An analysis on the flow resistance in the entrance region of diffuse laminar flow between two parallel spherical surfaces

In this paper, V.V. Golubef method is first extended to the diffuse laminar flow between two parallel spherical surfaces. With the boundary layer motion equation in spherical coordinates, we derive the momentum integral equation together with the energy integral equation for the laminar boundary layer of the entrance region between two parallel spherical surfaces. And then by applying Picards gradually approaching method for the momentum integral equation, we get the approximate expression which the entrance region length varies with the thickness of boundary layer. In the end every coefficient of entrace region effect is analyzed and calculated. Projects Suported by the Science Fund of the Chinese Academy of Sciences.     

Elastic impact on finite Timoshenko beam

In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The initial values of the impact forces are exactly determined by the momentum conservation law. The propagation of the longitudinal and transverse waves along the beam, especially, the effects of boundary conditions on the characteristics of the reflected waves, are investigated in detail. Some results are compared with those by MSC/NASTRAN.     

Elastic waves in a Timoshenko beam with boundary damping

In this paper we consider a Timoshenko beam with a damping moment applied to one of the endpoints. The elastic waves that develop from two sets of localised initial disturbances in the beam, are simulated and it is shown that properties of the spectrum adequately explain the features of the waves. We also show, using energy calculations, that the boundary moment is significantly more effective in reducing vibrations in the beam in one of the cases under consideration. The so-called second spectrum of the Timoshenko beam plays a prominent part in explaining this phenomenon.     

Chaotic behavior in the nonlinear elastic beam

The dynamic response of the non-linear elastic simply supported beam subjected to axial forces and transverse periodic load is studied. Melnikov method is used to consider the dynamic behavior of the system whose post-buckling path is steady. The effect of the higher order terms in the controlling equation is taken into account. It is found that the fifth-order terms have a great influence on the dynamic behavior of the system. The result shows that there exist either homoclinic orbits or heteroclinic orbits in the system. In this paper, the critical values of the system entering chaotic states are given. The diagram of an example is shown. The project is supported by the National Natural Sciences Foundation of China.     

Complex modes and traveling waves in axially moving Timoshenko beams

Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.     

Impact-induced flexural waves in a Timoshenko beam—Shearographic detection and analysis

The propagation of a flexural wave generated by transverse point impact on a beam between clamped supports is observed using pulsed-laser shearography—the rapid double exposure of holographic film to a laser-illuminated object at different instants of its deformation. Sequential images are captured via a camera which has a thin, wedge-shaped prism placed in front of half its lens. This prism produces another image which is sheared in the direction of the wedge, with a net result that two images—sheared with respect to each other—are recorded by the camera. When the structure is deformed and a second exposure is superimposed on the first, a fringe pattern is produced; this becomes visible when observed via high-pass optical Fourier filtering. The fringes represent loci of constant out-of-plane displacement derivatives with respect to the direction of image shear. These are integrated to yield the instantaneous out-of-plane displacement induced by the flexural wave. Adjustment of the delay between the first and second laser pulses produces sequential displacement data on the propagating wave. This is compared with theoretical results based on Timoshenko beam analysis employing the method of characteristics to compute displacement values. The experimental data compare favorably with theoretical predictions, substantiating the use of sheariographic techniques for stress-wave studies.     

Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

Considering the effect of crack gap,the bending deformation of the Timoshenko beam with switching cracks is studied.To represent a crack with gap as a nonlinear unidirectional rotational spring,the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function.A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks.Three examples of bending of the Timoshenko beam are presented.The influence of the beam’s slenderness ratio,the crack’s depth,and the external load on the crack state and bending performances of the cracked beam is analyzed.It is revealed that a cusp exists on the deflection curve,and a jump on the rotation angle curve occurs at a crack location.The relation between the beam’s deflection and load is bilinear,each part corresponding to an open or closed state of crack,respectively.When the crack is open,flexibility of the cracked beam decreases with the increase of the beam’s slenderness ratio and the decrease of the crack depth.The results are useful in identifying non-destructive cracks on a beam.     

Modeling and bifurcation analysis of the centre rigid-body mounted on an external Timoshenko beam

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Analysis method of chaos and sub-harmonic resonance of nonlinear system without small parameters

The Melnikov method is important for detecting the presence of transverse homoclinic orbits and the occurrence of homoclinic bifurcations.Unfortunately,the traditional Melnikov methods strongly depend on small parameters,which do not exist in most practical systems.Those methods are limited in dealing with the systems with strong nonlinearities.This paper presents a procedure to study the chaos and sub-harmonic resonance of strongly nonlinear practical systems by employing a homotopy method that is used ...     

Active control of flexural waves in a phononic crystal beam with staggered periodic properties

The band gaps of a phononic crystal beam with staggered periodic structure are investigated. The periodic system consists of a pure elastic (i.e. PMMA) matrix beam and some piezoelectric (i.e. PZT) patches with coupling between the mechanical–electrical components. The PZT patches connected by negative capacitance circuits are applied to function as the active control system. Based on the condition at the interface between adjacent unit cells, the transfer matrix and localization factor are derived. The influence of the degree of interlacing and negative capacitance circuits are discussed. The numerical results show that another band gap can be generated by the staggered periodic structure of PZT patches. The widths and locations of the band gaps can be changed by the degree of interlacing.     

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.     

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is investigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.     

Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆｒａｃｔｉｏｎａｌｄｅｒｉｖａｔｉｖｅｃｏｎｓｔｉｔｕｔｉｖｅｍｏｄｅｌｓｏｆａｖｉｓｃｏｅｌａｓｔｉｃｍａｔｅｒｉａｌｗｅｒｅｐｒｏｐｏｓｅｄｂｙＧｅｍｅｎｔａｔｆｉｒｓｔｉｎ 1 93 0’ｓ[1].Ｓｉｎｃｅ 1 980’ｓ,ｔｈｅｍｏｄｅｌｓｈａｖｅｒｅｃｅｉｖｅｄｉｎｃｒｅａｓｉｎｇａｔｔｅｎｔｉｏｎ[2 ,3].Ｏｎｌｙａｆｅｗｐａｒａｍｅｔｅｒｓａｒｅｃｏｎｔａｉｎｅｄｉｎｔｈｅｍｏｄｅｌｓａｎｄｔｈｅｍｏｄｅｌｓｃａｎｄｅｓｃｒｉｂｅｔｈｅｍｅｃｈａｎｉｃａｌｃｈａｒａｃ…     

Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model

This paper deals with the free vibration analysis of circular alumina （Al2O3） nanobeams in the presence of surface and thermal effects resting on a Pasternak foun- dation. The system of motion equations is derived using Hamilton＇s principle under the assumptions of the classical Timoshenko beam theory. The effects of the transverse shear deformation and rotary inertia are also considered within the framework of the mentioned theory. The separation of variables approach is employed to discretize the governing equa- tions which are then solved by an analytical method to obtain the natural frequencies of the alumina nanobeams. The results show that the surface effects lead to an increase in the natural frequency of nanobeams as compared with the classical Timoshenko beam model. In addition, for nanobeams with large diameters, the surface effects may increase the natural frequencies by increasing the thermal effects. Moreover, with regard to the Pasternak elastic foundation, the natural frequencies are increased slightly. The results of the present model are compared with the literature, showing that the present model can capture correctly the surface effects in thermal vibration of nanobeams.     

任意边界条件下Timoshenko梁及其修正理论的自振特性分析

提出一种求解任意边界条件下经典Timoshenko梁以及修正Timoshenko梁自振频率和振型的新方法。利用改进的傅立叶级数消除传统傅立叶级数的边界不收敛问题,然后通过Rayleigh-Ritz法导出Timoshenko梁的拉格朗日泛函,根据Hamilton原理将原问题转化为求解矩阵广义特征值问题。通过与解析解对比,本文采用的方法具有较好的收敛性以及较高的计算精度;通过数值计算发现,经典Timoshenko梁的自振频率略高于修正的Timoshenko梁,随着振型阶数的提高,经典Timoshenko梁的计算结果逐渐偏离文献解和有限元结果,而修正的Timoshenko梁能够保持较好的一致性;对于不同边界条件下修正Timoshenko梁的计算结果均能与有限元的计算结果吻合得很好。最后运用MATLAB编程软件将程序设计为App,对于不同情形的梁只需要修改参数即可,可为实际工程提供高效便捷的计算方案和可靠理论依据。     

Quantitative guided wave testing by applying the time reversal principle on dispersive waves in beams

A method for the localization and characterization of defects in waveguide-like structures is presented in this paper. In contrast to traditional ultrasound and guided wave techniques, a broadband signal is used to enforce strong dispersion of the flexural wave mode. Since dispersion is well compensated in a time reversal experiment we use a time reversal numerical simulation to identify the origin and the original shape of the flexural wave excited at a local non-uniformity due to mode conversion. Limitations of the time reversal process for broadband signals due to multimode and evanescent behavior of guided waves are discussed and eliminated using a Timoshenko beam model. The resulting novel process which uses both flexural and longitudinal wave information allows detection, localization and size estimation of several defects in a beam with only a single measurement. The method proposed is experimentally validated on rectangular solid beams and cylindrical hollow beams with notches of different sizes and positions. Up to three notches could be localized from one measurement, with a maximum error of 3% with respect to the propagation distance. The size was accurately predicted for notches as small as 0.5 mm depth or 8.3% of the cross section, using a generalized spring model of a crack.     

Nonlinear faraday waves in a parametrically excited circular cylindrical container

IntroductionIn 1 83 1 ,Faraday[1]reportedhisexperimentalobservationofsurfacewavesindifferentfluidscoveringahorizontalplatesubjectedtoaverticalvibration ,andheobservedthesurfacestandingwavesoffluidsliketheteethofaveryshortcoarsecomb .Heremarksthatthesesurfacewaveshaveafrequencyequaltoonehalfthatoftheexcitation .ThisisthefamousFaradayexperiment.WedesignatethosefluidsurfacewavesformedbyverticallyexcitationandhaveafrequencyequaltoonehalfthatoftheexcitationasFaradaywaves.FollowingthisproblemMatth…     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)