Python Tool for Classical Mechanics PDF Print E-mail
Written by Luke   
Tuesday, 17 March 2009 15:56

On Friday I headed up to University of Nevada, Reno to meet with Ondrej Certik, the project leader of the SymPy project.  The goal for the weekend was to:

  1. Develop some python/sympy tools to generate equations of motion for multibody mechanical systems
  2. Go on a bike ride
  3. Drink some beer

We were successful in all three endeavors!

 The basic functionality that we implemented were some classes for symbolic unit vectors, reference frames, sympy expressions involving unitvectors, rotations, angular velocities, expressing  a vector expression in different frames, etc.  So far, we have derived the equations of motion for two systems, the simple point mass pendulum, and a torus rolling on the plane without slip.  A lot more work is needed to polish the user interface, but the core functionality is there and seems to be working very well.

 A few really exciting things about this project so far:

  1. Pretty printing of vector expressions -- unitvectors in bold with subscripts
  2. LaTeX output of symbolic expressions
  3. Generation of C/C++/Cython code for the equations of motion or whatever quantities you want
  4. Really easy to animate things using the functionality of Visual Python

The goal is essentially to be a one stop shop place to study the dynamics of mechanical systems, especially multi body systems and those with nonholonomic constraints.

 The project will be part of the Sympy project, but currently it is available at:

git://github.com/hazelnusse/pydy.git

If you have git installed, just go to the terminal and type:

git clone git://github.com/hazelnusse/pydy.git

and you'll have a working copy that you can play with and run.

 

Last Updated on Tuesday, 17 March 2009 15:57
 
Steady Turning Velocity and Steer Torque Level Curves PDF Print E-mail
Written by Luke   
Sunday, 01 March 2009 23:48

At last, some results!

 

 

 This was a tough cookie.  I need to automate this, otherwise it will never be done again.  It was really time consuming. 

Green lines on the left and right are the minimum and maximum lean curves.  Solid thick black lines are boundary curves to the feasible steady turning motions.  Solid thin black lines are the level curves of velocity.  Thick dashed lines are the steer torque level curves corresponding to zero torque.  Red lines are positive steer torque level curves, blue lines are negative steer torque level curves.

There is more to say about this figure than I have time for at the moment....

More to follow soon.

 

 
Last Updated on Monday, 02 March 2009 00:15
 
General Circular Motions of the benchmark bicycle PDF Print E-mail
Written by Luke   
Tuesday, 17 February 2009 16:00

My current work is focused on investigating the general circular motions of the benchmark bicycle model.  The parameters for this model are described by Meijaard et al., and what I am doing is generating steady turning level curves of yaw rate (or one of the wheel rates) and steer torque in the lean - steer plane.  In a steady turn, lean and steer are fixed (hence their rates are zero, rear frame pitch is determined from the holonomic constraint, and rear frame pitch rate is zero), but the wheel(s) are spinning and the bicycle is yawing.  In certain special cases, one of the wheels may spin about its axle while the other doesn't spin about its axle but instead spins only about the contact point.  For this reason, I am convinced that using the yaw rate as a parameter of the circular motions is better than using either of the wheel rates, because in any configuration besides upright zero steer, there will be a yaw rate.  The upright zero steer configuration has been well studied by Meijaard et al. for a set of benchmark parameters, my goal is to study motions away from this configuration, but using the same parameters.

In the interest of better serving the research community that is interested in bicycle dynamics, I set out to do create the following software tools and make them (and their source code) publicly available:

  • Holonomic constraint solver.  Lean, frame pitch, and steer are not independent coordinates if both wheels are required to touch the ground.  This tool will allow you to numerically specify two out of the three, then calculate the third.
  • Nonholonomic constraint solver. 
  • Equations of motion.  Be able to numerically specify two independent coordinates (lean and steer, lean and pitch, or pitch and steer) and three independent generalized speeds (really only the time derivatives of generalized coordinates, the simplest case) that you are considering to be independent, and solve for the time derivatives of these three independent generalized speeds.  Also will allow for a flag that allows you to specify if you want the time derivatives of the dependent generaliized speeds.
  • Eigenvalues of equations of motion when linearized about an arbitrary configuration.  Five independent quantities need to be specified:  two independent coordinates and three rates.
  • Animations of nonlinear simulation results that show the four rigid bodies of the bicycle:  Rear wheel, front wheel, rear bicycle frame with rigidly attached rider, and front fork/handlebar assembly.  Include ground reaction forces.
  • Parameter converter between the parameters used by Meijaard et al., and those I use (I don't use wheelbase, head angle complement, and trail; instead I use those that Franke et al. used).
  • Generalized coordinates converter between Basu-Mandal and Meijaard and my coordinates.

Each small subtask will be programmed as a C/C++ function.  SWIG interface files to Python will be created in order to allow Python scripts to utilize compiled code to quickly perform numerical studies.  These functions will be part of a C/C++ library that I will make available here.  My goal is to have many of these tasks complete by the end of February 2009.

Last Updated on Tuesday, 17 February 2009 16:40
 

pfizer viagra online buy levitra online canada levitra or viagra sildenafil citrate over the counter cheap generic vardenafil viagra south africa viagra liquid form tadalafil 10mg discount cialis viagra by Intas Pharma viagra equivalents cheapest viagra price viagra pro order now oral jelly vardenfil viagra buying online generic cialis soft tabs cheap generic viagra online viagra next day uk buy sildenafil online low cost cialis cialis street price buy soft viagra viagra generico cialis discount generic cialis soft tabs generic viagra mastercard cost of viagra in uk viagra without ed sildenafil trade names viagra pro reviews how to get viagra prescription online viagra types vardenafil side effects indian vardenafil order cheap viagra online viagra best prices where to buy levitra viagra price canada cialis without prescription canada viagra brands in india best generic viagra prices 25mg viagra online buy viagra over the counter viagra professional price herbal viagra products 100mg of viagra generic viagra super active generic cialis super active viagra picture of pill ranbaxy viagra buy real viagra online without prescription viagra online safely cialis ordering canada cialis generic viagra canada online pharmacy sildenafil no prescription cialis discount cialis women use vardenafil professional viagra sale buy cost of viagra in australia cialis online prescription cialis canadian pharmacy female cialis tadalafil soft tablets 20mg cialis india indian sildenafil cialis uk prescription kamagra for less generic viagra wholesale jelly viagra cialis promise program cialis effective dose discount cialis canada cheapest viagra prices indian levitra vardenafil levitra pharmacy buy viagra pills viagra gel cipla sildenafil buy cheap levitra is 25mg viagra enough cheap generic viagra online genuine viagra sales ordering viagra without a prescription viagra india buy generic levitra price cialis professional 20 mg viagra prescribing UK suppliers of viagra online prescription viagra tadalafil uk cialis drug prices viagra generic online kamagra oral jelly paypal cialis patent expiration sildenafil low cost brand viagra online without prescription kamagra 100 chewable tablet generic cialis safe natural viagra gnc tadalafil professional viagra fast shipping is cialis over the counter? cheap soft viagra ordering viagra in canada cialis sale 50mg viagra cialis cost per pill viagra indian pharmacy viagra order generic cialis 10 mg cialis soft pills generic viagra purchase generic viagra mexico how to buy viagra safely cheap viagra prices sublingual viagra sildenafil levitra pricing buy viagra prescription online viagra free pill green kamagra tablets cheap kamagra tablets cheap levitra tablets online viagra sales tadalafil daily use buy viagra by the pill generic levitra tablets drug prescription viagra cialis eli lilly viagra canada no prescription cialis dublin vardenafil maximum dosage viagra mail sildenafil citrate liquid order sildenafil viagra online cialis tablets 20mg cialis 20mg generic buy viagra in singapore viagra legitimate cheap viagra overnight cialis for sale in uk get viagra in canada viagra natural alternatives cialis 20mg generic viagra super force viagra chemical name levitra online sales kamagra price buy tadalafil in new zealand levitra prescribing information cialis once a day review where can i buy kamagra in london reliable viagra online how to buy viagra safely purchase vardenafil viagra purchase online viagra tablets online is generic cialis effective why is viagra blue viagra compare prices generic levitra canada online viagra canada generic cialis mastercard cheapest kamagra online gel viagra viagra generic names fast viagra viagra manufacturer viagra trials viagra herbal street value of cialis cheap kamagra online split cialis pills alternative viagra generic viagra soft cialis tablets side effects cialis singapore tadalafil 5mg buy viagra in britain viagra 50 mg tablets cialis 40 mg dosage buy tadalafil 20mg buy levitra in australia herbal viagra canadian pharmacy cialis get cialis prescription online cialis samples tadalafil generic cialis substitute of viagra buy levitra pills levitra mg vardenafil prescription sildenafil citrate tablets price buy cialis in australia generic viagra toronto buy levitra online online prescription viagra brand viagra canadian cialis cialis prescription viagra discounters buy viagra jelly viagra shops cialis overnight delivery viagra gel caps viagra over the counter in canada cheap viagra no prescription kamagra oral jelly cialis daily use review buy online viagra securely tadalafil manufacturer cialis chewable generic viagra without prescription cialis alternatives cialis 100mg buy viagra prescription online generic cialis 10mg online viagra pharmacy generic cialis overnight shipping what does generic cialis look like cheapest generic cialis viagra india buy canadian viagra generic sample viagra purchase cialis online without prescription cialis manufacturer generic sildenafil citrate 100mg cheap kamagra oral jelly uk levitra with no prescription cialis 20 mg reviews order viagra no prescription viagra 150 mg cialis best prices buy viagra no prescription order viagra cheap kamagra online buy cialis thailand buy cialis online without a prescription order levitra without prescription order cheap viagra online tadalafil cialis from india online cialis generic viagra online shop cialis drug viagra substitutes viagra levitra comparison kamagra for sale 5mg cialis online indian generic cialis buy viagra 25mg sildenafil vs tadalafil viagra belfast generic levitra without prescription kamagra suppliers uk buy generic viagra in australia online viagra prescription vardenafil hcl 10mg buy viagra no prescription liquid tadalafil name brand cialis viagra medicare buy viagra alternative OTC viagra substitutes tadalafi from india where to buy sildenafil citrate buy legal viagra buy levitra 20 mg discount levitra online kamagra tablets online tadalafil 20 mg cheap cialis soft tabs sildenafil tablets sildenafil for sale sildenafil citrate soft cialis soft tabs online kamagra suppliers viagra softtabs cialis prices canada liquid cialis research online tadalafil discount viagra usa viagra copies how to get viagra prescription online price of viagra at walgreens kamagra next day viagra canada no prescription online generic viagra viagra costs cheapest cialis online viagra pfizer buy cipla generic viagra buy 25mg viagra buy cialis cheap buy cialis in canada cialis online cialis how often viagra 20mg viagra sellers viagra pills for men kamagra oral jelly wholesale dosage of cialis cialis paypal viagra active ingredient discounts on viagra order online viagra indian generic viagra female sildenafil buy kamagra online uk how much does levitra cost tadalafil price kamagra pills viagra 100 mg purchase sildenafil citrate buy bayer levitra purchase viagra on line purchase levitra online viagra overnight viagra without erectile dysfunction generic levitra 20mg natural viagra for men generic viagra online UK buy cialis mastercard online viagra without prescription kamagra 100 online viagra UK viagra street price viagra 100mg generic vardenafil how long does it last viagra india price tadalafil buy viagra equivalent for women viagra paypal cialis compare viagra online viagra prescriptions how to get viagra without a prescription canada online viagra cialis multiple erection buy cheap tadalafil buy viagra meds online kamagra oral jelly australia over the counter viagra substitute cheap levitra pills order viagra by phone buying viagra online sildenafil soft tablets 100mg genuine cialis kamagra india viagra women buy viagra soft tabs online viagra without prescription in australia cheapest cialis buy viagra spray cialis trial pack cialis 20mg price discount viagra online genuine viagra uk viagra online generic cialis without prescription sale viagra brand name of viagra viagra tabs viagra sample pack levitra sale buy viagra cheap viagra cheapest cialis cost cost of viagra per pill viagra online uk viagra sales online real viagra canada cialis tadalafil buy generic viagra in singapore buy viagra american express buy kamagra uk buy sildenafil sildenafil pills generic viagra sildenafil medicine cialis viagra to buy cheap generic viagra cost levitra usage purchase viagra online no prescription sildenafil dose cialis made in india levitra 10mg cialis pills for sale viagra pricing pramil sildenafil 50mg vardenafil 20 mg order cialis kamagra oral viagra cialis cheap cheap cialis super active cheapest kamagra online generic viagra super active cheap original viagra generic tadalafil online viagra buy in UK online viagra cost in australia cialis daily use price tadalafil without a prescription cialis professional online generic levitra online viagra female cialis gels cialis price per pill buy levitra no prescription indian levitra viagra over counter generic viagra 50mg free shipping viagra purchase viagra online no prescription viagra cheap online sildenafil nhs prescription viagra vs cialis woman viagra cialis from canada buy cialis overnight buy sildenafil online cialis 20mg daily viagra pay with paypal mail order levitra levitra viagra comparison where to buy viagra in montreal buy cialis 10mg generic tadalafil uk how to buy viagra in australia generic levitra 5 mg cheap cialis levitra medication levitra now online viagra suppliers in UK sildenafil generic order cialis without prescription generic cialis dosage kamagra gel review kamagra soft buy cialis without prescription cheap sildenafil citrate cialis 2.5mg cialis pharmacy prices cialis 5 mg online generic levitra price viagra professional generic buy viagra in toronto viagra tablets sale buy viagra 50 mg buy vardenafil tablets how do i get cialis viagra female effects buy viagra online cheap viagra online order kamagra over the counter color of viagra pill levitra experiences cost of viagra in new zealand cialis australia viagra to buy online prescription viagra canada cialis online viagra next day air buy online viagra securely viagra 200mg levitra for women where to buy viagra in bangkok viagra cipla generic viagra generic levitra canada purchase cialis canada generic viagra purchase viagra sildenafil 50mg viagra in canada viagra india online generic online viagra viagra samples pfizer viagra liquid cialis next day generic cialis mexico cheap soft viagra viagra 50mg side effects buy levitra uk is kamagra legal indian viagra reviews viagra next day shipping buy cialis daily use sildenafil citrate 100mg india buy generic kamagra online viagra without prescription generic levitra 20mg what is the drug viagra kamagra jellies sildenafil 100 mg sildenafil citrate 25mg viagra from india generic viagra US sildenafil heart how much is viagra in australia levitra mexico cialis online pharmacy sildenafil citrate best price viagra UK no prescription buy viagra USA buy kamagra cheap cialis active ingredient vardenafil hcl 10mg tab levitra vs viagra vs cialis levitra sample buy viagra without prescription canada viagra pharmacy buy online viagra securely cialis tablets for sale viagra supplier cialis price costco buy viagra without prescription uk rx viagra pfizer coupons for viagra tadalafil soft viagra canada online oral jelly sildenafil tadalafil for sale viagra without a prescription canada viagra cost per pill overnight cialis delivery kamagra 100 gold cialis generic no prescription vardenafil otc viagra cialis compare viagra half pill viagra cost UK cheap viagra order online cheap viagra for sale viagra england viagra information buy cialis no prescription viagra tips buy viagra safely online generic viagra pills levitra without a prescription vardenafil online generic cialis online does kamagra work on women generic cialis price genuine kamagra cheap viagra generic sildenafil citrate 50mg sildenafil powder cialis prescribing viagra pfizer price buy cialis online usa compare viagra cialis vardenafil cheap normal dosage of cialis viagra prescription drug