
Journal of Water Resources & Environmental Engineering - No. 87 (12/2023)
25
Tire-road separation: Problem and solving method
Nguyen Dang Quy
1
, Vu Ngoc Tuan
1*
, Nguyen Truong Sinh
1
, Vu Van Tan
2
Abstract: The paper discusses the movement of a quarter-
car concerning whether the tires are in contact
with the road or not. A set of equations is formulated to highlight the two different states, and a new
approach is p
resented to solve these equations while also considering the separation of the tires. A
comparison between the new and old models shows that previous studies lacking this consideration
must be reevaluated. A simulation is conducted to demonstrate how the c
haracteristics of the road and
suspension affect vertical displacement. Ultimately, this research provides more accurate outcomes that
can be used to inform future suspension design.
Keywords: Quarter-car vibrations, vehicle vibrations, vibration dynamics, tire-road separation, v
ehicle
safety, ride comfort.
1. Introduction
*
Previous research on vehicle vibrations has
traditionally assumed that the vehicle remains in
contact with the road (Jazar, 2017). However, in
real-world scenarios, cars can experience instances
where they lose contact with the road due to
various factors (Jazar, 2023). Recent studies in
separation dynamics have demonstrated that
accounting for the possibility of separation between
the wheel and the ground leads to a more realistic
representation of the system (Nguyen, 2022a),
(Nguyen, 2021), (Nguyen, 2021a), (Nguyen,
2021b), (Nguyen, 2022b), (Khazaie, 2018).
To observe the normal vibration dynamics,
researchers have employed vibrating half-car
models or more comprehensive models such as
the full-car model (Gillespie, 1992). However,
solving the governing equations for these
models, particularly in their discontinuous states
due to wheel and suspension constraints, can be
more complex (Nguyen, 2021b). A quarter-car
model, which focuses on studying the vertical
1
Le Quy Don Technical University, Hanoi, Vietnam
2
University of Transport and Communications, Hanoi,
Vietnam
*
Corresponding author; Email: tuan.vungoc@lqdtu.edu.vn
Received 20
th
Sep. 2023
Accepted 23
rd
Oct. 2023
Available online 31
st
Dec. 2023
responses and investigates a single corner of the
vehicle with body bounce, is widely used and
popular (Jazar, 2017). Moreover, the quarter-car
model is also suitable for vibration optimization
purposes (Wong, 2008).
Therefore, this paper aims to investigate how
the progress of separation affects the vertical
dynamics of a 2-degree-of-freedom (2DOF)
quarter-car model and validates the ride comfort
of the system under specific conditions. By
focusing on the tire-road separation with 2DOF,
this study provides a solid foundation for
understanding the basics of vibration dynamics
and serves as a valuable resource for gaining
insights into ride comfort.
2. Vehicle modeling
This section, the focuses on modeling the
progress of separation in a quarter-car system
and the numerical method employed to solve the
dynamic equation system. The separation
phenomenon is incorporated into the model to
capture its effects on the system's behavior.
2.1. Dynamic Equations of Motion
In order to obtain the dynamic equations of
motion, we analyze the free-body diagram of
the masses at a particular moment, as shown in
Figure 1. Note that the forces from the gravity
of the masses cancel with the static spring