Calculation of the lateral force of a trailer coupling

[Semat10]

TL;DR Summary

I have a vehicle towing a trailer. I calculated the longitudinal forces on the hitch of the trailer. Now i fail to calculate the lateral force on the hitch.

Hello,

i´ve got a problem with the calculation of the forces on the hitch of a vehicle and a trailer.
I do need the lateral force on the hitch for the calculation of my single track model of the vehicle and trailer.

I have a longitudinal model of the vehicle and the trailer. My input signal is the velocity of my vehicle. I transform the velocity of the vehicle, with a coordinate transformation into the velocity of the trailer. I got the longitudinal force F_zeta (see image) with the equation of motion.

(image of the cutting forces on the hitch of the trailer and the hitch of the vehicle; because of cornering right, the trailer turns away from the longitudinal axes of the vehicle with the angle gamma)

I need a rotation matrix to transform the longitudinal force on the hitch of the trailer F_zeta to the longitudinal force of the vehicle F_x.

F_x= cos(gamma)*F_zeta + sin(gamma)*F_eta
F_y=-sin(gamma)*F_zeta + cos(gamma)*F_eta

To do this i need the lateral force on the trailer hitch F_eta, but i don´t know how to calculate it.

kind regards,
semat

 

[jack action]

To do this i need the lateral force on the trailer hitch F_eta, but i don´t know how to calculate it.

What does the free body diagram of your trailer look like?

 

[Semat10]

It looks like this for my trailer. But i need the lateral force "F_eta_hit" as my input signal for my single track model to calculate my jaw rate and slip angle.

 

[jack action]

I'm not sure I understand the problem. In the FBD you presented, you have $F_{\eta , hit}$, so what do you mean by "but i don´t know how to calculate it"?

 

[Semat10]

The force F_eta_hit is the same force as the F_eta_hit in the first picture with the cutting forces. To calculate my single track model, I need to know the value of F_eta_hit (F_eta_hit is unknown). I have everything for the single track model of my vehicle and trailer except F_eta_hit and F_y_hit. Now I don´t know where I can get F_eta_hit from or how to calculate it.

 

[jack action]

But you have $F_{\eta , tra}$ and $\psi_{tra}$ in your FBD, you must be able to figure $F_{\eta , hit}$ with those.

Taking a second look at your FBD I noticed that you have $F_{\eta , tra}$ at both the wheel and the center of gravity. That is a wrong assumption. They are not opposite and equal. The hitch will take loads just like a front wheel would do.

 

[Semat10]

Yes the forces in the center of gravity and on the wheels are wrong. The force on the wheel is the lateral wheel force and in the center of gravity the d´alembert inertial force.

But I need the hitch force F_eta_hit as a input signal, because its the only force which forces the trailer on a circular path. I thought i can get the F_eta_hit from the first picture about angular relationships or something.

 

[.Scott]

The lateral force will be the sum of these three values:
1) As you change the angular momentum of the trailer - with either a change in speed or rate of turn, that change occurs through the hitch.
2) If the trailers center of gravity is not positioned directly over the axis of the wheels in the fore/aft direction, then during a turn the centrifugal force applied at the center of gravity will cause a pivot that can only be corrected through a lateral force from the hitch.
3) If the trailer's center of gravity is not aligned left/right with the hitch, then all accelerations and decelerations will result in trailer pivot that can only be corrected through the hitch.

Basically, you need to separate the motion of the trailer into two components:
1) The forward/reverse acceleration.
2) The rotational control of the trailer's center of gravity about the midpoint of its wheel axis.

 

[Semat10]

I did the seperation. The results are two differential equations for forward/reverse acceleration and rotational control of the trailer's center of gravity. After rearrange of the equation I have a state-space for the yaw angle acceleration and the slip angle speed. Now I have 2 differential equations with 3 unknown quantities (yaw angle acceleration, slip angle speed and the lateral hitch force F_eta_hit) the rest is known.
Thats the point where I can´t solve my equations cause I don´t have the lateral hitch force F_eta_hit.

 

[.Scott]

Your introductory statement talks about the forces on the hitch. But you have mentioned "slip angle". Are you looking for the lateral force on the hitch or the lateral forces on the tires - or both?

I would think that you would want to express the longitudinal and lateral hitch forces in terms of velocity, acceleration, the path, distance from the hitch to the midpoint of the tire axis, location of the trailer center of gravity relative to the midpoint of the tire axis, and the mass of the trailer. To avoid dealing with the cumulative effects of the path, you should probably make the hitch forces a function of the lateral offset of the trailer.

Which brings up another issue. If you are looking for the longitudinal and lateral forces on the hitch, you need to specify whether you are talking about the hitch socket (truck coordinate system) or post (trailer coordinate system).

@Semat10 : Also...
You say you want to calculate the "Jaw rate". What is that?
Also, you want the "slip angle" - I am guessing this is the side-slip angle of trailer - thus the difference between where the trailer is pointing versus the direction it is going. Is this what you meant?

 

[.Scott]

And ... I almost forgot the trailer's "moment of inertia" (aka "rotational inertia").

Is this your basic scenario? :

I'm also assuming that the truck gets to apply whatever force is required to pull and turn the trailer - and to control all reactions the trailer might present.

If so, notice that:
When the trailer's coordinate system is used, all longitudinal motion/acceleration on the hitch move the trailer forward or backwards. All lateral motion/acceleration on the hitch change the angular momentum. If the center of gravity is not along the X-axis, longitudinal acceleration will result in an additional lateral (reaction) force from the trailer. On the other hand, if the center of gravity is not on the Y-axis, any change in angular velocity will result in an additional longitudinal (reaction) force from the trailer.

So I'm thinking that what you are looking for is something that will take all the starting conditions and the hitch location as a function of t (t>=0) and turn those into the resulting hitch forces as a function of t.

 

[Semat10]

I simplified the model to a single track model. That means the center of gravity is in the X-axis. I´am looking for a formula which calculates me in every simulation step the lateral force on the hitch point. With that force I will calcualte my quantities in the trailer (slip angle, yaw rate, and so on) and with the right lateral force on the hitch I get the right hitch angle results. Because in the single track model the only quantities (input signal in the trailer) which forces the trailer on a circular path is the lateral hitch force.

 

[jack action]

I´am looking for a formula which calculates me in every simulation step the lateral force on the hitch point.

I will calcualte my quantities in the trailer (slip angle, yaw rate, and so on)

I get the right hitch angle results.

You should obtain all of those values by decoupling the trailer from the vehicle and do a free body diagram on each. This should give you a certain number of equations with an equal amount of unknowns. The hitch (pin joint) will have its own equations, imposing its own conditions to link the two. You then have to solve that set of equations.

Or am I misunderstanding your problem?

 

[Semat10]

I did the free body diagramm of the trailer and the vehicle. For both I got 2 equations (one for the lateral forces and one for the moment around the center of gravity).
That results in 2 equations and 3 unknown (unkown: slip angle, yaw rate, lateral hitch force --> in the vehicle system) for the vehicle and 2 equations and 3 unknown (unknown: slip angle, yaw rate, lateral hitch force --> in the trailer system). But the lateral forces on the hitch like you see in the picture above are a cutting force. That means the lateral hitch force of the vehicle and trailer must be the same value. Because of the cornering the trailer is twisted against the vehicle. So I did a coordinate transformation. The X-axis of the trailer vertical and Y-axis horizontal and the zeta and eta twisted with the articulation angle gamme.

The result to your comment:
I want the slip angle and the yaw rate for the vehicle and trailer (4 quantities/6 unknown), but I have one unknown quantitie too much and this is the lateral hitch force. So I search a way to calculate the lateral hitch force for my trailer and vehicle.

Sorry if it´s really incomprehensible and confusing.

 

[hutchphd]

I did the free body diagramm of the trailer and the vehicle. For both I got 2 equations (one for the lateral forces and one for the moment around the center of gravity).
That results in 2 equations and 3 unknown (unkown: slip angle, yaw rate, lateral hitch force --> in the vehicle system) for the vehicle and 2 equations and 3 unknown (unknown: slip angle, yaw rate, lateral hitch force --> in the trailer system)

Apologies this may be a stupid question. In 2 dimensions there should be one rotational and two translational equations for each free body (its a vector equation). Why do you only have one?

 

[jack action]

Show us your FBD and your equations, it will be a lot easier for us to see what is missing.

 

[Semat10]

Because I separate my vehicle in the longitudinal part and the lateral part in my simulation. I already have the longitudinal calculation with all forces. Now I calculate the lateral part and in the literature for the linear single track model you have the 2 equations... the forces in lateral direction and the moments around the COG.

 

[hutchphd]

You will need to show mathematically what you have done. Your vehicle has a hinge in the middle and it is not clear what you have done.

 

[jack action]

Because I separate my vehicle in the longitudinal part and the lateral part in my simulation.

How can you separate the two? One necessarily influences the other. The sum of the moments must necessarily include forces in both directions.

 

[hutchphd]

I do not see what you are attempting here. Your #19 lists 4 equations and 4 unknowns (??)
Perhaps you need to start again with a clean statement of the problem. List the givens (masses, velocities, External Forces, physical constraints) and what it is you wish to know. You need to flesh out the opening statement with the exact conditions (turning acceleratiing speed etcc)

 

[Semat10]

Okay.
Unknowns: psi_vehicle, psi_trailer, beta_trailer, beta_vehicle, F_eta_hit, F_y_hit

My model has the given equations of a single track model. It‘s right that the longitudinal forces should be in the equation of the moments, but the linear single track model is a simplified model to discribe the lateral dynamic of the vehicle. One assumption is that the longitudinal forces can be ignored.
The problem is I need to calculate my psi_vehicle, psi_trailer, beta_trailer and beta_vehicle (the important physical quantities of the lateral dynamic). That is the aim. But I have the lateral hitch forces which are also unknown. To get to my aim which is the calculation of the lateral dynamic. I need the lateral hitch force as a input signal. So I search a way how I can calculate my lateral hitch forces. Then when I have my lateral hitch forces I can put them in my 4 equations and simulate my model.

 

[jack action]

OK, I understand what you mean, because all the longitidunal forces are going through the CG.

But in your case, you have an angle between your trailer and your vehicle, thus the longitidunal forces on the trailer will have an effect on the lateral components of the vehicle, and vice versa.

I don' think you can treat them separately. The forces acting on the trailer will necessarily influence the vehicle (and vice versa).