Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Dynamics Analysis of a Parallel Mill-turn Tool Spindle Head Driven by Dual-linear Motors Using Extended Transfer Matrix Method

The hybrid dynamics of multi-rigid-body and multi-flexible-body system becomes the mainstream of multi-body dynamics.Currently there lacks a compact approach to model the hybrid dynamics,especially in modern machine tool application,due to the difficulty of solving the hybrid equations or the limitation of current software when dealing with the hybrid dynamics.The extended transfer matrix method（E-TMM）,which extends elements in three-dimensional space with higher matrixes,is proposed to simplify the modeling process of the hybrid dynamics.The E-TMM modeling approaches of 3 basic elements including 3D vibrant rigid body,joint and flexible body are studied in details.A parallel mill-turn tool spindle head unit driven by dual-linear motors is chosen as a plant to demonstrate the E-TMM modeling process.By using E-TMM,the spindle head unit is simplified as a topological network consisting of the three types of element,i.e.,3D vibrant rigid body,joint and flexible body,including 11 rigid bodies,14 joints and 1 3D-Timoshenko beam.Then the dynamic model of the system can be easily obtained by deducing the element-network by means of state vector transformation.The dynamic characteristics of the spindle head,such as natural frequencies,dynamic flexibility,etc.can be predicted by solving the obtained model.Experiment verification indicates that the E-TMM is valid with enough accuracy in the dynamic analysis of the parallel mill-turn tool spindle head.The E-TMM is capable of modeling the dynamics of machine tool structure with no requirements of deducing and solving the sophisticated differential equations.Moreover,the E-TMM provides a simple and elegant tool for hybrid dynamic analysis in future dynamic design of machine tools.

Transfer matrix method for determination of the natural vibration characteristics of elastically coupled launch vehicle boosters

The analysis of natural vibration characteristics has become one of important steps of the manufacture and dynamic design in the aerospace industry. This paper presents a new scenario called virtual cutting in the context of the transfer matrix method of linear multibody systems closed- loop topology for computing the free vibration characteristics of elastically coupled flexible launch vehicle boosters. In this approach, the coupled system is idealized as a triple-beam system-like structure coupled by linear translational springs, where a non-uniform free-free Euler-Bemoulli beam is used. A large thrust-to-weight ratio leads to large axial accelera- tions that result in an axial inertia load distribution from nose to tail. Consequently, it causes the development of significant compressive forces along the length of the launch vehicle. Therefore, it is important to take into account this effect in the transverse vibration model. This scenario does not need the global dynamics equations of a system, and it has high computational efficiency and low memory requirements. The validity of the presented scenario is achieved through com- parison to other approaches published in the literature.

Carbon matrix effects on the micro-structure and performance of Pt nanowire cathode prepared by decal transfer method

High performance cathode for polymer electrolyte membrane fuel cell was prepared by depositing Pt nanowires in a carbon matrix coated on a substrate, and using decal transfer method to fabricate the membrane electrode assembly. The effects of carbon and ionomer contents on the electrode micro-structure and fuel cell performance are investigated by physical characterization and single cell testing. The Pt nanowires are gradient distributed across the cathode thickness, and more Pt exists near the membrane. Both the carbon and ionomer contents can affect the Pt nanowires distribution and aggregation. In addition, the carbon loading dominates the transport distance of gas and proton, and the ionomer content affects the triple phase boundaries and porosity in the cathode. The optimal structure of Pt nanowire cathode is obtained at 0.10 mg·cm^-2 carbon loading and 10 wt% ionomer.

Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space

In this paper, by defining new state vectors and developing new transfer matrices of various elements mov- ing in space, the discrete time transfer matrix method of multi-rigid-flexible-body system is expanded to study the dynamics of multibody system with flexible beams moving in space. Formulations and numerical example of a rigid- flexible-body three pendulums system moving in space are given to validate the method. Using the new method to study the dynamics of multi-rigid-flexible-body system mov- ing in space, the global dynamics equations of system are not needed, the orders of involved matrices of the system are very low and the computational speed is high, irrespec- tive of the size of the system. The new method is simple, straightforward, practical, and provides a powerful tool for multi-rigid-flexible-body system dynamics.

Vibration analysis of fluid- structure interaction in water hammer based on transfer matrix method

In consideration of the problem that the effect of conduit structure on water hammer has been ignored in the classical theory,the Poisson coupling between the fluid and the pipeline was studied and a fourteen-equation mathematical model of fluid-structure interaction(FSI)was developed.Then,the transfer matrix method(TMM)was used to calculate the modal frequency,modal shape and frequency response.The results were compared with that in experiment to verify the correctness of the TMM and the results show that the fluid-structure coupling has a greater impact on the modal frequencies than the modal shape.Finally,the influence on the response spectrum of different damping ratios was studied and the results show that the natural frequency under different damping ratios has changed little but there is a big difference for the pressure spectrum.With the decreasing of damping ratio,the damping of the system on frequency spectrum is more and more significant and the dispersion and dissipation is more and more apparent.Therefore the appropriate damping ratio should be selected to minimize the effects of the vibration of the FSI.The results provide references for the theory research of FSI in the transient process.

Analysis of the flexural vibration of ship's tail shaft by transfer matrix method

A ship's tail shaft has serious flexural vibration due to the cantilevered nature of the propeller's blades.Analysis of the nature frequency of flexural vibration is vital to be able to provide effective shock absorption for a ship's tail shaft.A mathematic model of tail shaft flexural vibrations was built using the transfer matrix method.The nature frequency of flexural vibration for an electrically propelled ship's tail shaft was then analyzed,and an effective method for calculating it was proposed:a genetic algorithm(GA),which calculates the nature frequency of vibration of a system.Sample calculations,with comparisons by the Prohl method under conditions bearing isotropic support,showed this method to be practical.It should have significant impact on engineering design theory.

基于混合FTMM-SEA法的高速列车真空玻璃车窗隔声特性预测与分析

真空玻璃在轻质高隔声方面的性能优势尤为突出,将其用于高速列车车窗是轻量化高速列车车内噪声控制的新型解决方案。将声波在真空玻璃中的传播分解为沿真空层和沿声桥结构的两条独立传播路径,分别使用有限传递矩阵法(FTMM)和统计能量分析法(SEA)计算两条路径的声功率传递系数,建立基于混合FTMM-SEA法的真空玻璃车窗结构隔声量预测模型,并通过实测真空玻璃混响隔声量验证模型的准确性。进一步调查分析声桥结构刚度对耦合板间能量传递、真空玻璃隔声特性的影响。结果表明:减小声桥结构刚度是提高真空玻璃车窗隔声性能的一种有效的方法。与刚性声桥相比,柔性声桥具备更好的能量吸收能力与隔振特性。

The analytical transfer matrix method for quantum reflection

In this paper, the analytical transfer matrix method （ATMM） is applied to study the properties of quantum reflection in three systems： a sech2 barrier, a ramp potential and an inverse harmonic oscillator. Our results agree with those obtained by Landau and Lifshitz [Landau L D and Lifshitz E M 1977 Quantum Mechanics （Non-relativistic Theory） （New York： Pergamon）], which proves that ATMM is a simple and effective method for quantum reflection.

Energy eigenvalues from an analytical transfer matrix method

A detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems. The derivation is strict and complete. The energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method. The anharmonic oscillator potential and the rational potential are two important examples. Checked by numerical techniques, the results for the two potentials by the present method are proven to be exact and reliable.

Dynamic characteristics of a WPC-comparison of transfer matrix method and FE method

To find the difference in dynamic characteristics between conventional monohull ship and wave penetrating catamaran (WPC), a WPC was taken as an object; its dynamic characteristics were computed by transfer matrix method and finite element method respectively. According to the comparison of the nature frequency results and mode shape results, the fact that FEM method is more suitable to dynamic characteristics analysis of a WPC was pointed out, special features on dynamic characteristics of WPC were given, and some beneficial suggestions are proposed to optimize the strength of a WPC in design period.

Controlling the Bandgaps of One-Dimensional TiO2/SiO2, TiO2/SnO2, and SiO2/SnO2 Photonic Crystals Using the Transfer Matrix Method

One-dimensional photonic crystals (1D PhCs) have a unique ability to control the propagation of light waves, however certain classes of 1D oxides remain relatively unexplored for use as PhCs. Specifically, there has not been a comparative study of the three different 1D PhC structures to compare the influence of layer thickness, number, and refractive index on the ability of the PhCs to control light transmission. Herein, we use the transfer matrix method (TMM) to theoretically examine the transmission of 1D PhCs composed of layers of TiO2/SiO2, TiO2/SnO2, SiO2/SnO2, and combinations of the three with various top and bottom layer thicknesses to cover a substantial region of the electromagnetic spectrum (UV to NIR). With increasing layer numbers for TiO2/SiO2 and SiO2/SnO2, the edges became sharper and wider and the photonic bandgap width increased. Moreover, we demonstrated that PhCs with significantly thick TiO2/SiO2 layers had a high transmittance for a wide bandgap, allowing for wide-band optical filter applications. These different PhC architectures could enable a variety of applications, depending on the properties needed.

TRANSFER-MATRIX METHOD FOR SURFACE SPIN WAVES IN A SEMI-INFINITE MAGNETIC SUPERLATTICE WITH A SINGLE-ION ANISOTROPY

Surface spin waves in a semi-infinite magnetre super lattice with a single-ion uniaxial anisotropy are investigated through the transfer mafrix method.The dispersion equations of surface spin wavs are obfained.We find that not all the magnetic superlattice structures can excite the surface spin waves,and that the anisotropy term need not be favorable to the excitation of surface spin wavs,but surely influences the values of the energy of the excited surface spin waves.

Enabling the transfer matrix method to model serial-parallel compliant mechanisms including curved flexure beams

Compliant mechanisms with curved flexure hinges/beams have potential advantages of small spaces,low stress levels,and flexible design parameters,which have attracted considerable attention in precision engineering,metamaterials,robotics,and so forth.However,serial-parallel configurations with curved flexure hinges/beams often lead to a complicated parametric design.Here,the transfer matrix method is enabled for analysis of both the kinetostatics and dynamics of general serial-parallel compliant mechanisms without deriving laborious formulas or combining other modeling methods.Consequently,serial-parallel compliant mechanisms with curved flexure hinges/beams can be modeled in a straightforward manner based on a single transfer matrix of Timoshenko straight beams using a step-by-step procedure.Theoretical and numerical validations on two customized XY nanopositioners comprised of straight and corrugated flexure units confirm the concise modeling process and high prediction accuracy of the presented approach.In conclusion,the present study provides an enhanced transfer matrix modeling approach to streamline the kinetostatic and dynamic analyses of general serial-parallel compliant mechanisms and beam structures,including curved flexure hinges and irregular-shaped rigid bodies.

Radiation heat transfer model for complex superalloy turbine blade in directional solidification process based on finite element method

For the sake of a more accurate shell boundary and calculation of radiation heat transfer in the Directional Solidification(DS) process, a radiation heat transfer model based on the Finite Element Method(FEM)is developed in this study. Key technologies, such as distinguishing boundaries automatically, local matrix and lumped heat capacity matrix, are also stated. In order to analyze the effect of withdrawing rate on DS process,the solidification processes of a complex superalloy turbine blade in the High Rate Solidification(HRS) process with different withdrawing rates are simulated; and by comparing the simulation results, it is found that the most suitable withdrawing rate is determined to be 5.0 mm·min^(-1). Finally, the accuracy and reliability of the radiation heat transfer model are verified, because of the accordance of simulation results with practical process.

The Effects of Interfaces on Stress Transfer in Short Fiber Reinforced Metal Matrix Composites

In this paper, the effects of interface properties on the stress transfer between matrix and fiber in short fiber reinforced metal matrix composites (SFRMMCs) is studied with the method of the elasto plastic finite element. The interface properties include Young’s modulus, thickness and elasto plastic performances. In the calculation an interfacial layer with given thickness is introduced into the single fiber model. It is shown that, for a soft interface, the variation in interfacial properties influences the stress transfer greatly.

基于传递矩阵法的指向式旋转导向工具主轴振动特性

指向式旋转导向工具应用于井眼轨迹控制时,其主轴的动力学特性将直接影响到井眼轨迹控制效果及作业安全,而在钻进工况下,工具钻压传递机构在主轴上的位置对主轴的动力学特性起着重要的作用。因此,为了分析钻压传递机构的位置对主轴动力学特性的影响,利用多体系统传递矩阵法建立了指向式旋转导向工具外壳-轴承-主轴系统的动力学模型。通过推导主轴上任意节点的几何和力学方程,该研究能够评估钻压传递机构的位置对主轴动态性能的影响规律。结合实例运算分析了钻压传递机构的安装位置对主轴固有频率和振型的影响,探讨了工具造斜时,钻压传递机构安装位置对主轴动态位移的影响。结果表明,随着钻压传递机构安装位置距离钻头越远,主轴固有频率和动态位移均增大,因此在满足工作要求下,推荐钻压传递机构安装位置尽可能靠近钻头端。为验证基于传递矩阵法建立的主轴动力学模型的正确性以及钻压传递装置安装位置对主轴动态位移影响规律,在指向式旋转导向工具试验台架上进行了钻进工况下主轴测点偏移量测试试验,结果表明,试验测得的数据与理论计算的数据基本吻合,误差最大仅为6.47%。

压电复合材料梯度分布声学特性研究

目前,压电复合材料在吸声降噪领域应用广泛,但其低频吸声性能并不理想。为解决此问题,以0-3型压电复合材料为研究对象,提出使压电陶瓷体积含量呈梯度分布以提高其低频吸声系数的方法。首先,基于声电类比法和传递矩阵法,推导多梯度分布的理论公式,并通过文献构型对其进行验证;其次,将均匀分布与梯度分布的吸声系数进行比较,发现梯度分布会改变材料的声阻抗,从而改变吸声系数。最后,进行梯度分布压电复合材料数值分析,结果表明,同等条件下,当压电陶瓷体积含量呈梯度递减,且梯度差和第一梯度的厚度增加时,可明显提高低频吸声系数。

平面柔性微定位平台的设计分析与优化

为设计具有良好静动态特性的柔性微定位平台,提出了两种基于柔性梁与圆弧形柔性铰链所组成的柔性模块的柔性运动副。首先,基于该运动副设计了新型平面2自由度微定位平台结构;其次,采用柔度矩阵法建立了平台力-位移关系及输出位移的理论模型,基于传递矩阵法建立了平台固有频率与瞬时动力响应的理论模型,并通过有限元仿真验证了理论模型的正确性。理论研究与仿真验证表明,平台具有较高的静动态性能,其输入输出完全解耦,在0.3mm输入位移下丢失运动不超过1.8%;1阶固有频率为320.34 Hz,在0.1 s时间内动力响应与仿真趋势一致。最后,进行了平台静、动态性能的灵敏度分析和参数优化。结果表明,优化后的平台在同样驱动下丢失运动和固有频率分别优化提升了1.95%和43.9%,进一步提升了平台的动态性能。

基于微波光子晶体的温度传感方法研究

针对基于电子技术的测温传感器灵敏度低、尺寸大和易受环境影响等缺陷,提出一种基于镜像型一维微波光子晶体模型的温度测量方法。利用传输矩阵法计算了一维缺陷微波光子晶体在不同环境温度条件下的带隙和缺陷峰频谱,建立了缺陷峰波长或缺陷峰偏波长移量与环境温度之间的关系;通过自由空间法测量缺陷峰波长在待测温度条件下的缺陷峰波长或其偏移量,则可以反演环境温度。结果表明,该温度测量方法具有较好的温度传感灵敏度和较大的温度测度范围,为设计微波传感器提供了一种新理论和实践案例。