Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

Microcantilevers have recently received widespread attentions due to their extreme applicability and versatility in both biological and non-biological applications. Along this line, this paper undertakes the non-linear vibrations of a piezoelectrically driven microcantilever beam as a common configuration in many scanning probe microscopy and nanomechanical cantilever biosensor systems. A part of the microcantilever beam surface is covered by a piezoelectric layer (typically ZnO), which acts both as an actuator and sensor. The bending vibrations of the microcantilever beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in the piezoelectric materials. The non-linear terms appear in the form of quadratic expression due to presence of piezoelectric layer, and cubic form due to geometrical non-linearities. The Galerkin approximation is then utilized to discretize the equations of motion. In addition, the method of multiple scales is applied to arrive at the closed form solution for the fundamental natural frequency of the system. An experimental setup consisting of a commercial piezoelectric microcantilever attached on the stand of a state-of-the-art microsystem analyzer for non-contact vibration measurement is utilized to verify the theoretical developments. It is found that the experimental results and theoretical findings are in good agreement, which demonstrates that the non-linear modeling framework could provide a better dynamic representation of the microcantilever than the previous linear models. Due to microscale nature of the system, excitation amplitude plays an important role since even a small change in the amplitude of excitation can lead to significant vibrations and frequency shift.     

考虑竖向地震作用的振动控制及其优化研究

目前,MRD结构的分析通常只考虑水平地震动而不考虑竖向地震动的影响,地面的水平运动和竖向运动具有相关性,从而影响控制效果．因此对双向地震耦合作用下MRD结构的理论进行研究,建立动力分析模型并得出运动微分方程．以8层MRD结构为例进行地震反应分析,研究表明,考虑和不考虑竖向地震作用,MRD对结构均有良好的控制作用,但是竖向地震作用的存在,使结构各层的层间位移峰值有不同程度的增加,其增量随着竖向地震作用的增加而增加,因此建议在高烈度地区的建筑物考虑竖向地震作用对结构的影响．建立MRD结构优化设计模型,采用IHGA程序对结构进行优化设计．结果表明, MRD结构在各种工况下的各项地震反应均得到更好的控制．     

Gyromat 2000陀螺经纬仪强制对中装置的研制

Gyromat2000陀螺经纬仪是目前自动化程度和精度最高的电子陀螺经纬仪，为了保证其在强制对中方位边上高精度的测量，必须对仪器的强制对中装置进行研制，该文结合作者实践过程，对该装置的研制和测试工作进行阐述。     

MAXIMAL ELEMENTS OF A FAMILY OF GB-MAJORIZED MAPPINGS IN PRODUCT FC-SPACES AND APPLICATIONS

A new family of GB-majorized mappings from a topological space into a finite continuous topological spaces （in short, FC-space） involving a better admissible set-valued mapping is introduced. Some existence theorems of maximal elements for the family of GB-majorized mappings are proved under noncompact setting of product FCspaces. Some applications to fixed point and system of minimax inequalities are given in product FC-spaces. These theorems improve, unify and generalize many important results in recent literature.     

The Structure Functions of the Velocity and Temperature Fields from the Perspective of Dimensional Scaling

In this paper, dimensional scaling is used to describe the turbulence structure of the velocity and temperature fields in the inertial range and the far dissipation range as well as the intermediate transition range under locally isotropic conditions at sufficiently large Reynolds numbers. This kind of scaling is expressed in a strictly mathematical manner employing dimensional π -invariants analysis. It is shown that in the case of the asymptotic solutions for either the inertial range or the far dissipation range only one π number occurs that has to be considered as a non-dimensional universal constant. This π number may be determined theoretically or/and empirically. In the case of the transition range two π numbers occur. Consequently, a universal function is established that has to be derived theoretically or/and empirically, too. Here, Batchelor's [7] classical interpolation formula for the turbulence structure of the velocity field and the empirical one of Stolovitzky et al. [59], both may serve as universal functions, are compared with the results provided by numerical solutions of Kolmogorov′s [32] structure equation for the velocity field. It is shown that these interpolation formulae match not only the asymptotic solutions of the inertial range and the far dissipation range, respectively, but also these numerical results in an excellent manner. The former may be considered as necessary condition and the latter as sufficient condition. In the case of the temperature field results of the corresponding universal function are predicted using Yaglom's [63] structure equation. These results also match the corresponding asymptotic solutions of both the inertial range and the far dissipation range. However, in contrast to the case of the velocity field, the predicted universal function for the temperature field may notably overshoot its asymptotic solution for the inertial range. This overshooting occurs in the transition range and may be considered as an analogue to the so-called Hill ‘bump’ that usually occurs in the high-wave number portion of the temperature spectrum.     

Geometric Singular Perturbations for Multiple Turning Points: Invariant Manifolds and Exchange Lemmas

In this paper, we consider singularly perturbed systems with multiple sets of turning points. Two types of related results contributing to geometric singular perturbation theory are obtained. The first result establishes a new class of invariant manifolds. The second result consists of several exchange lemmas that characterize the evolution of an invariant manifold passing through the vicinity of the slow manifold.Partially supported by NSF Grants DMS-0071931 and DMS-0406998.     

Numerical analysis of free vibrations of a beam with oscillators

The problem of free vibrations of a beam with free ends of variable cross section and mass, from which point masses (oscillators) are suspended by bars, is considered. It is shown that parametric resonances can occur in this oscillating system. Numerical examples showing the efficiency of the calculation method proposed are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 135–144, July–August, 2006.     

Model equations for the Eiffel Tower profile: historical perspective and new results

Model equations for the shape of the Eiffel Tower are investigated. One model purported to be based on Eiffel's writing does not give a tower with the correct curvature. A second popular model not connected with Eiffel's writings provides a fair approximation to the tower's skyline profile of 29 contiguous panels. Reported here is a third model derived from Eiffel's concern about wind loads on the tower, as documented in his communication to the French Civil Engineering Society on 30 March 1885. The result is a nonlinear, integro-differential equation which is solved to yield an exponential tower profile. It is further verified that, as Eiffel wrote, “in reality the curve exterior of the tower reproduces, at a determined scale, the same curve of the moments produced by the wind”. An analysis of the actual tower profile shows that it is composed of two piecewise continuous exponentials with different growth rates. This is explained by specific safety factors for wind loading that Eiffel & Company incorporated in the design of the free-standing tower. To cite this article: P. Weidman, I. Pinelis, C. R. Mecanique 332 (2004).     

基于反应谱值分析的爆破震动破坏评估研究

采用反应谱方法研究了爆破震动信号的振动速度和频率特征,提出了采用反应谱曲线积分值来评估爆破震动破坏效应。通过对289组爆破地震波数据的统计表明,在反应谱曲线的积分值小于5时,爆破地震动不会对所研究的普通民房结构物产生震动破坏效应。