Natural frequencies and mode shapes for axisymmetric vibrations of shells in turning-point range

In the present paper, the natural frequencies and modes in turning-point frequency range where the governing equations have turning points are studied for revolution shells under various boundary conditions by applying the uniformly valid solutions obtained in a previous paper. Due to the presence of the turning points in this frequency range, several novel features emerge from the analytic and computational results for vibration of shells, in which the most basic nature is the coupling of bending and membrane solutions for frequencies and modes. And simple expressions for the bending-edge-condition effect and frequency spacing are presented.     

Influence of end impedances on the natural frequencies of an acoustic resonator

Journal of Applied Mechanics and Technical Physics -     

An evolutionary search technique to determine natural frequencies and mode shapes of composite Timoshenko beams

Natural frequencies and mode shapes of composite Timoshenko beams are determined by a diversity guided evolutionary algorithm (DGEA) with different boundary conditions. After applying boundary conditions, frequency equation is obtained in determinant form. Then, natural frequencies and consequently mode shapes are obtained using DGEA where the absolute value of determinant is the subject of optimization. Advantages of employing DGEA are: first, all natural frequencies are produced in a simple run, second, its simplicity for implementation and third, the procedure is not computationally prohibitive. Results clearly show the applicability of the proposed method for obtaining natural frequencies and mode shapes.     

Natural frequencies and mode shapes of statically deformed inclined risers

We investigate numerically the linear vibrations of inclined risers using the Galerkin approach. The riser is modeled as an Euler–Bernoulli beam accounting for the nonlinear mid-plane stretching and self-weight. After solving for the initial deflection of the riser due to self-weight, we use a Galerkin expansion employing 15 axially loaded beam mode shapes to solve the eigenvalue problem of the riser around the static equilibrium configuration. This yields the riser natural frequencies and corresponding exact mode shapes for various values of inclination angles and tension. The obtained results are validated against a boundary-layer analytical solution and are found to be in good agreement. This constitutes a basis to study the nonlinear forced vibrations of inclined risers.     

The finite element technique for predicting the natural frequencies,mode shapes and damping values of filamentary composite plates

This article presents the numerical method for predicting the natural frequencies, mode shapes and damping values of filamentary composite plates. This method is based on finite element technique, using damped element and allowing transverse shear deformation. For the example of this technique, the theoretical results comparing with experimental values of carbon fibre and glass fibre reinforced plastics plates (mid-plane symmetric) are provided. The dynamic properties of these laminates are discussed. Finally, a simple graphic technique to estimate the natural frequencies and damping values is suggested.     

Mass defect influence on the longitudinal vibration frequencies and mode shapes of a beam

The problem of the influence of complicated-shape defects on the natural frequencies and mode shapes of elastic systems is considered. The beam is considered as an example to discover the general properties of defects of various physical nature such as of mass, elasticity, and cross-section. Some notions of damage and a criterion that permits defectoscopy by nondestructive inspection techniques are introduced. The problem is solved in more detail for a free beam with a mass defect. The resonance method is used experimentally to determine the mass defect with high accuracy. The efficiency of the developed numerical-analytic and experimental approaches to studying the properties of elastic systems with defects is confirmed.     

Determining the resonance frequencies of forced vibrations of turbogenerators with foundations

The model method for determination of dynamic properties of framed foundations on plastic scale models is elaborated. Position shifting of resonance peaks has been found by evaluation of different response curves, measured on models. Theoretical and experimental analysis of forced vibrations of complex damped mechanical system with several degrees of freedom has shown that resonance peaks, measured at different points of the same system, correspond to various frequencies of exciting force. To locate eigen frequencies and dangerous resonances from results of measurements, it is necessary to plot the envelope of the set of curves and to determine the positions of its peaks, or, at least, to find the concentration of resonance maxima of all individual curves.     

On the strategy for processing modal shapes with the clustered natural frequencies

In this paper, it is pointed out that when several natural frequencies of a structure are clustered in a narrow band the individual modal shapes are very hard to measure or calculate with acceptable accuracy, whereas the invariant subspace spanned by them can be easily determined. It is recommanded that the sine of the included angle of two subspaces of equal dimensions may be used as an index for measuring the difference of the subspaces. A simple method is proposed for calculating the cosine of the included angle.The project was supported by National Natural Science Foundation of China.     

Experimental determination of acoustic mode shapes in a reverberant room

Experimental Techniques -     

Geometrically exact planar beams with initial pre-stress and large curvature: Static configurations,natural frequencies,and mode shapes

Within this paper, an analytical formulation is provided and used to determine the natural frequencies and mode shapes of a planar beam with initial pre-stress and large variable curvature. The static configuration, mode shapes, and natural frequencies of the pre-stressed beam are obtained by using geometrically exact, Euler–Bernoulli beam theory. The beam is assumed to be not shear deformable and inextensible because of its slenderness and uniform, closed cross-section, as well as the boundary conditions under consideration. The static configuration and the modal information are validated with experimental data and compared to results obtained from nonlinear finite-element analysis software. In addition to the modal analysis about general static configurations, special consideration is given to an initially straight beam that is deformed into semi-circular and circular static configurations. For these special circular cases, the partial differential equation of motion is reduced to a sixth-order differential equation with constant coefficients, and solutions of this system are examined. This work can serve as a basis for studying slender structures with large curvatures.     

Identifying mode shapes and modal frequencies by operational modal analysis in the presence of harmonic excitation

Operational modal analysis techniques allow us to extract the modal properties of structures based on their response to non-measured stationary white noise, i.e., by considering only the system response to operational excitations. In this paper we outline a procedure to deduce modal parameters from operational response measurements. In particular, we discuss a novel approach to analyze operational responses due to unknown harmonic excitation in addition to noise. Structural eigenfrequencies and modal damping are computed using a modified least-squares complex exponential method. Once the poles of the system are identified, mode shapes are obtained by post-processing. The robustness and accuracy of the approach are illustrated by performing tests on a plate structure.     

On the free unsteady expansion of an ideal gas

The asymptotic laws governing the unsteady flow of an ideal (i.e., inviscid and non-heat-conducting) gas expanding freely out of a plane, cylindrical or spherical volume are studied within the framework of the continuum model. In the case of expansion into a vacuum, the laws obtained are valid in a certain neighborhood of the plane, axis or center of symmetry for time values at which the gas density (or specific enthalpy) is much smaller than its initial value. The characteristic feature of this region is its elongatedness in the direction of the timet-axis in thex, t-plane (x being the spatial variable). This region expands in thex-direction as well, but not as fast as in thet direction. For large values oft the same laws govern the flow induced by two symmetrically expanding plane, axisymmetric or spherical pistons, in the neighborhood of the plane, axis or center of symmetry. Similar laws governing steady-state hypersonic gas flows were studied in [1–6].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 155–163, May–June, 1993.The author is grateful to V. S. Vostretsova for assisting with this work.