The dynamical mechanisms controlling the rheology of dense suspensions close to jamming are investigated numerically, using simplified models for the relevant dissipative forces. We show that the velocity fluctuations control the dissipation rate and therefore the effective viscosity of the suspension. These fluctuations are similar in quasi-static simulations and for finite strain rate calculations with various damping schemes. We conclude that the statistical properties of grain trajectories-in particular the critical exponent of velocity fluctuations with respect to volume fraction φ-only weakly depend on the dissipation mechanism. Rather they are determined by steric effects, which are the main driving forces in the quasistatic simulations. The critical exponent of the suspension viscosity with respect to φ can then be deduced, and is consistent with experimental data.