Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

Abstract

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

Global stability of the Euler- Bernoulli beams excited by multiplicative white noises

“…The dynamics of the beams are important problem in the elastokinetics [1]. Tajik [2] proposed the stability analysis of motion equations of unbalanced spinning pre-twisted beam. Dai [3] investigated the limit point bifurcations and jump of cantilevered microbeams according to Galerkin method and modal truncation.…”

“…Let ∥u∥ ≡ ∥u∥ L 2 (D) ,∥u∥ s ≡ ∥u∥ H s 0 (D) ,(u, v) ≡ (u, v) L 2 (D) ,(u, v) s = (u, v) H s 0 (D) , where H s (D), H s 0 (D), s ∈ R are the usual Sobolev Spaces,for more detailed, see [26]. A = ∆ 2 with boundary condition (2), then D(A) = {u|u ∈ H 4 (D) H 1 0 (D), ∆u = 0}, and then A is self-adjoint, positive, unbounded linear operators and A −1 ∈ L (L 2 (D)) is compact. then, their eigenvalues {λ i } i∈N satisfy 0 < λ 1 ≤ λ 2 ≤ • • • → ∞ and the corresponding eigenvalues {e i } ∞ i=1 form an orthonormal basis in L 2 (D).…”