Stability and Nonlinear Density Waves in a Mass–Spring–Dashpot Damped-Oscillator Traffic Flow Chain
- 1School of Transportation and Logistics, Shandong Jiaotong University, Jinan 250023, China
- 2School of Rail Transit, Shandong Jiaotong University, Jinan 250023, China
Abstract
We formulate a chain of damped harmonic oscillators as a mechanical analogy for a single-lane vehicle platoon and show that it reproduces, and yields analytical insight into, the congestion patterns of car-following traffic. Each follower is treated as a point mass m coupled to its leader by a spring of stiffness k2 (spacing-error feedback) and a dashpot of coefficient c (relative-speed feedback), with an additional optimal-velocity (OV) force k1[V(Δx)−v] acting as a non-conservative driving input. The resulting second-order equation is a damped-oscillator chain with headway-tunable effective stiffness K(h)=k1V′(h)−k2, for which the local damping ratio partitions the parameter plane into an active (negatively damped) regime, an overdamped regime, and an underdamped window that supports ringing oscillations. Long-wavelength string-stability analysis yields a closed-form stable band for the OV slope, whose boundaries trace a tongue in the parameter plane. A weakly nonlinear multiple-scale analysis derives amplitude-dependent frequency shifts, centre drift, and nonlinear envelope decay, and recovers a double-well potential that saturates density-wave growth at a finite amplitude. Ring road simulations validate the predictions on two levels: a systematic numerical sweep shows that the zero-growth contour coincides with the analytic band along the whole tongue, and unbiased cycle-resolved measurements follow the analytic amplitude–frequency curve. Although developed for traffic, the construction is generic for dissipative coupled-oscillator chains with non-monotone driving laws.
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BibTeX
@article{zhang2026stability,
title = {{Stability and Nonlinear Density Waves in a Mass–Spring–Dashpot Damped-Oscillator Traffic Flow Chain}},
author = {Lidong Zhang and Lichao Duan and Liping Feng and Qinglian Li},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193600},
url = {https://www.mdpi.com/2227-7390/14/19/3600},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Stability and Nonlinear Density Waves in a Mass–Spring–Dashpot Damped-Oscillator Traffic Flow Chain
AU - Lidong Zhang
AU - Lichao Duan
AU - Liping Feng
AU - Qinglian Li
PY - 2026
DA - 2026/10/03
JO - Mathematics
DO - 10.3390/math14193600
UR - https://www.mdpi.com/2227-7390/14/19/3600
AB - We formulate a chain of damped harmonic oscillators as a mechanical analogy for a single-lane vehicle platoon and show that it reproduces, and yields analytical insight into, the congestion patterns of car-following traffic. Each follower is treated as a point mass m coupled to its leader by a spring of stiffness k2 (spacing-error feedback) and a dashpot of coefficient c (relative-speed feedback), with an additional optimal-velocity (OV) force k1[V(Δx)−v] acting as a non-conservative driving input. The resulting second-order equation is a damped-oscillator chain with headway-tunable effective stiffness K(h)=k1V′(h)−k2, for which the local damping ratio partitions the parameter plane into an active (negatively damped) regime, an overdamped regime, and an underdamped window that supports ringing oscillations. Long-wavelength string-stability analysis yields a closed-form stable band for the OV slope, whose boundaries trace a tongue in the parameter plane. A weakly nonlinear multiple-scale analysis derives amplitude-dependent frequency shifts, centre drift, and nonlinear envelope decay, and recovers a double-well potential that saturates density-wave growth at a finite amplitude. Ring road simulations validate the predictions on two levels: a systematic numerical sweep shows that the zero-growth contour coincides with the analytic band along the whole tongue, and unbiased cycle-resolved measurements follow the analytic amplitude–frequency curve. Although developed for traffic, the construction is generic for dissipative coupled-oscillator chains with non-monotone driving laws.
SN - 2227-7390
ER -