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Thread: moment diagram with weight transfer

  1. #121
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    Just a moment, Please...

    The problem with this analogy is that nowhere in traditional vehicle dynamics straight line or cornering equations are displacement constraints (other than where is the ground). Vehicles initiate a turn because of a moment imbalance and continue to yaw and sideslip until this moment (whether from a front or rear steer angle or a wind gust or a tire induced imbalanced, etc.) is nulled. Depending in your point of view, the final resting trim can either be favorable or unfavorable relative to walls, trees, lakes, rivers or snow banks.

    As an added observation, may I point out that REAL train's couplers are body mounted, not truck mounted (as your model trains are assembled). There is a huge advantage to this when negotiated turns at speeds well above a posted speed limits set by bank angles. Yes, train wheel flanges are a displacement constraint and the resulting cornering g levels from flanges plus coupler moments can be almost unbelievable. GM has an instrumented automobile carrier car that often rides along in coast to coast deliveries. The dynamic forces read were eye openers and resulted in major changes to batteries, wheel bearings, and vehicle tie down methods and systems. They are often higher than those experienced during abusive driving! GM uses the results to coax rail owners to repair portions of the route or avoid some portions altogether thank you.

  2. #122
    Quote Originally Posted by BillCobb View Post
    The problem with this analogy is that nowhere in traditional vehicle dynamics straight line or cornering equations are displacement constraints (other than where is the ground). Vehicles initiate a turn because of a moment imbalance and continue to yaw and sideslip until this moment (whether from a front or rear steer angle or a wind gust or a tire induced imbalanced, etc.) is nulled. Depending in your point of view, the final resting trim can either be favorable or unfavorable relative to walls, trees, lakes, rivers or snow banks.

    As an added observation, may I point out that REAL train's couplers are body mounted, not truck mounted (as your model trains are assembled). There is a huge advantage to this when negotiated turns at speeds well above a posted speed limits set by bank angles. Yes, train wheel flanges are a displacement constraint and the resulting cornering g levels from flanges plus coupler moments can be almost unbelievable. GM has an instrumented automobile carrier car that often rides along in coast to coast deliveries. The dynamic forces read were eye openers and resulted in major changes to batteries, wheel bearings, and vehicle tie down methods and systems. They are often higher than those experienced during abusive driving! GM uses the results to coax rail owners to repair portions of the route or avoid some portions altogether thank you.
    I knew you would respond, correct on every count by the way, BillCobb. I got to thinking about Z's posts on the long wheelbase bus, then the kinematics of a bicycle model on rails that he suggested and that is when I decided, well heck, to get myself started thinking in the right direction about the simple kinematics I'll start playing with my toy trains.

    It did not take long to realize that this was folly when compared to a pneumatic tired vehicle, but it was a good exercise none the less. However, I was fully aware of how far away from road vehicle dynamics I had strayed with this crude example.

    It does reenforce the 'whose doing what to whom' approach that must be taken in any kinematics problem. Those old words 'With Respect To' can not be glossed over.

    What can I say, had a kitchen floor that looks like graph paper, had track, had a long wheelbase flat car, cut some paper coordinate systems and play around on a rainy Saturday morning.

    Thanks for the description of, and the data related to, real railway cars which I was unaware of and find very interesting.

    Ralph
    Last edited by rwstevens59; 10-01-2015 at 02:59 PM.

  3. #123

    Off Topic...

    Quote Originally Posted by rwstevens59 View Post
    Attached (I hope) are a few photos of a very crude example of the kinematics I believe Z was trying to describe on page 10 of this thread.

    After reading his post I had a look in the attic and in a very short time produced this 'model' to play with to more fully understand what Z was trying to describe.

    Worked for me.
    Attachment 763Attachment 764Attachment 765Attachment 766

    The object of the last photo using two different radii is that in the world of oval track skewing of the solid rear axle wrt the car centerline is a common practice and I just wanted to have a look at what 'crabbing' down the track might look like.

    Again, Sorry for the toys.

    Ralph
    Mmm, multi-track drifting....

    https://www.youtube.com/watch?v=86PUB4u2s2A

  4. #124
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    Beltway Boogie.

    Instead, get out your biggest belt sander and turn it upside down. Then take a soft tired model car and put a string on it for restraint. Yes, Rembrandt, the car needs to fit on the sander belt and the restraint string has to be properly located (exam question). Then cut 'er loose. You can push on a front spindle with a straw to generate some initial side force. Now watch for the moment reaction and corresponding sideslip activity. You should be able to immediately see that the rear wheels go the wrong way first, then catch up to the rest of the motion.

    For best results use a very fine paper (This is not Darlington), and some All Season tires. Getting a team together with a buzz on, a pizza and a strobe light will make for a REALLY cool evening science project. Video on demand.

    Extra credit if the roll DOF is excitable.

  5. #125
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    Quote Originally Posted by BillCobb View Post
    ... generate some initial [front] side force. Now watch for the moment reaction and corresponding sideslip activity. You should be able to immediately see that the rear wheels go the wrong way first, then catch up to the rest of the motion.
    Bill,

    I will cover this below (next week?) under "The School Bus... (2) Starts Entering the Corner.".

    (Spoiler alert: The "Sports Bus" behaves as per above quote. The "Slow Coach" does not, so is slower...)

    Z

  6. #126
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    Quote Originally Posted by BillCobb View Post
    Instead, get out your biggest belt sander and turn it upside down. ...
    Once you have the belt sander and model car, you can also constrain the model--see RCVD Figure 8.1. This was done c.1970, the model car has a wheelbase of about a foot (I still have this model). The "belt sander" was a proof-of-concept scale model of the belt installation for the Calspan tire tester, TIRF (where TTC data is measured). We didn't have a good photo, so worked with our artist to create the line drawing.

  7. #127
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    Please make yourselves comfortable. Eight longish posts follow, covering YMDs (aka MMMs), the Acceleration Pole, and this whole business of transient cornering.
    ~o0o~

    Better Yaw-Moment Diagrams.
    ========================


    First I would like to clarify some of the criticisms I made earlier regarding the general use of YMDs.

    The main problem I see with these types of diagrams, especially when they are presented in the manner given by Silente and Jpusb on this thread, is that too much of the useful information that could/should be displayed is NOT THERE. In short, a rainbow of pretty colours, even when drawn with computer precision, does NOT maketh good communication of information.

    At the very least, these diagrams should include ALL the information necessary for their construction. So for these YMDs, the Velocity of Car, Radius of Corner, Details of Car (eg. is it a "bicycle model", or does it have width and weight-transfer), Details of Tyres, UNITS of X,Y-Axes!, etc., etc., should all appear somewhere in the diagram.

    And this applies even if such diagrams are only for your own personal use. This is because in a few years time you might be working on a YMD problem and you remember that you solved a similar problem a few years back. So you get out your old YMDs (or open that file on your i-3000-whatsit...). And then ... you spend the next week trying to figure out what units you used for the axes!
    ~o0o~

    More importantly, these STATIC diagrams might have been the bee's knees back in 1950, but much more can be done today.

    Interestingly, Doug Milliken (an M in MMM) gave a very useful link on the "Pitch Axis" thread.
    "Here is an interesting website that takes a novel (to me) approach to computing and displaying dynamic systems, http://worrydream.com/KillMath/ note the animations attached to the text. This sub-page uses a very simple car model as an example, http://worrydream.com/LadderOfAbstraction/ ."

    I reckon those links give a really good approach to understanding these problems.

    For example, you might start with the YMD in one corner of your computer screen. In another corner you have a plan-view of the car showing its major dimensions, corner-radius, etc., and also a display of each tyre's "Friction Circle" potential (as in Figure 8.25 of RCVD, or the "FRC" diagram on front cover). Elsewhere you have all the other important information.

    The YMD/MMM has a cross-hair on it that can be moved about to highlight any given point on the diagram. As this is done the tyre-force arrows on the Friction Circle Diagram automatically update to show how much of their Friction Circle potential the car is actually using. And a whole bunch of other useful stuff can also be updated, such as all four tyre-slip angles, total force vector acting on car (as a single Force-vector acting along some LoA, or Force + Couple drawn at CG), and so on.

    If you are not interested in certain information, then you simply ignore it.

    But the more information that you have immediately available, then the more useful is this whole process to solving your given problem.

    The following posts give a hint of some of the things I would like to see alongside any YMD I might draw...

    Z
    Last edited by Z; 10-06-2015 at 11:11 PM.

  8. #128
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    Sketch 1/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
    ================================================== ==============


    These posts and sketches all relate to my earlier comments on the "Acceleration Pole", and the fact that all this YMD stuff, and, in fact, all of Vehicle Dynamics generally, is about UNDERSTANDING "TRANSIENTS".

    So get the idea out of your heads that you only have to understand the Steady-State stuff, "...because "transients" are just the brief and unimportant "joining-up" bits". Wrong!

    EVERYTHING IS TRANSIENT!!!

    Learn transients, and all that boring SS stuff becomes a doddle...
    ~o0o~

    PRELIMINARIES - If I was doing this as a proper paid job, say, like Claude does, then I would start by listing ALL DEFINITIONS and ASSUMPTIONS (aka Axioms) needed to make sense of what follows. And what follows would be very rigorously built-up using ONLY those initial Definitions and Assumptions. But these posts are just interweb-waffle, so I will skip much of that rigour.

    However, I will stress here that what follows is taken from a small corner of the idealised world known as "Mechanics". This world exists inside the similarly imaginary universe of "Applied Mathematics". So, the "Kinematics" discussed below deals only with ideal bodies that are perfectly rigid, perfectly smooth, perfectly massless, perfectly without forces, and moving in perfect Euclidian space. And later on there is also a pinch of perfect "Dynamics" thrown in, and a few other unrealistic things, such as all motions are only in perfectly parallel planes.

    Some practical Engineers might say "But..., real racecars are not perfectly rigid, and you have to know all of their real compliances to understand them. So all that 'idealisation' makes the theoretical approach useless!".

    Well, the fact is that any useful engineering analysis has NO CHOICE but to build from the above very solid, albeit idealised, foundations. To take just one example, in order to measure the compliances of your floppy racecar, YOU MUST FIRST ASSUME some sort of Euclidian reference-frame that the floppy bits move "with respect to". The assumed motion of the floppy-bits wrt some reference-frame, namely the "Kinematics" of the situation, is a necessary prerequisite for measuring the compliances. If no clearly postulated Kinematic foundation, then no measurements!

    Anyway, enough of this deep and meaningful stuff. The following simple idealised descriptions are a good starting point for understanding how your cars corner. To ease this understanding, in this and the next four posts I cover the SPECIFIC case of the aforementioned School Bus going around a corner. In the last post below I cover GENERAL Kinematic Accelerations, albeit still in the idealised Planar world (= Flatland).

    All this is done in a very brief and non-rigorous way, but hopefully good enough to give you a taste of how transient cornering works. For those of you interested in the more general case of fully 3-D Accelerations, well, some hints in the last post. Or you might ask Claude at one of his many seminars?
    ~o0o~

    STRAIGHT AND STEADY - This first sketch shows the School Bus travelling (almost) straight and steady in a Northerly direction. The main thing of interest shown here is the "Velocity-vectors", with respect to "Inertial Space", of selected points on the bus.

    Note that most of this and the following sketches has been worked out with real numbers, so is "to scale". For example, this particular bus has a wheelbase of 5 metres, width of 2.5 metres, CG slightly aft of mid-wheelbase (55R%), and the Velocity-vectors here represent a speed of 15 metres/second (= 54 kph, ~34 mph). (I have these numbers written down on numerous scraps of paper, and over too many weeks now, so I hope they all still add up...).



    Because all the Velocity-vectors of different points on the bus are aligned North-South, it follows that the "n-lines" (ie. lines of NO motion) of all those points are aligned East-West. IF all the Velocity-vectors are perfectly N-S, then all the n-lines will also be perfectly parallel E-W, and so all the n-lines will "intersect at infinity".

    But we assume that the Vs are NOT quite perfectly parallel, and the n-lines actually intersect at a Velocity Pole some distance out to the right (East) of the bus, perhaps just out past Pluto. So the bus is not actually travelling in a perfectly straight line, but is in fact cornering, albeit around a very large radius corner. This gives the bus a very small Rotational Velocity W (= "Omega", and more commonly called "Angular Velocity"). In fact, it is an extremely small W, because W = V/R, and while V = 15, R is extremely large.

    Likewise the slow change in direction of all the Velocity-vectors amounts to "centripetal" (= "centre-seeking") accelerations of all the points toward the right. But these accelerations are also extremely small, because A.centripetal = V^2/R = 225/a-very-big-number. So I have not bothered drawing any Acceleration-vectors in this sketch. No pencil is sharp enough!

    More coming...

    Z
    Last edited by Z; 10-07-2015 at 06:55 PM.

  9. #129
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    Sketch 2a/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
    ================================================== ==============


    ENTERING THE CORNER - Now things start to get interesting. The school kids are shouting louder than usual, which wakes the bus driver from his slumber, and he suddenly realises he must turn into a side-street. So he yanks hard-right on the steering-wheel, performing what Vehicle Dynamicists call a "step-steer". This puts a "steer-angle" (or a "slip-angle", as some VDs call it) between the direction of motion of the two front-wheelprints (= the V-vectors at the front-wheelprint centres) and the centreplanes of the wheels, sometimes called the wheels' "heading-angles" and shown on the sketch as "centrelines".

    After a very short time known as the "tyre relaxation length" (= distance tyre rolls before its sidewalls distort and "tense" up) the two front-wheels develop axial-forces Flf and Frf, in the direction of their axles. These are often referred to as the tyre's "Fy" forces, but note that they do NOT point in the bus's Y-coordinate direction. That is, they are NOT PURELY SIDEWAYS to the bus, but also point a bit backwards.

    The two front-wheel axial-forces intersect some distance to the right of the bus and add together as a resultant force Ft ("t" for total, because "r" for resultant gets confused with "r" for rear). We also assume that all the upward ground-to-wheel forces cancel out all the downward gravity-forces, and there are no other forces acting on the bus, such as from aero, etc. So Ftotal is indeed the total external force acting on the School Bus as a "free-body".

    The big question is, what does this assumed sum total of all forces acting on the School Bus, Ftotal, do to the bus?

    The answer is easily found with some old-school geometry. The force Ft is slid along its Line-of-Action (LoA) to the point where the bus's CG is perpendicular to the LoA. Then, using the construction shown in the sketch, a semi-circle is drawn on the line through Ftotal and CG, such that the semi-circle also passes a distance "K" from the CG. ("K" = Radius-of-Gyration of the bus in Yaw. See below for more details...)

    BINGO! This Planar Dynamics problem is now fully solved.



    The above method tells us that the Acceleration Pole (AP) of the bus has now moved to the point M2, just in front of, and slightly to the left of the differential. We also know that all points on the bus are accelerating clockwise about the AP, with the magnitude of the A-vectors increasing in direct proportion to their distance from the AP, and with all their directions PERPENDICULAR to radii drawn from the AP.

    We can even calculate that the front numberplate of the bus is accelerating rightward at about 28 m/s/s, or ~2.8 G, the front-wheels are accelerating rightward at ~20 m/s/s (~2 G), and the bus's CG is accelerating rightward at just under 7 m/s/s (~0.7 G). All this comes from a slightly optimistic front-tyre Mu = 1.5, and the other assumed dimensions.

    Importantly, in the "Low-K Sports Bus" (main drawing at left of sketch), we see that the cool kids sitting in the rear seats are accelerating LEFTWARD! Likewise, the rear-wheels of this bus are also accelerating leftward. This is good, because it means the rear-wheels quickly develop the sort of slip-angles that produce a rightward axial-force, which is the direction the driver wants to go.

    By contrast, in the smaller "High-K Slow Coach" sketch (bottom-centre), and at this very early phase of corner entry, the Acceleration Pole is some distance BEHIND the bus. So ALL points on the bus are accelerating RIGHTWARD. This is bad, because the rear-wheels initially move rightward while they still point due North, so they develop tyre-axial-forces that push the rear of the bus LEFTWARD, and out of the corner. This is NOT the direction the driver wants to go. It follows that High-K (= large-Yaw-Inertia) does NOT maketh a sporty bus.

    Also worth noting that at this very early phase of "transient corner entry", the bus is still travelling essentially straight-ahead. (I gave it a very slight deviation from North at this early time-step.) And because all the V-vectors are still pointing close to due North, the Velocity Pole is still a long, long, way away to the right.
    ~o0o~

    More coming...

    Z
    Last edited by Z; 10-07-2015 at 07:20 PM.

  10. #130
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    Sketch 2b/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
    ================================================== ===============


    EQUIVALENT MASS SYSTEMS.
    =========================
    The old-fashioned Planar Dynamic Analysis done above is useful for solving a large range of problems. These problems just have to be close to "planar", where the Body can be "real 3-D", but all its motions must be in parallel planes. A similar approach also works for more general 3-D problems, and is very useful to understanding them! See Centre-of-Percussion, Compound-Pendulum, +++, for examples. The following is just a brief introduction to all this.

    The gist of the method is that the real body, whatever its "true" nature, is first assumed to be a "continuum mass distribution". That is, the School Bus is assumed to be made of idealised stuff that has Inertial Mass properties (ie. it resists acceleration per N's Laws), but this stuff can be indefinitely subdivided, and may also be of variable density. Various Integral Calculus methods are then used to find things like the Total-Mass, Centre-of-Mass, and Moments-of-Inertia. Check your school books, or ask your teachers.

    The next step is to replace the original complicated shaped body with a much simpler body, but one that has EXACTLY THE SAME Dynamic properties as the original body. This new, simpler, body then allows much easier calculations. Fortunately, Nature has been incredibly kind to us, making this process very easy!

    All we need do is make sure that the first three "Mass-Moments-of-Inertia" of the differently shaped bodies are equal. No higher order MoIs are needed! Each "MoI" = Sum-over-all-i's-of-(Mi x Ri^N), where Mi is the elemental mass, Ri is the position vector to Mi in some reference frame, and each Ri is raised to the power N, which is the "order" of the MoI.

    So, the first three MoIs are;
    N = 0, the "zeroth MoI", which gives the Total-Mass of the body (note that R^0 always = 1),
    N = 1, the "first MoI", which gives the CG position (or more correctly, the CoM) of the body, and
    N = 2, the "second MoI", which gives the Second-Mass-Moment-of-Inertia of the body.



    One "simpler body" that can be used to represent the original body is a circular ring, as shown at left of sketch. This ring has the same Total-Mass Mt as the School Bus, the CGs of ring and bus are coincident, and the radius K of the ring gives an equal 2nd-MoI-in-Yaw (= Mt x K^2) to that of the bus. Unfortunately, this ring does NOT simplify any calculations, because it is still a distributed mass, so it is still subject to Euler's Rigid Body Equations, just like the original School Bus. (Some deep-and-meaningful stuff here, but moving on...)

    So an even "simpler body" is to reduce the bus down to just TWO POINT-MASSES connected by a "massless rod". Yippee! Now we can use Newton's Laws of Motion directly, which is significantly simpler than using Euler's RBEs. There are two infinities of ways of choosing our two point-masses, which is good because it gives us a lot of freedom in solving the problem. Very briefly, (in this planar case) we choose the position (= X,Y coords) of one point-mass in any way we want, and then the size of both point-masses and the position of the second point-mass are automatically determined. This is shown, rather non-rigourously, at bottom-right of the sketch.

    One simple choice of the two point-masses is to have two masses, each Mt/2, at each end of a diameter of the circular ring of radius K. This gives a "dumb-bell" of length 2 x K that accurately represents the whole bus. Note that for these planar problems the orientation of the dumb-bell is irrelevant. It can be aligned along the centreline of the bus, or it can be lateral to the bus, or at any other angle, but the mid-point of the "massless rod" connecting the two half-masses must be at the bus's CG.

    Here, a more useful choice is to have the first point-mass M1 sitting on the LoA of Ft, which lies a distance "a" from the CG. The location of the second point-mass M2, at a distance "b" on the other side of the CG, is then determined by the semi-circle method, namely "K^2 = a.b". Use Pythagoras to verify the little drawing at bottom-right of sketch. The sizes of M1 and M2 are found from the "0th & 1st MoIs", namely M1 = Mt x b/(a+b), and M2 = Mt x a/(a+b).

    The last step of this Dynamic Analysis is to note that the force Ft can only act on M1 but NOT on M2 (<- important!), because, although the "massless rod" connecting M1 and M2 is capable of carrying forces (perfectly well!), at this instant there is NO COMPONENT of the force Ft directed along the massless rod. Therefore, while Ft accelerates M1 to the right, the second mass M2 just sits there minding its own business, as per NI (aka Galileo's Law of Inertia).

    Therefore, at this instant, M2 has no acceleration whatsoever, and is thus the Kinematic Acceleration Pole of the School Bus.

    Interestingly, note that the Sports Bus has both Ft and M1 in front of, and OUTSIDE, its K-ring, which puts M2, and thus the Acceleration Pole, INSIDE the K-ring. However, the Slow Coach has Ft and M1 inside its K-ring, putting M2 and the AP outside the K-ring, and thus a long way rearward. This greatly affects the transient performance of the two buses, as noted in previous post.

    Of course, all the above is only the case for the very first instant of this Dynamic problem, because as each "dT" time-step passes the AP moves to a new location, and other things happen...

    So, where does the Acceleration Pole move to next?

    More coming...

    Z
    Last edited by Z; 10-07-2015 at 07:23 PM.

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