A train of axle loads crossing a simply supported span: modal response, damping and the resonance speeds set by the axle spacing.
The science
The span is an Euler-Bernoulli beam, simply supported, with bending stiffness EI and mass per length mu. Its mode shapes are sin(n pi x / L) with frequencies w_n = (n pi / L)^2 sqrt(EI / mu). Each axle is a moving point force P; projected on mode n the equation is q_n'' + 2 zeta w_n q_n' + w_n^2 q_n = 2P / (mu L) sin(n pi x_p / L), solved with RK4 for five modes. Midspan deflection is sum q_n sin(n pi / 2).
A single slow axle gives the static deflection P L^3 / 48 EI. A train is a periodic load: axles spaced d apart excite the span at v / d, so resonance occurs near v = f1 d (and f1 d / 2, f1 d / 3 ...) where f1 is the first bending frequency. The dynamic factor in the sweep is largest there and damping lowers it. The speed that would resonate with one moving load alone, (pi / L) sqrt(EI / mu), is usually far above real train speeds.
Simplifications: moving forces without vehicle-bridge interaction (no suspension, no track irregularity), a uniform span, ten identical axles, only five modes in the time history. Checks: the seven-mode static sum and the RK4 period of the first mode.