Hygro–thermo–magnetically induced vibration of nanobeams with simultaneous axial and spinning motions based on nonlocal strain gradient theory

Size-dependent vibrational behavior of embedded spinning tubes under gravitational load in hygro-thermo-magnetic fields

“…The work variation due to environmental conditions can be presented as follows 36,58 in which NM, NH, and NT are applied loads by magnetic, moisture, and thermal fields, respectively, and are determined as 36 where β , Δ H , α, Δ T, μ , and B specify the humidity expansion coefficient, moisture changes percentage, thermal expansion coefficient, temperature gradient, magnetic permeability coefficient, and magnetic field strength, respectively. 59…”

“…Work done by environmental condition. The work variation due to environmental conditions can be presented as follows 36,58 δW e ¼…”

“…Geometric and physical properties of the considered system. 36,46,54 Property Value amplitude of the system response enhances with time. Thus, it can be concluded that compared with the static instability, the system is more vulnerable in the dynamical instability state.…”

“…34,35 For instance, absorption of water molecules and the appearance of degradation conditions induced by humid environments accelerate vibrational instabilities in the nanostructures. 36,37 Therefore, comprehension of the mechanism of environmental attacks in vibrational stability of micro and nanotubes conveying fluid is of great importance and is the subject of numerous scientific reports over recent years. Within this context, Sadeghi-Goughari et al 38 characterized the dynamical behavior of nanoscale cantilevered pipes conveying fluid subjected to a longitudinal magnetic field.…”

“…(a) Real and (b) imaginary parts of the first four eigenvalues of the P-P system without foundation versus the fluid velocity when V=5, τ=θ=B=ΔH=ΔT=γ=q=Kn=0.Table1. Geometric and physical properties of the considered system 36,46,54.…”