Free vibrations of flexible shallow shells of complex shape

A method is proposed for studying the free vibrations of flexible shallow shells with a complex planform. The method is based on variational and R-function methods. The R-function method allows constructing a system of basis functions in an analytic form. This makes it possible to reduce the Donnell-Mushtari-Vlasov equations to Duffing equations. The amplitude-frequency characteristics of shallow shells with a complex planform are given for different curvatures and boundary conditions. The results obtained are compared with published results for simply supported square shells to demonstrate the reliability and efficiency of the method __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 99–109, April 2007.     

Dynamical modelling of wind-induced vibrations of overhead lines

A dynamical model is presented for the wind-induced vibrations of overhead lines due to Karman vortex shedding. Because of the complexity of the excitation mechanism the model for the continuous structure is based on an appropriate non-linear model of a circular rigid cylinder that is oscillating transversally in a flowing fluid. The exciting non-linear forces acting on the cylinder are deduced from known experimental data. It is shown that the observed well-known “lock-in”-effect can be described in satisfactory manner by this model. The transition from the rigid body model to the continuous, flexible structure leads to a non-linear boundary value problem. Approximate solutions are derived by perturbation theory. The model is able to predict the observed vibration frequency and the corresponding vibration mode (depending on the wind velocity) as well as the observed vibration amplitude. An important consequence of the model consists in the self-restricting behaviour of the aerodynamic forces. On this basis, a new conception of the vortex shedding mechanism on a vibrating cable will be presented.     

Nonlinear interaction between self- and parametrically excited wind-induced vibrations

Nonlinear Dynamics - The aeroelastic behavior of a planar prismatic visco-elastic structure, subject to a turbulent wind, flowing orthogonally to its plane, is studied in the nonlinear field. The...     

Non-linear vibrations of sandwich viscoelastic shells

This paper deals with the non-linear vibration of sandwich viscoelastic shell structures. Coupling a harmonic balance method with the Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. The latter are determined by solving a classical eigenvalue problem and two linear ones. This permits to get the non-linear frequency and the non-linear loss factor as functions of the displacement amplitude. To validate our approach, these relationships are illustrated in the case of a circular sandwich ring.     

Analysis of the nonaxisymmetric vibrations of flexible ellipsoidal shells discretely reinforced with transverse ribs under nonstationary loads

The problem of forced nonaxisymmetric vibrations of reinforced ellipsoidal shells under nonstationary loads is formulated. A numerical algorithm of solving it is developed and the results obtained are analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 63–73, October 2008.     

Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities

A general low-order fluid–structure interaction model capable of evaluating the multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures is presented. Cross-flow motions due to unsteady lift forces of inclined sagged cables and tensioned beams in uniform currents are investigated. In contrast to a linear equation governing the transverse motion of straight beams or cables typically considered in the literature, coupled horizontal/vertical (axial/transverse) displacements and geometric nonlinearities of curved cable (straight beam) are accounted for. A distributed nonlinear wake oscillator is considered in the approximation of space–time varying hydrodynamics. This semi-empirical fluid force model in general depends on the mass-damping parameter and has further been modified to capture both the effects of varying initial curvatures of the inclined cylinder and the Reynolds number. Numerical simulations are performed in the case of varying flow velocities and parametric results highlight several meaningful aspects of vortex-induced vibrations of long flexible cylinders. These comprise multi-mode lock-in, sharing, switching and interaction features in the space and time domains, the estimated maximum modal and total amplitudes, the resonant nonlinear modes of flexible cylinders and their space–time modifications, and the influence of fluid/structure parameters. A shortcoming of single-mode or linear structural model is underlined. Some quantitative and qualitative comparisons of numerical/experimental results are discussed to demonstrate the validity and required improvement of the proposed modelling and analysis predictions.     

Low-frequency free vibrations of viscoelastic cylindrical shells

Vitebsk State Pedagogical Institute, Belarus. Translated from Prikladnaya Mekhanika, Vol. 28, No. 9, pp. 50–55, September, 1992.     

Non-linear vibrations of doubly curved shallow shells

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain-displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.     

Non-linear beam-type vibrations of long cylindrical shells

This paper presents a closed-form solution of the problem of free beam-type vibration of a long cylindrical shell subject to uniform axial tension, uniform internal pressure, and elastic axial restraint. The shell is assumed to be clamped at the ends. The bending moment-curvature relationship employed in this paper is non-linear due to the effect of ovalization (flattening) of initially circular cross sections. Another non-linear effect taken into account is the stretching of the shell axis which is caused by the axial restraint.The analysis results in a cubic differential equation; the frequency of the solution of this equation is found exactly using elliptic integrals.     

Free vibrations of long pretwisted cylindrical shells

Tbilisi Mathematical Institute, Academy of Sciences of the Georgian SSR. Translated from Prikladnaya Mekhanika, Vol. 26, No. 9, pp. 54–60, September, 1990.     

Wake structures and vortex-induced vibrations of a long flexible cylinder—Part 1: Dynamic response

Results showing the dynamic response of a vertical long flexible cylinder vibrating at low mode numbers are presented in this paper. The model had an external diameter of 16 mm and a total length of 1.5 m giving an aspect ratio of about 94, with Reynolds numbers between 1200 and 12 000. Only the lower 40% of its length was exposed to the water current in the flume and applied top tensions varied from 15 to 110 N giving fundamental natural frequencies in the range from 3.0 to 7.1 Hz. Reduced velocities based on the fundamental natural frequency up to 16 were reached. The mass ratio was 1.8 and the combined mass–damping parameter about 0.05. Cross-flow and in-line amplitudes, x–y trajectories and phase synchronisation, dominant frequencies and modal amplitudes are reported. Cross-flow amplitudes up to 0.7 diameters and in-line amplitudes over 0.2 were observed with dominant frequencies given by a Strouhal number of 0.16.