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.

Determination of initial cable force of cantilever casting concrete arch bridge using stress balance and influence matrix methods

Cantilever casting concrete arch bridge using form traveller has a broad application prospect.However,it is difficult to obtain reasonable initial cable force in construction stage.In this study,stress balance and influence matrix methods were developed to determine the initial cable force of cantilever casting concrete arch bridge.The stress balance equation and influence matrix of arch rib critical section were established,and the buckle cable force range was determined by the allowable stress of arch rib critical section.Then a group of buckle cable forces were selected and substituted into the stress balance equation,and the reasonable initial buckle cable force was determined through iteration.Based on the principle of force balance,the initial anchor cable force was determined.In an engineering application example,it is shown that the stress balance and influence matrix methods for the determination of initial cable force are feasible and reliable.The initial cable forces of arch rib segments only need to be adjusted once in the corresponding construction process,which improves the working efficiency and reduces the construction risk.It is found that the methods have great advantages for determining initial cable force in cantilever casting construction process of concrete arch bridge.

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.

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.

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.

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.

2022 Multiple-country Monkeypox Outbreak and Its Importation Risk into China:An Assessment Based on the Risk Matrix Method

Objective To analyze the 2022 multiple-country monkeypox outbreak and assess its importation risk into China.Methods Data was from United States Centers for Disease Control and Prevention.We described the global epidemic and calculated concentration index to measure economic-related inequality.Importation risk into China was evaluated and ranked by using risk matrix method and Borda count method,respectively.Results As of July 29,2022,of 79 countries or territories,39(49.37%,39/79),17(21.52%,17/79),6(7.59%,6/79),12(15.19%,12/79),and 5(6.33%,5/79)country or territories identified cases<10,10-,51-,101-,and>1,000.There were economic-related health disparities exist in the distribution of cases(the concentration index=0.42,P=0.027),and the inequality disadvantageous to the rich(pro-poor).There were 12(15.38%,12/78),15(19.23%,15/78),6(7.69%,6/78),and 45(57.69%,45/78)countries or territories with extremely high,high,moderate,and low importation risk.United States and France ranked first with the highest Borda points of 156,and counts of zero.Conclusion Of 78 countries or territories,the key attention need be paid to the United States and France,relatively.As the epidemic progresses,preparing prevention and control measures to further reduce importation risk was crucial.

Global Epidemic of Ebola Virus Disease and the Importation Risk into China: An Assessment Based on the Risk Matrix Method

Objective To analyze the global epidemic status of the Ebola virus disease(EVD) and assess the importation risk into China.Methods Data from World Health Organization reports were used. We described the global epidemic status of EVD from 1976–2021, and assessed and ranked the importation risk of EVD from the diseaseoutbreaking countries into China using the risk matrix and Borda count methods, respectively.Results From 1976–2021, EVD mainly occurred in western and central Africa, with the highest cumulative number of cases(14,124 cases) in Sierra Leone, and the highest cumulative fatality rate(85%) in the Congo. Outbreaks of EVD have occurred in the Democratic Republic of the Congo and Guinea since 2018. The importation risk into China varies across countries with outbreaks of disease.The Democratic Republic of the Congo had an extremely high risk(23 Borda points), followed by Guinea and Liberia. Countries with a moderate importation risk were Nigeria, Uganda, Congo, Sierra Leone,Mali, and Gabon, while countries with a low importation risk included Sudan, Senegal, and Co te d’Ivoire.Conclusion China is under the risk of EVD importation with the globalization and severe epidemic status of EVD. Key attention need to be paid to the Democratic Republic of the Congo, Guinea, and Liberia. Therefore, it is necessary to prevent and prepare in advance for importation risk in China.

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.

Pseudo-spectrum Matrix Method and Its Application

A base function expressed with Chebyshev polynomials is reached. The relationship between the coefficients of the partial differential equation and the base function is deduced. Using the relationship, one can obtain nearly the same results as those calculated by Fast Fourier Transformation (FFT). The pseudo-spectral matrix method is applied in this paper to simulate numerically the incompressible laminar boundary flow on a plate. The simulation proves to be precise and efficient.

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.

Generation and classification of the translational shape-invariant potentials based on the analytical transfer matrix method

For the conventional translational shape-invariant potentials （TSIPs）, it has demonstrated that the phase contribution devoted by the scattered subwaves in the analytical transfer matrix quantization condition is integrable and independent of n. Based on this fact we propose a novel strategy to generate the whole set of conventional TSIPs and classify them into three types. The generating functions are given explicitly and the Morse potential is taken as an example to illustrate this strategy.

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.

Electron impact excitation of Ni-like gold studied by Dirac R-matrix method

We calculate the electron impact excitation of Ni-like gold by using the Dirac R-matrix theory, and the cor- responding collision strengths and effective collision strengths are obtained. In the calculations of the level energy, （1sZ2sZ2p6）3sZ3p63d10, 3s23p63d94/, 3s23p53d104/, and 3s3p63d104/（l = 0, 1,2,3） configurations are included and 107 fine-structure levels are generated. In the calculations of the collision strengths, only the first 59 levels are included. Com- parisons are made with the distorted wave （DW） results of Zeng et al. for both collision strengths and effective collision strengths. For the collision strengths, the two sets of calculations are in excellent agreement for most of the transitions. However, because of the inclusion of the resonances, our effective collision strengths are generally several times larger than those of Zeng et al.. The accuracy of our calculations is assessed.

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.

Application of the Magnetic Surface Based PARK-Matrix Method in the HCOOH Laser Interferometry System on HL-2A

A new electron density profile reconstruction procedure based on the PARK-matrix method has been firstly exploited for the multi-chord formic acid（HCOOH, λ=432.5 μm） laser interferometry system on the HL-2A tokamak. According to the geometric coordinates of the magnetic surfaces reconstructed by the CF（current fitting） code and the assumption that the electron density between two adjacent magnetic surfaces is a constant, the local electron density is calculated layer by layer, and the electron density profile n_e（Z） can be determined, as well as the density profile n_e（R）. The simulation result indicates that the error of the PARK-matrix method is acceptable for the four-chord HCOOH laser interferometer. In the applications, it shows that the reconstructed electron density profile agrees well with the microwave reflectometry measurement,and the sawtooth reversion radius is consistent with that deduced from the soft X-ray signals.Meanwhile, the electron density profiles with electron cyclotron resonance heating（ECRH） and supersonic molecular beam injection（SMBI） are also reconstructed and analyzed.

Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method

The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the eff ects of the Winkler coeffi cient, Pasternak coeffi cient and damping coeffi cient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary diff erential equations with variable coeffi cients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verifi ed through two numerical examples. The infl uences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with diff erent taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.

Decoupling Control Research on Test System of Hydraulic Drive Unit of Quadruped Robot Based on Diagonal Matrix Method

The mathematical model of hydraulic drive unit of quadruped robot was built in this paper. According to the coupling characteristics between position control system and force control system, the decoupling control strategy was realized based on diagonal matrix method in AMESim?. The results of simulation show that using diagonal matrix method can achieve the decoupling control effectively and it can achieve the decoupling control more effectively with the method of not offset pole-zero in the S coordinate. This research can provide theoretical basis for the application of test system of hydraulic drive unit.

Recursive Asymptotic Hybrid Matrix Method for Acoustic Waves in Multilayered Piezoelectric Media

This paper presents the recursive asymptotic hybrid matrix method for acoustic waves in multilayered piezoelectric media. The hybrid matrix method preserves the numerical stability and accuracy across large and small thicknesses. For discussion and comparison, the scattering matrix method is also presented in physics-based form and coherent form. The latter form resembles closely that of hybrid matrix method and helps to highlight their relationship and distinction. For both scattering and hybrid matrix methods, their formulations in terms of eigenwaves solution are provided concisely. Making use of the hybrid matrix, the recursive asymptotic method without eigenwaves solution is described and discussed. The method bypasses the intricacies of eigenvalue-eigenvector approach and requires only elementary matrix operations along with thin- layer asymptotic approximation. It can be used to determine Green’s function matrix readily and facilitates the trade-off between computation efficiency and accuracy.