Simple vibration modeling of structural fuzzy with continuous boundary by including two-dimensional spatial memory

Many complicated systems of practical interest consist basically of a well-defined outer shell-like master structure and a complicated internal structure with uncertain dynamic properties. Using the "fuzzy structure theory" for predicting audible frequency vibration, the internal structure is considered as one or more fuzzy substructures that are known in some statistical sense only. Experiments have shown that such fuzzy substructures often introduce a damping in the master which is much higher than the structural losses account for. A special method for modeling fuzzy substructures with a one-dimensional continuous boundary was examined in a companion paper [L. Friis and M. Ohlrich, "Vibration modeling of structural fuzzy with continuous boundary," J. Acoust. Soc. Am. 123, 718-728 (2008)]. In the present paper, this method is extended, such that it allows modeling of fuzzy substructures with a two-dimensional continuous boundary. Additionally, a simple method for determining the so-called equivalent coupling factor is presented. The validity of this method is demonstrated by numerical simulations of the vibration response of a master plate structure with fuzzy attachments. It is revealed that the method performs very well above a nondimensional frequency of 500 of the master, and it is shown that errors below this frequency are caused mainly by simplifying assumptions concerning the shape of the master vibration displacement.     

Vibration of continuous systems with compliant boundaries

A theoretical study of two types of continuous systems with a general form of compliant boundary conditions is presented. The systems considered are elastic beams and circular plates with elastic damped edge constraints. Beam studies are restricted to those with identical boundary conditions at each end. The method of solution consists of formulating the edge condition of the system in terms of the impedance of the compliant boundary material and of using classical solution techniques to solve the equations of motion. The result of matching the boundary conditions of the system with constraining conditions is the system frequency equation in terms of the constraint impedances.A discussion is presented giving the influence of the compliant material on the vibration of the structure. The models give numerically the effect of elasticity and damping of the supports on the resonant frequencies of the systems. Parameters are obtained which indicate when one may assume simply supported or clamped boundaries for the actual case of elastic damped constraints without introducing large errors in the natural frequencies.     

Vibration of rectangular plates with time-dependent boundary conditions

A general method of solution for the vibration of rectangular plates with any type of time-dependent boundary conditions is developed by an extension of the method of Mindlin and Goodman [1]. For illustration, the problems of a plate with different time-dependent boundary conditions are solved and the closed form solutions for the transverse deflections of the plate are obtained. The non-dimensionalized transverse deflections, ($\text{w}\text{a}$) at the middle of the plate are evaluated numerically for different dimensions of the plate and different forcing functions. These are presented graphically against the non-dimensionalized time, T, for three cases and tabulated for other cases.     

Vibration analysis of rectangular plates with general elastic boundary supports

In this investigation, the Rayleigh-Ritz method is used to determine the modal characteristics of a rectangular plate with general elastic supports alone its edges. Each of the admissible functions here is composed of a trigonometric function and an arbitrary continuous function that is introduced to ensure the sufficient smoothness of the so-called residual displacement function at the edges. As a result, a drastic improvement of the convergence can be expected of the solution expressed as a series expansion in terms of the admissible functions. Perhaps more importantly, this study has developed a general approach for deriving a complete set of admissible functions that can be universally applied to various boundary conditions. Several numerical examples are given to demonstrate the accuracy and convergence of the current solution.     

Vibration modeling of magnetic tape with vibro-impact of tape-guide contact

Next-generation, high track-density linear tape drives will require improved tape guiding and dimensional stability in order to achieve better performance and higher storage capacities. Drive vibrations, tape degradation, debris formation, and tape guiding all contribute to unwanted lateral tape motion (LTM), which can lead to track-misregistration errors during reading of the tape. Understanding the nature of LTM is of prime interest. A nonlinear vibration model of a tape-guide system with vibro-impact is presented in this study. The guides are modeled as rigid constraints, while the reels are modeled as clamped ends. Free vibration (due to initial velocity) and forced vibration (due to moving boundaries) were performed for various test cases to determine the effect of tape tension and thickness, tape-guide separation distance, reel tilt amplitude, and tape speed. The model results are compared to actual experimental LTM data, though with the absence of tape damping and tape-guide frictional forces in the model, only limited comparisons may be made.     

Vibration characteristics of thin rotating cylindrical shells with various boundary conditions

An analysis is presented for the vibration characteristics of thin rotating cylindrical shells with various boundary conditions by use of Fourier series expansion method. Based on Sanders’ shell equations, the governing equations of motion which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotating are derived. The displacement field is expressed as a product of Fourier series expressions which represents the axial modal displacements and trigonometric functions which represents the circumferential modal displacements. Stokes’ transformation is employed to derive the derivatives of the Fourier series expressions. Then, through the process of formula derivation, an explicit expression of the exact frequency equation can be obtained for a thin rotating cylinder with classical boundary conditions of any type. Once the frequency equation has been determined, the frequencies are calculated numerically. To validate the present analysis, comparisons between the results of the present method and previous studies are performed and very good agreement is achieved. Finally, the method is applied to investigate the vibration characteristics of thin rotating cylindrical shells under various boundaries, and the results are presented.     

Dynamic modeling of beams with non-material,deformation-dependent boundary conditions

In conventional problems of structural mechanics, both kinematic boundary conditions and external forces are prescribed at fixed material points that are known in advance. If, however, a structure may move relative to its supports, the position of the imposed constraint relations generally changes in the course of motion. A class of problems which inherently exhibits this particular type of non-material boundary conditions is that of axially moving continua. Despite varying in time, the positions of the supports relative to the material points of the body have usually assumed to be known a priori throughout the deformation process in previous investigations. This requirement is abandoned in the present paper, where the dynamic behavior of a structure is studied, which may move freely relative to one of its supports. As a consequence, the position of such a non-material boundary relative to the structure does not only change in time but also depends on the current state of deformation of the body. The variational formulation of the equilibrium relations of a slender beam that may undergo large deformations is presented. To this end, a theory based on Reissner's geometrically exact relations for the plane deformation of beams is adopted, in which shear deformation is neglected for the sake of brevity. Before a finite element scheme is developed, a deformation-dependent transformation of the beam's material coordinate is introduced, by which the varying positions of the constraint relations are mapped onto fixed points with respect to the new non-material coordinate. By means of this transformation, additional convective terms emerge from the virtual work of the inertia forces, whose symmetry properties turn out to be different from what has previously been presented in the literature. In order to obtain approximate solutions, a finite element discretization utilizing absolute nodal displacements as coordinates is subsequently used in characteristic numerical examples, which give an insight into the complex dynamic behavior of problems of this type. On the one hand, the free vibrations of a statically pre-deformed beam are investigated, on the other hand, an extended version of the sliding beam problem is studied.     

Vibration control of a structure with ATMD against earthquake using fuzzy logic controllers

In this paper, fuzzy logic and PD controllers are designed for a multi-degree-of freedom structure with active tuned mass damper (ATMD) to suppress earthquake-induced vibrations. Fuzzy logic controller (FLC) is preferred because of its robust character, superior performance and heuristic knowledge use effectively and easily in active control. A fifteen-degree-of-freedom structural system is modeled with two types of actuators. These actuators are installed on the first storey and fifteenth storey which has ATMD. The system is then subjected to Kocaeli Earthquake vibrations, which are treated as disturbances. In control, linear motors are used as the active isolators. At the end of the study, the time history of the storey displacements and accelerations, ATMD displacements, control voltages, frequency responses of the both uncontrolled and the controlled structures are presented. Performance of the designed FLC has been shown for the different loads and disturbances using ground motion of the Kobe Earthquake. The results of the simulations show a good performance by the fuzzy logic controllers for different loads and the earthquakes.     

Estimation of fuzzy structure parameters for continuous junctions

The fuzzy structure theory introduced 15 years ago is designed to predict the frequency response functions of structures and structural acoustic systems with structural complexity, in the low- and medium-frequency ranges. This paper constitutes a first validation of the fuzzy structure theory for continuous junctions between the master structure and the fuzzy substructures. In addition, a method to estimate the fuzzy structure parameters introduced in the fuzzy structure theory is presented and it is validated for the case of continuous junctions. This validation obtained by numerical simulations opens the field of experimental identifications.     

Thermodynamics and structural aspects of grain boundary segregation

Physical and chemical properties of solid materials are strongly. influenced by the chemical composition of internal interfaces, One of the crucial parameters affecting interfacial chemistry is the atomic structure of the interface. Due to its importance. a considerable amount of work was done to elucidate the relationship between structure and chemical composition of interfaces. This article reviews the present understanding of an important and fundamental part of this relationship, namely, the structural aspects of grain boundary segregation. After a brief outline of grain boundary structure and geometry. thermodynamic approaches to describe grain boundary segregation are summarized and their application to materials is discussed. covering particular sites at a single grain boundary as well as the role of interfaces in polycrystals. Both the experimental evidence of grain boundary segregation anisotropy and the theoretical results of computer simulations of grain boundary segregation are summarized. Useful methods of predicting grain boundary segregation are presented. Finally, segregation behavior of solutes at grain boundaries is compared with that at free surfaces, and examples of chemical composition of intexphase boundaries are given.     

Vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment

The free vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment is carried out by the 3D elasticity theory and the Chebyshev–Ritz method. The material properties are assumed to be temperature dependent and graded in the thickness direction. The mixed boundary conditions which include upper and lower surfaces partially fixed, inner side partially fixed and outer side partially fixed are considered, respectively. The accuracy of the present approach for solving the free vibration of the plates with different boundary conditions is validated by comparing the present numerical results with the results available. The effects of the different mixed boundary conditions, the temperature rise, the material graded index and the geometrical parameters on the eigen-frequencies are studied.     

Vibration of a membrane having a circular outer boundary and an eccentric circular inner boundary

In this paper, a method of solving vibration problems for a membrane having a circular outer boundary and an eccentric circular inner boundary is presented. The frequency equation for the membrane is given, and the dependence of the natural frequency on the eccentricity is obtained. It is shown that the result obtained from this analysis in the limit of the eccentricity being zero coincides with that of a membrane with a concentric circular inner boundary.     

Propagation of sound at continuous structural phase transitions

Structural phase transitions of second order can be divided into two groups: (i) distortive phase transitions, with a soft (ultimately overdamped) optic mode, and (ii) elastic phase transitions, with an acoustic soft mode or no soft phonon for shear or isostructural transitions, respectively. The propagation of sound shows significantly different features in these two cases. We consider the theory of the critical variation of the velocity of ultrasonic modes as well as the damping and dispersion near transitions of second order.Talk given at the Conference on Transport and Propagation in Nonlinear Systems, Los Alamos, May 21–25, 1984.     

Poroelastic modeling of seismic boundary conditions across a fracture

Permeability of a fracture can affect how the fracture interacts with seismic waves. To examine this effect, a simple mathematical model that describes the poroelastic nature of wave-fracture interaction is useful. In this paper, a set of boundary conditions is presented which relate wave-induced particle velocity (or displacement) and stress including fluid pressure across a compliant, fluid-bearing fracture. These conditions are derived by modeling a fracture as a thin porous layer with increased compliance and finite permeability. Assuming a small layer thickness, the boundary conditions can be derived by integrating the governing equations of poroelastic wave propagation. A finite jump in the stress and velocity across a fracture is expressed as a function of the stress and velocity at the boundaries. Further simplification for a thin fracture yields a set of characteristic parameters that control the seismic response of single fractures with a wide range of mechanical and hydraulic properties. These boundary conditions have potential applications in simplifying numerical models such as finite-difference and finite-element methods to compute seismic wave scattering off nonplanar (e.g., curved and intersecting) fractures.     

Vibration influence on a region with gas under the adiabatic and isothermal boundary conditions

The influence of vibration on the behavior of the perfect viscous gas inside a rectangular cavity is investigated numerically. The heat and mass transfer processes in gas are compared in the region under the isothermal and adiabatic boundary conditions. The problem is solved in the one-dimensional statement.     

Analysis of structural - acoustic coupling of elastic rectangular enclosure with arbitrary boundary conditions

The structural acoustic coupling characteristics of a rectangular enclosure con- sisting of two elastic supported flexible plates and four rigid plates are analyzed.A general formulation considering the full coupling between the plates and cavity is developed by using Hamiltonian function and Rayleigh-Ritz method.By means of continuous distributions of ar- tificial springs along boundary of flexible plates,a wide variety of boundary conditions and structure joint conditions are considered.To demonstrate the validity of the analytical model, the responses of sound pressure in the cavity and plate velocity are worked out.The analytical results coincides well with Kim's experimental results.The result is satisfactory.Finally,an- alytical results on the structure vibration and the sound field inside the cavity are presented. These results indicate that the coupling of the combined structure is relatively weak,so the internal cavity sound is controlled by plate directly excited,and the translational stiffness affects the sound more than the rotational stiffness does.     

Higher‐Dimensional Unification with continuous and fuzzy coset spaces as extra dimensions

We first review the Coset Space Dimensional Reduction (CSDR) programme and present the best model constructed so far based on the , 10‐dimensional E₈ gauge theory reduced over the nearly‐Kähler manifold with the additional use of the Wilson flux mechanism. Then we present the corresponding programme in the case that the extra dimensions are considered to be fuzzy coset spaces and the best model that has been constructed in this framework too. In both cases the best model appears to be the trinification GUT .