We present a new formulation to calculate the response of a system containing rolling-element bearings operating under a radial clearance and a dominant radial load. The nonlinear bearing force- and stiffness-displacement characteristics in combination with the bearing clearance necessitate an advanced numerical analysis. The response of a shaft-bearing-housing assembly can be unstable in the transient regions, e.g., at the start of a system run-up or when passing the critical speed of a system. This can lead to long computational times or even to non-converged solutions. In this paper, a new analytical bearing-stiffness model is presented that is capable of overcoming these problems by smoothing the nonlinear bearing force- and stiffness-displacement characteristics in the discontinuous regions. The smoothing is implemented on the deformation scale. The proposed model is modular, allowing us to define a specific value of the smoothing to each rolling element that comes into contact. A simple case study that involves two bearings of different types (ball and cylindrical roller) is presented. They support an unbalanced rotor, subjected to a constant angular acceleration. We show that a small smoothing value can significantly enhance the numerical calculation of the chosen system in terms of speed and stability.