Exact analytical treatment of the rocking-factor integral in the two-wheel vibration model

Develop an exact closed-form evaluation of the wavenumber-integrated power dissipation for the two-wheel bicycle model with rocking motion under the IRI spectrum, by computing the integral obtained when the rocking factor R_ω = (1 − ξ_M)^2 + 2(1 − ξ_M)ξ_M cos(ωL_W/v) + ξ_M^2 is inserted into the response integral; derive an explicit speed-dependent analytic expression for the correction to roughness resistance without resorting to numerical quadrature or averaging.

Background

Extending the one-mass vertical model to a two-wheel configuration introduces a rocking motion, which modifies the forcing via a speed-dependent phase term and the rocking factor R_ω. The resulting integral for power dissipation becomes analytically challenging.

The author reports no exact treatment for this integral and uses numerical quadrature and an averaging approximation for practical purposes. An exact solution would clarify the non-monotonic speed dependence and improve predictive accuracy for roughness resistance in realistic two-wheel scenarios.