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Stability and Nonlinear Density Waves in a Mass–Spring–Dashpot Damped-Oscillator Traffic Flow Chain

Year: 2026
Type: article
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.
Source: Mathematics
DOI: 10.3390/math14193600
Language: English
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7220456243
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Cites: 34
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Related to: 10