A study on small scale thermal dynamic instability of rotating GPL-reinforced microbeams under principal parametric resonance stimulation of axial and transversal modes regarding the proportional damping

A study on the principal parametric stimulation of rotating microbeams strengthened by means of graphene platelet is presented. The microbeam is exposed to a temperature gradient. On the basis of the assumptions of the hypothesis of the Timoshenko for beams alongside the modified couple stress hypothesis the nonlinear motion equations are achieved. It is supposed that the rotating speed of the microbeam varies harmonically about a mean quantity. The frequency of the harmonic term of the rotating velocity is presumed to be almost two times an axial or a transversal natural frequency. In such situation, the principal parametric agitation is stimulated. Assuming a proportional damping, a least square scheme is employed to define the Rayleigh's coefficients. The impacts of the rotating speed, the weight fraction as well as the scattering pattern of the graphene platelets, and the damping coefficient on the presented results are examined. The results illuminate that as a consequence of the addition of the damping coefficient, the critical incitement amplitude constant increases more for reinforced microbeams rather than not-reinforced microbeams. Moreover, by means of enlarging the weight fraction of the graphene platelets, the critical incitement amplitude constant develops more for a rotating X microbeam rather than an O microbeam. Additionally, at moderate to high magnitudes of the damping coefficient, the instability area border aligned with the fundamental axial mode is impressed by the graphene platelet scattering pattern although it is invariant for small values of the damping coefficient. Moreover, by the development of the temperature the instability area border aligned with the fundamental transversal mode gets broader, while the critical incitement amplitude coefficient decreases; the both more for on O scattering pattern for the graphene platelet. Furthermore, the instability region boundary is more influenced by the graphene platelet weight fraction, while the critical incitement amplitude coefficient is more impressed by the damping coefficient design value.