Mathematical Modeling of an articulated vehicle (trailer)

[Palguna Kumar]

Hello,

I have an problem with mathematical modelling of a special kind of articulated trailer (please have a look at the images: 1st image is the yaw plane and the second is the ZX plane). I have two methods of deriving equations of motion namely Lagrange and Newton-euler to analyse their paths or trajectories. Do you think the system has non holonomic constraints because the side slip of the first body cannot be defined since it has no wheels ? Which method is preferrable lagrange or Newton euler? How to model the system in minimal coordinates ( generalized coordinates) to avoid lagrange multipliers? I have tried a lot but system is entering singularity (maybe because I did not consider non holonomic constraints). Is there a better way where I can model the system ?

 

[Baluncore]

Welcome to PF.
I am having trouble reading your attached pictures and identifying the degree of freedom of the couplings between the two trailer parts and the tractor.

The front hitch could be the two arms of a 3 point linkage or a ball.
Can you please provide a sketch that shows the two trailer bodies and the tractor or tow vehicle.
Can you show a sketch of the couplings with links, pin or ball joints identified.
How is the mass of body 2 supported.
The linkage between body 1 and body 2 looks like a 4 bar trapezoidal linkage with 4 vertical pins.

If the front hitch is a single ball joint then it will be impossible to reverse the combination in a controlled way.

 

[Palguna Kumar]

Thank you for your reply. Sorry for the bad pictures. This is the diagram of my whole system.
The whole model is on XY Plane, which means I have built a lateral dynamic model of tractor trailers.
I am not modeling the system for reversing. It is only for forward motion, that my second trailer must follow the path of the tow vehicle.
All the links or hitches are simple cylindrical joints on 2D plane.
The 2nd trailer's trajectory is controlled through two hydraulic cylinders Fh1 and Fh2 attached to both Trailer 1 and the four bar linkage ABCD as shown in figure.
delta is the steering angle
Fy1 Fy2 Fy3 are lateral tire forces.
theta1 is the relative angle between the tower and the trailer 1
theta2 is the relative angle between the trailer1 and the trailer 2

How can I include the kinematic loop equation for ABCD into my dynamic equations ?
How can I model the hydraulic forces? as spring forces or as just stiffness to rotation ?

 

[Baluncore]

How can I model the hydraulic forces? as spring forces or as just stiffness to rotation ?

Hydraulics rigidly fix the geometry of the structure because hydraulic cylinders are operated without compressible fluid in the system.
In this case, hydraulic cylinder position will be a modelling parameter.
For the purpose of steering path analysis you can assume that the hydraulic cylinders are fixed length links.

 

[Wes Tausend]

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Palguna Kumar,

Baluncore seems to have already described, "The linkage between body 1 and body 2 looks like a 4 bar trapezoidal linkage with 4 vertical pins", which is my take on part of your mechanism.

I am familiar with a type of hitch that uses the 4-bar to stabilize bumper-pull trailers. I used one on my camper and it is extraordinarily stable and sway resistant in gusty wind conditions compared to all other weight-distributing hitches or fifth-wheel configurations. However your diagram does not seems to use it configured quite that common way. On the chance that you are attempting to understand such 4-bar principles, there is a vast thread about it on this camper forum: http://www.rv.net/forum/index.cfm/fuseaction/thread/tid/15531727.cfm

Wes
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