Detecting cracks in a longitudinally vibrating beam with dissipative boundary conditions

This paper focuses on detecting a small open crack in an axially vibrating beam with viscous boundary conditions by using non-destructive dynamical measurements. The damage is simulated by an equivalent linear elastic spring. It is shown that the measurement of the changes in a suitable pair of eigenvalues leads to the solution of the diagnostic problem, namely identification of crack location and severity. Results apply to uniform beams under various sets of boundary conditions.     

Ramond-Neveu-Schwarz string with new boundary conditions

For the Ramond-Neveu-Schwarz string in D space-time dimensions we seek boundary conditions which preserve Poincaré invariance in d dimensions, d<D. We obtain twisted closed and twisted open strings preserving Gervais-Sakita supersymmetry. Covariant BRST quantization yields D=10. For some boundary conditions, partition functions exhibit space-time supersymmetry.     

A new boundary condition for simulations of systems with charge-charge interactions

We introduce a new boundary condition for simulation of charged systems which is a generalization of periodic boundary conditions and reduces the long-range correlations inherent in periodic boundary conditions. An effective pair potential is calculated for the new condition.     

Asymptotic analysis of a vibrating cantilever with a nonlinear boundary

Nonlinear vibration of a cantilever in a contact atomic force microscope is analyzed via an asymptotic approach. The asymptotic solution is sought for a beam equation with a nonlinear boundary condition. The steady-state responses are determined in primary resonance and subharmonic resonance. The relations between the response amplitudes and the excitation frequencies and amplitudes are derived from the solvability condition. Multivaluedness occurs in the relations as a consequence of the nonlinearity. The stability of steady-state responses is analyzed by use of the Lyapunov linearized stability theory. The stability analysis predicts the jumping phenomenon for certain parameters. The curves of the response amplitudes changing with the excitation frequencies are numerically compared with those obtained via the method of multiple scales. The calculation results demonstrate that the two methods predict the same varying tendencies while there are small quantitative differences. Supported by the National Outstanding Young Scientists Fund of China (Grant No. 10725209), the Shanghai Leading Academic Discipline Project (Grant No. S30106), and Shandong Jiaotong University Science Foundation (Grant No. Z200812) Recommended by LIAO ShiJun     

A hexahedron element formulation with a new multi-resolution analysis

A multiresolution hexahedron element is presented with a new multiresolution analysis(MRA)framework.The MRA framework is formulated out of a mutually nesting displacement subspace sequence,whose basis functions are constructed of scaling and shifting on element domain of a basic node shape function.The basic node shape function is constructed from shifting to other seven quadrants around a specific node of a basic isoparametric element in one quadrant and joining the corresponding node shape functions of eight elements at the specific node.The MRA endows the proposed element with the resolution level(RL)to adjust structural analysis accuracy.As a result,the traditional 8-node hexahedron element is a monoresolution one and also a special case of the proposed element.The meshing for the monoresolution finite element model is based on the empiricism while the RL adjusting for the multiresolution is laid on the solid mathematical basis.The simplicity and clarity of shape function construction with the Kronecker delta property and the rational MRA enable the proposed element method to be more rational,easier and efficient in its implementation than the conventional mono-resolution solid element method or other MRA methods.The multiresolution hexahedron element method is more adapted to dealing with the accurate computation of structural problems.     

A mortar contact formulation using scaled boundary isogeometric analysis

正Contact analysis is recognized as being the most challenging problem in computational mechanics,because the functional system of contact problems is nonlinear and non-smooth and the convergence and accuracy of contact algorithms are difficult to guarantee.In the traditional finite element method(FEM)-based contact analysis[1,2],the contact body is spatially discretized,and the contact boundary is described using a low-order Lagrange interpolation polynomial.While     

Bosonic string theories with new boundary conditions

We show that the classical Nambu-Goto string inD dimensions admits Poincaré invariance ind dimensions (d⩽D) if (i)d − 2 of the transverse co-ordinatesx ⁱ are periodic and the rest quasi-periodic involving a real orthogonal matrix with (D − d) (D − d − 1)/2 free parameters, or if (ii)d − 2 ofx ⁱ obey Neumann and the rest obey a boundary condition involvingN free parameters, whereN=(D − d)²/2 ifD − d is even, andN=[(D − d)² − 1]/2 ifD − d is odd.     

A generalized Dunkerley-Graeffe procedure for complex vibrating systems

The combined use of the principle underlying Dunkerley's rule for the approximate determination of the gravest eigenfrequency of a multi-degree-of-freedom elastic system and of the root-squaring process suggested by Graeffe provide a generalized procedure for the approximate solution of complex frequency equations, i.e., for the determination of any number of eigenvalues at the desired accuracy level, provided that the eigenfunctions involved can be expressed in series expansion. By simple algebraic means, the method yields solutions in cases for which sophisticated computing facilities would otherwise be necessary and provides the means for checking complicated computer outputs as well as approximate results for preliminary design purposes.     

Driven nonequilibrium lattice systems with shifted periodic boundary conditions

We present the first study of a driven nonequilibrium lattice system in the two-phase region, withshifted periodic boundary conditions, forcing steps into the interface. When the shift corresponds to small angles with respect to the driving field, we find nonanalytic behavior in the (internal) energy of the system, supporting numerical evidence that interface roughness is suppressed by the field. For larger shifts, the competition between the driving field and the boundary induces the breakup of a single strip with tilted interfaces into many narrower strips with aligned interfaces. The size and temperature dependences of the critical angles of such breakup transitions are studied.     

A skew ray tracing approach for the error analysis of optical elements with paraboloidal boundary surfaces

Using the error analysis methodology developed by the current author in previous studies for optical systems comprising elements with flat boundary surfaces, this study examines the errors induced in a light ray's path as it is reflected or refracted at a paraboloidal boundary surface. In analyzing the light path, two principal sources of error are considered, namely (1) translational errors (Δx_i, Δy_i and Δz_i) and rotational errors (ΔΓ_i, ΔΨ_i and ΔΦ_i), which collectively determine the deviation of the light path at each boundary surface, and (2) the differential changes induced in the incident point position and unit directional vector of the refracted/reflected ray as a result of differential changes in the position and unit directional vector of the light source. The validity of the proposed approach is verified using a generic parabaloidal boundary surface for illustration purposes. Overall, the results show that the proposed error analysis methodology provides a straightforward means of analyzing the performance of optical systems characterized by paraboloidal boundary surfaces such as headlight reflectors, optical telescope mirrors, flashlights and so forth.     

Global evolution of general-relativistic gaseous systems with controllable boundary conditions

The solutions of the general-relativistic kinetic equation for bounded gaseous systems are studied. It is shown that the processes induced by the field of a gravitational wave are irreversible. A method for describing theoretically the process of controlling the behavior of the system as a whole is given for the case of diffuse reflection of particles at the boundaries.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 63–68, October, 1990.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions

In this paper, the influences of nonideal boundary conditions (due to flexibility) on the primary resonant behavior of a piezoelectrically actuated microbeam have been studied, for the first time. The structure has been assumed to treat as an Euler–Bernoulli beam, considering the effects of geometric nonlinearity. In this work, the general nonideal supports have been modeled as a the combination of horizontal, vertical and rotational springs, simultaneously. Allocating particular values to the stiffness of these springs provides the mathematical models for the majority of boundary conditions. This consideration leads to use a two-dimensional analysis of the multiple scales method instead of previous works' method (one-dimensional analysis). If one neglects the nonideal effects, then this paper would be an effort to solve the two-dimensional equations of motion without a need of a combination of these equations using the shortening or stretching effect. Letting the nonideal effects equal to zero and comparing their results with the results of previous approaches have been demonstrated the accuracy of the two-dimensional solutions. The results have been identified the unique effects of constraining and stiffening of boundaries in horizontal, vertical and rotational directions. This means that it is inaccurate to suppose the nonideality of supports only in one or two of these directions like as previous works. The findings are of vital importance as a better prediction of the frequency response for the nonideal supports. Furthermore, the main findings of this effort can help to choose appropriate boundary conditions for desired systems.     

Natural frequency intervals for vibrating systems with polytopic uncertainty

We consider the problem of locating the natural frequencies of uncertain systems whose describing matrices are functions of an unknown parameter vector which is included in an assigned bounding set. We face what we call the weak frequency interval detection problem of determining the smallest interval which includes all possible frequencies. We show that if the system matrices depend affinely on the parameter vector, whose bounding set is a compact polyhedron, then this problem requires the solution of a finite number of eigenvalue problems associated with the vertices of such a polyhedron. Unfortunately, detecting the intervals associated with all the natural frequencies (strong frequency interval detection problem), cannot rely on this property, so that one must resort to Monte Carlo methods or numerical optimization to find them. We show that the strong version is solvable “exploring the vertices only” under some stronger assumptions. In the case in which the uncertainty bounding set is not defined by linear inequalities, not even the extremal frequencies can be associated with the vertices of the admissibility domain. Then again, numerical approach is necessary unless we accept to merge the original system in a larger one of an “affine nature”. Finally, we present as an application the study of structures with uncertain mass distribution.