Coupling between high-frequency modes and a low-frequency mode: Theory and experiment

An analytical and experimental investigation into the response of a nonlinear continuous system with widely separated natural frequencies is presented. The system investigated is a thin, slightly curved, isotropic, flexible cantilever beam mounted vertically. In the experiments, for certain vertical harmonic base excitations, we observed that the response consisted of the first, third, and fourth modes. In these cases, the modulation frequency of the amplitudes and phases of the third and fourth modes was equal to the response frequency of the first mode. Subsequently, we developed an analytical model to explain the interactions between the widely separated modes observed in the experiments. We used a three-mode Galerkin projection of the partial-differential equation governing a thin, isotropic, inextensional beam and obtained a sixth-order nonautonomous system of equations by using an unconventional coordinate transformation. In the analytical model, we used experimentally determined damping coefficients. From this nonautonomous system, we obtained a first approximation of the response by using the method of averaging. The analytically predicted responses and bifurcation diagrams show good qualitative agreement with the experimental observations. The current study brings to light a new type of nonlinear motion not reported before in the literature and should be of relevance to many structural and mechanical systems. In this motion, a static response of a low-frequency mode interacts with the dynamic response of two high-frequency modes. This motion loses stability, resulting in oscillations of the low-frequency mode accompanied by a modulation of the amplitudes and phases of the high-frequency modes.     

两跨输电线非线性主共振响应分析

研究两跨输电线非线性共振响应问题,应用Hamilton变分原理推导了两跨输电线的振动微分方程以及对应的边界条件。利用Galerkin离散方法和多尺度法,得到了单模态主共振响应。研究结果表明:幅频响应曲线表现出软、硬弹簧性质,随着外激励幅值的增大,输电线系统响应由软弹簧性质向硬弹簧性质转换;系统阻尼减小或外激励幅值增大时,系统幅值个数也随之发生变化,表现出多值和跳跃现象。     

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

Resistance of a circular pipe with pulsatory variation of the flow rate

The results of an experimental investigation of the hydraulic resistance of a circular pipe for turbulent flow with periodic flow rate fluctuations are presented. The presence of resonance phenomena in the pipe is revealed. It is established that, for hydrodynamic nonstationarity, the pipe resistance is a nonmonotonous function of the frequency of the imposed flow rate fluctuations and differs from the pipe resistance in the stationary flow regime. Under the conditions considered, to find the pipe resistance it is necessary to take into account the variation of the flow kinetic energy with respect to the phase of the imposed flow rate fluctuations due to the deformation of the velocity profile.     

梁的动力稳定性分析的有限元方法

提出了对梁进行动力学稳定性分析的有限元方法──给出了单元质量矩阵，抗弯刚度矩阵，几何刚度矩阵及相应的Ｍａｔｈｉｅｕ方程，通过坐标变换消除了方程的动力与静力耦合，然后说明了由这种具有参数激励耦合的多自由度系统的Ｍａｔｈｉｅｕ方程求得系统一般参数共振及组合参数共振的过渡曲线的约束参数方法与多尺度方法。最后作为算例求出了均匀简支梁受简谐轴向力作用时的过渡曲线。     

Nonlinear random coupled motions of structural elements with quadratic nonlinearities

The second-order closure method is used to analyze the nonlinear response of two-degree-of-freedom systems with quadratic nonlinearities. The excitation is assumed to be the sum of a deterministic harmonic component and a random component. The case of primary resonance of the second mode in the presence of a two-to-one internal (autoparametric) resonance is investigated. The method of multiple scales is used to obtain four first-order ordinary-differential equations that describe the modulation of the amplitudes and phases of the two modes. Applying the second-order closure method to the modulation equations, we determine the stationary mean and mean-square responses. For the case of a narrow-band random excitation, the results show that the presence of the nonlinearity causes multi-valued mean-square responses. The multi-valuedness is responsible for a jump phenomenon. Contrary to the results of the linear analysis, the nonlinear analysis reveals that the directly excited second mode takes a small amount of the input energy (saturates) and spills over the rest of the input energy into the first mode, which is indirectly excited through the autoparametric resonance.     

Octahedral and Cubic Gold Nanoframes with Platinum Framework

Herein, we report a general synthetic pathway to various shapes of three‐dimensional (3D) gold nanoframes (NFs) embedded with a Pt skeleton for structural rigidity. The synthetic route comprises three steps: site‐specific (edge and vertex) deposition of Pt, etching of inner Au, and regrowth of Au on the Pt framework. Site‐specific reduction of Pt on Au nanoparticles (NPs) led to the high‐quality of 3D Au NFs with good structural rigidity, which allowed the detailed characterization of the corresponding 3D metal NFs. The synthetic method described here will open new avenues toward many new kinds of 3D metal NFs.     

Synthesis and Evaluation of GdIII‐Based Magnetic Resonance Contrast Agents for Molecular Imaging of Prostate‐Specific Membrane Antigen

Magnetic resonance (MR) imaging is advantageous because it concurrently provides anatomic, functional, and molecular information. MR molecular imaging can combine the high spatial resolution of this established clinical modality with molecular profiling in vivo. However, as a result of the intrinsically low sensitivity of MR imaging, high local concentrations of biological targets are required to generate discernable MR contrast. We hypothesize that the prostate‐specific membrane antigen (PSMA), an attractive target for imaging and therapy of prostate cancer, could serve as a suitable biomarker for MR‐based molecular imaging. We have synthesized three new high‐affinity, low‐molecular‐weight Gd^III‐based PSMA‐targeted contrast agents containing one to three Gd^III chelates per molecule. We evaluated the relaxometric properties of these agents in solution, in prostate cancer cells, and in an in vivo experimental model to demonstrate the feasibility of PSMA‐based MR molecular imaging.     

Methoxy and Methyl Group Rotation: Solid‐State NMR 1H Spin‐Lattice Relaxation,Electronic Structure Calculations,X‐ray Diffractometry,and Scanning Electron Microscopy

We report solid‐state ¹H nuclear magnetic resonance (NMR) spin‐lattice relaxation experiments, X‐ray diffractometry, field‐emission scanning electron microscopy, and both single‐molecule and cluster ab initio electronic structure calculations on 1‐methoxyphenanthrene ( 1 ) and 3‐methoxyphenanthrene ( 2 ) to investigate the rotation of the methoxy groups and their constituent methyl groups. The electronic structure calculations and the ¹H NMR relaxation measurements can be used together to determine barriers for the rotation of a methoxy group and its constituent methyl group and to develop models for the two coupled motions.     

Arrival time distributions of product ions reveal isomeric ratio of deprotonated molecules in ion mobility–mass spectrometry of hyaluronan‐derived oligosaccharides

Hyaluronic acid is a naturally occurring linear polysaccharide with substantial medical potential. In this work, discrimination of tyramine‐based hyaluronan derivatives was accessed by ion mobility–mass spectrometry of deprotonated molecules and nuclear magnetic resonance spectroscopy. As the product ion mass spectra did not allow for direct isomer discrimination in mixture, the reductive labeling of oligosaccharides as well as stable isotope labeling was performed. The ion mobility separation of parent ions together with the characteristic fragmentation for reduced isomers providing unique product ions allowed us to identify isomers present in a mixture and determine their mutual isomeric ratio. The determination used simple recalculation of arrival time distribution areas of unique ions to areas of deprotonated molecules. Mass spectrometry data were confirmed by nuclear magnetic resonance spectroscopy. Copyright © 2015 John Wiley & Sons, Ltd.