# Just a trailer: Trailer Tow Vehicle Dynamics Modeling

## Vehicle dynamics model of articulated trailer vehicles and their yaw stability states

What is an articulated vehicle ? It normally looks like the figure below. It has a pivot joint that allows it to turn sharply. Articulation is particularly helpful in case of longer vehicles as it helps with the turn but it also poses its own challenges. Handling Behavior of an articulated vehicle is more complex and unpredictable than a non articulated vehicle. That poses greater risk for the drivers who are not aware of the dynamic complexities. Moreover, you can add as many articulations as you want but in this article, I am only going to discuss modeling and stability of single articulation vehicle. Single articulation vehicles are the most common ones with examples being the tractor-semitrailer and car-caravan.

## Lateral Dynamics of Articulated vehicles

Only the yaw handling behavior of an articulated vehicle is discussed neglecting the roll. Roll would add extra complexity to an already complex model. In one of my previous blogs, I discussed the bicycle model for a car which is simple yet powerful tool for analysing the lateral dynamics of vehicles. From yaw modeling perspective, articulated vehicles can be seen as the Bicycle towing a unicycle. The articulation joint is called hitch joint and the angle which the trailer makes with the vehicle or trailer is called articulation angle.

The tractor is shown in black and trailer in red. The lateral velocity is** **shown as** v**. The yaw rate (rate of change of angular velocity around Z axis) for tractor is **r** and for trailer is **rs. δ f**

*is the front steering angle and*

*ψa**is the articulation angle.*

Distance from the center of the mass to the front and rear axle respectively is **a, b** for tractor and **c, d** for trailer. Also, **e** is the offset of hitch joint from the rear axle. Please note that depending on the type of vehicle **e **can be positive or negative.

Also length of the tractor : **L** = a+b, length of the trailer:** Ls**= c+d

Taking the velocities at the hitch point, trailer_cg and trailer wheel

Also lets go back to the bicycle model. In steady state handling, the simple geometry would yield the front and rear slip angles as

where αf and αr are the front and rear sideslip angles. Please note that for above we have assumed the vehicle to be front steered only and the assumption of small angle holds such that tan(angle) = angle

The above equations can be re written as

Where** R** is the radius of the turn

A similar equation for the trailer can be written for the trailer slip angle αs which can be further simplified for small angle assumption to arrive at the equation for articulation angle.

The thing to note here is *ψa *is in opposite direction to the vehicle due to the negative sign. Also the first term in the equation above is modified ackermann and the second term is dynamic affected by axle stiffness.

Next what were are going to do is to draw the FBD and try to balance the lateral and vertical forces for steady state turning.

**In Y direction**

The FBD of the system can be drawn as shown below for the forces in Y direction. Here **m,ms** are the masses of tractor and trailer respectively. **Fyf, Fyr** are the tractor forces at front and rear axle, **Fys** is the trailer force and **Fyh** is the force at the hitch joint.

For Tractor

For Trailer

**In Z direction**

Repeating the same exercise in Z direction, **Fzf, Fzr** are the tractor forces at front and rear axle, **Fzs** is the trailer force and **Fzh** is the force at the hitch joint.

For Tractor

For Trailer

Solving all the steady state equations in Y and Z direction gives:

**Understeer Gradient**

Using the equations from above, the steer angle for tractor can be written as:

**Kust** is the understeer coefficient of tractor. Notice how the lateral forces are replaced by vertical forces in the cornering stiffness term( as they have the similar ratio as derived from steady state Y, Z equations). Also note how it is almost similar to a vehicle without trailer. However, the difference comes in the loading as the presence of trailer would mean different axle loads than a non trailer vehicle. A positive Kust is understeer and negative Kust is oversteer.

We can also write the articulation angle equation as:

Here** Kuss **is the trailer understeer coefficient. It could be positive or negative.

## Instabilities

The task of controlling the vehicle becomes the most difficult when the system becomes unstable. Articulated vehicles may experience two types of instability in the yaw plane:

**Jackknifing**

The first one is a divergent instability such as jackknifing, in which the articulation angle increases without experiencing oscillations. This occurs when the understeer gradient of a vehicle becomes negative and the speed is above a critical velocity. It may be caused by mechanical failure or improper braking or slippery road conditions.

**Trailer Sway**

The second type is dynamic in nature and may lead to oscillatory response of trailer with increasing amplitude known as sway. This happens when an outer force like wind or road disturbance acts in a perpendicular direction to vehicle motion.

## Articulation angle gain

To better understand these instabilities, it is crucial to understand the maths behind it. Let’s define a metric — articulation angle gain which is change in articulation angle per unit change in steer angle. Using the equations above, it can be mathematically represented as:

Now there can be five different vehicle behaviour scenarios :

**Both the tractor and trailer are understeer**

Kust >0 and Kuss > 0

The articulation angle gain is finite and positive for all values of forward speed. The response starts at Ls/L and settles to Kust/Kuss. The tractor-trailer is directionally stable.

Please note that the offset **e **has been neglected in the figures above

2. **The tractor is understeer, while the trailer is oversteer**

Kust >0 , whereas Kust < 0

The articulation angle gain remains finite for all values of forward speed. However, when the forward speed U is greater than Uct given below, the articulation angle gain changes the sign:

This indicates that when the forward speed U approaches Uct the articulation angle approaches zero, and when U> Uct, the orientation of the trailer with respect to the tractor will be opposite.

3. **The tractor is oversteer, while the trailer is understeer**

Kust < 0 and Kuss >0.

It can be seen that when the forward speed approaches the critical speed Ucrit given below, the denominator in the gain equation approaches zero and the articulation angle gain approaches infinity. This instability is jackknifing.

Note the difference between Uct and Ucrit. Ucrit depends on the tractor properties only while Uct depends on trailer properties.

When the forward speed U approaches Ucrit,, the tractor longitudinal axis becomes increasingly oriented towards the center of the turn, resulting in jackknifing. This instability is just the result of tractor oversteer.

4. **Both the tractor and trailer are oversteer**, **and the ratio of the understeer coefficient of the trailer to that of the tractor is less than the ratio of the trailer wheelbase to the tractor wheelbase.**

Kust < 0, Kuss< 0, and (Kuss/Kust) < (Ls/L)

The variation of the articulation angle gain with forward speed is shown in the figure. Similar to case 3, when the forward speed approaches the critical value, jackknifing will occur.

5. **Both the tractor and trailer are oversteer**, **and the ratio of the understeer coefficient of the trailer to that of the tractor is greater than the ratio of the trailer wheelbase to the tractor wheelbase**

Kust < 0, Kuss< 0, and (Kuss/Kust) > (Ls/L)

The variation of the articulation angle gain with forward speed is shown in the figure. It can be seen that the articulation angle gain decreases with increasing forward speed.

When the forward speed U approaches Uct, the gain approaches zero. With a further increase in the forward speed, the gain becomes negative and approaches minus infinity as the forward speed approaches Ucrit . In this case, the trailer longitudinal axis becomes increasingly oriented towards the center of the turn, resulting in trailer sway.

In short, for any form of directional instability (jackknifing or trailer swing) to occur, the tractor must be oversteer. Jackknifing can occur when the trailer is either understeer or oversteer. However, for trailer swing to occur, in addition to the condition that the trailer must be oversteer, it is required that the ratio of the understeer coefficient of the semitrailer to that of the tractor be greater than the ratio of the semitrailer wheelbase to the tractor wheelbase.

The individual component properties can be deduced from these equations afterwards. For example, Jackknifing only occurs when tractor is oversteer. What affects that ? Tractor properties and hitch load. What affects hitch load ? The trailer mass and and location of center of mass of trailer. All of this information can be used to control the system to keep it stable.

## References

“Vehicle Dynamics Course Notes “ — University of Michigan

“Stability and Control Considerations of Vehicle-Trailer Combination” — Aleksander Hac, Daniel Fulk and Hsien Chen

“Theory of Ground Vehicles” — J. Y. Wong

*Written while listening to **The Lumineers*