Shell vibrations at high-frequency

The free vibration is called high-frequency when the frequency parameter is limited by the inequalities >max{R ₂ ^–2 (s)} and O(h ⁰). In this case there is only one boundary layer type of solution in the neighbourhood of any edge which is not sufficient to satisfy the two non-tangential boundary conditions to be dropped by the membrane equations at the edge, and is called non-complete.An asymptotic approach is presented in this paper, by means of which we find that there are two types of principal modes to be operative over the whole range of the shell surface, when the shell vibrates axisymmetrically at high frequency. One of the principal modes is a membrane type (¦u¦¦w¦, and the index of variation is zero) and the other is a quasi-transverse one with quick variation (¦u¦¦w¦, and the index of variation is equal to 1/2). Correspondingly, the set of frequency parameters can also be divided into two subsets, one of which corresponds to the membrane modes as their eigenvectors, while the other subset corresponds to the quasi-transverse modes with quick variation as their eigenvectors.     

Electroelastic vibrations of conical elements

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 8, pp. 53–59, August, 1990.     

On vibrations of rectangular parallelepipeds

An exact, closed-form solution of the three-dimensional equations of elasticity is given for vibrations of rectangular parallelepipeds with traction-free faces.     

Chaotic vibrations in high-speed milling

A large number of literatures are devoted to the stability of the milling process and various control methods for chatter suppression. But chaotic dynamics beyond the stable region has not been considered extensively. Moreover, modeling issues for chaotic motion need more challenge for accurate prediction of its complex dynamical behavior. This paper presents a detailed two-degree-of-freedom mechanics based model for the study of chaotic vibrations in milling. Segmental multiple regenerative effect that is the principle feature of nonlinear vibrations in milling processes besides two state dependent time delays has been considered. Exact geometrical formulation of multiple regenerative effects by considering simultaneously different numbers of delayed tool positions over the cutting zone is presented for the first time. Phase portrait, bifurcation diagram, largest Lyapunov exponent, and surface profile were calculated for a given machine tool and workpiece parameters. The simulation results show positive values of the largest Lyapunov exponent corresponding to the existence of chaos in high-speed milling operations. Also, investigation of the machined surface of the workpiece formed by the helical mill demonstrates an irregular pattern on the surface.     

Oscillation pattern of stick-slip vibrations

This paper studies the stick-slip oscillations of discrete systems interacting with translating energy source through a non-linear smooth friction curve. The stick-slip limit cycle oscillations of a single degree-of-freedom model are examined by means of numerical time-integration and analytical methods. Similar approaches are also applied to the model of the coupled friction oscillator. Particularly, it is found that the steady-state response of the coupled oscillator can be divided into two different forms of oscillation (mode-merged and mode-separated oscillations) according to the frequency separation of two modes. The oscillation pattern of the steady-state response is shown to depend on system parameters such as detuning factor, energy source speed, and normal contact load.     

Transversal vibrations of double-plate systems

This paper presents an analytical and numerical analysis of free and forced transversal vibrations of an elastically connected double-plate system. Analytical solutions of a system of coupled partial differential equations, which describe corresponding dynamical free and forced processes, are obtained using Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one-mode vibrations correspond to two-frequency regime for free vibrations induced by initial conditions and to three-frequency regime for forced vibrations induced by one-frequency external excitation and corresponding initial conditions. The analytical solutions show that the elastic connection between plates leads to the appearance of two-frequency regime of time function, which corresponds to one eigenamplitude function of one mode, and also that the time functions of different vibration modes are uncoupled, for each shape of vibrations. It has been proven that for both elastically connected plates, for every pair of m and n, two possibilities for appearance of the resonance dynamical states, as well as for appearance of the dynamical absorption, are present. Using the MathCad program, the corresponding visualizations of the characteristic forms of the plate middle surfaces through time are presented.The English text was polished by Keren Wang.     

A perturbation method for nonlinear vibrations of imperfect structures: Application to cylindrical shell vibrations

A perturbation method is used to analyse the nonlinear vibration behaviour of imperfect general structures under static preloading. The method is based on a perturbation expansion for both the frequency parameter and the dependent variables. The effects on the linearized and nonlinear vibrations caused by geometric imperfections, a static fundamental state, and a nontrivial static state are included in the perturbation procedure.The theory is applied in the nonlinear vibration analysis of anisotropic cylindrical shells. In the analysis the specified boundary conditions at the shell edges can be satisfied accurately. The characteristics of the analysis capability are shown through examples of the vibration behaviour of specific shells. Results for single mode and coupled mode nonlinear vibrations of shells are presented. Parametric studies have been performed for a composite shell.     

Transverse vibrations of cylindrical helical springs

Summary In this paper we have calculated the natural frequencies of the transverse vibrations of cylindrical helical springs, with respect to the variation in slenderness and relative compression. The calculation was made by considering the spring as a constant-section beam, and by applying Timoshenko theory to the latter. The theoretical section of this paper is an extension of Haringx's work on the same subject, not so much because it studies natural modes higher than the first (a problem which was treated by Haringx), but rather because its more general approach to the transverse vibrations phenomenon raises problems of results' interpretation that this Author did not face. The calculated values of natural frequencies and the corresponding deflection curves have been compared with experimentally determined values. In the course of experimentation Thickness-shear modes were visualized, which Timoshenko theory takes into account, but which, as far as beams are concerned, are not detectable.

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Sommario Sono state calcolate le frequenze naturali delle vibrazioni laterali delle molle ad elica cilindrica con estremi incastrati, al variare della snellezza e del precarico. Il calcolo è stato effettuato assimilando la molla ad una trave a sezione costante, ed applicando a questa la teoria di Timoshenko.La parte teorica del presente studio è una estensione del lavoro di Haringx sullo stesso argomento, non tanto perché in esso sono stati considerati modi di vibrare superiori al primo (problema questo trattato dall'Haringx), quanto per il fatto che una visione più generale del fenomeno delle vibrazioni laterali pone problemi di interpretazione dei risultati, non affrontati da questo Autore.I valori calcolati delle frequenze naturali e le corrispondenti linee elastiche sono stati confrontati con quelli determinati sperimentalmente. Nel corso della sperimentazione è stato possibile visualizzare modi di vibrare caratteristici (Thickness-shear modes) previsti dalla teoria di Timoshenko, ma non osservabili nel caso delle travi.

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Research sponsored by the National Research Committee (C.N.R.). Paper presented during the: Incontro di studio «Metodi classici e moderni per lo studio delle vibrazioni di sistemi dinamici», Università di Padova, maggio 1976.     

Forced vibrations of laterally loaded piles

An analytical approach for the determination of the response of a single circularcylindrical pile subjected to a lateral dynamic load is presented. The kinetic and the potentialenergies of the pile-foundation system are minimized by variational principle to obtain thegoverning field equations of the pile-foundation system along with the appropriate boundaryconditions. A non-dimensional parameter γ, associated with the characteristics of the pile, thefoundation and the loading is used to represent the elastic medium. This parameter γ can bedetermined by using an iterative procedure. The classical finite difference method is used tosolve the governing field equations of the pile-foundation system. The validation of the proposedmodel is demonstrated by applying to several published field pile load tests. Parametric studieswith regard to the frequency response of the pile head and the resonant frequency of the pile-foundation system are presented.     

Free vibrations of a piezoelectric body

We present a systematic analysis of the eigenvalue problem associated with free vibrations of a finite piezoelectric body. The analysis is based on an abstract formulation of the three-dimensional theory of piezoelectricity. A series of fundamental properties of free vibrations of a piezoelectric body are proved concisely. The problem of free vibrations of a piezoelectric plate governed by the two-dimensional plate equations due to Mindlin is treated in a similar manner.Address after September 1, 1994: Department of Mechanical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA.Address after August 16, 1994: Clifton Garvin Professor, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA.