R (resistance) b (viscosity)

1/C (inverse of capacitance) k (spring constant) L (inductance) m (mass)

V = Lq'' + Rq' + q/C resonance at SQRT(1/LC) F = mx'' + bx' + kx resonance at SQRT(k/m)

Joe

Sure, but what's the mechanical analogue of a series inductor - which doesn't have one end earthed/grounded? Isn't that the sort of thing that demands transformation from series to shunt, which complicates the problem beyond the normal electrical solution?

Chris

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Mathematically, it's just another term of a loop equation in x''. Physically it might take up a lot of room!

I'm no advocate of mechanical analogues - I've already suggested that the O.P. should learn the electrical ones.

-- Joe

May I suggest another tool instead of LTSpice. I'm using NL5 (http:// nl5.sidelinesoft.com) for simulating quite complex heat transfer processes (same heat-to-electricity analogy). Two reasons: 1) NL5 deals with ideal componants, is very fast, zero learning curve. 2) I'm an author (sorry for self-adverising :)

Thank you, Alexei

You mean with one end open? Then i (current) has to always equal zero, which is a constant. Therefore, di/dt also equals zero. So the mechanical equivalent would be a massless mass... correct?

Dave

You mean with one end open? Then i (current) has to always equal zero, which is a constant. Therefore, di/dt also equals zero. So the mechanical equivalent would be a massless mass... correct?

Dave

The given 'pi' circuit has each terminal of the inductor connected to a capacitor, whose other terminal is earthed, and to either the output or input port. There would be some current flowing through the inductor in this case when the circuit was excited.

Chris

IIRC, the mappings work like this: Resistance Dashpot, Capacitance Springs, Inductance Mass; but i do not quite remember how the connections map. That said, the circuit shown is intended for fixed frequency PWM use; and the inductor and the two capacitors form a tank circuit tuned to the PWM carrier frequency. Try looking up m-derived filters.

Perhaps:

__R___ C L C

----_____|---////---|||||---////------+ | load | ============================================= reference bar Notice that the C is not connected to reference, thus my comment about not remembering how connections map.