Estimation of structural wave numbers from spatially sparse response measurements

A method is presented for estimating the complex wave numbers and amplitudes of waves that propagate in damped structures, such as beams, plates, and shells. The analytical basis of the method is a wave field that approximates response measurements in an aperture where no excitations are applied. At each frequency, the method iteratively adjusts wave numbers to best approximate response measurements, using wave numbers at neighboring frequencies as initial estimates in the search. In comparison to existing methods, the method generally requires far fewer measurement locations and does not require evenly spaced locations. The number of locations required by the method scales with the number of waves that propagate in the structure, whereas the number of locations required by existing methods scales with the minimum wavelength. In addition, the method allows convenient inclusion of the analytic relationships between wave numbers that exist for flexural vibrations of beams and plates. Advantages of the method are illustrated by an example in which a beam is excited by a transverse force at one end. Using analytic data and experimental measurements, the method produces a wave field that matches response measurements to within 1 percent. One interesting feature of the new method is that, when applied to analytic data, it supplies more robust wave number estimates using responses at unevenly spaced locations.     

New multiplexed system for synchronous measurement of out-of-plane deformation and two orthogonal slopes

We propose a novel system for synchronous measurement of out-of-plane deformation and two orthogonal slopes using a single camera. The linearly polarized reference beam introduced by an optical fiber interferes with the unpolarized object beam to measure the out-of-plane deformation. A modified Mach—Zehnder interferometer is used to measure the two orthogonal slopes of the out-of-plane deformation. One of the object beams of the Mach—Zehnder interferometer is an unpolarized beam, and the other object beam is split into two orthogonal linearly polarized object beams by a polarizing prism. The two beams are orthogonally polarized. Hence, they will not interfere with each other. The two polarized beams respectively interfere with the unpolarized beam to simultaneously measure the two orthogonal slopes of the out-of-plane deformation. In addition, the imaging lens and apertures are respectively placed in three optical paths to independently control the carrier frequencies and shearing amounts. The effectiveness of this method can be proved by measuring two pressure-loaded circular plates.     

Residual vibration reduction of a flexible beam deploying from a translating hub

This paper presents a method for reducing the residual vibration of a flexible beam deployed from a translating hub. Whereas previous studies have discussed reducing vibration in translating constant-length beams, this study investigates a vibration reduction method for translating beams of variable length. The partial differential equation of motion for a translating beam is derived and transformed into a variational equation. Based on the discretized equations from the variational equation, the dynamic responses of the flexible beam under translation are analyzed. A vibration reduction method is proposed that is effective for both constant- and variable-length deploying translating beams.     

An analytical method for free vibration analysis of functionally graded beams with edge cracks

In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler–Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures.     

Structural synthesis of sandwich beams with outer layers of box-section

It is proved by model measurements that, for sandwich beams constructed from two rectangular tubes and a damping layer glued between them, the following calculation methods can be applied. Static bending and shear stresses as well as deflections of simply supported beams may be calculated by Allen's formulae for sandwich beams with flexurally stiff faces. The first eigenfrequency and the loss factor can be determined by using the diagrams given in reference [1]. For the loss factors Ungar's formula gives a suitable approximation. A minimum cost design procedure is presented for a sandwich beam with constant cross-section. The unknown dimensions of the cross-section are determined which satisfy the design constraints and minimize the material costs. In a numerical example, constraints relating to the maximal dynamic stresses and deflection as well as local buckling of plate elements are considered. In the optimization the backtrack combinatorial discrete programming method is applied. A numerical comparison shows that the material costs of a sandwich beam are lower than those of a homogeneous box one.     

用线偏振光产生可调矢量椭圆空心光束

提出了利用π相位板产生矢量空心光束的新方案,两个偏振方向互相垂直的线偏振光波分别通过π相位板调制后进行强度叠加,得到椭圆空心光束,用矩形光阑调节相位板的几何尺寸,可以实时调节椭圆的离心率;调节π相位板的方位,能够实现径向矢量空心光束到角向矢量空心光束的转换。分析、讨论了方案的可行性和在原子光学中的潜在应用,结果表明:本方案在原子光学中有很好的应用前景。     

Phase singularities of nonparaxial cosh-Gaussian vortex beams diffracted by a rectangular aperture

Analytical expression for the propagation of nonparaxial cosh-Gaussian (ChG) beams diffracted by a rectangular aperture is derived based on the vector Rayleigh-Sommerfeld diffraction integrals and expansion of the aperture window function into a finite sum of complex Gaussian functions, and used to study the phase singularities of nonparaxial diffracted ChG vortex beams. The pair creation, annihilation, motion of phase singularities in the diffracted field and the dependence of position and number of phase singularities on the aperture and beam parameters, as well as on the beam nonparaxiality are illustrated by numerical examples.     

Transverse vibrations of pre-twisted sandwich beams

A set of equations of motion governing the bending and extensional displacements of a pre-twisted sandwich beam of rectangular cross-section are derived by using Hamilton's principle. The middle viscoelastic core is assumed to deform mainly through the classical shearing mechanism. The eigenvalues and loss factors of simply supported pre-twisted sandwich beams are computed by using the variational method. Analysis of the results revealed that pre-twisting the beam increases the real part of the eigenvalue by as much as 20% while reducing the loss factor by as much as 30 %. The loss factor of very soft, thickcored beams is especially sensitive to even small angles of pre-twist: e.g., a 22· 5° pre-twist may reduce the loss factor by as much as 80%. The effect of pre-twist is, however, shown not to be appreciable for soft, thin-cored beams. In any case, pre-twisting of the beam has a detrimental effect on the maximum loss factor that one can obtain for a specific size of the beam when only the shear parameter of the beam is changed.     

Flexural wave motion in beam-type disordered periodic systems: Coincidence phenomenon and sound radiation

This paper deals with flexural wave motion in uniform beam-type periodic systems whose repeating units are identical finite beams with multiple beam-length disorders. A general expression derived for the propagation constants has been employed to study its variation with frequency for a beam system having 4-span disordered repeating units. This is helpful in understanding flexural wave motion in disordered periodic beams. Free flexural waves have been studied as wave groups consisting of a large number of harmonic components of different wavelengths, phase velocities and directions. Phase velocities have been computed and plotted for different frequencies in the propagation zones in which the free waves progress without attenuation. This has been found to be useful in understanding and predicting the coincidence phenomenon in disordered periodic beams under convected pressure field loading. The excitation of wave groups in disordered periodic beam-type systems by a slow (subsonic) convecting pressure field can include fast (supersonic) moving flexural wave components which can radiate sound. It has been pointed out that sound radiation from a disordered periodic beam (or plate) can be quite different as compared to that from a periodic beam under similar convected pressure field loading.     

Design of phase plates for shaping partially coherent beams by simulated annealing

Taking the Gaussian Schell-model beam as a typical example of partially coherent beams, this paper applies the simulated annealing （SA） algorithm to the design of phase plates for shaping partially coherent beams. A flow diagram is presented to illustrate the procedure of phase optimization by the SA algorithm. Numerical examples demonstrate the advantages of the SA algorithm in shaping partially coherent beams. An uniform flat-topped beam profile with maximum reconstruction error RE 〈 1.74% is achieved. A further extension of the approach is discussed.     

基于非球面透镜的光分束器设计

 针对目前532 nm波长光的分束技术在光电测控系统（如激光干涉测速）等领域的广泛应用,提出一种光分束器的设计方法,此分束器可将光纤入射的532 nm绿光分成接近等比的多束光（包括一分二、一分三、一分四）。此系统首先用非球面透镜将光纤入射的光束准直,再通过分光片分光后用同样的透镜将光束耦合进光纤,达到了77%的通光效率,附加损耗约1 dB。详细介绍了非球面透镜的设计、分束器的结构、装配及其实验结果,并对实验结果以及研制过程中影响分束器效率的各种因素进行分析。     

Structural waveguide behaviour of a beam–plate system

Structures which have a constant cross-section normal to a longitudinal axis can be considered as waveguides in which vibration can propagate in the form of various waves in the longitudinal direction. The dynamic behaviour of such systems can be found by using a Fourier transform approach in terms of wavenumbers in the longitudinal direction. Analytical solutions are available for simple, infinitely long waveguides, whereas for more complex waveguides numerical approaches have been developed using finite element techniques to describe the cross-section. In the present paper an analytical Fourier transform approach is used to find the dynamic behaviour of a system consisting of two parallel beams coupled by a plate, when a point force is applied to one of the beams. Multiple waves occur in the longitudinal direction, the number of waves depending on the number of modes of the equivalent cross-section. However, the motion of the driven beam is shown to be dominated by the contribution from only one or two waves at each frequency, these having wavenumbers closest to that of the uncoupled beam. The motion of the plate is also shown to be dominated by these wavenumbers for excitation on the beam. Experimental results are obtained on beam–plate–beam systems with identical and non-identical beams, which show good agreement with the predictions. In particular, these confirm that the plate response is dominated by waves with wavenumbers in the beam direction that follow those of the excited beam.     

Resonance frequencies and stability of two flexible permanent magnetic beams facing each other

This paper aims at investigating the interaction of two flexible permanent magnet beams facing each other. The governing equations of motion are obtained based on the Euler–Bernoulli beam model along with Hamilton's principle. Assuming that the beams' tips are far enough, each magnet beam is considered as a series of dipole segments and the external force and moment distributions over each beam due to the magnetic field of the other one is calculated in the deformed configuration. The transverse deflections of the beams are written as series expansions of the mode shapes of an unloaded cantilever beam and the Galerkin method is applied to determine the stability and resonance frequencies. Using the obtained model, the stability regions of the beams for both cases of opposite poles and same poles facing each other are obtained. Also the effect of magnet's strength and flexibility of the beams on the stability boundaries are illustrated.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Equilibrium and stability of off-axis periodically focused particle beams

A general equation for the centroid motion of free, continuous, intense beams propagating off axis in solenoidal periodic focusing fields is derived. The centroid equation is found to be independent of the specific beam distribution and may exhibit unstable solutions. A new Vlasov equilibrium for off-axis beam propagation is also obtained. The properties of the equilibrium and the relevance of centroid motion to beam confinement are discussed.     

POWER FLOW ANALYSIS OF REINFORCED BEAM-PLATE COUPLED STRUCTURES

A power flow analysis method was developed to predict the vibrational responses of reinforced beam-plate coupled structures in frequencies ranging from medium to high. This analysis method was successfully applied to simply supported rectangular plates reinforced with multi-parallel beams by utilizing the power flow coupling relationships at the beam-plate junctions and the zero intensity conditions at the plate edges. Through numerical simulations, the power flow energy density and intensity fields of two different plates, consisting of single beam and eight beams, were compared with those of classical displacement solutions, and they show good agreement in terms of the global decay and the attenuation patterns of the energy density.     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

Vibration transmission through plate/beam structures typical of lightweight buildings: Applicability and limitations of fundamental theories

This paper examines the modelling of vibration transmission through plate/beam structures typical of lightweight buildings. Key experiments have been carried out on simple structures to identify the applicability and limitations of fundamental theories. The systems tested included a single plate connected along its centre to a beam, two parallel plates attached along their centre to a beam (plates opposite or offset), and four plates connected along their edges to a beam. The analysis focused in particular on the applicability of modelling a beam as a one-dimensional element in point connected systems (widely spaced screws in terms of bending wavelength). Statistical energy analysis (SEA) was the framework of analysis used for all predictions, but the theories examined were independent from SEA. The results obtained indicate that simple point models are only applicable to the single plate and beam system, and to the parallel opposite plates connected along their centre to a beam; even then, the applicability of such models is limited to low and mid frequencies (below 2 kHz for the structures tested). Transmission between two parallel plates connected to a beam with screws closely spaced was also examined, and it was found that rigid and pinned line predictions can provide limits for transmission between panels on the same side of a wall (where junctions with shallow beams tend to behave rigidly, whilst junctions with deep beams are better modelled as pinned).     

贝塞尔-高斯涡旋光束相干合成研究

基于相干合成技术,提出了对特定离散空间分布的高斯光束阵列加载离散涡旋相位生成二阶贝塞尔-高斯(Bessel-Gaussian,BG)涡旋光束的方案.利用干涉法、桶中功率和相关系数对合成BG涡旋光束的拓扑荷、光束质量进行了定量评价及参数优化.结果表明:基于相干合成技术能够产生特定的目标BG涡旋光束,阵列子光束紧密排布时合成BG光束的光束质量更高.该方法的提出对于其他涡旋光束的产生或者涡旋光束功率的提高具有一定的参考意义.     

基于双开口谐振环超表面的宽带太赫兹涡旋光束产生

提出了一种基于双开口谐振环单元结构超表面的太赫兹宽带涡旋光束产生器.该结构由金属-电介质两层构成,位于顶层的是基于双开口谐振环单元结构的超表面,底层为介质层.对单元结构阵列进行数值仿真,圆偏振的入射光可以被转换成相应的交叉偏振透射光,通过旋转表层金属谐振环,可以控制交叉偏振透射光具有相同的振幅和不同的相位.这些单元结构按照特定的规律排列,可以形成用以产生不同拓扑荷数的涡旋光束的涡旋相位板.以拓扑荷数1和2为例,设计了两种涡旋相位板,数值分析了圆偏振波垂直入射到该涡旋相位板生成交叉圆偏振涡旋光束的特性.结果表明,在1.39—1.91 THz的频率范围内产生了比较理想的不同拓扑荷数的涡旋光束,且透过率高于20%,最高可达到24%,接近单层透射式超表面的理论极限值.