Hi all. I'm planning an oral presentation for my calculus class. Since I want to do physical modeling, I thought this would be a good opportunity to try to do synthesis with Mass Spring Damper models. Right now, I think I have a hold of basic harmonic motion. What I'd like is some resources that help me to link up numerous springs and masses in multiple dimensions to make a complex form (like a drum head.) Is this learn-able in an afternoon or two? I've done this in software (PMPD) but I didn't need to do the math... Any suggestions? Google search terms?
Going farther with Mass Spring Damper.
Started by ●December 5, 2010
Reply by ●December 5, 20102010-12-05
On 05/12/2010 18:09, Matthew Aidekman wrote:> Hi all. I'm planning an oral presentation for my calculus class. > Since I want to do physical modeling, I thought this would be a good > opportunity to try to do synthesis with Mass Spring Damper models. > > Right now, I think I have a hold of basic harmonic motion. > > What I'd like is some resources that help me to link up numerous > springs and masses in multiple dimensions to make a complex form (like > a drum head.) > > Is this learn-able in an afternoon or two? I've done this in software > (PMPD) but I didn't need to do the math... > > Any suggestions? Google search terms?It's a big subject! The canonical reference is the CORDIS-AMIMA package by Cadoz and others at ACROE: http://www-acroe.imag.fr/produits/logiciel/logiciel_en.html (also described in the MIT book "Representations of Musical Signals", if you can find a copy - long out of print). But it doesn't appear that the software is readily available (the website says you have to phone them). But you will find many references online for it. The modern approach is basically via waveguides and waveguide meshes, with lots of work also on direct finite difference modelling. PMPD is indeed probably your best bet especially if you want to get on top of things (?) in an afternoon or two, and want to use the graphics. There are some good waveguide-based physical models in Perry Cook's STK (some of which have also been ported to Csound), but you will have to dig down into the code if you want to break it up into underlying components: https://ccrma.stanford.edu/software/stk/index.html Richard Dobson
Reply by ●December 5, 20102010-12-05
On 12/05/2010 10:09 AM, Matthew Aidekman wrote:> Hi all. I'm planning an oral presentation for my calculus class. > Since I want to do physical modeling, I thought this would be a good > opportunity to try to do synthesis with Mass Spring Damper models. > > Right now, I think I have a hold of basic harmonic motion. > > What I'd like is some resources that help me to link up numerous > springs and masses in multiple dimensions to make a complex form (like > a drum head.) > > Is this learn-able in an afternoon or two? I've done this in software > (PMPD) but I didn't need to do the math... > > Any suggestions? Google search terms?Really solving mass-spring-damper problems is a subject for a differential equations class, but it's well within reach of the information you've been taught in calculus. Just doing a presentation on all the permutations of a simple one-mass, one-spring, one-damper system will be impressive in your situation. Complete solutions to a drum head model are an advanced diff eq topic, because the system is continuous instead of having discrete states, and they're two-dimensional on top of that. You could maybe do an interesting report on the results of a simulation with that, but if you limit yourself to the tools available to you in a calculus class you couldn't explain what's going on. So if you're looking for a way to reduce the problem to a few equations and solve them -- no, you're not going to do that in a couple of afternoons starting with a calculus class as a base. At least, not unless you're a heck of a lot smarter than we are here! -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●December 5, 20102010-12-05
Matthew Aidekman <matthewaudio@gmail.com> wrote:> Hi all. I'm planning an oral presentation for my calculus class. > Since I want to do physical modeling, I thought this would be a good > opportunity to try to do synthesis with Mass Spring Damper models.What level of calculus class is this? This is about the simplest second order linear differential equation, which should come at the end of the first year or early second year of a college math series for engineers.> Right now, I think I have a hold of basic harmonic motion.Stories are that when people were first learning about RLC resonant circuits they used analogies to mechanical mass, spring, frictional damper systems. Now, they usually use the electrical circuits to help understand the mechanical systems. Anyway, the undamped resonator is a simple second order differential equation, damping adds one more term, and so is a little harder to understand.> What I'd like is some resources that help me to link up numerous > springs and masses in multiple dimensions to make a complex form (like > a drum head.)First you learn about the mass and spring resonator. Next go on to the wave equation for a taught string with a specified mass per unit length, as a solution to a partial differential equation. For a square drum head, you get a nice separable partial differential equation, which isn't so hard to understand if you can do the mass spring resonator. For a circular drum head (more popular with musicians), you need Bessel functions, as solutions to Bessel's equation.> Is this learn-able in an afternoon or two? I've done this in software > (PMPD) but I didn't need to do the math...> Any suggestions? Google search terms?Most likely it is well explained in wikipedia. -- glen
Reply by ●December 5, 20102010-12-05
Matthew Aidekman <matthewaudio@gmail.com> wrote:> Hi all. I'm planning an oral presentation for my calculus class. > Since I want to do physical modeling, I thought this would be a good > opportunity to try to do synthesis with Mass Spring Damper models.> Right now, I think I have a hold of basic harmonic motion.(continuing on after a previous post) OK, there is one way to do it that works pretty well for a presentation, though not the way you will usually find it in introductory books. You start out by assuming that the solution is an exponential with an unknown constant, that is, exp(A t) for unknown A. Put that into the differential equation, factor out exp(A t), and solve for A. You can do that for the undamped mass/spring (imaginary A), or damped mass/spring (complex A). That either requires understanding the complex exponential, or can be used as an introduction to complex exponentials. (You know the properties it needs to have, such that one can be the solution to the given equation.)> What I'd like is some resources that help me to link up numerous > springs and masses in multiple dimensions to make a complex form (like > a drum head.)As I said before, not so bad for square drums, but not easy at all for circular drums. -- glen
Reply by ●December 5, 20102010-12-05
On Dec 5, 8:21�pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:> Matthew Aidekman <matthewau...@gmail.com> wrote: > > Hi all. �I'm planning an oral presentation for my calculus class. > > Since I want to do physical modeling, I thought this would be a good > > opportunity to try to do synthesis with Mass Spring Damper models. > > What level of calculus class is this? �This is about the simplest > second order linear differential equation, which should come at > the end of the first year or early second year of a college math > series for engineers. > > > Right now, I think I have a hold of basic harmonic motion. > > Stories are that when people were first learning about RLC > resonant circuits they used analogies to mechanical mass, > spring, frictional damper systems. �Now, they usually use > the electrical circuits to help understand the mechanical > systems. > > Anyway, the undamped resonator is a simple second order > differential equation, damping adds one more term, and so > is a little harder to understand. >I think you are mistaking here. A second order DE can be underdamped, overdamped, critically damped or no damping. In all cases the solution is two exponential expressions. The over damped solution is two real exponentials, the critically damped is a double solution, the underdamped is two conjugate complex exponentials and the no damping is two conjugate imaginary exponentials.> > What I'd like is some resources that help me to link up numerous > > springs and masses in multiple dimensions to make a complex form (like > > a drum head.) > > First you learn about the mass and spring resonator. �Next go > on to the wave equation for a taught string with a specified > mass per unit length, as a solution to a partial differential > equation. > > For a square drum head, you get a nice separable partial > differential equation, which isn't so hard to understand if > you can do the mass spring resonator. �For a circular drum > head (more popular with musicians), you need Bessel functions, > as solutions to Bessel's equation. > > > Is this learn-able in an afternoon or two? �I've done this in software > > (PMPD) but I didn't need to do the math... > > Any suggestions? �Google search terms? > > Most likely it is well explained in wikipedia. > > -- glen
Reply by ●December 5, 20102010-12-05
brent <bulegoge@columbus.rr.com> wrote: (snip, I wrote)>> Anyway, the undamped resonator is a simple second order >> differential equation, damping adds one more term, and so >> is a little harder to understand.> I think you are mistaking here. A second order DE can be underdamped, > overdamped, critically damped or no damping. In all cases the > solution is two exponential expressions. The over damped solution is > two real exponentials, the critically damped is a double solution, the > underdamped is two conjugate complex exponentials and the no damping > is two conjugate imaginary exponentials.The important word in that statement was "simple." That is, with no damping (y') term. If you progress through the math, adding more terms as you understand the solutions to the previous ones, you should start with the undamped form as simpler, and then procede to the damped form. Try to remember when you first learned about such equations, having never seen them before. The simplest second order differential equations are y''=y and y''=-y. Understand them first, then add damping. -- glen
Reply by ●December 6, 20102010-12-06
On 12/05/2010 05:40 PM, glen herrmannsfeldt wrote:> brent<bulegoge@columbus.rr.com> wrote: > (snip, I wrote) > >>> Anyway, the undamped resonator is a simple second order >>> differential equation, damping adds one more term, and so >>> is a little harder to understand. > >> I think you are mistaking here. A second order DE can be underdamped, >> overdamped, critically damped or no damping. In all cases the >> solution is two exponential expressions. The over damped solution is >> two real exponentials, the critically damped is a double solution, the >> underdamped is two conjugate complex exponentials and the no damping >> is two conjugate imaginary exponentials. > > The important word in that statement was "simple." That is, with > no damping (y') term. If you progress through the math, adding > more terms as you understand the solutions to the previous ones, > you should start with the undamped form as simpler, and then > procede to the damped form. > > Try to remember when you first learned about such equations, > having never seen them before. The simplest second order > differential equations are y''=y and y''=-y. Understand them > first, then add damping.And if you want to be perverse, add the sort of dampening that you'd get from a _real_ mechanical damper, who's action is going to be mostly sign(v) * v^2, with some coulombic friction, and (depending on who you buy your damper from) maybe some significant laminar flow drag which will actually be proportional to velocity. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●December 6, 20102010-12-06
Tim Wescott <tim@seemywebsite.com> wrote: (snip on mass/spring/damper systems)>> Try to remember when you first learned about such equations, >> having never seen them before. The simplest second order >> differential equations are y''=y and y''=-y. Understand them >> first, then add damping.> And if you want to be perverse, add the sort of dampening that you'd get > from a _real_ mechanical damper, who's action is going to be mostly > sign(v) * v^2, with some coulombic friction, and (depending on who you > buy your damper from) maybe some significant laminar flow drag which > will actually be proportional to velocity.I wonder sometimes why were were lucky enough to have nice linear (most of the time) resistors to build circuits with. The mechanical engineers aren't so lucky, as you note. Capacitors are usually reasonably close to ideal (except at higher frequencies), inductors not quite as good. -- glen
Reply by ●December 6, 20102010-12-06
On 12/05/2010 10:02 PM, glen herrmannsfeldt wrote:> Tim Wescott<tim@seemywebsite.com> wrote: > (snip on mass/spring/damper systems) > >>> Try to remember when you first learned about such equations, >>> having never seen them before. The simplest second order >>> differential equations are y''=y and y''=-y. Understand them >>> first, then add damping. > >> And if you want to be perverse, add the sort of dampening that you'd get >> from a _real_ mechanical damper, who's action is going to be mostly >> sign(v) * v^2, with some coulombic friction, and (depending on who you >> buy your damper from) maybe some significant laminar flow drag which >> will actually be proportional to velocity. > > I wonder sometimes why were were lucky enough to have nice linear > (most of the time) resistors to build circuits with.I don't know, but I'm not complaining! Even though it's so simple, Ohm got in trouble with "the establishment" for publishing: http://en.wikipedia.org/wiki/Ohm%27s_Law#History (this reminds me a bit of some of Rune's comments on cultural bias in engineering projects).> The mechanical > engineers aren't so lucky, as you note. Capacitors are usually > reasonably close to ideal (except at higher frequencies), inductors > not quite as good.-- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
