Determining the Velocities of a Polygon in a Curve

[VicenteMMOS]

Hello, everyone!

My question is really simple, in fact I even feel a bit ambarrassed of asking it. :x Imagine that a car is making a constant radius turn, of a given radius R. For the purposes of this question is enough to say that the car may be thought as an isosceles trapezium, or even as a rectangle, for the sake of simplicity. My knonwns are the vehicle's center of mass longitudinal velocity, lateral velocity, and yaw velocity (the angular speed at which the vehicle changes orientation during the turn). Given also that the trapezoid has front track Tf, rear track Tr (<Tf), and a wheelbase (distance between front and rear tracks) of l = a+b, where a and b are respectively the distances with respect to the position of the center of mass of the front and rear tracks, I want to find out the resultant analytical velocities of each wheel center. I hope my description had been enough precise in order to propose the question. I've been trying to solve this question for some time now, without coming to a conclusion. I hope you guys can help me with it. I'm aware that this is a very trivial question, but I don't know where to find the answer to it. It would really help me a lot in this moment to have at least the explanation of how to find the solution.

Thank you!

 

[mfb]

You just need the distance to the center of the curve the car drives, you can calculate this with a bit of geometry (did you draw a sketch?). Every part of the car will follow a curve with the same angular velocity and center of rotation but with a different radius.

It is possible to derive it if you add the velocity from the motion of the center of mass with velocities induced by rotation of the car around its center of mass, but I guess that is more complicated.

 

[VicenteMMOS]

Hi!

Thanks for the prompt answer. You see, the problem really is a bit more complicated than how I first stated it. I started by explaining it with a constant radius because for practical reasons, it is roughly constant. But really, there are variations to the radius of curvature, that for now I'm neglecting. I do need to know though how to find the analytical expressions of the resultant velocities of each wheel center of the vehicle, by only knowing its longitudinal and lateral velocities, vx and vy [m/s], and its yaw rate, psi' [rad/s]. And yes, I have drawn the sketch to this problem a few times, but I never actually managed to figure out the equations of motion. I believe there must be some kinematic relationship between the three velocities, so that the rotation and translation of the vehicle may be perfectly described by only 2 of these inputs, and that's all there is to it, knowing the values of the velocities of each wheel center, when knowing the vehicle's CG velocities. Actually, I just realized I posed the wrong question all along. I do have the correct equations of the resultant velocities of each wheel! Here is what they look like. Taking the nomenclature I used, and considering a left-hand turn, I defined the velocity of the front-left wheel as:

V_fl = V_flt + Vx (vectorial sum)

where V_flt = psi'*(a^2+Tf^2/4)^0.5 is the tangential component of V_fl, with respect to the rotation along the CG.

so, the modulus of this sum would be "Pithagoras":

(Vx^2+V_flt^2 -2*Vx*V_flt*cos(theta))^0.5, being theta the angle between Vx and V_flt, theta = arctan(2*a/Tf)

I first thought these equations were wrong because when I computed the values for each wheel, they seemed different from the ones my professor got. But if I substitute just these equations for V_fl, V_fr, V_rl, V_rr, in the professor's code, they result exact in his program. So, I'm deducing something with my inputs is wrong. See? I told you there was more to it! =) My input variables are the longitudinal and yaw velocities, and the angle at the steering wheel. When I compare my inputs to the one of the professor's code, I see that his Vx is somewhat varying. His psi' input is a ramp-up, starting at 2 s of simulation time. Namely, this input is telling this piece of code to linearly increase psi' from 0 indefinitely until the end of the simulation. But this shows an effect of a small variation in Vx, making it not really constant. Also, the input in the steering wheel is put to a similar condition, where it starts increasing from 0 at 2s up to a maximum, and stays there until the end of the simulation. But the thing to notice is that Vx is dependent on psi', and so my question should be different. How can I find such a relationship? I took a bit too long to notice the equations for the resultant velocities were correct, and that I was getting wrong results because my inputs were linearly independent when they really are not.

Thank you for the support!

 

[mfb]

I don't understand what is right and what is wrong now.

 

[VicenteMMOS]

xD Ok, let me explain better. I realize I confused anyone reading the question, because actually, I took long to see that the equations describing the resultant velocities at the wheel centers were right all along. The results I were getting were wrong, but not the equations. I know so because when I copied and pasted my equations into the professor's code, they worked. So the error were in the inputs, that I cannot reproduce just yet. I am building a full-vehicle dynamic model and so, having to start somewhere, I started here. I have finished describing the equations for the tire model, but they require inputs like wheel velocities, in the wheel ref. frame, which on their hand are dependent on the chassis kinematics and dynamics, and those, in a loop, are due to the interactions tire/road. =) So, having to start somewhere, I started at the tires. It went all fine, up to the point where I couldn't just input the tire specific independent variables and get the forces as the outputs, I had to keep implementing more equations to the system. There being 4 tires, I cannot neglect the fact that a car under a given acceleration transfers weight across the wheels, and those imply different tire forces. Now I'm at a point were I need the wheel hub velocities in order to determine the tire forces later, but these depend on the CG velocities, which seem to be dependent on each other. So, going back to the sketch I described, how could I try and find a relationship between psi' and Vx? What would you recommend me do? I need to find an equation that accounts for both, as apparently I can't just input both as constants, and expect for correct results.

Thank you.

 

[jack action]

V_fl = V_flt + Vx (vectorial sum)

where V_flt = psi'*(a^2+Tf^2/4)^0.5 is the tangential component of V_fl, with respect to the rotation along the CG.

so, the modulus of this sum would be "Pithagoras":

(Vx^2+V_flt^2 -2*Vx*V_flt*cos(theta))^0.5, being theta the angle between Vx and V_flt, theta = arctan(2*a/Tf)

That looks wrong to me because you do not consider Vy. Assuming clockwise positive for psi:

V_flx = Vx + psi * Tf /2
V_fly = Vy + psi * a
V_fl = (V_flx² + V_fly²)^0.5

V_frx = Vx + psi * Tf /2
V_fry = Vy - psi * a
V_fr = (V_frx² + V_fry²)^0.5

V_rlx = Vx - psi * Tr /2
V_rly = Vy + psi * b
V_rl = (V_rlx² + V_rly²)^0.5

V_rrx = Vx - psi * Tr /2
V_rry = Vy - psi * b
V_rr = (V_rrx² + V_rry²)^0.5

For any wheel XX : tan( thetaXX ) = V_XXx / V_XXy

The velocities are a result of their corresponding rotational or linear acceleration over time. The accelerations are the result of their corresponding resultant force or moment created by the tires and aerodynamic. That is where you should start. Once you initiate the steered wheel angle, the vehicle longitudinal velocity will translate into a lateral and longitudinal velocities for the tire frame of reference such that you can find the initial tire force.

 

[VicenteMMOS]

Interesting point. I see the difference in the equations, and it does make sense. But still, your system of equations bring Vx, Vy and psi' as linearly independent variables, when they are actually not. This is the same reasoning I made myself before posting the 1st question, but then by checking the values in the code, I saw that Vx wasn't responding in a "predictable" fashion so as to be considered as an independent variable. Granted, it is an input for computing the components at each wheel, but I figure that due to the fact the vehicle is in a circle, and that psi' is such that the vehicle ref. frame is roughly always tangent to this trajectory circle at any given time, there might be some mathematical correlation between these three, Vx, Vy and psi'. It is correct, though, that it is first the longitudinal velocity that is present, and then the lateral and yaw velocities take place, due to the steer input. Still, the professor's code shows only a dependence on Vx and psi'. That was the the doubt that got me thinking and searching for a physics forum =) Either way, how would you suggest I verify whether one or the other method is wrong?

Thank you so much, that was really helpful.

 

[VicenteMMOS]

The velocities are a result of their corresponding rotational or linear acceleration over time. The accelerations are the result of their corresponding resultant force or moment created by the tires and aerodynamic. That is where you should start. Once you initiate the steered wheel angle, the vehicle longitudinal velocity will translate into a lateral and longitudinal velocities for the tire frame of reference such that you can find the initial tire force.

Ok, perhaps I understand better the point you were trying to make. You are telling me to separate these velocities because they actually are independent, although they being dependent is just a particular case of the motion? I mean, they won't necessarily always be dependent on each other, a physical object could theoretically describe a circular motion without rotating, or with a rotation that is not following the path of curvature, right? So, is it more likely to what you were trying to say? It sure got me thinking. Thank you anyway.

 

[jack action]

Vx, Vy and psi are independent by definition. They represent the 3 degree of freedom in a 2-dimension space.

Obviously, when in a straight line, Vy and psi are equal to zero, leaving only a value for Vx. Once turning will be initiated by a single resultant force, Vy and psi will increase and the geometry of the car should link both of them together in a relative fashion.

If the resultant force happens at the CG, then Vy increases and psi is not affected. If it happens anywhere else, then there is a moment about the CG created and psi increases in addition to the Vy increase.

 

[VicenteMMOS]

Vx, Vy and psi are independent by definition. They represent the 3 degree of freedom in a 2-dimension space.

Obviously, when in a straight line, Vy and psi are equal to zero, leaving only a value for Vx. Once turning will be initiated by a single resultant force, Vy and psi will increase and the geometry of the car should link both of them together in a relative fashion.

If the resultant force happens at the CG, then Vy increases and psi is not affected. If it happens anywhere else, then there is a moment about the CG created and psi increases in addition to the Vy increase.

Wow! I just felt like my mind blew off! xD

 

[VicenteMMOS]

Ok, I understand that the longitudinal, lateral and angular velocities are actually independent from one another. But now, for practical reasons, I want to ellaborate the equations of a uniform circular motion. I want to find out the equations that correlate these 3 velocities of a material point along a circle of radiu R, supposing that the material point has a local fixed ref. frame on itself. How can I describe the equations of this motion? I can suppose a given angular velocity around the center of the circle, omega, and thus Vx would be omega*R, but how about Vy and psi (the rate of change of the orientation of the point's local ref. frame, as it goes around the circle)? Will psi be equal to omega? I'm not sure of it. How can I determine these equations?

 

[jack action]

The local velocities will depend on the acceleration, which in turn depend on the forces and moments acting on your object.

And omega*R is related to the tangential velocity relative to the path. This velocity may be the same as Vx in your local reference frame but it may also be a combination of Vx and Vy. Heck, it may even be Vy alone (a car skidding sideways in a curve, for example).

How can I determine these equations?

Do a free body diagram.