Bernhard Holzmayer wrote:> > Rune Allnor wrote: > > > Bernhard Holzmayer <holzmayer.bernhard@deadspam.com> wrote in > > message news:<1674026.D71KQjRvpg@holzmayer.ifr.rt>... > > >> .... Except for saturation effects, you should be able to find a > >> linear model which is good enough to model the behaviour - and > >> much easier to be understood. It should even reveal the > >> subharmonics if it's properly done. > > > > Do you know any examples of linear models that generate > > subharmonics? Or did you mean that the subharmonic terms should be > > included as auxilliary source terms? > > > > I thought of oscillators which are pumped by sources which are not > running at the resonance frequency but at a higher frequency. > But after some thinking I'm not quite sure if this would work if the > excitation would be a signal which consists of only exact > harmonics. If I reason that such a harmonic excitation signal > injects power only at perfectly orthogonal frequencies, therefore I > guess that it's impossible that power is transferred to a > subharmonic in a completely linear system. > In practice, this will probably work only because systems are not > perfect and because the excitation is not purely harmonic. > > Examples would have been: tuning fork excited with a pulse. > Or my guitar. Exciting a flageolett tone (this is cause a fractional > swing of the string) will cause it to emit a harmonic tone at > first, which fades away until you hear the fundamental tone (which > is then a subharmonic) in the end. > But this may be a misinterpretation, because I might have excited > both tones at first without noticing the lower tone at first - I'm > not sure. > > But now take a gear box: > exciting one (smaller) wheel with a certain rotation frequency would > cause a subharmonic rotation on the other side (of the bigger > wheel). What is this? - Clearly power at one frequency is > transferred into power at another frequency which has at least a > fractional relationship - could be a subharmonic. Has it to do with > it? Or is the direct coupling something different, because the > system is completely stiff? > > Bernhard > > -- > before sending to the above email-address: > replace deadspam.com by foerstergroup.deBernhard, Your agile mind usually delights me. Sometimes I find it hard to follow your path, but it's worth the effort. Sometimes, like now, you stimulate me to think beyond what I have taken for granted. I take as given that subharmonics will not appear in linear, time- invariant systems. In the example of pumping a pendulum with a double- frequency vertical drive, the drive actually couples to the double- frequency vertical motion in the pendulum itself. Even if the pendulum were perfectly linear (as a cycloidal pendulum is), the vertical motion would exist. Electronic parametric amplifiers exploit non-linearities and work in sufficiently similar ways so that the pendulum is often used to explain them. I find your question about how a gear train fits into the scheme of things disturbing, hence interesting. An ideal gear train is certainly linear. A gear train can produce not only subharmonics, but any integer ratios. What puts it outside the the paradigm that we normally use when discussing linear systems? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
speaker modeling
Started by ●July 1, 2003
Reply by ●July 7, 20032003-07-07
Reply by ●July 8, 20032003-07-08
Jerry Avins wrote:> Bernhard, > > Your agile mind usually delights me. Sometimes I find it hard to > follow your path, but it's worth the effort. Sometimes, like now, > you stimulate me to think beyond what I have taken for granted.;-> Oh thanks, makes me happy that I've contributed something beyond confusion... It's just that I try to create an image of the problem in my mind: everything is possible there, sometimes it emits solutions, sometimes such awkward things creep out ... Concerning the gear train:> What puts it outside the the paradigm that we normally use when > discussing linear systems?At first I thought that it contains no storage elements, therefore having a trivial transfer function. However, the rotating wheels do contain rotation energy, thus the system does store energy. But since the system contains no other storage elements, energy cannot flow between different storages so there should be no ringing, instead the system will just keep the rotational speed until damping (or external load) has consumed all energy. Probably this system is not interesting enough to be mentioned. But another example creeps into my mind, which is as easy to imagine, and might be closer to the topic of this thread: Imagine a spring-mass system. The transfer function would be 1/(ms^2+k) with m the mass and k the stiffness of the spring, so it's obviously a linear system. Now let's imagine an excitation with a rectangular pulse. The system receives energy which cannot leave the system afterwards. Therefore it must start swinging with its resonance frequency. Now let's assume that it's excited with a sine of double resonance frequency but which is applied only for a short time like the pulse before (maybe one sine period). Certainly, after some initial irregularities, the system will swing with its proper frequency as soon as the excitation is away. Even if the sine excitation is applied forever, the system will show its resonance frequency during its movement. (a simple Matlab or Simulink proves it). Obviously energy is transferred from one frequency (excitation) to the other (resonance of system). I'd have expected this because I'm used to think of filters which always show both: properties of the input signal and properties of the filter itself which convolve. Nevertheless, maybe I've been following a wrong path. Or this is simply another issue. Because, if it isn't this would mean that a speaker which is excited with a harmonic wave, would certainly show subharmonic components in its response. Without taking any nonlinearities into account. Besides: regarding the spring-mass example demonstrates the strength of impulse response technique in system evaluation. Bernhard -- before sending to the above email-address: replace deadspam.com by foerstergroup.de
Reply by ●July 8, 20032003-07-08
Bernhard Holzmayer wrote:>...> > Imagine a spring-mass system. > The transfer function would be 1/(ms^2+k) with m the mass and k the > stiffness of the spring, so it's obviously a linear system. > Now let's imagine an excitation with a rectangular pulse. > The system receives energy which cannot leave the system afterwards. > Therefore it must start swinging with its resonance frequency. > > Now let's assume that it's excited with a sine of double resonance > frequency but which is applied only for a short time like the pulse > before (maybe one sine period). > Certainly, after some initial irregularities, the system will swing > with its proper frequency as soon as the excitation is away. > > Even if the sine excitation is applied forever, the system will show > its resonance frequency during its movement. > (a simple Matlab or Simulink proves it). > > Obviously energy is transferred from one frequency (excitation) to > the other (resonance of system). >... I don't think that's right. When the off-resonance excitation is first applied, it creates a "transient" which causes an oscillation at the resonant frequency. The "transient" never decays in a lossless system, which is the reason for the quote marks. Introduce the merest loss, and eventually, the only response will be at the exciting frequency Computing with lossless elements can lead to counter-intuitive results. Apply a sinusoidal excitation to a lossless inductor, starting at a zero crossing. The initial current is what would be inferred from the voltage's polarity. Is it obvious that the current never reverses? (Don't try to patent this rectifier. It deserves a place alongside the binomial antenna array. It does serve to show that a linear process can be a rectifier if rectification is defined too broadly.) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 9, 20032003-07-09
Bernhard Holzmayer <holzmayer.bernhard@deadspam.com> wrote in message news:<8531044.EL7ZbHhPU4@holzmayer.ifr.rt>...> Imagine a spring-mass system. > The transfer function would be 1/(ms^2+k) with m the mass and k the > stiffness of the spring, so it's obviously a linear system. > Now let's imagine an excitation with a rectangular pulse. > The system receives energy which cannot leave the system afterwards. > Therefore it must start swinging with its resonance frequency. > > Now let's assume that it's excited with a sine of double resonance > frequency but which is applied only for a short time like the pulse > before (maybe one sine period). > Certainly, after some initial irregularities, the system will swing > with its proper frequency as soon as the excitation is away. > > Even if the sine excitation is applied forever, the system will show > its resonance frequency during its movement. > (a simple Matlab or Simulink proves it). > > Obviously energy is transferred from one frequency (excitation) to > the other (resonance of system). >I think you will find that that is just not the case. Because you have gated a sine wave it's spectrum is no longer pure and will have energy at the resonant frequency of your mass spring system. There is no transfer of energy from one frequency to another. Any man made signal, whether real or in a model, will of necessity have a start (ie. a time before which the signal was zero) and hence any test you do will be with a signal that has energy at a range of frequencies including the resonance of your system. You don't need the excitation frequency to be harmonically (sub or toherwise) to the resonant frequency for this behaviour to occur. It should happen with any frequency, albeit to a lesser or greater extent. To prove your point with MatLab you would have to see that as time continues the magnitude of the system movement should get bigger and bigger indefinitely. If their is no growth in movement over time then there is no net energy transfer. Just the energy stored in the system from the transient that set it in motion in the first place. Does you simple MatLab test show this? Regards, Paavo Jumppanen Author of AtSpec : A 2 channel PC based FFT spectrum analyzer http://www.taquis.com
Reply by ●July 9, 20032003-07-09
Paavo Jumppanen wrote:> I think you will find that that is just not the case. Because you > have gated a sine wave it's spectrum is no longer pure and will > have energy at the resonant frequency of your mass spring system. > There is no transfer of energy from one frequency to another. > > Any man made signal, whether real or in a model, will of necessity > have a start (ie. a time before which the signal was zero) and > hence any test you do will be with a signal that has energy at a > range of frequencies including the resonance of your system. You > don't need the excitation frequency to be harmonically (sub or > toherwise) to the resonant frequency for this behaviour to occur. > It should happen with any frequency, albeit to a lesser or greater > extent. > > To prove your point with MatLab you would have to see that as time > continues the magnitude of the system movement should get bigger > and bigger indefinitely. If their is no growth in movement over > time then there is no net energy transfer.All right, I can agree on that.> Just the energy stored in the system from the transient > that set it in motion in the first place. > Does you simple MatLab test show this?It shows no growth over time if the excitation is stationary. Thanks for this contribution - helped my intuition to align with reality :-) Bernhard -- before sending to the above email-address: replace deadspam.com by foerstergroup.de
