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The influence of tire transient property on vehicle behavior
Published in Johannes Edelmann, Manfred Plöchl, Peter E. Pfeffer, Advanced Vehicle Control AVEC’16, 2017
The concepts of this model are, 1) to calculate the lateral deflection at the leading edge of tire string model by solving the differential equation 1(Pacejka 2012, 336–338), 2) to define the equivalent slip angle by that deflection and the intersection length (Equation 2), 3) to introduce the intersection length dependency on the equivalent slip angle and the relaxation length at slip angle 0 (Equation 7). Finally, the side force and the aligning moment due to lateral deflection are calculated by using the Magic Formula (Equation 3, 4). The gyroscopic couple due to the tire lateral deformation (Equation 5) is added to the total aligning moment (Equation 6). In the numerical calculation of Equation 1 and 5, the state variable x (distance travelled) will be transformed by using the relation, dx = Vdt.
Prediction of yaw natural frequency taking roll motion into account
Published in Maksym Spiryagin, Timothy Gordon, Colin Cole, Tim McSweeney, The Dynamics of Vehicles on Roads and Tracks, 2018
Let the relaxation length of front tire be σf. σf is defined by the following equation (Pajejka 2012). () σf≡Kfkf=lrl⋅h2mCfKx
Modelling transient response using PAC 2002-based tyre model
Published in Vehicle System Dynamics, 2022
Aashish Shaju, Ashok Kumar Pandey
As mentioned earlier, the Magic formula-based PAC 2002 model can only predict steady state tyre forces and moments for a given slip quantity, and it does not take into account the time dependency of the forces that results due to the tyre compliance. Modelling the actual buildup of true forces in the simulation was done by lagging the slip quantities. Figure 1(a) shows a schematic of this situation showing the lagged and instant slip angles and α, respectively. The rationale for this idea is that the delay in the development of forces after wheel is steered or accelerated/decelerated is caused by lagging within the tyre carcass, i.e.between the carcass and the wheel. Since, tyres are made of rubber, they can be modelled as a combination of spring and damper system both in lateral and longitudinal directions as shown in Figure 2. Both these quantities captures the tyre deformation and dissipation caused by lateral and longitudinal forces originating from the tyre contact patch. To incorporate this simple model for vehicle dynamics, the concept of relaxation length is introduced, which is defined as the distance a tyre needs to travel before it finishes deformation to accommodate the new forces applied by the motion of the vehicle. It can be implemented into the vehicle model as a delay in the slip quantities given by Equations (7) and (9). Figure 1(b) shows the block diagram of the modified PAC 2002 model with the extension to include transient effects.
On the influence of tyre and structural properties on the stability of bicycles
Published in Vehicle System Dynamics, 2018
Alberto Doria, Sergio Daniel Roa Melo
The overturning moment (about the longitudinal axis parallel to the ground) is proportional to the camber angle as shown in Equation (3), where and are the front and rear overturning moments and is the overturning moment stiffness. Some authors have shown that the transient tyre behaviour plays an important role in stability [2,3]. For including the transient phenomena and the delay in the development of tyre forces, transient side-slip angle and transient camber angle are included in the model. The transient side-slip angle is described by Equation (4) in which and are the transient side-slip angles of front and rear tyres, respectively, and is the relaxation length due to side-slip (the same for both tyres). The transient camber angle is described by Equation (5) in which and are the transient camber angles of front and rear tyres, respectively, and is the relaxation length due to camber (the same for both tyres), which in most cases is a small fraction of the relaxation length due to side-slip [3]. In the following, is set to .
Hopf bifurcation and energy transfer of automobile shimmy system with consideration of road roughness excitation
Published in Vehicle System Dynamics, 2022
Heng Wei, Jianwei Lu, Hangyu Lu, Lei Shi, Bofu Wu, Ping Jiang
The constraint between the slip angle and the shimmy angle [12] can be given by where is the relaxation length of the tire; a is the length of the contact patch.