Nonlinear response and stability of flexoelectric nanotube conveying fluid under temperature field using nonlocal strain gradient theory

In this article, the method of multiple scales is presented for solving nonlinear free and forced vibration equations of flexoelectric nanotube conveying viscous fluid under a temperature field located on a nonlinear elastic foundation using nonlocal strain gradient theory. By assuming simple Euler-Bernoulli beam theory and the nonlinear geometry of Van Carmen, the differential equation governing nonlinear vibrations was derived. An electrical voltage was applied to the upper surface of the nanotube, which created the electric field conditions of the closed circuit. Finally, the effect of different parameters such as temperature changes, electrical voltage, etc. on the real and imaginary parts of natural frequencies was investigated. Also, the effect of flexoelectric coefficient on primary, subharmonic and super harmonic resonance was investigated. The results show that the flexoelectric coefficient at the primary and super harmonic resonance initially causes the hardening behavior in the system and the jump phenomenon is quite clear; But as it increases, the system shows softening behavior.