We propose an orthogonal collocation method (CM) for solving Cosserat rod Dirichlet-Neumann boundary value problems in static and dynamic modes. We interpolate the internal loading and collocate the strong form of the differential equations. The method uses Chebyshev polynomials in order to minimize Runge’s phenomenon. The time derivatives are implicitly discretized using the backward differentiation formula BDF-α. We compare our method with the shooting method (SM), multiple shooting method (MSM) and two isogeometric CM against three static and one dynamic applications. The results show that our CM is more stable than SM and faster than MSM.