Domains Of Dynamic Instability Research Articles

SPATIAL PROBLEMS OF DYNAMIC STABILITY OF FRAME STRUCTURES

Periodic longitudinal forces in structural elements caused by operational or seismic influences, at certain values of the parameters of these forces can cause the occurrence and growing of transverse oscillations of these elements. This phenomenon is called parametric resonance or loss of dynamic stability. In the works of N. M. Belyaev, N. M. Krylov, М. М. Bogolyubov, E. Mettler, V. N. Chelomey, V. V. Bolotin flat problems of dynamic stability of frame structures were investigated. In this paper the modified Bolotin’s method, proposed to solve flat problems of dynamic stability of frames, is used. Instead of the deformation method used by V. V. Bolotin to construct analytical expressions of deflections of frame rods, in the modified method the numerical-analytical method of boundary elements is used. The article proposes a method for constructing domains of dynamic instability of frames in the space of parameters (frequency and amplitude) of seismic and operational dynamic influences that cause longitudinal forces in the rods, which periodically change over time and lead to unlimited growth of transverse oscillations amplitudes in the domains of instability. The proposed method is demonstrated in example, which considers the spatial problem of dynamic stability of a П-shaped frame with two concentrated masses located on it, which are under the action of vertical periodic forces. These forces create periodic longitudinal forces in the vertical rods of the frame. Areas of dynamic instability of the frame were constructed. Taking into account the destructive effect of oscillations is important for practical application. The most dangerous destructive effect of oscillations is observed in earthquakes and explosions. The study of this action makes it possible to avoid undesirable consequences of oscillations by limiting their level and to solve important practical problems of the dynamics of structures. Solving dynamics problems is a difficult problem. Dynamic calculation of structures provides their bearing capacity under the combined action of static and dynamic loads.

To Calculation of Rectangular Plates on Periodic Oscillations

Geometrically nonlinear mathematical model of the problem of parametric oscillations of a viscoelastic orthotropic plate of variable thickness is developed using the classical Kirchhoff-Love hypothesis. The technique of the nonlinear problem solution by applying the Bubnov-Galerkin method at polynomial approximation of displacements (and deflection) and a numerical method that uses quadrature formula are proposed. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. Parametric oscillations of viscoelastic orthotropic plates of variable thickness under the effect of an external load are investigated. The effect on the domain of dynamic instability of geometric nonlinearity, viscoelastic properties of material, as well as other physical-mechanical and geometric parameters and factors are taken into account. The results obtained are in good agreement with the results and data of other authors.

Dynamic stability of a viscoelastic plate

The paper considers the problem of stability of a viscoelastic plate subjected to intermittent load. A differential equation was designed, which describes the dynamic stability of the plate stressed by permanent and alternating loads across the plate’s plane. A solution in the form of a series with separated variables has been examined. Linear differential equations of the third order were obtained for the time function. A critical frequency equation was obtained as well. The major domains of dynamic instability were investigated. Another analysis was dedicated to the way the position of the prime instability domain is affected by relaxation time as well as correlation between long-term and instantaneous moduli of elasticity.

Domains of dynamic instability of FGM conical shells under time dependent periodic loads

On the basis of the dynamic version of linear Donnell type equations and with deformations before instability taken into account, the dynamic instability of simply supported, functionally graded (FG) truncated conical shells under static and time dependent periodic axial loads is analyzed. Appling Galerkin’s method, the partial differential equations are reduced into a Mathieu-type differential equation describing the dynamic instability behavior of the FG conical shell. The domains of principal instability are determined by using Bolotin’s method. Validation of numerical results was done with those available from previous researches. The influences of various parameters like static and dynamic load factors, volume fraction index, FG profiles and shell characteristics on the domains of dynamic instability of conical shell were investigated.

Identifying the domains of dynamic instability for inhomogeneous shell systems under periodic loads

The paper outlines an approach to identifying the principal dynamic-instability domain for systems composed of shells of revolution with different shapes under axisymmetric periodic loading. The original problem is reduced to one-dimensional eigenvalue problems with respect to the meridional coordinate. Results of calculations for a specific shell system are presented

Short-term flutter-type instability of undamped tdof system with randomly varying bifurcation parameter

A model problem of a two-degrees-of-freedom (tdof) flutter is considered for an undamped system with random temporal variations of its bifurcation parameter. The nominal system, i.e. one with the mean value of the bifurcation parameter is assumed to be stable; however the above variations may occasionally bring the system temporarily into its domain of dynamic instability. A procedure for predicting probability density function (PDF) of the peaks of the corresponding intermittent response is outlined as based on parabolic approximation for the parameter variation in the vicinity of its peaks. For the case of relatively slow parameter variations the equations of motion are reduced by Krylov–Bogoliubov averaging to those describing static instability of the response amplitude. The basic relation between peak values of the bifurcation parameter and of the corresponding response outbreak for the reduced system is therefore available from previous studies of short-term static instability in a sdof system.

Short-term Dynamic Instability of a System with Randomly Varying Damping

A single-degree-of-freedom system with temporal random variations of its total apparent damping is considered. It is demonstrated that the dynamic response of the system exhibits spontaneous transient outbreaks, which are induced by brief periods when the damping coefficient becomes negative. The analysis is based on a parabolic approximation for the random temporal variations of the damping coefficient during these excursions into the domain of dynamic instability, together with the asymptotic Krylov-Bogoliubov method of averaging over the response period. It results in an explicit relation for the response amplitude in terms of the peak value of the damping coefficient during the corresponding downcrossing of the zero level. In this way the reliability analysis for the system as based on a solution to the first-passage problem for the response or on evaluating the probability density (pdf) of the response peaks is reduced to the corresponding problems for the damping coefficient the accuracy of the derived expression for the above pdf is verified by direct Monte-Carlo simulation of the basic equation. Furthermore, the above analytical solution is used also to derive a simple identification procedure for the system from its measured (on-line) response. Specifically, the mean value of the damping coefficient can be estimated, as can its standard deviation and the mean frequency of its temporal variations.

Vibration of a rotating shaft with randomly varying internal damping

A simple Jeffcott rotor is considered with both external and internal damping. Coefficient of internal damping is subject to temporal random variations which may occasionally bring the rotor into the domain of dynamic instability. The corresponding sporadic outbreaks in the rotor's vibrational response (whirl) are studied by applying the Krylov–Bogoliubov averaging method to the complex equation of motion and using parabolic approximation for the random coefficient of the internal damping. This results in an explicit analytical solution for the radius of whirl which may be used for predicting reliability of the rotor. Furthermore, a convenient procedure is described for interpreting measured on-line test data for the rotor. Namely, the mean value of the coefficient of internal damping as well as its standard deviation and mean frequency of temporal variations may be estimated directly from the trace of whirl radius which exhibits spontaneous random outbreaks in response.

Parametric Vibrations of a Three-Layer Conical Piezoshell

Based on the Kirchhoff-Love hypotheses and adequate supplementary hypotheses for the distribution of electric field quantities, a model for parametric vibrations of composite shells of revolution made of a passive (without a piezoeffect) middle layer and two active (with a piezoeffect) surface layers under the action of harmonic mechanical and electric loads is developed. The dissipative material properties are taken into account by linear viscoelastic models. Since the vibrations on the boundary of the main domain of dynamic instability (MDDI) are harmonic, the investigation of this domain, in a first approximation, is reduced to generalized eigenvalue problems, which are solved by the finite-element method. The problem on parametric vibrations of a three-layer conical shell under harmonic mechanical loading is considered. The influence of the shell thickness, dissipation, and electric boundary conditions on the MDDI is investigated. Two limiting cases of electric boundary conditions are considered, where the electrodes are short-circuited or not. The curves presented show a considerable influence of the electric boundary conditions on the characteristics of the MDDI, namely on its width and position on the frequency axis and on the critical parameter of excitation.

Nonlinear dynamics of a machining system with two interdependent delays

The dynamics of turning by a tool head with two rows, each containing several cutters, is considered. A mathematical model of a process with two interdependent delays with the possibility of cutting discontinuity is analyzed. The domains of dynamic instability are derived, and the influence of technological parameters on system response is presented. The numeric analysis show that there exists specific conditions for given regimes in which one row of cutters produces an intermittent chip form while the other row produces continuous chips. It is demonstrated that the contribution of parametric excitation by shape roughness of an imperfect (unmachined) cylindrical workpiece surface is not substantial due to the special filtering properties of cutters that are uniformly distributed circumferentially along the tool head.

Static and Dynamic Buckling of a Fiber Embedded in a Matrix With Interface Debonding

Analyses were performed for static and dynamic buckling of a continuous fiber embedded in a matrix in order to determine effects of interfacial debonding on the critical buckling load and the domain of instability. A beam on elastic foundation model was used for the study. The study showed that a local interfacial debonding between a fiber and a surrounding matrix resulted in an increase of the wavelength of the buckling mode. An increase of the wavelength yielded a decrease of the static buckling load and lowered the dynamic instability domain. In general, the effect of a partial or complete interfacial debonding on the domain of dynamic instability was more significant than its effect on the static buckling load. For dynamic buckling of a fiber, a local debonding of size 10 to 20 percent of the fiber length had the most important influence on the domains of dynamic instability regardless of the location of debonding and the boundary conditions of the fiber. For static buckling, the location of a local debonding was critical to a free, simply supported fiber, but not to a fiber with both ends simply supported.