Some nontrivial effects are investigated, which can occur if strongly damped mechanical systems are subjected to strong high-frequency (HF) excitation. The main result is a theoretical prediction, supported by numerical simulation, that for such systems the (quasi-)equilibrium states can change substantially with the level of damping. For example, a strongly damped pendulum, with a hinge vibrated at high frequency along an elliptical path with horizontal or vertical axis, will line up along a line offset from the vertical; the offset vanishes for very light or very strong damping, attaining a maximum that can be substantial (depending on the strength of the HF excitation) for finite values of the damping. The analysis is focused on the differences between the classic results for weakly damped systems, and new effects for which the strong damping terms are responsible. The analysis is based on a slightly modified averaging technique, and includes an elementary example of an elliptically excited pendulum for illustration, alongside with a generalization to a broader class of strongly damped dynamical systems with HF excitation. As an application example, the nontrivial behavior of a classical optimally controlled nonlinear system is investigated, illustrating how HF excitation may cause the controller to leave the system in an unexpected equilibrium state, quite different from the setpoint. The effects can be interesting for specialists in control of mechanical systems and structures. However the obtained results are more general. Similar effects could be expected first of all for microsystems where damping forces are typically dominating over inertia forces.