An invariance principle for a class of ordinary differential equations with discontinuous right-hand side is developed. Based on this principle, asymptotic stability of one-degree-of-freedom mechanical oscillators with Coulomb friction is studied. The system is shown to be asymptotically stabilizable via a static feedback of the position, unlike those systems with no friction, whose stabilization requires a dynamic feedback when the position is the only available measurement. Along with this development, a velocity observer is proposed. Theoretical results of the paper are supported by some numerical simulations which, in addition, carry out a finite-time convergence of the controller and the observer proposed. [S0022-0434(00)00804-2]

This paper introduces a modern version of the classical Huygens’ experiment on synchronization of pendulum clocks. The version presented here consists of two monumental pendulum clocks—ad hoc designed and fabricated—which are coupled through a wooden structure. It is demonstrated that the coupled clocks exhibit ‘sympathetic’ motion, i.e. the pendula of the clocks oscillate in consonance and in the same direction. Interestingly, when the clocks are synchronized, the common oscillation frequency decreases, i.e. the clocks become slow and inaccurate. In order to rigorously explain these findings, a mathematical model for the coupled clocks is obtained by using well-established physical and mechanical laws and likewise, a theoretical analysis is conducted. Ultimately, the sympathy of two monumental pendulum clocks, interacting via a flexible coupling structure, is experimentally, numerically and analytically demonstrated.

In this paper, hybrid control synthesis is proposed for a class of 2-degrees-of-freedom (DOF) underactuated mechanical systems with Coulomb friction in the joints. The control objective is to regulate both actuated and unactuated joints to desired positions. The proposed synthesis combines second-order sliding-mode (SOSM) control methods and a pure technical solution of imposing a relatively stronger dry friction on the unactuated joint as compared to that of the actuated joint. This design feature allows us to decouple the hybrid synthesis procedure into two steps. At the first step, a SOSM control algorithm is utilized to drive the unactuated link to the desired position in finite time. Once this task is achieved, the control input is switched to another SOSM algorithm which drives the actuated link to its desired position. As the amplitude of the controller used at the second step is smaller than the Coulomb friction level in the unactuated link, but it is greater than that in the actuated link, the unactuated link remains at rest, whereas the actuated link is regulated to the endpoint of interest. Performance issues of the developed synthesis, including robustness features of the closed-loop system, are illustrated in an experimental study made for a laboratory horizontal 2-DOF pendulum.