Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement,Part II: Experiment

To compare with the theoretical results of Part I, detailed experimental results have been obtained on the non-linear response of a clamped beam under a uniformly distributed periodic load. A duralumin test specimen with breadth 26 mm, thickness 0·5 mm and length 150 mm was used and tests were conducted by pulsating the base frame with a constant peak accerelation and measuring the relative displacement of the beam to the frame. It is found that the theoretical results are generally is reasonable agreement with the experimental ones, the effects of initial axial displacements being included. Besides the results for comparison with theoretical predictions, various non-linear responses were observed in connection with the internal resonance, combination resonance and dynamic snap-through phenomena, for which the present experimental results seem to provide effective data facilitating further theoretical analyses.     

Estimation of the stresses in a coating growing on a flat plate under nonisothermal conditions

A mathematical model of a coating which grows during magnetron sputtering has been formulated and investigated. To estimate the average mechanical stresses, both the thermal and the chemical diffusion contribution are taken into account. It has been shown that the kinetics of the reaction on the surface is no less important in the evolution of stresses than the relationship between the mechanical properties of the growing coating and of the substrate.     

Non-linear vibrations of a simple-simple beam with a non-ideal support in between

A simply supported Euler-Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted.     

Non-linear vibrations of a harmonically excited autoparametric system

Harmonic forced vibration of a spring-mass-damper system with a parametrically excited pendulum hinged to the mass is investigated. Two types of restoring forces on the pendulum are considered. The method of harmonic balance is used to evaluate the system response. The results are also verified by numerical integration. Non-periodic system responses are possible if the excitation parameter is large. The performance of the pendulum as an absorber is also studied.     

Non-linear vibrations of general structures

Based on the finite element and perturbation procedures, an analytical-numerical approach to non-linear vibrations is developed. In particular, the overall solution employs the finite element method to handle spatial dependencies while the constrained version of the perturbation procedure is used to treat the temporal behavior. Due to the generality of the method developed, any combination of structure and boundary restraints can be treated as well as the possibility of conservative and non-conservative situations wherein the system can have any number of frequency branches. Furthermore, the procedure is not restricted to excitations in the neighborhood of specific frequencies, but rather applies to the full range. To demonstrate the capabilities of the solution approach, the results of several numerical studies are included along with experimental verification.     

Non-linear vibrations of cables in three dimensions,part I: Non-linear free vibrations

Analysis and results for non-linear free vibrations of both horizontal and inclined cables in three dimensions are presented. Sag-to-span ratios of the cables are not limited to being small. Computed results are presented for various geometrical and material parameters. The major findings are that the geometrical non-linearity may be of the stiffening type or the softening type, depending on the sag-to-span ratio, and the stiffness of out-of-plane vibrations is affected by the corresponding in-plane vibration near a resonant frequency due to non-linear coupling between out-of-plane and in-plane vibrations.     

Forced vibrations of a rectangular plate with non-uniform thickness

The dependence on frequency of the maximum deflection and surface stresses of a simply supported rectangular plate subjected to a uniformly distributed sinusoidal excitation is discussed and simple formulae are proposed for estimating the deflection and surface stresses. The thickness of the plate varies linearly in one direction parallel to a side of the plate.     

Free vibrations of a plate with an inner support

The title problem is tackled by using a simple polynomial co-ordinate function and the Rayleigh-Schmidt method. It is assumed that the inner support is parallel to the free edge. When the support coincides with the free edge the frequency equation degenerates properly into the case of a simply supported edge. Numerical results are presented for the situation where two opposite edges are simply supported and the edge parallel to the free edge is either clamped or simply supported.     

Non-linear free vibrations of inextensible beams

Non-linear free vibrations of inextensible clamped-free and free-free beams are analyzed by using Galerkin's method and the harmonic balance method.     

Non-linear vibrations of cables considering loosening

A cable cannot resist the axial compressive force that may be induced during large amplitude vibrations. In this paper, the effect of cable loosening on non-linear vibrations of flat-sag cables is discussed by using the finite difference method that can express cable loosening. In the present method, flexural rigidity and damping of the cable are considered in the equations of motion of a cable in order to handle the numerical instability. The effect of cable loosening is evaluated explicitly in the present paper. Furthermore, non-linear vibration properties are evaluated for various parameters under periodic and step vertical loading. The effect of cable loosening on response under vertical periodic time-varying load is small and it is possible for the sag-to-span ratio to roughly equal the ratio for modal transition. The loosening under the vertical step loading in the direction opposite to the gravity appears at almost the same sag-to-span ratio.     

Thermally induced vibrations of a viscoelastic plate

A method for the study of thermally induced vibrations of a viscoelastic plate of arbitrary shape is proposed. The method is based upon the concept of iso-amplitude contour lines on the surface of the plate. It is shown that the time behaviour can be found by assuming a normal mode expansion in terms of the eigenfunctions for the associated elastic plate problem, and the deflection is obtained by using the elastic-viscoelastic analogy. As an illustration of the technique, the vibration of a viscoelastic circular plate under a thermal shock at its centre is discussed, all details of which are illustrated by graphs.