Effect of size dependency on in-plane vibration of circular micro-rings

In this paper, based on the modied couple stress theory, the size-dependent dynamic behavior of circular rings on elastic foundation is investigated. The ring is modeled by Euler-Bernoulli and Timoshenko beam theories, and Hamiltonʹs principle is utilized to derive the equations of motion and boundary conditions. The formulation derived is a general form of the equation of motion of circular rings and can be reduced to the classical form by eliminating the size-dependent terms. On this basis, the size-dependent natural frequencies of a circular ring are calculated based on the non-classical Euler-Bernoulli and Timoshenko beam theories. The ndings are compared with classical beam theories. Response of the micro-ring under application of static and dynamic loads is investigated and compared with the classical theories. Results show that when the thickness of the ring is in the order of the length scale of the ring material, the natural frequencies evaluated using the modied couple stress are considerably more than those predicted based on the classical beam theories, while the de ection and natural frequencies of the classical and non-classical beam theories approach one another for the rings with thickness much larger than the material length scale.