Modeling and Free Vibration Behavior of Rotating Composite Thin-walled Closed-section Beams with SMA Fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy（SMA） fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton＇s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin＇s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.

Nonlinear coupling modeling and dynamics analysis of rotating flexible beams with stretching deformation effect

Dynamic coupling modeling and analysis of rotating beams based on the nonlinear Green-Lagrangian strain are introduced in this work.With the reservation of the axial nonlinear strain,there are more coupling terms for axial and transverse deformations.The discretized dynamic governing equations are obtained by using the finite element method and Lagrange’s equations of the second kind.Time responses are conducted to compare the proposed model with other previous models.The stretching deformation due to rotating motion is observed and calculated by special formulations under dynamic equilibrium.The stretching deformation and the change of the associated equilibrium position are taken into account to analyze the free vibration and frequency response of the rotating beams.Analytical and numerical comparisons show that the proposed model can provide reliable results,while the previous models may lead to imprecise results,especially in high-speed conditions.

Novel Method for Spatial Angle Measurement Based on Rotating Planar Laser Beams

Spatial angle measurement, especially the measurement of horizontal and vertical angle, is a basic method used for industrial large-scale coordinate measurement. As main equipments in use, both theodolites and laser trackers can provide very high accuracy for spatial angle measurement. However, their industrial applications are limited by low level of automation and poor parallelism. For the purpose of improving measurement efficiency, a lot of studies have been conducted and several alternative methods have been proposed. Unfortunately, all these means are either low precision or too expensive. In this paper, a novel method of spatial angle measurement based on two rotating planar laser beams is proposed and demonstrated. Photoelectric receivers placed on measured points are used to receive the rotating planner laser signals transmitted by laser transmitters. The scanning time intervals of laser planes were measured, and then measured point＇s horizontal/vertical angles can be calculated. Laser plane＇s angle parameters are utilized to establish the abstract geometric model of transmitter. Calculating formulas of receiver＇s horizontal/vertical angles have been derived. Measurement equations＇ solvability conditions and judgment method of imaginary solutions are also presented after analyzing. Proposed method for spatial angle measurement is experimentally verified through a platform consisting of one laser transmitter and one optical receiver. The transmitters used in new method are only responsible for providing rotating light plane signals carrying angle information. Receivers automatically measure scanning time of laser planes and upload data to the workstation to calculate horizontal angle and vertical angle. Simultaneous measurement of multiple receivers can be realized since there is no human intervention in measurement process .Spatial angle measurement result indicates that the repeatable accuracy of new method is better than 10＂. Proposed method can improve measurement＇s automation degree and speed while ensuring measurement accuracy.

Coupled flapwise-chordwise-axial-torsional dynamic responses of rotating pre-twisted and inclined cantilever beams subject to the base excitation

A rotating pre-twisted and inclined cantilever beam model (RPICBM) with the flapwise-chordwise-axial-torsional coupling is established with the Hamilton principle and the finite element (FE) method. The effectiveness of the model is verified via comparisons with the literatures and the FE models in ANSYS. The effects of the setting and pre-twisted angles on the dynamic responses of the RPICBM are analyzed. The results show that:(i) the increase in the setting or pre-twisted angle results in the increases in the first-order flapwise and torsional frequencies while the decrease in the first-order chordwise frequency under rotating conditions;(ii) a positive/negative setting angle leads to a positive/negative constant component, while a positive/negative pre-twisted angle leads to a negative/positive constant component;(iii) when the rotation speed is non-zero, the pre-twisted angle or non-zero setting angle will result in the coupled flapwise-chordwiseaxial- torsional vibration of the RPICBM under axial base excitation.

GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.

Dynamic analysis of a rotating tapered cantilever Timoshenko beam based on the power series method

The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to inves- tigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method(NS-PIM),for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection.This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations,but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method(FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.

Flutter analysis of rotating beams with elastic restraints

The aeroelastic stability of rotating beams with elastic restraints is investigated.The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling.The Greenberg aerodynamic model is used to describe the unsteady aerodynamic forces.The additional centrifugal stiffness effect and elastic boundary conditions are considered in the form of potential energy.A modified Fourier series method is used to assume the displacement field function and solve the governing equation.The convergence and accuracy of the method are verified by comparison of numerical results.Then,the flutter analysis of the rotating beam structure is carried out,and the critical rotational velocity of the flutter is predicted.The results show that the elastic boundary reduces the critical flutter velocity of the rotating beam,and the elastic range of torsional spring is larger than the elastic range of linear spring.

Propagation of rotating elliptical Gaussian beams from right-handed material to left-handed material

By applying the ABCD matrix method, we report the propagating properties of the rotating elliptical Gaussian beams（REGBs） from the right-handed material（RHM） to the left-handed material（LHM）. Based on the propagation equation, we obtain the intensity distributions of the REGBs during the propagation. It is found that the rotating direction of the REGBs is opposite in the RHM and the LHM, and the rotation angles tend to be π /2 as the propagation distance is long enough.Then we analyze the relationship between the refractive index and the rotating velocity. Furthermore, the energy flow and the angular momentum（AM） of the REGBs which can rotate are also obtained.

Nonlinear Free Vibration Analysis of Thin-walled Curved Beam with Non-symmetric Open Cross Section

A finite element formulation was presented for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies were derived by the virtual principle. The energy function includes the effect of fiexural-torsional coupling, the torsion warping and the shear centre location. For finite element analysis, cubic polynomials were utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem was solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.

A new finite element of spatial thin-walled beams

Based on the theories of Timoshenko＇s beams and Vlasov＇s thin-walled members, a new spatial thin-walled beam element with an interior node is developed. By independently interpolating bending angles and warp, factors such as transverse shear deformation, torsional shear deformation and their Coupling, coupling of flexure and torsion, and second shear stress are considered. According to the generalized variational theory of Hellinger-Reissner, the element stiffness matrix is derived. Examples show that the developed model is accurate and can be applied in the finite element analysis of thinwalled structures.

An advanced wind vector retrieval algorithm for the rotating fan-beam scatterometer

The rotating fan-beam scatterometer （RFSCAT） is a new type of satellite scatterometer that is proposed approximately 10 a ago. However, similar to other rotating scatterometers, relatively larger wind retrieval errors occur in the nadir and outer regions compared with the middle regions of the swath. For the RFSCAT with the given parameters, a wind direction retrieval accuracy decreases by approximately 9 in the outer regions compared with the middle region. To address this problem, an advanced wind vector retrieval algorithm for the RFSCAT is presented. The new algorithm features an adaptive extension of the range of wind direction for each wind vector cell position across the whole swath according to the distribution histogram of a retrieved wind direction bias. One hundred orbits of Level 2A data are simulated to validate and evaluate the new algorithm. Retrieval experiments demonstrate that the new advanced algorithm can effectively improve the wind direction retrieval accuracy in the nadir and outer regions of the RFSCAT swath. Approximately 1.6 and 9 improvements in the wind direction retrieval are achieved for the wind vector cells located at the nadir and the edge point of the swath, respectively.

Elastic buckling analysis and optimization of thin-walled bamboo-like pipe beam subjected to pure bending

In this paper,a stiffening design imitating the bamboo node is proposed for weight reduction of the long composite pipe beam subjected to bending load.The distribution of bamboo nodes can efficiently suppress the ovalization of the section,thus significantly improving the bending resistance of the bamboo.Based on this principle,ring stiffeners are proposed to be fixed to the pipe beam,making the long beam equivalent to the combination of a series of short pipes that suffered less section ovalization.A database of the optimal laminate orientations for different normalized lengths is obtained through optimizations,where the discreteness of the ply count is considered.Based on this database,weight optimizations are conducted,and the optimal designs of beams with and without stiffeners are obtained and compared.The comparison results show that the proposed bamboo-like stiffened beam not only regains a near-linear load–displacement relationship,but also reduces the weight by up to 16%under the same buckling load.In addition,it is found that for the pipe beams with radius-to-thickness ratios of more than 18,increasing the radius leads to a decrease in elastic buckling resistance when the weight remains a constant,which is opposite to the design for strength and stiffness.The model and database developed in this paper can provide a reference for weight reduction design and weight estimation for composite pipe beams.

Thickness distribution of large scale metallic sheet by EB-PVD on rotating substrate

A two-dimensional kinetic Monte-Carlo(KMC) method was used to approach the thickness distribution of large scale metallic sheet deposited by electron beam physical vapor deposition(EB-PVD) on rotating substrate. The KMC model involves incident atom attachment, adatom diffusion and adatom detachment. The effective deposition rate and effective incident angle along substrate radial were studied as influencing factors of sheet thickness distribution. The KMC simulation results indicate that incident angle is a very important factor to affect the sheet thickness distribution as well as theory deposition mass. The experiments results show that the KMC model can predict the thickness distribution of large scale sheet deposited by EB-PVD on rotating substrate.

Theoretical analysis of the optical rotational Doppler effect under atmospheric turbulence by mode decomposition

The optical rotational Doppler effect associated with orbital angular momentum provides a new means for rotational velocity detection.In this paper,we investigate the influence of atmospheric turbulence on the rotational Doppler effect.First,we deduce the generalized formula of the rotational Doppler shift in atmospheric turbulence by mode decomposition.It is found that the rotational Doppler signal frequency spectrum will be broadened,and the bandwidth is related to the turbulence intensity.In addition,as the propagation distance increases,the bandwidth also increases.And when C_(n)^(2)≤5×10^(-15)m^(-2/3)and 2z≤2 km,the rotational Doppler signal frequency spectrum width d and the spiral spectrum width d_(0)satisfy the relationship d=2d_(0-1).Finally,we analyze the influence of mode crosstalk on the rotational Doppler effect,and the results show that it destroys the symmetrical distribution of the rotational Doppler spectrum about 2l·Ω/2π.This theoretical model enables us to better understand the generation of the rotational Doppler frequency and may help us better analyze the influence of the complex atmospheric environment on the rotational Doppler frequency.

A nonlinear explicit dynamic GBT formulation for modeling impact response of thin-walled steel members

A nonlinear explicit dynamic finite element formulation based on the generalized beam theory(GBT)is proposed and developed to simulate the dynamic responses of prismatic thin-walled steel members under transverse impulsive loads.Considering the rate strengthening and thermal softening effects on member impact behavior,a modified Cowper-Symonds model for constructional steels is utilized.The element displacement field is built upon the superposition of GBT cross-section deformation modes,so arbitrary deformations such as cross-section distortions,local buckling and warping shear can all be involved by the proposed model.The amplitude function of each cross-section deformation mode is approximated by the cubic non-uniform B-spline basis functions.The Kirchhoff s thin-plate assumption is utilized in the construction of the bending related displacements.The Green-Lagrange strain tensor and the second Piola-Kirchhoff(PK2)stress tensor are employed to measure deformations and stresses at any material point,where stresses are assumed to be in plane-stress state.In order to verify the effectiveness of the proposed GBT model,three numerical cases involving impulsive loading of the thin-walled parts are given.The GBT results are compared with those of the Ls-Dyna shell finite element.It is shown that the proposed model and the shell finite element analysis has equivalent accuracy in displacement and stress.Moreover,the proposed model is much more computationally efficient and structurally clearer than the shell finite elements.

Distortional buckling analysis of steel-concrete composite box beams considering effect of stud rotational restraint under hogging moment

Restrained distortional buckling is an important buckling mode of steel-concrete composite box beams(SCCBB)under the hogging moment.Rotational and lateral deformation restraints of the bottom plate by the webs are essential factors affecting SCCBB distortional buckling.Based on the stationary potential energy principle,the analytical expressions for the rotational restraint stiffness(RRS)of the web upper edge as well as the RRS and the lateral restraint stiffness(LRS)of the bottom plate were derived.Also,the SCCBB critical moment formula under the hogging moment was derived.Using twenty specimens,the theoretical calculation method is compared with the finite-element method.Results indicate that the theoretical calculation method can effectively and accurately reflect the restraint effect of the studs,top steel flange,and other factors on the bottom plate.Both the RRS and the LRS have a nonlinear coupling relationship with the external loads and the RRS of the web’s upper edge.Under the hogging moment,the RRS of the web upper edge has a certain influence on the SCCBB distortional buckling critical moment.With increasing RRS of the web upper edge,the SCCBB critical moment increases at first and then tends to be stable.

Reference Coordinate System of Nonlinear Beam Element Based on the Geometrically Exact Formulation under Large Spatial Rotations and Deformations

Analysis of slender beam structures in a three-dimensional space is widely applicable in mechanical and civil engineering. This paper presents a new procedure to determine the reference coordinate system of a beam element under large rotation and elastic deformation based on a newly introduced physical concept: the zero twist sectional condition, which means that a non-twisted section between two nodes always exists and this section can reasonably be regarded as a reference coordinate system to calculate the internal element forces. This method can avoid the disagreement of the reference coordinates which might occur under large spatial rotations and deformations. Numerical examples given in the paper prove that this procedure guarantees the numerical exactness of the inherent formulation and improves the numerical efficiency, especially under large spatial rotations.

Tunable Optical Rotation with an M-Type Atomic System Using Vortex Beam

We consider the optical rotation of the polarization of a linearly polarized probe field passing through an M-type atomic system by using the interaction between two vortex control fields and optical transitions. We investigate theoretically to generate the spatially dependent structured light with the atoms acting as a spatially varying circular birefringent medium. We show that the polarization and intensity distributions of the vector beam spatially vary by changing the orbital angular momentum (OAM) of the vortex control field.

Experimental Investigation of “Scale Influence on Plastic Rotational Capacity of Reinforced Concrete Beams”

The importance of the geometrical effect in practical design has been evaluated, showing that an overestimation of the actual member rotation is very likely if the available rotation capacity is based on the evaluation of the behavior of the reference members within a limited size range. The increase of ductility with decreasing member size has been interpreted in fracture mechanics of reinforced concrete. In fracture mechanics it’s seen that beams with higher dimensions are brittle, while those with small dimensions are ductile, so it’s important here to clarify if the same material and design concepts can be applied for reinforced concrete beams with different scales. Three point bending test was executed on 20 reinforced concrete beams varying scale and slenderness ratio (where steel ratio being kept constant). The experimental results obtained varying beam slenderness and beam depth will be used to analyze the structural response for a practical construction, taking in consideration the size effect, these beams are normally designed in such a way that the distribution of their internal forces over the transversal section has been calculated as per elastic beam theory, while the beam dimension will be designed as per the ultimate limit state to obtain a ductile response of the reinforced concrete beams which is necessary to guarantee the structural safety [1].