Vibration characteristics of curved beams

Abstract

The paper presents a concise framework studying the coupled vibration of curved beams, whether the curvature is built-in or is caused by loading. The governing equations used are both geometrically exact and fully intrinsic, with a maximum degree of nonlinearity equal to two. For beams with initial curvature, the equations of motion are linearized about the reference state. For beams that are curved because of the loading, the equations of motion are linearized about the equilibrium state. A central difference spatial discretization scheme is applied, and the resulting linearized ordinary differential equations are cast as an eigenvalue problem. Numerical examples are presented, including: (1) validation of the analysis for both in-plane and out-of-plane vibration by comparison with published results, and (2) presentation of results for vibration of curved beams with free-free, clamped-clamped, and pinned-pinned boundary conditions. For coupled vibration, the numerical results also exhibit the low-frequency mode transition or veering phenomenon. Substantial differences are also shown between the natural frequencies of curved beams and straight beams, and between initially curved and bent beams with the same geometry.

Phononic Band Gaps in Periodic Cellular Materials

“…These results agree with the findings of Perkins [16] and Addessi et al [17] It is interesting to notice that a similar observation was also reported in a previous theoretical work, [17][18][19][20] which also reported the occurrence of the crossover between the first symmetric and anti-symmetric modes of the curved beams with hinged-hinged boundary conditions. [24] revealed that for struts with hinged ends, the first mode undergoes the transition from the flexural mode of a straight strut into an extensional stage for increasing amplitude. [22,23] Ref.…”

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

“…Hodges (2003) has introduced a set of complete intrinsic equations for the dynamics of initially curved and twisted geometrically exact beams and has shown the advantages of fully intrinsic formulation for a beam under non-conservative transverse follower force. The latter approach which is based on a finite difference scheme has been later used by Chang and Hodges (2009a) and Chang and Hodges (2009b) for the free vibration and stability analysis of curved beams. Khouli et al (2009) and Ghorashi and Nitzsche (2009) also have used finite difference schemes for the spatial discretization of the intrinsic formulation presented by Hodges (2003) for the dynamic analysis of helicopter rotor blades.…”

A unified approach for nonlinear vibration analysis of curved structures using non-uniform rational B-spline representation

“…Moreover, significant differences were reported between the natural frequencies of the curved and straight beams. Interestingly enough, it was shown that there is a substantial difference between the nonlinear natural frequency of the initially curved and that of the bent beam even with the same beam geometry [16]. The anti-symmetric response to the symmetric sinusoidal excitation of a clamped-clamped curved beam was examined by Lee et al [17].…”