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.

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.

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.

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.

MatDEM-fast matrix computing of the discrete element method

Discrete element method can effectively simulate the discontinuity,inhomogeneity and large deformation and failure of rock and soil.Based on the innovative matrix computing of the discrete element method,the highperformance discrete element software MatDEM may handle millions of elements in one computer,and enables the discrete element simulation at the engineering scale.It supports heat calculation,multi-field and fluidsolid coupling numerical simulations.Furthermore,the software integrates pre-processing,solver,postprocessing,and powerful secondary development,allowing recompiling new discrete element software.The basic principles of the DEM,the implement and development of the MatDEM software,and its applications are introduced in this paper.The software and sample source code are available online(http://matdem.com).

Numerical Methods for a Class of Quadratic Matrix Equations

Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.

Kohn-Sham Density Matrix and the Kernel Energy Method

The kernel energy method(KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries.KEM breaks a molecule into smaller subsets,called kernels,for the purposes of calculation.The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy.A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals.In this study,the kernel expansion for the density matrix is examined in the context of density functional theory(DFT) Kohn-Sham(KS) calculations.A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined,and is then converted into a normalizedprojector by using the Clinton algorithm.Such normalized projectors are factorizable into linear combination of atomic orbitals(LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis.Both straightforward KEM energies and energies from a normalized,idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion.Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules.In the case of the proof-of-concept system,calculations were performed using the STO-3 G and6-31 G(d,p) bases over a range of atomic separations,some very far from equilibrium.The water cluster was calculated in the 6-31 G(d,p) basis at an equilibrium geometry.The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases.In the case of the water cluster,the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result.The KS density matrices of this study are applicable to quantum crystallography.

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.

一种用于深空网天线组阵的基于归一化权的Matrix-Free Power Method

对深空网天线组阵中的几种信号相关算法作简要介绍,重点研究无需形成矩阵的幂的方法——Matrix-Free Power Method。通过理论分析,提出一种改进算法,即基于归一化权的Matrix-Free Power Method。理论计算和软件仿真结果均表明,基于归一化权的Matrix-Free Power Method信噪比合成性能略优于原Matrix-Free Power Method。

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.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Types and Correction Methods of Matrix Effect during Zircon LA-ICP-MS U-Pb Dating

Matrix effect primarily impacts the accuracy and precision of zircon LA-ICP-MS U-Pb data.This paper describes three types of matrix effect in zircon LA-ICPMS U-Pb dating,i.e.,the element matrix effect,high Ddpa or uranium matrix effect and alpha dose matrix effect,and illustrates the correction of these three effects.In addition,we point out the limitation and possible problems of the existing correction methods.

A procedure of the method of reverberation ray matrix for the buckling analysis of a thin multi-span plate

A procedure of the method of reverberation ray matrix(MRRM)is developed to perform the buckling analysis of thin multi-span rectangular plates having internal line supports or stiffeners.A computation algorithm for the reverberation ray matrix in the MRRM is derived to determine the buckling loading.Specifically,the analytical solutions are presented for the buckling of the structure having two opposite simply-supported or clamped-supported edges with spans,while the constraint condition of two remaining edges may be in any combination of free,simply-supported,and clamped boundary conditions.Furthermore,based on the analysis of matrices relating to the unknown coefficients in the solution form for the deflection in terms of buckling modal functions,some recursive equations(REs)for the MRRM are introduced to generate a reduced reverberation ray matrix with unchanged dimension when the number of spans increases,which promotes the computation efficiency.Several numerical examples are given,and the present results are compared with the known solutions to illustrate the validity and accurateness of the MRRM for the buckling analysis.

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.

Revealing the Hidden Mathematical Beauties of the Cayley-Hamilton Method

The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.

Long noncoding RNAs HAND2-AS1 ultrasound microbubbles suppress hepatocellular carcinoma progression by regulating the miR-873-5p/tissue inhibitor of matrix metalloproteinase-2 axis

BACKGROUND Increasing data indicated that long noncoding RNAs(lncRNAs)were directly or indirectly involved in the occurrence and development of tumors,including hepatocellular carcinoma(HCC).Recent studies had found that the expression of lncRNA HAND2-AS1 was downregulated in HCC tissues,but its role in HCC progression is unclear.Ultrasound targeted microbubble destruction mediated gene transfection is a new method to overexpress genes.AIM To study the role of ultrasound microbubbles(UTMBs)mediated HAND2-AS1 in the progression of HCC,in order to provide a new reference for the treatment of HCC.METHODS In vitro,we transfected HAND2-AS1 siRNA into HepG2 cells by UTMBs,and detected cell proliferation,apoptosis,invasion and epithelial-mesenchymal transition(EMT)by cell counting kit-8 assay,flow cytometry,Transwell invasion assay and Western blotting,respectively.In addition,we transfected miR-837-5p mimic into UTMBs treated cells and observed the changes of cell behavior.Next,the UTMBs treated HepG2 cells were transfected together with miR-837-5p mimic and tissue inhibitor of matrix metalloproteinase-2(TIMP2)overexpression vector,and we detected cell proliferation,apoptosis,invasion and EMT.In vivo,we established a mouse model of subcutaneous transplantation of HepG2 cells and observed the effect of HAND2-AS1 silencing on tumor formation ability.RESULTS We found that UTMBs carrying HAND2-AS1 restricted cell proliferation,invasion,and EMT,encouraged apoptosis,and HAND2-AS1 silencing eliminated the effect of UTMBs.Additionally,miR-873-5p targets the gene HAND2-AS1,which also targets the 3’UTR of TIMP2.And miR-873-5p mimic counteracted the impact of HAND2-AS1.Further,miR-873-5p mimic solely or in combination with pcDNA-TIMP2 had been transformed into HepG2 cells exposed to UTMBs.We discovered that TIMP2 reversed the effect of miR-873-5p mimic caused by the blocked signalling cascade for matrix metalloproteinase(MMP)2/MMP9.In vivo results showed that HAND2-AS1 silencing significantly inhibited tumor formation in mice.CONCLUSION LncRNA HAND2-AS1 promotes TIMP2 expression by targeting miR-873-5p to inhibit HepG2 cell growth and delay HCC progression.

BOUNDARY ELEMENT-COMPLIANCE MATRIX METHOD FOR 3D ELASTIC FRICTIONAL CONTACT ANALYSIS

Boundary element compliance matrix (BEM) method to deal with 3D elastic frictional contact problem was presented, and its calculation method and convergence criteria were discussed. It has the advantages of both boundary element method and compliance matrix methods: small number of inputting data, simple and convenient model, precise solution, short calculating time and requirements for a small quantity of computer memory. In comparison to the other BEM with friction problems, we obtain more precise solution and less iteration times. The effect of friction coefficient on contact area. contace state, and relative displacement, normal and tangential stress was analyzed by two examples. And because of the quickness of calculation of program and efficient method, we visualize the result in virtual reality (VR) environment. We grant the real time of VR and provide more immersion to users who ware the VR device.

Optimization of the gas separation performance of polyurethane–zeolite 3A and ZSM-5 mixed matrix membranes using response surface methodology

In the present work, the response surface method software was used with five measurement levels with three factors.These were applied for the optimization of operating parameters that affected gas separation performance of polyurethane–zeolite 3A, ZSM-5 mixed matrix membranes.The basis of the experiments was a rotatable central composite design(CCD).The three independent variables studied were: zeolite content(0–24 wt%), operating temperature(25–45 ℃) and operating pressure(0.2–0.1 MPa).The effects of these three variables on the selectivity and permeability membranes were studied by the analysis of variance(ANOVA).Optimal conditions for the enhancement of gas separation performances of polyurethane–3A zeolite were found to be 18 wt%, 30 ℃ and 0.8 MPa respectively.Under these conditions, the permeabilities of carbon dioxide, methane, oxygen and nitrogen gases were measured at 138.4, 22.9, 15.7 and 6.4 Barrer respectively while the CO_2/CH_4, CO_2/N_2 and O_2/N_2 selectivities were 5.8, 22.5 and 2.5, respectively.Also, the optimal conditions for improvement of the gas separation performance of polyurethane–ZSM 5 were found to be 15.64 wt%, 30 ℃ and 4 bar.The permeabilities of these four gases(i.e.carbon dioxide, methane, oxygen and nitrogen) were 164.7, 21.2, 21.5 and 8.1 Barrer while the CO_2/CH_4, CO_2/N_2 and O_2/N_2 selectivities were 7.8, 20.6 and 2.7 respectively.

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.

Methods for Derivation of Density Matrix of Arbitrary Multi-Mode Gaussian States from Its Phase Space Representation

We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation.An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values.As an example,we apply our method to the derivation of three-mode symmetric continuous variable entangled state.Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.