VBI Visualizer: Vehicle-Bridge Interaction

Bridge

Span L [m] Flexural rigidity EI [GN·m²] Mass per length m̄ [kg/m] Modal damping ζ [%] Number of modes N

Vehicle

Sprung mass m_s [kg] Unsprung mass m_u [kg] Suspension stiffness k_s [kN/m] Suspension damping c_s [kN·s/m] Tire stiffness k_t [kN/m] Tire damping c_t [kN·s/m] Speed v [m/s]

Road roughness

ISO 8608 class Random seed Time step Δt [ms]

Live animation

Bridge deflection scaled for visibility. Vehicle position is shown in real time as the load crosses the span.

Mid-span deflection

Bridge dynamic deflection at x = L/2 (downward positive). The static moving-load deflection is overlaid for reference.

Vehicle vertical acceleration

Vertical acceleration of the sprung mass — the quantity actually measured by smartphone or onboard sensors in indirect bridge monitoring.

Frequency content

FFT magnitude of the vehicle acceleration. The first few bridge natural frequencies (computed analytically) are marked for comparison — the principle behind indirect modal identification.

Method

The bridge is modeled as a simply supported Euler-Bernoulli beam with mode shapes φ_n(x) = sin(nπx/L) and natural frequencies ω_n = (nπ/L)² √(EI/m̄). The beam response is expanded as w(x,t) = Σ q_n(t) φ_n(x).

The vehicle is a quarter-car: a sprung mass m_s coupled to an unsprung mass m_u by a parallel spring-damper (k_s, c_s), with the unsprung mass connected to the contact point through tire spring-damper (k_t, c_t). The contact-point elevation is w(p(t), t) + r(p(t)), with p(t) = vt and r a ISO 8608 random roughness profile generated from a discrete inverse-FFT of the displacement PSD G_d(n) = G_d(n₀) · (n/n₀)^-2.

The coupled equations include the convective term v · ∂w/∂x at the contact point. The full mass/damping/stiffness matrices are time-varying through φ(p(t)) and are integrated with the constant-average-acceleration Newmark-beta scheme (γ = 1/2, β = 1/4).

The FFT panel illustrates the indirect identification principle: peaks in the sprung-mass acceleration spectrum cluster around the bridge natural frequencies, which is the core observation underlying crowdsensing-based bridge monitoring.

Related publications

Talebi-Kalaleh, M.; Gül, M.; Mei, Q. (2025). Crowdsensing-based Bridge Vibration Monitoring Using a Sparse Network of Random Mobile Sensors. Journal of Sound and Vibration, 618B. DOI
Talebi-Kalaleh, M.; Mei, Q. (2026). An Open-Source Framework for Coupled Vehicle-Bridge Interaction Analysis Using OpenSees. arXiv:2605.09175. arXiv
Talebi-Kalaleh, M.; Mei, Q. (2023). A Mobile Sensing Framework for Bridge Modal Identification. Sensors, 23(11), 5154. DOI