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.

Accurate machine learning models based on small dataset of energetic materials through spatial matrix featurization methods

A large database is desired for machine learning(ML) technology to make accurate predictions of materials physicochemical properties based on their molecular structure.When a large database is not available,the development of proper featurization method based on physicochemical nature of target proprieties can improve the predictive power of ML models with a smaller database.In this work,we show that two new featurization methods,volume occupation spatial matrix and heat contribution spatial matrix,can improve the accuracy in predicting energetic materials' crystal density(ρ_(crystal)) and solid phase enthalpy of formation(H_(f,solid)) using a database containing 451 energetic molecules.Their mean absolute errors are reduced from 0.048 g/cm~3 and 24.67 kcal/mol to 0.035 g/cm~3 and 9.66 kcal/mol,respectively.By leave-one-out-cross-validation,the newly developed ML models can be used to determine the performance of most kinds of energetic materials except cubanes.Our ML models are applied to predict ρ_(crystal) and H_(f,solid) of CHON-based molecules of the 150 million sized PubChem database,and screened out 56 candidates with competitive detonation performance and reasonable chemical structures.With further improvement in future,spatial matrices have the potential of becoming multifunctional ML simulation tools that could provide even better predictions in wider fields of materials science.

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.

PROBLEMS AND METHODS IN MATRIX VALUED RATIONAL INTERPOLATION

A variety of matrix rational interpolation problems include the partial realizationproblem for matrix power series and the minimal rational interpolation problem for generalmatrix functions.Several problems in circuit theory and digital filter design can also be re-duced to the solution of matrix rational interpolation problems[1—4].By means of thereachability and the observability indices of defined pairs of matrices,Antoulas,Ball,Kang and Willems solved the minimal matrix rational interpolation problem in[1].On

GENERALIZED INVERSE RATIONAL EXTRAPOLATION METHODS FOR MATRIX SEQUENCES

Assume that a convergent matrix sequence{An}:An→A（n→∞）, An,A∈C3×3.We want to form a new matrix sequence {Hn}, derived from {An}, which has also A aslimit and whose convergence is faster than the of {An}. Three rational extrapolation meth-ods for accelerating the convergence of matrix sequences {An} are presented in this paper.The underlying methods are based on the generalized inverse for matrices which is

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.

Corrigendum to“Atomic-scale electromagnetic theory bridging optics in microscopic world and macroscopic world”

The signs of the electric field markers in Figs.2 and 4 of the paper[Chin.Phys.B 32104211(2023)]have been corrected.These modifications do not affect the results derived in the paper.

Exact Tail Asymptotics for a Queueing System with a Retrial Orbit and Batch Service

This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.

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.

THE STRESS SUBSPACE OF HYBRID STRESS ELEMENT AND THE DIAGONALIZATION METHOD FOR FLEXIBILITY MATRIX H

The following is proved: 1) The linear independence of assumed stress modes is the necessary and sufficient condition for the nonsingular flexibility matrix; 2) The equivalent assumed stress modes lead to the identical hybrid element. The Hilbert stress subspace of the assumed stress modes is established. So, it is easy to derive the equivalent orthogonal normal stress modes by Schmidt's method. Because of the resulting diagonal flexibility matrix, the identical hybrid element is free from the complex matrix inversion so that the hybrid efficiency, is improved greatly. The numerical examples show that the method is effective.

Method of reverberation ray matrix for static analysis of planar framed structures composed of anisotropic Timoshenko beam members

Based on the method of reverberation ray matrix（MRRM）, a reverberation matrix for planar framed structures composed of anisotropic Timoshenko（T） beam members containing completely hinged joints is developed for static analysis of such structures.In the MRRM for dynamic analysis, amplitudes of arriving and departing waves for joints are chosen as unknown quantities. However, for the present case of static analysis, displacements and rotational angles at the ends of each beam member are directly considered as unknown quantities. The expressions for stiffness matrices for anisotropic beam members are developed. A corresponding reverberation matrix is derived analytically for exact and unified determination on the displacements and internal forces at both ends of each member and arbitrary cross sectional locations in the structure. Numerical examples are given and compared with the finite element method（FEM） results to validate the present model. The characteristic parameter analysis is performed to demonstrate accuracy of the present model with the T beam theory in contrast with errors in the usual model based on the Euler-Bernoulli（EB） beam theory. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the framed structures.

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.

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.

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.

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.

Finding vibrations of inclined cable structures by approximately solving governing equations for exterior matrix

In this paper, how to compute the eigenfrequencies of the structures composed of a series of inclined cables is shown. The physics of an inclined cable can be complicated, so solving the differential equations even approximately is difficult. However, rather than solving the system of 4 first-order equations governing the dynamics of each cable, the governing equations are instead converted to a set of equations that the exterior matrix satisfies. Therefore, the exterior matrix method （EMM） is used without solving the original governing equations. Even though this produces a system of 6 first-order equations, the simple asymptotic techniques to find the first three terms of the perturbative solution can be used. The solutions can then be assembled to produce a 6 x 6 exterior matrix for a cable section. The matrices for each cable in the structure are multiplied together, along with the exterior matrices for each joint. The roots of the product give us the eigenfrequencies of the system.