Abstract. The theoretical analyses and practical experience show that only the damping effect adaptable to the current operating conditions makes it possible to achieve optimum performance of damping devices inserted in the supports of rotating machines. This is offered by magnetorheological squeeze film dampers. The magnetorheological oils are liquids sensitive to magnetic induction. Their response to the change of a magnetic field is not instantaneous, but it is a process called the delayed yielding. The research was focused on enhancement of the mathematical model of the magnetorheological squeeze film damper by considering the delayed yielding phenomenon and on its application for the study of the influence of the delayed yielding on the force transmission between the rotor and its stationary part. The results of the computational simulations show that rising value of the delayed yielding time constant that characterizes the delayed yielding process reduces the damping effect and efficiency of the magnetorheological damping devices.