It is now well established that discrete energy conservation/dissipation plays a key-role for the unconditional stability of time integration schemes in nonlinear elastody-namics. In this paper, from a rigorous conservation analysis of the Hilber-Hughes-Taylor time integration scheme [1], we propose an original way of introducing a controllable energy dissipation while conserving momenta in conservative strategies like [2–5]. Moreover, we extend the technique proposed in [3] to provide energy-controlling time integration schemes for frictionless contact problems enforcing the standard Kuhn-Tucker conditions at time discretization points. We also extend this technique to viscoelastic models. Numerical tests involving the impact of incompressible elastic or viscoelastic bodies in large deformation are proposed to confirm the theoretical analysis.