1,721,068 research outputs found
Free vibrations of shallow inextensible cables: a perturbation approach
No full textThe statics and dynamics of catenary cables are classic matters of theoretical and applied mechanics. The paper systematizes a methodological strategy to achieve analytical solutions for the nonlinear static problem and linearized free dynamic problem of inextensible shallow cables. First, the one-dimensional continuum model of elastically extensible, perfectly flexible cables is revisited to state the nonlinear differential equations governing the static and dynamic equilibria as parametric expressions of the cable shallowness and extensibility. Second, an original hierarchical generalization of the Force Method is presented as methodological solution strategy. The key is the systematic application of perturbation schemes to equilibrium equations, indeformability constraints and geometric compatibility conditions. As a principal point of strength, the proposed strategy allows the unified and consistent treatment of the static and dynamic problems, while requiring the sole assumption of cable shallowness as postulate a priori. As major achievements, fully analytical high-order solutions are obtained for the asymptotic approximation of (i) the catenary configuration in the static field, and (ii) the natural frequencies and classical modal forms in the linearized dynamic field. Parametric analyses of the results highlight that high-order terms determine significant qualitative and quantitative effects on the modal solutions, including competing softening or hardening effects in the natural frequencies
Catenary configuration and geometric stiffness matrix of inextensible cables: Analytical high-order asymptotic solutions for parametric design
No full textThe analytical and geometric study of catenary curves is a classic matter of applied mathematical modeling. The paper systematizes a methodological strategy to achieve fully analytical solutions for the mechanical problem of determining the equilibrium configuration assumed by inextensible inclined cables under gravitational loads. By developing a two-step perturbation scheme, the statically indeterminate equilibrium problem is asymptotically formulated, and asymptotic solutions are obtained in the form of convergent polynomial series of terms with increasing orders of smallness. As a major achievement, the asymptotic analytical solutions are explicit functions of the governing parameters and automatically satisfy the integral compatibility condition, which traditionally requires a numerical solution to assess the hyperstatic unknowns. The mathematical conditions for the existence of admissible solutions and asymptotic consistency of the perturbation scheme are provided, while the characteristic properties of the asymptotic series are recognized or demonstrated. Parametric analyses successfully verify the high approximation accuracy achievable by high-order asymptotic series, truncated to a large but finite number of terms. With the aim of extending the methodology to the largest possible variety of cable structure applications, a fully analytical, although asymptotic, expression of the geometric stiffness matrix of inextensible inclined cables is determined. Parametric analyses confirm that high-order asymptotic expressions can accurately approximate the exact stiffness matrix assessed numerically. This achievement opens up viable opportunities for analytically studying and parametrically designing complex cable structures (including collaborations with other structural elements), within the framework of the direct stiffness method. Finally, the feasibility and effectiveness of the asymptotic direct stiffness method are successfully verified by solving in a fully-analytical way a paradigmatic static problem for the catenary cable-stayed beam
Modal interactions in flexible structures with energy transfer from low- to high-frequency
No full textMulti-parameter perturbation methods for the eigensolution sensitivity in discrete systems exhibiting multiple frequency veering
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Parametric interactions in the nonlinear sectional dynamics of suspended and cable-stayed bridges
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Nonlinear interactions in the planar dynamics of cable-stayed beam
No full textAn analytical model is proposed to study the nonlinear interactions between beam and cable dynamics in stayed-systems. The integro-differential problem, describing the in-plane motion of a simple cable-stayed beam, presents quadratic and cubic nonlinearities both in the cable equation and at the boundary conditions. Mainly studied are the effects of quadratic interactions, appearing at relatively low oscillation amplitude. To this end an analysis of the sensitivity of modal properties to parameter variations, in intervals of technical interest, has evidenced the occurrence of one-to-two and two-to-one internal resonances between global and local modes. The interactions between the resonant modes evidences two different sources of oscillation in cables, illustrated by simple 2dof discrete models. In the one-to-two global-local resonance, a novel mechanism is analyzed, by which cable undergoes large periodic and chaotic oscillations due to an energy transfer from the low-global to high-local frequencies. In two-to-one global-local resonance, the well-known parametric-induced cable oscillation in stayed-systems is correctly reinterpreted through the autoparametric resonance between a global and a local mode. Increasing the load the saturation of the global oscillations evidences the energy transfer from high-global to low-local frequencies, producing large cable oscillations. In both cases, the effects of detuning from internal and external resonance are presented
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