The Heresy of the T/S Parameter Bl

[smowry]

Quasi-dynamic AC Model

Note the AC FEA solution is for one frequency only; however, the solution can be for any frequency. Typically, several AC solutions are obtained at various frequencies. Dr. Klippel seems to have missed that Fr also varies with frequency. Anyone who has performed AC FEA will attest to that. However, his transducer model is the most detailed that I have ever seen and is very helpful in understanding transducer AC phenomenon.

Try writing an equation for a closed form solution for the above model! Good luck captain. That's why folks pay US$40,000 for a Klippel DA. DC plus AC (restarted from DC) FEA can simulate the complete electromantic problem.

Having said all that, the above illustrates why Almost Air Core and Steallus are so desirable, μ = μo, the permeability of air. The problem with that is that 10x or more NdFeB magnet volume is required.

 

[NTK]

1. The solution of interest is Beta not Force.
2. The model I am looking for is static and i = 0.
3. If a current is applied, i(t), then there will be an AC magnet field also, Total Flux density(x,t) = BDC(x) + BAC(x,t)
4. For the Purifi coil rc is not a constant and the OD changes as in and out steps.
5. FEA simulation starts with a static DC model and Beta and/or Bl is acquired with a command file. Then an AC current density is applied with frequency, f.
6. Empirical motor evaluation, typically manually sweeps the coil from -x to x and dlambda/dx = Beta is returned. Below is my command file but this will not work with the Purifi coil due to the OD steps. I would need to sweep several command files and sum the results. I had previously simulated the TC Sounds LMS motor like that.

1. A lot of the simulation models I've seen from Klippel, MATLAB/Simulink examples, Andre Bezzola of Samsung Audio Lab in LA (a few miles from Harman Northridge), etc. are simulations in the time domain. In those cases computing the motor force is a critical step, since the motor force drives the diaphragm, and the diaphragm motion drives the sound radiation. I haven't seen β used much. I don't know if that kind of "abstraction" is beneficial to these simulations where many of the pertinent parameters (e.g. B, Le, Cms, Rms, Sd, etc.) are not constants anymore and can be dependent the voicecoil/diaphragm position/velocity, time, etc. They are in essence "brute force" methods that all the messy physics are thrown into the computer and let it crunch the numbers out. As someone has said, mathematics is the art of avoiding arithmetic (or computations). Since computations is cheap and easy now, so skip the math.
2. Large signal (big i) is where the fun is now
3. I don't know much at all about electromagnetics. But I guess the AC magnetic field from the signal current can be modeled by FEM.
4. We should be able to accommodate that by changing rc to a function of z and move it inside the integral.

Try writing an equation for a closed form solution for the above model! Good luck captain. That's why folks pay US$40,000 for a Klippel DA.

We'll just setup the problem in FEM and give it to the computer. Or may be in the not-too-distant future, just tell the whatever popular at the moment AI chatbot to go at it.

 

[smowry]

@NTK

You sent me a picture of a DC static FEM model with no solution.

Along with the DC static FEM model above, you sent AC equations that indicated how that used cylindrical coordinates. There is no current; there is no force!
Where:
r = rc
z = z
θ = 2π

Although, the equations are controversial, are they AC or DC? Can you tell? Well I can't. You claim time domain. Where do you see that. N(z), voice coil packing density is key the, parameter and it new to me. I learned something from you again. Thank you.

Then with respect to the picture that you sent me contained above, I wrote the following equation. Frankly, I was stuck and could have not done it without you.

The equation above uses the parameters contain in the picture that you sent me. If you feel that I am incorrct, please feel free to correct me. However, I believe that my Beta equation is correct. You then referred to Beta as an abstraction. Well I disagree but that's subjective. Below is the equation that was included and implied to relate to the picture.

Unfortunately this equation is effectively meaningless. There is no current. Are you following me?
So I will correct that equation assuming the same limits of integration..

Bl(z) = 2πrc∫ B(z)N(z)dz kg/s This a the DC simulation of BDCl(z).

Now if the coil has a constant ID and OD, N will be known and we can write the following equation with the same limits of integration.

Bl(z) = 2πrc∫ B(z)Ndz kg/s

Sadly, although correct, the equation above is Heresy!

To prove this, I will go back to the Purifi 10 inch woofers examples.

Voice coil for transducers A & B.

rc = 0.026 m
H = 0.0345 m

Transducer A:

Bl(z) = 2πrc∫B(z)NAdz N/A
Bl(0) = 12.8

Transducer B:

Bl(z) = 2πrc ∫B(z)NBdz N/A
Bl(0) = 15.7

Which transducer has the more efficient motor A or B?
Not only can't you tell, referring to Bl(0) or Bl, to identify motor performance is misleading!
Let's try using the my abstraction equation.

Transducer A:

β(z) = 2πrcσ ∫ B^2(z) AA NA dz kg/s
β(0) = 38.5
Where σCCA ≈ 38 x 10^6 1/(Ωm)

Transducer B:

β(z) = 2πrcσ ∫ B^2(z) AB NB dz kg/s
β(0) = 42.0

I was using terms like Heresy and meaningless but abstraction seems to fit Bl also. β is the true figure of motor merit regardless on Impedance.

What's it going to take?

NTK wrote, "I don't know much at all about electromagnetics. But I guess the AC magnetic field from the signal current can be modeled by FEM."

I explained this. With FEA we start with a DC static Model. You sent me a picture of an unsolved DC model. Below is a DC FEA solution that illustrates both flux Distribution and flux density.

Then we apply an AC current density to the voice coil section and restart the AC simulation from the DC permeability. Below is a 100 Hz example.

You can find the complete DC and AC simulations here. https://pearl-hifi.com/06_Lit_Archi...Mowry_Steve/A_Small_Shielded_Motor_Design.pdf

NTK wrote, "1. A lot of the simulation models I've seen from Klippel, MATLAB/Simulink examples, Andre Bezzola of Samsung Audio Lab in LA (a few miles from Harman Northridge), etc. are simulations in the time domain. In those cases computing the motor force is a critical step, since the motor force drives the diaphragm, and the diaphragm motion drives the sound radiation. I haven't seen β used much. I don't know if that kind of "abstraction" is beneficial to these simulations where many of the pertinent parameters (e.g. B, Le, Cms, Rms, Sd, etc.) are not constants anymore and can be dependent the voicecoil/diaphragm position/velocity, time, etc. They are in essence "brute force" methods that all the messy physics are thrown into the computer and let it crunch the numbers out. As someone has said, mathematics is the art of avoiding arithmetic (or computations). Since computations is cheap and easy now, so skip the math."

The Klippel DA is a powerful measurement tool. Transducer Engineers cannot live without it. However, I tried SIM and it does not provide information on permeability. FEA is how motor assemblies are simulated. SIM is based on Klippel Model shown again below.

When I was practicing, I told Dr. Klippel that his DA was a great measurement tool but measurements do not add value. Measurements are key to transducer evaluation and verification of FEA simulation. So he went to work and developed SIM. He even sent me a trial license. It's ineffective for motor assembly design in my opinion. The Kippel DA is best used on prototypes that has been designed with FEA. I could not design with SIM. Can anyone post a motor assembly designed with SIM without FEA. I have designed more than 100 motors with Vector Fields OPERA, beginning at Bose. Back then B&W's R&D was using VF OPERA too. Today most folks use COMSOL and it works great. My claim is that SIM is ineffective and cumbersome. First a transducer prototype's parameters are acquired with the Klippel DA. Then and only then can Sim simulate changes in those parameters. There are no contour plots to provide a road map of where to go. Take a minute to review my FEA examples. https://pearl-hifi.com/06_Lit_Archive/14_Books_Tech_Papers/Mowry_Steve/Prof_Klippel_Interview.pdf I have no experience with MatLab Simulink.

Below is a patent drawing of a motor assembly that I designed for Ilpo before he passed. It uses only one magnet but contains 2 x shorting rings to control AC flux modulation. I used VF OPERA DC and restart AC FEA to design the magnetic assembly. Subsequently, prototype samples were created and then evaluated using the Klippel DA. Did they use SIM to tweak the design. I really don't know. However, I don't think so. Obviously, I cannot show you my Genelec simulations but I can reference a Patent.

US20180324527A1 - Diaphragm Assembly, Transducer and Method of Manufacture - Google Patents

A novel diaphragm assembly, transducer and manufacturing method is here in proposed making use of a diaphragm assembly including a diaphragm having a first diaphragm component and a second diaphragm component. Both diaphragm components extend between respective inner perimeter and outer rim. The...

patents.google.com

Here's their first coaxial transducer motor assembly that I was tasked to replace. https://patentimages.storage.googleapis.com/2e/f9/51/9578c0ab9b3b33/US8660279.pdf

The performance was okay but they could not manufacture it. The Patent does not even expire until 2028. When I asked the MD at Genelec, Siamak, how many units they could build per day, he quickly replied 6 but only one technician was qualified to do assembly! He wanted it fixed so they could manufacture the transducer assembly in Finland.
The motor topology that Ilpo and I designed is used in all the Genelec coaxials to this day. Carsten from Purifi was working with the Genelec R&D team at that time but when they needed a new coaxial, Ilpo came to Phuket to recruit me on the way back from a trip to China. My daughters called him Santa Claus. I told my daughters back then that Ilpo was not Santa but Santa did live in northern Finland.
Having traveled from Phuket to Finland three times, I have a joke. What's the difference between a Russian and a Finn?
Answer: Not much!

To Beta or not to Beta, that is the question or is it the abstraction? Beta is the true figure of merit for motor assemblies regardless of impedance.
Thanks again for your help. I am also addicted to fun and learning, even at 75.

 

[Snickers-is]

The opening post with It's headline suggests that there is a problem with the Bl parameter.

It is like suggesting that measuring the length of a car is some kind of scam since it does not tell us anything about the weight of the car. The length is still the length.

Bl^2/Re is useful for comparing drivers with different Re, but for most other use cases, Bl is the interesting parameter.

I also think it is important to point out that the l in Bl does not refer to the length of the entire coil. It neither points at the length within the gap. It is the fraction of the Bl product that defines the current path within the useful magnetic field. If we include the entire coil, we need to calculate the average normal magnetic flux density over the entire length. One must also take into account that not all force components are caused by the permanent magnet.

So the point is that the effective length and the effective flux density both vary as function of position, signal level, polarity and frequency.

The discussion about externalizing parts of Re is an interesting one. For multi coil drivers, one must include the same volume of coil active in order to achieve the same effective force factor (or Bl^2/Re). For a 2 * 4 ohm (Re) driver connected in parallel, the windings are run as parallel, meaning Bl will be reduced by 50%. Since the driver is now a 2 ohm driver, it's effective force factor is still the same. But if we connect a 6 ohm resistor in series, we have now kept Bl, but reduced the effective force factor to 50% (the same current, but 50% as many turns in the coil since they run two in parallel).

In the example with alu vs copper, connecting a resistor in series with the copper coil to make them the same Re for the same winding geometry would keep Bl and Bl^2/Re the same, but mass would be reduced. That affects other parameters, and that is also why we need to include mass, mechanical loss and Fs to cross calculate between Bl and Qts.

 

[smowry]

No one has presented any evidence that Bl is more "interesting" than Beta. It's been all subjective rhetoric. I posted this on 15 January and I stand by my claim.

For example as one of just 3 R&D Transducer Engineers back then, Dr. Bose would occasionally stop by my desk and chat about project status and other project related topics. If one ever was so ignorant to even use the term Bl, Dr. Bose would have been insulted. At Bose and in his Acoustics class at MIT, it must have been Beta if one wished to survive. It may seem like a small thing now but back then it was a big thing. All electromagnetic motor measurements results and simulations were in terms of Beta at Bose. Having said that, Dr. Bose out ranks anyone I have meet and/or worked with in the Loudspeaker Industry. If you did not give Dr. Bose the respect that he had earned, then beware your time of employment and/or advancement at Bose was doomed. And clearly that included not using the term Bl.

Frankly. I place Dr. Bose above, Klippel, Small, Thiele, Geddes, Floyd, Button and certainly any member of the ASR community that I have met to date. However, more important than all that, I strongly agree with Dr Bose. My motive of starting and participating in this thread, is to hopefully impact young and up coming engineers in spreading Dr. Bose's message. A term for the Transducer Engineers at Bose in 1995 was "Beta Freaks". The primary transducer design objective was to maximize Beta within the projects constraints. Don't forget that the 901 and 802 transducers had an Re = 0.7 ohms and Bl was relatively low compared to competitors and 901 and 802 used the highest packing factor voice coils in the history of transducers. Only TAD Pro continues to use single layer rectangular anodized Al magnet wire today. While competitors focused on the misleading and effectively meaningless Bl = B2(pi)rN, Bose focused on Beta = (B^2)Volume of conductor(Conductivity), the parameters that really count. In a relative sense, I have shown that the more turns, N the higher the Bl but the lower Beta. So take it for what it is worth but if you want my respect and that of prospective employers. start using the terms Beta, transducers and loudspeakers and forget Bl, drivers, and speakers especially at job interviews. One would not dare to use those terms in the presence of Dr. Bose and his R&D staff. Raise your standards and follow Dr. Bose's lead. Understand that the Loudspeaker Industry is developmentally delayed by decades. So any claims that invoke the idea that this is what the industry typically does are suspect to say the least.

Sincerely,

Steve Mowry

 

[Snickers-is]

It is still a claim. Bose's or your opinion does not change that, it is only an opinion. Bl is widely used among loudspeaker engineers all over the world.

Could you simply be wrong?

 

[smowry]

Let's try another simple example. The Dual voice coil woofer.

8 Ω Nominal ​

Bl(0) = 21.10 N/A

2 Ω Nominal
Bl(0) = 10.55 N/A

Which is more efficient?
Answer: Neither

β+(0) = β||(0) = 60.16 kg/s
`
Bl is misleading.
Whereas β is not misleading and is the true figure of merit for motor efficiency.

Don't get this wrong on your job interview, ouch!

 

[smowry]

I linked our discussion to LinkedIn where I have 7,676 followers.

You're welcome! @amirm

 

[smowry]

Why is it that nobody shows any equations or proofs that disprove my claims? While I post example after example of why Bl is misleading and Bl considered alone as a parameter is essentially meaningless. As I indicated Bl(x) does indicate motor linearity; however, β(x) can also indicate linearity and efficiency. Is that all you got is rhetoric? Johnny picks his nose, so you pick your nose. People all over the world pick their noses.

 

[Snickers-is]

Bl has never been intended to be a way to quantify efficiency. It sounds like you expect something from Bl that Bl was never intended to be.

 

[smowry]

η0 is the efficiency of the transducer.
ρ/2πc is a constant and can be replaced by the value 5.445×10−4 (m^2)s/kg for dry air.
Bl is the Force Factor. N/A
Sd is the surface area of the cone. m^2
Mms is the total moving mass. kg
Re is the DC resistance. Ω
So then η0 goes as the square of Bl, Sd, and 1/Mms.
Well not really, be careful, Bl is misleading.
η0 goes as β and the square of Sd and 1/Mms.

β is the true figure of merit for motor analysis. kg/s
Where

Don't get that wrong on your job interview.

 

[IPunchCholla]

Well not really, be careful, Bl is misleading.
η0 goes as β and the square of Sd and 1/Mms.

My math is super rusty, and nowhere in this statement do you seem to have essette (my phone won’t make the damn symbol) so these two sentences are opaque. To me the imply that B and Bl are equal.

If you are trying to get engagement for your idea (not just show off your math skills) you might give all us lay folk who just use the damn parameters to simulate speakers plain explanations of what you are proposing and why it matters.

 

[smowry]

My math is super rusty, and nowhere in this statement do you seem to have essette (my phone won’t make the damn symbol) so these two sentences are opaque. To me the imply that B and Bl are equal.

If you are trying to get engagement for your idea (not just show off your math skills) you might give all us lay folk who just use the damn parameters to simulate speakers plain explanations of what you are proposing and why it matters.

Your point is valid and I have updated my post with a definition of Beta.

However, the tone of your post is insulting and adds nothing to an intelligent discussion. I have reported your post to @RickS. Let's see what he wants to do.

 

[IPunchCholla]

Your point is valid and I have updated my post with a definition of Beta.
View attachment 506278
However, the tone of your post is insulting and adds nothing to an intelligent discussion. I have reported your post to @RickS. Let's see what he wants to do.

Ok. Thank you. I still don’t understand why B is a useful intellectual tool since we already have that in the commonly used parameters. Why should I pay attention to B rather than (Bl)squared/Re?

 

[Snickers-is]

@IPunchCholla you are absolutely spot on. B has no real function as it varies so much over the entirety of the coil. Bl is the only useful part of this as it takes into account the dynamic character of B, and actual force is the only interesting part.

The only difference between Bl and Beta is that one is force as a function of Current, while the other one is force as a function of power. Since Re and current also gives the power, it is not important which one we start with. However most equations and lumped element models are built around the current weighted force, meaning that Beta would just add another step in the calculation.

If you are interested in doing maths and/or simulations, Bl is the king. If you are interested in bragging about motor force, then I guess Beta is nice to have.

So Beta is also useful, but mostly for one to one comparison of motors. However, for anyone who has done this for a while, it is far easier to go via Q and Fs as it more clearly points towards expected tuning range at the same time.

A crucial part of this is that Beta just calculates against Re. But Re is not the only part of the impedance, so actual efficiency is not really determined by Beta either. We see this in ultra high Bl designs like the IPAL drivers that are used primarily in their reactive range.

So Beta is not a precise measure of the actual relationship between power and force. Bl, on the other hand, is a precise measure of the relationship between current and force.

 

[smowry]

For those late arrivals, @Lars Risbo referred to 1/β in terms of damping.

So again we see that if one considers Bl alone for an indication of electrical damping, then it is misleading and in the limit, it's essentially meaningless. Whereas Qes goes as 1/β.

Then looking at the mechanical damping. we see that 1/Rms and 1/β are like sister and brother and together are responsible for all the damping at fs.

It follows that we can evaluate the relationship between Qts and Mms.

1/Qts = 1/Qes + 1/Qms

1/Qts = β/(2πfs Mms) + Rms/(2πfs Mms)

Multiply both sides by 2πfs and combine.

2πfs/Qts = (β + Rms)/Mms Hz or 1/s

or Qts = 2πfs Mms/(β + Rms)

I have always used β for transducer related evaluation; however, Lars pointed out system considerations based on the above.

Where β = (B^2)(Volume of conductor)(Conductivity of the conductor) These quantities are intuitively relatable to the motor design and clearly not misleading.

Note that the unit of β is N^2/W or kg/s.

Then β represents the square of force relative to power without considering current as in static DC FEA,

 

[IPunchCholla]

@IPunchCholla you are absolutely spot on. B has no real function as it varies so much over the entirety of the coil. Bl is the only useful part of this as it takes into account the dynamic character of B, and actual force is the only interesting part.

The only difference between Bl and Beta is that one is force as a function of Current, while the other one is force as a function of power. Since Re and current also gives the power, it is not important which one we start with. However most equations and lumped element models are built around the current weighted force, meaning that Beta would just add another step in the calculation.

If you are interested in doing maths and/or simulations, Bl is the king. If you are interested in bragging about motor force, then I guess Beta is nice to have.

So Beta is also useful, but mostly for one to one comparison of motors. However, for anyone who has done this for a while, it is far easier to go via Q and Fs as it more clearly points towards expected tuning range at the same time.

A crucial part of this is that Beta just calculates against Re. But Re is not the only part of the impedance, so actual efficiency is not really determined by Beta either. We see this in ultra high Bl designs like the IPAL drivers that are used primarily in their reactive range.

So Beta is not a precise measure of the actual relationship between power and force. Bl, on the other hand, is a precise measure of the relationship between current and force.

Thank you. That clarified things for me mightily.

 

[Snickers-is]

So again we see that if one considers Bl alone for an indication of electrical damping,

I doubt anyone do. Bl was never intended to be. It is just a measure of Lorenz force.

 

[smowry]

This thread is coming to a close, so then I will make my closing statements. I will begin by simply listing the Fundamental Linear Parameters. My primary source is Wiki. That's about as general public as it gets.

Then below are the derivations of the T/S Parameters.

I see Bl but only as it relates to β. By substituting β for (Bl^2)/Re, Bl vanishes. The parameters above can be used to formulate the specifications for the transducer design project. However, one cannot use the above to design a moving coil audio transducer. These parameters are acquired from a physical transducer. The acquisition is a measurement. Measurements add no value to the Simulation and Design Process. Measurements are key to the Simulation Verification and Design Evaluation in the Development Process. Measurements facilitate a data base of parameters for existing designs. With regards to transducers, one is dealing with materials and geometry. Without FEA, the Simulation and Design Process becomes stochastic. Where is @René - Acculution.com? The way a Transducer Engineer typically utilizes FEA is to start with a 2D DC Static Model (simple and convenient but powerful) or what is sometimes called a "Boxer" because one will see a bunch of boxes that represent the cross-sections of parts. @NTK posted the simple example contained below. Unfortunately, he then went on to relate it to force in the time domain.

Rather than (x,y) representation, cylindrical coordinates are utilized, where x = r, y = z, and θ = 2π. Then due to the cross-sectional representation, outputs are in densities. For example flux density, B. There is no current and there is no force. Key inputs are BH Curves that define permeability, μ = B/H N/A^2. In the picture above the voice coil does nothing and is just a place holder. It's treated as air. Ferromagnetic materials have nonlinear permeability, while nonferrous materials including air have linear permeability. The solution of the 2D simulation contains B and H and thus the DC permeability, μDC is simulated. A command file can be conveniently used to acquire β(z). Where β(z) = B(z)^2(Volume of conductor)(Conductivity of conductor). Then Bl(z) can be calculated from β(z), Bl(z) = (β(z)^0.5)Re Tm. Tell me that's not a paradox.

The next step in FEA is AC simulation restarted from the DC permeability. The input to AC FEA is the voice coil's section current density, J A/m^2 at a specified frequency, f (Hz). The AC solution identifies the operating permeability,
μ = μDC + μAC = (B + BAC) / (H + HAC). The DC + AC FEA allows for identification for the electromagnetic nonlinear parameters within the moving coil transducer model contained below before any parts or assemblies are created.

These include Bl(x)i, Le(f,x,i), and Fr(f,x,i). What's Fr? Fr(f,x,i) = BACl(f,x)i N. The AC solution is also static and the command files must be used in a piecewise acquisition. Move the voice coil section and solve and acquire, repeat..... These acquisitions are again more convenient when based on βAC = (BAC^2)(Volume of conductor)(Conductivity of conductor) kg/s.

My conclusion is that it more convenient to simulate β with FEA, while it is more convenient to measure Bl with the Klippel DA. Additionally, most of the objections for using β as the true figure of motor merit are originating from folks that are not users of FEA but they do perform good faith measurements.

Then at the end of the day, how is it that I can only find one company that specifies β within their data sheets. That company is not Purifi, nor Seas, nor Scan, nor Radian, nor Dayton. It is Peerless by Tymphany (now part of Eastech) and I claim that this was influenced by Bose.

Something must be wrong. The Loudspeaker Industry is developmentally delayed and is decades behind the other associated AV industries. Not recognizing, specifying, nor using β is not the cause of this. Rather it is a symptom of the Loudspeaker Industry's retardation (pun intended). Finally, rhetoric, metaphor, and satire are subjective and useless for the purpose of audio transducer design. One needs to use FEA. Personably, I proudly admit that I am a "Beta Freak"!

P.S. @amirm the ASR user interface is Johnson.

 

[NTK]

Before this thread get locked (just in case), I'll give an example of time domain simulation.

In this thread I showed the frequency domain derivation of the loudspeaker transfer functions.

Mathematical Model of Dynamic Loudspeaker Driver and Physics of Thiele/Small Parameters

There are discussions in another thread regarding the merits of maximizing the amplifier “damping factor” by reducing the amplifier output impedance (which is usually assumed to be purely resistive) to essentially zero or even negative. The theory is that this will maximize the electrical...

www.audiosciencereview.com

Here, the loudspeaker input to output transfer function is computed using a time domain simulation. This is basically a numerical analog of the typical measurement process. An input is given to the model, and the sound pressure output is computed at many time steps. The transfer function (i.e. frequency response) is the FFT of the output divided by the FFT of the input. It arrived at the same results as the frequency domain method. I kept Bl constant in this simulation but it doesn't have to be.

The big difference is that in time domain simulations, the factors Bl, Kms, Sd, Le, etc., can be made dependent on the diaphragm displacement and velocity, time, temperature, etc. Frequency domain analyses require the system to be LTI (linear time invariant), and time domain analyses don't. If you want to deal with nonlinearities, you'll need to simulate in the time domain, and since F = ma is (one of) the fundamental physically law in loudspeaker physics, you have to calculate force. Software like Loudsoft FINEMotor Pro let you simulate your motor design and provide you with the Bl(x) and Le(x) functions. Or you can perform your own FEA.

[Edit] Added the plots of the time domain simulation results (input, displacement, velocity, acceleration, current). Updated the ZIP file contents.

Figure from the previous frequency domain study.

@olieb complained that I used too much Mathematica, so this time I use FOSS (free open source software) I thought of using Python first, since it is familiar to a lot of people. But I decided to go with Julia since it runs compiled code and should therefore be faster. The printout of the Julia "notebook" is HTML format in the ZIP file.