System Modeling and Equations of Motion

[jlangholzj]

All,

as kind of a side-project I've started working on getting a linear model of a car put together for use in matlab/simulink to do some basic analysis for things like system response and frequencies. My thought was to have the team use this for general design considerations and verification. Things like making sure the ride frequencies of the rear/front are set correctly, helping to observe roll stiffness, etc. I realize that there's already equations out there for hand calculating all of this but in the wonderful world of matlab you can do some awesome batch sims, root analysis of the system and characterization. On top of that I feel like its a pretty good exercise in engineering thought process.

Anyway I had a few thoughts and some general feedback on a few things to make sure I'm treading down the right path. Also please excuse my scribbling....it can be kind of bad at times!

First the views:

Top View

http://i.imgur.com/qV4xBMz.png

My thought here was to break up the frame into two separate masses to help include the torsional stiffness of the frame. In reality I'd suspect this should be broken up once more to include two different torsional stiffness values as not all parts of the frame are created equal.


Front View

http://i.imgur.com/ZQR8A99.png

This one I had a few questions on. The unsprung masses and tire spring coefficients are all pretty straight forward. So is the ARB torsional spring constant. My concern here was how to properly model the spring-damper system. I was thinking about using the distance (Lmr) or the Motion-ratio distance as a way to model the motion ratio we see in bellcranks. Another additional thought that I had was to model the MR as a rack-pinion system with respectable radii to model the motion ratio. The latter just gets a little more complex to draw but I think it would work none-the-less.

My second question with that is the placement of the high and low speed dampers on the model. I'm assuming that these should be in parallel?? The thought process behind that was that they're effectively the equivalent damping of the shock but having them in parallel should show the different poles associated with high and low speed damping. Another reason for the thought behind this was if you look at a simple damping force equation (Fd = -c dx/dt) this dictates that both damping coefficients should see the same displacement and lend to the overall damping force.

One additional thing that I see I forgot to label was the length of the ARB connecting arm (from the a-arm to the torsional spring). Whoopsies.

I'm a little out of depth here in the VD world (i know just about enough to talk to someone and understand) but the modeling bit I'm okay with. Just wanted a little feedback or some other thoughts on anything I may have missed. Thanks guys.

[slicktop]

I am no expert either, but, I think maybe you could just use a lookup table to model the damper. xdot in to the lookup block and Fd out.

[BillCobb]

Your description is one of a traditional Simulink application. One approach I tend to use is to model each of the 'parts' of the car as a collection of Simulink or User-Defined-Functions (UDFs) functions. (no that's not functional redundancy). Your Simulink program amounts to a time based solution for the interaction of these parts subjected to some form of input, just the way the real machine does.

For example, to model a car with hydraulic power assisted steering, you would have a steer angle or steer torque input to a steering shaft which moments a rotary converter to a rack displacement (or force, as you choose to venue). This displacement steers some tires which generate their own forces and moments. These moments are balanced back to the rack which has a rack force amplifier on it and that feeds back to the steering wheel as a moment or a displacement (the opposite of the input). Meanwhile, you have the vehicle dynamics to contend with (either as a simplified s-plane Bode form transfer function or more complicated dynamics reacting to the tire forces and moments. This is best solved as an iterative loop, especially if you don't have all the coefficients for solving all the actual inertial differential equations involved.

In this case, to be relevant, there are nonlinearities to model in the steering shaft, the hydraulic assist booster and perhaps even the tires. Then there is friction, either complicated friction or simple friction. You application of engineering talent comes into play when you decide what features move you toward a solution you can understand. Too many people throw in the kitchen sink and don't understand why the sewer gets plugged. Since a car is actually operated as a moment balancing act, there are one or more recursive loops in the system necessary to solve for the displacements of the torsion bar in the hydraulic valve and perhaps the balancing moment in the rubber steering shaft isolator if you have one. Some geeks use integrators for this. Some of us who wish to make progress use iteration techniques. But, modeling the System as a collection of parts (with their constitutive equations) and relationships (with their multiple input, multiple output characteristics) becomes a language for all future work you will now want to do. As ever, starting with a simple linear deal and adding complexity at each step will teach you how your machine works. This process will also make it simple to collect the characteristics of the parts you model, since the tests would be designed to produce the data for your component model's constitutive relations ship(s). That's why lap times are not a good way to start off a handling simulation. As in traditional System Engineering work, getting the open loop gain, frequency response and steady state response to be right are the small steps in play to a winning thesis.

When you start explaining your findings to others (and describing how the machinery works), in Simulink terms and diagrams, you will know you are 'there'.

[BillCobb]

I hope the di-lithium crystals can take this...

Note that this model passes a collection of data back to Matlab for post-processing. This data is the same as would be generated during an actual vehicle road test. So, the simulated test metrics can be compared directly with the road test results.

Obviously, Matlab is used to generate much of the preliminary material necessary to start this baby off. But, you get the picture. (No, really !)

[BillCobb]

If you think this looks like a tire test machine, you could be right. Hook it up to a force feedback steer controller and decide what camber gets you the best FY (and btw, what is the MX at this moment ? (Hey, that's a pretty darned good pun...)

[jlangholzj]

Bill if I haven't said it before....you're now officially my hero :)

Using UDF's is actually the route I was planning on going once I got a few things figured out in a modeling sense. I can't really run before I can walk!

Here's an example (albeit an open-loop solution) that I was working on for my capstone project. We were doing some research into a variable valve control system and wanted to get a traditional system modeled first to write some control software written. The reason for all the scopes all over the place was because this was run in parallel with another simulink simulation to look at some tradeoffs that we designed into the software....thus giving us some side-by-side comparisons between the two.

http://i.imgur.com/P4pf8SK.png


Anyway, I'm hoping for a little more insight into this before dipping into it completely....but I'll get there darnit

[Z]

as kind of a side-project I've started working on getting a linear model of a car put together for use in matlab/simulink to do some basic analysis for things like system response and frequencies. My thought was to have the team use this for general design considerations and verification. Things like making sure the ride frequencies of the rear/front are set correctly, helping to observe roll stiffness, etc. I realize that there's already equations out there for hand calculating all of this but in the wonderful world of matlab you can do some awesome batch sims, root analysis of the system and characterization. On top of that I feel like its a pretty good exercise in engineering thought process.

Jlangholzj,

I'm baffled.

I can see what you are putting into the analysis (sketches are good - more like above please, everyone!), but I honestly have no idea what you expect to get out of it, or why??? A few numbers that give you certain frequencies, and..., ummm..., roots and poles?

How will these numbers help you? What will you do with them?

Specifically,
1. If your model is at all realistic, then you should not find any "ride frequencies of the rear/front". Or, the other way around, if you do get clear "rear/front" frequencies, then the model is unrealistic (hint: "Mrear & Mfront" right on the axle lines is unrealistic in FSAE).

2. Conventional dampers are decidedly non-linear (ie. different slope for high and low speed, and in bump and rebound). So modelling two linear dampers in parallel (which just equals one stiffer damper) is, again, quite unrealistic.

Please note, this is not intended as random criticism. Rather, I am genuinely puzzled as to what you expect to get out of this analysis, and how useful you expect it to be?

(I was going to post a much longer piece on one of the LapSim threads, regarding how to get the most useful results from the least VDSim effort, but maybe another time...)

Z

[jlangholzj]

I'll preface this with a few things.

First, VD has never been one of my strongest suits. I understand enough about it so I can talk with the guys that know all the ins and outs of design. I've picked up a bit here and there but I'm in no words a "wiz" at this stuff. Second as engineers we realize that there's very few things that are actually linear. I realize that almost nothing in this system is linear...but if we assume it to be such there's a few useful things that can be gained from it. Lastly I'm hoping that once this model is completed it can be expanded on and used for other things....what those other things are, I'm not exactly sure yet but once you've got the car modeled it can be changed and revised per need.

First lets back out here a bit and take a look at the whole system. One of the properties that can be observed is the natural frequency of the system. For a given spring and damper rate, how will the car oscillate when traveling over say....a 20mm heave. bumps. could be alternating (left and right) or it could be simply going over a bump in the surface. How does changing our springs and dampers affect that. How about in an (aero)-unloaded corner...ie a slow speed corner vs a straight.

With that in mind, another observable phenomenon is the front and rear ride frequencies. This is something that can be hand-calculated (i've watched my frame-susp guy do this) and is something pretty straight forward. From my understanding the higher frequency correlates to a stiffer ride but the trade-off to that is less mechanical grip. There's a sweet spot somewhere in there where the ride height isn't changing too much but you've got enough travel to have the best mechanical grip. Things I can think of where this would be important would be pitch in braking and acceleration. Obviously a car that floats and has "mushy" suspension isn't desired as it tends to be a very non-responsive car but if the setup doesn't have enough mechanical grip there's some serious problems there too. On a perfectly smooth track, theoretically the stiffest suspension available would be beneficial but as soon as the car goes over a heave, well....have fun with that.

to tie both of these thoughts into analysis now.....

Moving a pole or a zero of a system can have an effect on the phase of its response. Most importantly, if a pole of the system is outside of a normalized unit circle, this is an unstable system. Typically this isn't something that we'd see in a suspension design (assuming all the parameters are reasonable) but what we CAN do is move the zeros of the system around to change the phase response of the system.

Also I'm curious to hear why a "front and rear mass is unrealistic in fsae". Wouldn't the car move about its axis? There should be the ability to simply the situation down to masses that rotate about an axis that are linked via "beam". In all reality they were an additional step that I took that may not be necessary for a simplified system. I just viewed it as a way to incorporate in torsional rigidity as I stated earlier.

[SNasello]

jlangholzj,

it sounds to me that you are overcomplicating your analysis without having a clear definition of what you want to find.

First of, I would leave the Kt frame out the equation at the moment. The reason I say this, is because it is a flexible body, with a distributed mass, with other masses attached to it in some way. Definitely not in the way that you described them. Adding undamped compliance to your spring/damper mount on the chassis is a much better use of your time. But lets get back to what I THINK you are trying to find.

Looking at the sprung mass of the car, you have a body with six DOFs. Assuming that the lateral and longitudinal translation and yaw rotation DOFs are constrained by the suspension kinematics, you are now down to only 3 (Vertical, roll and pitch). All of them are important for your suspension analysis.

The Mass of the sprung mass is the "Inertia" in the z direction. What about roll and pitch? You CAN simplify this to two masses like a dumbbell, but does this make sense? I won't go into this in as much depth as Z.

Now, you have also drawn four other masses. Lets assume that they only have the one DOF (keeping it simple at first right?). Later you can add in the rotational DOF about some arbitrary (kinematically defined) axis.

Now you have 7 DOFs (all interesting and important) which you need to connect to one another with a series of springs, dampers, torsion bars, etc. If you can draw this system and come up with the equations of motion, then your are good step ahead of the rest. I think after that you will have a light bulb go off in your head and be able to complicate (advance) your model a bit more.

[jlangholzj]

The Mass of the sprung mass is the "Inertia" in the z direction. What about roll and pitch? You CAN simplify this to two masses like a dumbbell, but does this make sense? I won't go into this in as much depth as Z.


After reading your reply through a few times and sketching a few things out on a napkin over lunch it makes sense why you wouldn't.

If we REALLY wanted to back things out further, a simple diagram as such could be used as well:

http://i.imgur.com/6YSnzgm.jpg


This would also have the same number of DOF and is a very basic model that could be analyzed. The only problem that I had with this was the MR of the bellcrank. As it stands the damper will have the same amount of travel as the unsprung masses which....would not be correct. One way that this could be changed would be to crate a UDF for the spring/damper combination that would have a scaling factor...or rather a scaling equation for the inputs. Doing that in simulink wouldn't be terribly hard but I was looking for a way to physically draw it into the system.

Also one thought that I had is that you could account for the non-linearity of the damper rather easily with an equation in the UDF. Ohlin provides damper curves that could be easily fit with a function that would provide the most "realistic" response.

In an effort to clarify a few things here, the goal with this is to try and get all of the equations that would normally be used for say...front a rear ride frequencies into a simulation...via model. That way instead of having spreadsheets or programs to calculate these, a simulation could simply be run and observe the phenomenon. Another thing that I'd like to see is the effect of changing the damping rates and spring rates on the cars natural frequency. In addition it would be nice to verify those hand calculations with the simulation...or really...the other way around.

I've got some homework and other things to do along with a guest speaker coming today but I'm hoping to get at this a bit more at the end of the weekend or towards the beginning of next week.

[SNasello]

A linear system is defined such that the response of the system (in this case acceleration or velocity) is a first order function of the state variables (velocity and position). As soon as you make the damper force a function of velocity, i.e. not constant, you no longer have a linear system. The linear system response in matlab will no longer cut it for you.

Simulink is a simulation environment. Matlab is too. They are also both programs. The point I am trying to make is, if you want to simulate a linear system then keep your equations linear, or linearize them about some operating point.

Ill say it again, if you can simulate the above model as a linear system in matlab, for which you will need the correct equations of motion, then you can just as easily put it in simulink. You will then see that making things non-linear is actually not that hard. In this way you don't have to just simulate damping and spring rates, but you can look at the effect of changing your damping and spring functions, with respect to position or velocity, over a complete range of frequencies, instead of just the natural frequency of the system.

The only problem I have with your drawing above (other than the missing roll and pitch DOFs), is that you tied the tire spring to ground. Where is the input to the system that will cause it to vibrate at its "Natural Frequency"

[Z]

Jlangholzj,

Very briefly for now (I'm running out of credit!), read the Regarding the front and rear ride frequency (http://www.fsae.com/forums/showthread.php?5177-Regarding-the-front-and-the-rear-ride-frequency) thread (it picks up speed on page 2, then becomes a bit of a train wreck... :)), which goes some way to explaining the simplified side-view 2-D motions of a car body. Or google "Rowell and Guest Free Vibrations of an Autocar" (R & G separately wrote a number of papers on the subject in 1920s). The ancientness and simplicity of R&G's analysis, together with the fact that today it seems to be totally forgotten/ignored/irrelevant, is what worries me most.

Also, I really like your last sketch. More of the same, please, FSAEers! (If only the forum could have a simple facility for such sketching, sighhh...) Note that the main things missing from your sketch are the Sprung-Mass's MoIs (in 2-D side-view this amounts to the SM being either a longer or shorter dumb-bell, with same CG position wrt springs).

As soon as you add significant damping to the above, the "free vibrations of the SM on its suspension" almost disappear, although the car still bounces on its tyres. In this case your sketch with just the tyre springs (+ MoIs!) will still exhibit interesting behaviour (very different behaviour for different MoIs). In fact, this behaviour should be quite realistic, given that many racecars are effectively suspensionless. One of the main reasons for having at least some spring-damping on racecars (on typical smooth tracks) is to suppress this "bouncing on the tyres".

Z

[DougMilliken]

Rowell and Guest are hardly forgotten, you just have to look in the right place. Our "Chassis Design" has a ~50 page chapter on Ride including a detailed summary of their work...plus Maurice Olley's comments and a summary of his flat ride experiments in 1931-32. SAE R-206, (C) 2002.

Note for students outside N. America--if it's too expensive to purchase, perhaps ask your university library to add it to their collection?

[Claude Rouelle]

Doug,

How do you dare to make a commercial allusion to your book on this forum? Who are you to do that?? Would you be a member of the T(ire) party?

:)

[Z]

Rowell and Guest are hardly forgotten, you just have to look in the right place. Our "Chassis Design" has a ~50 page chapter on Ride including a detailed summary of their work...
Doug,

Yes, and I have referenced your Olley "Chassis Design" book most places where this subject has come up. Trouble is, that book is a historical piece about stuff that happened waaaaay back in them olden days. I'm guessing that most FSAEers think they have to buy a modern book, about modern car stuff, in order to design their modern FSAE racing car. Not many modern Vehicle Dynamics books cover this simple subject.

My main point is that this subject really is at the very simple end of your "ladder of abstraction" (I think that is what you call it in RCVD*).

For example, the simplest VD analysis might be of a single point mass, with a single coincident tyreprint, moving around in a horizontal "flatland". For vertical analysis, put a spring-damper under the point mass, and calculate "heave" frequencies...

The next step closer to reality is to add a second tyreprint (with different cornering stiffness to the first), with CG somewhere inbetween (at a/b, still in "flatland"), and call it all a "bicycle" model. Still a very simple model, but good for some interesting, and close(ish) to realistic, results.

Turn the point mass in the middle of this bicycle into a dumb-bell, add a spring at each wheel, and you can do the "bounce and pitch" analyses that R&G did back in the 1920s. I reckon this R&G-type bounce and pitch analysis can all be explained with one good diagram, and about one page of text.

And, most importantly, it takes the student a lot closer to reality than the current "front and rear ride frequencies" approach. For example, the advantages of longitudinally connected springing become much more apparent with this more realistic model.

Z

(*PS. Wow, plugs for two of your books in the one post. The usual brown paper bag under the table will do... :))