Non-uniform Rods Research Articles

Exciton localization and dissociation dynamics in CdS and CdS-Pt quantum confined nanorods: effect of nonuniform rod diameters.

One-dimensional colloidal multicomponent semiconductor nanorods, such as CdSe-CdS dot-in-rod, have been extensively studied as a promising class of new materials for solar energy conversion because of the possibilities of using the band alignment of component materials and the rod-diameter-dependent quantum confinement effect to control the location of electrons and holes and to incorporate catalysts through the growth of Pt tips. Here we used CdS nanorods as an example to study the effect of nonuniform diameters along the rod on the exciton localization and dissociation dynamics in CdS and (platinum tipped) CdS-Pt nanorods. We showed that, in CdS nanorods prepared by seeded growth, the presence of a bulb with a larger diameter around the CdS seed resulted in an additional absorption band lower in energy than the exciton in the CdS rod. As a result, excitons generated in the CdS rod could undergo ultrafast localization to the bulb region in addition to trapping on the CdS rod. We observed that the Pt tip led to fast exciton dissociation by electron transfer. However, excitons localized on the CdS bulb showed slower average ET rates than those localized in the rod region. Our findings suggested that the effect of rod morphology should be carefully considered in designing multicomponent nanorods for solar energy conversion applications.

The Method of Receptances for Continuous Rods

It is shown that it is possible in general to assign 2n poles of an axially vibrating nonuniform rod by sensing the state at n points along the rod and by using an actuator which applies the control force. Moreover, the assignment of poles may be done without continuous or discrete analytical modeling. In particular, the rigidity, the density, and the variable cross-sectional area of the rod may be considered unknown. The control gains are determined by the receptances between the point of actuation and the points of sensing, which may be measured experimentally. In the analytical arena, where the receptances are exact, the assignment of the desired poles is also exact and suffers from no discretization or model reduction errors. However, the control force affects in general the distribution of the poles which are not intended to be desirably placed or to remain unchanged. Although the analysis is carried out for simplicity on an axially vibrating rod, the above mentioned results are generally applicable to other linear elastic structures of higher dimensions.

Numerical Analysis of Free Longitudinal Vibration of Nonuniform Rods: Discrete Singular Convolution Approach

AbstractIn this paper, the problem of longitudinal free vibration of nonuniform rods is investigated for the first time by using the discrete singular convolution (DSC). This method is a relatively novel and powerful numerical approach especially in analyzing vibration problems. The governing equation of this problem is nonlinear and has analytical solutions only in a few cases, so the analysis uses DSC as a numerical method. The regularized Shannon delta kernel is used in the present algorithm. The nondimensional natural frequencies of rods with various-area cross sections have been studied in this paper. Results show good agreement with the previous analytical solutions. This study indicates the efficiency of the DSC in solving nonuniform rod problems.

Uniform versus Nonuniform Axial Power Distribution in Rod Bundle CHF Experiments

Rod bundle experiments with axially uniform and nonuniform heat fluxes are examined to explore the potential limitations of using uniform rod bundle CHF data for CHF correlation development of light water reactors with nonuniform axial power distribution (APD). The case of upstream burnout is presented as an example of unique phenomena associated with nonuniform rod bundle CHF experiments. It is a result from combined effect of axial nonuniform power shape and different interchannel mixing mechanisms. In addition, several key parameters are investigated with respect to their potential impacts on the thermal-hydraulic behaviors between rod bundles with uniform and nonuniform APDs. This type of misrepresentation cannot be amended or compensated through the use of correction factors due to the lack of critical information in the uniform rod bundle CHF testing as well as the fundamental difference in the underlining driving mechanisms. Other potential issues involved with the use of uniform rod bundle CHF data for nonuniform APD system applications also present strong evidence concerning the limitations and inadequacy of using uniform rod bundle CHF data for the correlation, prediction, and design limit calculation for safety analysis.

Dynamic substructure analysis of stress waves generated by impacts on non-uniform rod structures

The dynamic substructure method (DSM) has high computational efficiency, and has been successfully applied in analyzing the modal responses of complex flexible bodies and in computing the dynamic responses of flexible bodies undergoing strong motion. However, the use of this method to analyze the propagation of stress waves generated by impacts has yet to be validated. The object of this paper is to propose a dynamic substructure computational procedure for the analysis of impact-induced stress waves in a non-uniform flexible structure. The procedure is established based on fixed-interface modal synthesis theory, finite element theory and a local contact model. The application of the procedure is demonstrated using composite rods subjected to longitudinal impacts. The governing equations during the contact and separation phases are solved by the Newmark-β method, and then the stress waveforms are obtained. In order to verify the accuracy of the procedure, the numerical solutions are compared with the analytical solutions obtained by method of characteristics. Furthermore, the ‘succession collision’ phenomenon resulting from the elastic wave effect is captured. All investigations showed that DSM was sufficiently accurate to analyze wave motion problems in complicated mechanical structures.

Approximate Solution for Longitudinal Vibration of Non-Uniform Beams by Differential Transform Method (DTM)

In this paper, differential transform method is applied for longitudinal vibration analysis of beams with non-uniform cross section. Natural frequencies and corresponding normalized mode shapes are calculated for different cases in cross section and boundary conditions. Comparison of results with the previous solutions proves the accuracy and versatility of the presented paper.     2 0 sin A xA a bx  . Li (4) combined the transfer matrix method and closed form solution of one step non-uniform rod to obtain a single frequency equation for a multi-step non-uniform rod. In another papers, he derived an exact solution for free longitudinal vibrations of one-step non-uniform rods with classical and non-classical boundary conditions (5) and an analytical solution for determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar with several boundary conditions (6). Li, et al. (7) found closed form solution for the longitudinal vibration of non-uniform structures with lumped masses and spring supports whose cross-section varies as   1 n

Axial vibration analysis of nanocones based on nonlocal elasticity theory

Carbon nanocones have quite fascinating electronic and structural properties, whose axial vibration is seldom investigated in previous studies. In this paper, based on a nonlocal elasticity theory, a nonuniform rod model is applied to investigate the small-scale effect and the nonuniform effect on axial vibration of nanocones. Using the modified Wentzel-Brillouin-Kramers (WBK) method, an asymptotic solution is obtained for the axial vibration of general nonuniform nanorods. Then, using similar procedure, the axial vibration of nanocones is analyzed for nonuniform parameters, mode number and nonlocal parameters. Explicit expressions are derived for mode frequencies of clamped-clamped and clamped-free boundary conditions. It is found that axial vibration frequencies are highly overestimated by the classical rod model because of ignorance of the effect of small length scale.

Free Axial Vibrations of Non-Uniform Rods

Free axial vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the pro-posed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the proposed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.

Free longitudinal vibrations of non-uniform rods

Free longitudinal vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the proposed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the proposed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.

A modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform one-dimensional waveguides

In this article, using the wave propagation method, the natural frequencies and mode shapes of an arbitrary non-uniform one-dimensional waveguide are calculated. The non-uniform rods and beams are partitioned into several continuous segments with constant cross-sections, for which there exists an exact analytical solution. At the end of each segment, waves in positive and negative directions are obtained in terms of waves at initial segment and subsequently, the calculations of the mode shapes become simple. By satisfying the boundary conditions, the characteristic equation is obtained and natural frequencies are calculated for both the arbitrary non-uniform rod and beam. Also, by adding waves in positive and negative directions at the end point of each segment, the mode shapes are obtained. To verify the modified wave method presented here, the frequencies and mode shapes of the rod and the beam with a polynomial cross-section having an exact analytical solution are compared and have been proven to be of high accuracy. Besides, comparisons of finite element method are also included. Therefore, this method can also be used to calculate the natural frequencies and mode shapes of rods and beams with any arbitrary variable cross-section for which no analytical solution is available. For the ‘Modified Wave Approach’ developed here, dimensions of transmission matrix remain constant if the number of segments is increased, while in general wave propagation method, dimensions of transmission matrix increase upon increasing the number of segments. Besides this novelty, this method has the advantage that it gives all the natural frequencies and mode shapes, unlike other approximate methods such as weighted residual, Rayleigh–Ritz, and finite difference methods which have their own shortcomings such as limited number of natural frequencies. Also, since each segment has an exact analytical solution, in contrast to other approximate methods, much higher accuracy is obtained even with only a few number of partitions.

Wave motions in non-uniform one-dimensional waveguides

Wave approach is used in the analysis of wave motions in one-dimensional non-uniform waveguides. Under assumptions of constant wave velocity and no wave conversion, one class of non-uniform rods including four different profiles results. The obtained results indicate kinetic energy is preserved as constant and wave amplitude is inversely proportional to square root of the cross-section area of the rod. The obtained spectrograms and dispersion relations indicate that under certain conditions there exists a cut-off frequency, below which waves do not propagate along the non-uniform waveguides. The conclusions are similar for high frequency longitudinal wave motions of arbitrary non-uniform rods. For a Euler-Bernoulli beam with exponent variation in geometry and material variation, the bending wave motion is also analyzed including its cut-off frequencies.

Longitudinal waves in one dimensional non-uniform waveguides

Wave approach is used to analyze the longitudinal wave motion in one dimensional non-uniform waveguides. With assumptions of constant wave velocity and no wave conversion, there exist four types of non-uniform rods and corresponding traveling wave solutions are investigated. The obtained results indicate that the kinetic energy is preserved as a constant and the wave amplitude is inversely proportional to square root of the cross-sectional area of the rod. Under certain condition, there exists a cut-off frequency for the rod with variation in geometric or material properties, below which waves do not propagate along the non-uniform rod. For the rod with arbitrary variable cross-section, the conclusions are similar if the wave frequency is high enough. And a series solution of the wave motion is presented.

Analysis of the spectrum of elastic vibrations of a rod in a nonuniform elastic medium

The spectrum of transverse vibrations of a thin nonuniform rod in an elastic medium with a variable rigidity coefficient and arbitrary boundary conditions of elastic fastening is studied. It is proved that the external medium can lead to an anomalous effect of increasing eigenfrequencies of lower modes of oscilla� tions with elongation of the rod length. The very fine properties of this variation depending on the length, mode number, and boundary conditions are revealed. Vibrations with standard methods of fastening are studied separately, and examples are calculated. 1. The problem of eigenfrequencies ω and forms u(x) of flat transverse vibrations of a thin nonuniform rod (see [1–7], etc.) with variable rigidity p(x) > 0 and density r(x) > 0 in a medium with elasticity q(x) ≥ 0 is studied. Using the Fourier method (procedures of sep�

Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply

Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply

Exact solutions for the longitudinal vibration of non-uniform rods [J. Sound Vib. 207 (1997) 721–729

Referred to by R.I. Sujith; Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply Journal of Sound and Vibration, Volume 329, Issue 19, 13 September 2010, Page 4108

Exact solutions for forced vibration of non-uniform rods by Laplace transformation

Exact solutions for forced vibration of non-uniform rods by Laplace transformation

Optimized Design of Microcoil Resonators

Various methods to overcome the difficulties in the fabrication of high-quality (Q) optical nanofiber microcoil resonators are investigated. Modifications of the resonator geometry by wrapping the nanofiber over nonuniform rods and/or by varying the pitch alter the resonator Q-factor significantly. These effects are quantified, and geometries, which allow for enhanced tolerance to fabrication uncertainties, are identified. These simulations apply equally to the fabrication of high-Q microcoil resonators with nanofibers suspended in air or embedded in low-refractive-index materials.

Wave separation in non-uniform Hopkinson bars using redundant measurements

In this paper we generalize the BCGO-wave separation method (Bussac, 2002, J. Phys. Mech. Solids, 50, 321-349) based on the maximum of likelihood principle to non-uniform rods. To achieve this aim, the force and the particle velocity at any cross-section are expressed as functions of the force and the particle velocity at the origin instead of the expressing them as function of the forward and downward waves. Due to this generalisation, one can perform long duration and medium strain-rate (1-100/s) tests using Hopkinson bars not only in the case of compression loadings but also for example dynamic bending and high-temperature tests. The new method is checked numerically on a stepped elastic bar. It shows good behaviour against noise.

Stability, Paths, and Dynamic bending of a Blunt Body of Revolution Penetrating into an Elastoplastic Medium

The deep penetration of a thin body with a blunt nose and rear into a low‐strength medium is explored. The motion of the body is described by a system of autonomous integrodifferential equations using the physical model of a separated asymmetric flow over the body and the local‐interaction method. An analytical calculation of the Lyapunov stability boundary for straight‐line motion is performed for bodies with a parabolic meridian. The dependences of the dynamic stability of the body on various parameters are studied numerically. Curved motion paths are constructed in the region of instability, and the classification of paths proposed in previous studies of the motion of pointed bodies is confirmed. It is shown that an reverse ejection is possible when a blunt impactor enters a semi‐infinite target. It is established that there is a fundamental possibility of attaining a path close to a specified one and that there is a weak dependence of motion characteristics with a developed separation on the separation angle. Examples are given of calculations of the evolution of the lateral load, the transverse force and moment, and the strength margin of the body using the theory of dynamic bending of a nonuniform rod.

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.