Vibration and stability of elasticity supported circular plates under conservative and non-conservative loads

This paper deals with the vibration and stability of circular plates elastically restrained against translation and rotation along concentric intermediate circles as well as at the outer edge. The plate is subjected to a horizontal or tangential radial load at the periphery. Both axisymmetric and non-axisymmetric vibrations are considered. It is assumed that the plate is thin, elastic, and isotropic. The solutions are obtained in terms of Bessel functions for both the complete and annular plate sections. The effect of the support stiffnesses on the plate natural frequency and the divergence and flutter instability loads are studied in detail.     

Vibration analysis of circular segment shaped plates

Circular segment shaped plates are analyzed to determine their natural frequencies and mode shapes of vibration. The analysis is based on the finite element approach. The curved sided triangular plate bending element is used for solving the problem. The effect of variation of the size of the plate on the vibrational characteristics is studied and several important conclusions are made.     

Resonance striction parametric excitation under multifrequency pumping

Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 33, No. 4, pp. 417–424, April, 1990.     

Combinational parametric resonance under imperfectly periodic excitation

Conditions for mean square stability of a two-degree-of-freedom system have been obtained for the case of sum combinational resonance due to sinusoidal parametric excitation with constant amplitude and white-noise random temporal variations in the instantaneous frequency. Analysis is based on the asymptotic Krylov–Bogoliubov averaging method. The resulting set of ten deterministic differential equations for second-order moments of the four new state variables (inphase and quadrature modal responses) was solved analytically for its neutral stability boundary. The imperfections in periodicity may lead to either stabilization or destabilization of the system and the corresponding conditions have been clearly established from the solution obtained. Furthermore, conditions for almost sure stability have been obtained for a special symmetric case of identical modal damping factors and identical modal excitation amplitudes.     

Vibration analyses of an inclined flat plate subjected to moving loads

The object of this paper is to present a moving mass element so that one may easily perform the dynamic analysis of an inclined plate subjected to moving loads with the effects of inertia force, Coriolis force and centrifugal force considered. To this end, the mass, damping and stiffness matrices of the moving mass element, with respect to the local coordinate system, are derived first by using the principle of superposition and the definition of shape functions. Next, the last property matrices of the moving mass element are transformed into the global coordinate system and combined with the property matrices of the inclined plate itself to determine the effective overall property matrices and the instantaneous equations of motion of the entire vibrating system. Because the property matrices of the moving mass element have something to do with the instantaneous position of the moving load, both the property matrices of the moving mass element and the effective overall ones of the entire vibrating system are time-dependent. At any instant of time, solving the instantaneous equations of motion yields the instantaneous dynamic responses of the inclined plate. For validation, the presented technique is used to determine the dynamic responses of a horizontal pinned–pinned plate subjected to a moving load and a satisfactory agreement with the existing literature is achieved. Furthermore, extensive studies on the inclined plate subjected to moving loads reveal that the influences of moving-load speed, inclined angle of the plate and total number of the moving loads on the dynamic responses of the inclined plate are significant in most cases, and the effects of Coriolis force and centrifugal force are perceptible only in the case of higher moving-load speed.     

Optimal design of passive viscous dampers for controlling the resonant response of orthotropic plates under high-speed moving loads

In this article, the resonant response of plates traversed by moving loads is addressed being its main application the dynamic performance of railway bridges under high-speed traffic. An innovative alternative to reduce the deck inadmissible oscillations that may appear in short simply supported structures in resonant conditions is proposed, based on artificially increasing the superstructure damping by retrofitting the deck with fluid viscous dampers. A particular auxiliary structure transforming the deck vertical deflection into relative movement within the devices is envisaged, being the main objectives of the study to optimise the retrofitting system parameters and to prove its efficiency under the action of railway vehicles. For these purposes, the retrofitted deck behaviour is first investigated using an orthotropic plate model under harmonic excitation. On the basis of an analytical approach, a dimensionless version of the equations of motion is presented, the governing parameters are extracted and an intensive sensitivity analysis of the plate response is performed. Finally, analytical closed-form expressions for the optimal dampers constants are derived and their adequacy is numerically evaluated. To this end, an existing bridge belonging to the Spanish Railway network is analysed using a three-dimensional finite element code specifically programmed by the authors for this application. In the end the controlling effect of the retrofitting system and the applicability of the optimal parameters analytical expressions are proven for a wide range of circulating velocities.     

A global analysis of a non-linear system under parametric excitation

A strongly non-linear mechanical system consisting of a rigid damped bar subjected to a periodic parametric excitation is treated here in an exact manner. The emphasis is on the global behavior of this system which is carried out by using the point mapping and the cell-to-cell mapping methods. Even though the mechanical system is a simple one, yet it has a very complex global behavior. It is shown here that the newly developed theory of cell-to-cell mappings offers a tremendous advantage in obtaining the global domains of attraction of strongly non-linear dynamical systems.     

Vibration of L-shaped plates under a deterministic force or moment excitation: a case of statistical energy analysis application

Analytical and closed form solutions are presented in this paper for the vibration response of an L-shaped plate under a point force or a moment excitation. Inter-relationships between wave components of the source and the receiving plates are clearly defined. Explicit expressions are given for the quadratic quantities such as input power, energy flow and kinetic energy distributions of the L-shaped plate. Applications of statistical energy analysis (SEA) formulation in the prediction of the vibration response of finite coupled plate structures under a single deterministic forcing are examined and quantified. It is found that the SEA method can be employed to predict the frequency averaged vibration response and energy flow of coupled plate structures under a deterministic force or moment excitation when the structural system satisfies the following conditions: (1) the coupling loss factors of the coupled subsystems are known; (2) the source location is more than a quarter of the plate bending wavelength away from the source plate edges in the point force excitation case, or is more than a quarter wavelength away from the pair of source plate edges perpendicular to the moment axis in the moment excitation case due to the directional characteristic of moment excitations. SEA overestimates the response of the L-shaped plate when the source location is less than a quarter bending wavelength away from the respective plate edges owing to wave coherence effect at the plate boundary.     

Vibration and buckling of rectangular plates under in-plane hydrostatic loading

Numerical solutions are presented for the fundamental natural frequency and mode shape of a rectangular plate loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and load magnitudes. All six possible combinations of simply supported and clamped edges are considered. The limiting conditions of unloaded vibration and buckling are discussed in detail, with emphasis on the preferred mode shape. Design curves and approximate formulae are presented which provide a simple means of determining the fundamental frequency parameter.     

Axial and conical parametric emissions from potassium atoms under two-photon fs excitation

We report on the emissions near the 5P_3/2,1/2–4S_1/2 and 4P_3/2,1/2–4S_1/2 transitions of potassium atoms which are excited by a fs laser beam. The field at the transition 5P_3/2,1/2–4S_1/2 is mainly the result of a parametric process with an axial profile when the excitation frequency is tuned above resonance and a conical one below resonance. Similar but not identical far-field patterns were also observed for the 4P_3/2,1/2–4S_1/2 emission. No amplified spontaneous emission was observed for the fs case, in contrast to the ns excitation for the 4P_3/2,1/2–4S_1/2 transition.     

Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass-spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker-Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.     

Effect of anomalous photoabsorption in parametric X-ray radiation under asymmetric reflection conditions

Parametric X-ray radiation (PXR) of a relativistic electron crossing a monocrystalline plate is considered in the Laue scattering geometry. Expressions describing the spectral-angular distributions of PXR and diffracted transient radiation (DTR), which are formed in atomic planes arranged at an arbitrary angle δ to the surface of a crystalline plane (asymmetric reflection), are derived in the two-wave approximation of the dynamic diffraction theory. The conditions under which the effect of anomalously weak photoabsorption (Borman effect) is manifested most clearly are determined. This effect can considerably enhance the intensity of the sources of tunable quasi-monochromatic X rays, which are based on the PXR mechanism.