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.

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.

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.

Analysis of Vibration of the Euler-Bernoulli Pipe Conveying Fluid by Dynamic Stiffness Method and Transfer Matrix

The dynamic stiffness method and Transfer method is applied to study the vibration characteristics of the Euler-Bernoulli pipe conveying fluid in this paper. According to the dynamics equation of the pipe conveying fluid, the element dynamic stiffness is established. The vibration characteristic of the single-span pipe is analyzed under two kinds of boundary conditions. The results compared with the literature, which has a good consistency. Based on this method, natural frequency and the critical speed of the two types of multi-span pipe are deserved. This paper shows that the dynamic stiffness method and transfer matrix is an effective method to deal with the vibration problem of pipe conveying fluid.

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.

Forced Axial and Torsional Vibrations of a Shaft Line Using the Transfer Matrix Method Related to Solution Coefficients

This present paper deals with a mathematical description of linear axial and torsional vibrations. The normal and tangential stress tensor components produced by axial-torsional deformations and vibrations in the propeller and intermediate shafts, under the influence of propeller-induced static and variable hydrodynamic excitations are also studied. The transfer matrix method related to the constant coefficients of differential equation solutions is used. The advantage of the latter as compared with a well-known method of transfer matrix associated with state vector is the possibility of reducing the number of multiplied matrices when adjacent shaft segments have the same material properties and diameters. The results show that there is no risk of buckling and confirm that the strength of the shaft line depends on the value of the static tangential stresses which is the most important component of the stress tensor.

Dynamic Performance Analysis of the Coupling Shaft System in a Carpet Tufting Machine Utilizing the Transfer Matrix Method

The dynamic performance of the coupling shaft system in a carpet tufting machine is the most critical factor affecting the carpet tufting machine's efficiency,and the product quality of the tufted carpet. To determine how to avoid resonance produced by the coupling shaft system's vibration during the weaving process,the dynamic performance of a coupling shaft system in a carpet tufting machine was analyzed. Focusing on a DHGN801D-400 carpet tufting machine,a dynamic model of coupling shaft system was established by utilizing transfer matrix methodology. On the basis of this model,the natural frequencies and mode shapes of the coupling shaft system were obtained through simulations. The correctness of the theoretical model and the dynamic performance of the coupling shaft system were validated by experiments. The first order natural frequency of the coupling shaft system was close to 600 r / min. A conclusion can thus be drawn that operating the carpet tufting machine near this speed should be avoided as much as possible.

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.

RESEARCH ON TRANSFER MATRIX OF ROBOT WRIST FORCE SENSOR

The transfer matrix of wrist force sensor is important for decoupling theoutput signals and enhancing the precision of wrist force sensor.This paper solves thetransfer matrix from the realization of optimal approximation of polynomial.First,thegeneral transfer matrix algorithm in the sense of L^2 and two kinds of modified algorithmare proposed.Then,starting from uniform approximation,the optimal transfer matrixalgorithm is offered and solved by the use of linear programming.The results of experi-ment and computation prove that these algorithms are effective.

PERTURBATION TRANSFER MATRIX METHOD FOR EIGENDATA OF ONE_DIMENSIONAL STRUCTURAL SYSTEM WITH PARAMETER UNCERTAINTIES

A general method based on Riccati transfer matrix is presented to calculate the 2nd order perturbations of eigendatas for one_dimensional structural system with parameter uncertainties. The method is applicable to both real and complex eigendatas of any one_dimensional structural system. The formulas for calculating the sensitivity derivatives of eigendatas based on this method are also presented. The method is applied to the perturbation analysis for the eigendatas of a rotor with gyroscopic moment, and the differences between the perturbation results and the accurate calculating results are small.

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.

Effects of Matrix Features on Stress Transfer of Short Fiber Composites

By employing the elastic and elastic plastic finite element method(FEM), the effects of matrix feature on the stress transfer mechanisms of short fiber composites are studied. In the calculation, the variations in matrix modulus, yield strength and hardening modulus are considered. It is concluded that large deformation of matrix is harmful to the improvement of the mechanical performances of the composites.

The rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method

We study the eigenvalues of the rotating Morse potential by using the quantization condition from the analytical transfer matrix（ATM） method.A hierarchy of supersymmetric partner potentials is obtained with Pekeris approximation,which can be used to calculate the energies of higher rotational states from the energies of lower states.The energies of rotational states of the hydrogen molecule are calculated by the ATM condition,and comparison of the results with those from the hypervirial perturbation method reveals that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially in the framework of supersymmetric quantum mechanics.

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.

New Kinetic Computerized Model for Multicomponent Mass Transfer in Bi-Functional Matrix of NanoComposites

The aim of this theoretical investigation is the description of the multicomponent mass transfer process in the Nano- Composites (NC)—novel materials with the bi-functional matrix. The new theoretical NC Model is assigned for the modern theoretical investigations of the multicomponent mass transfer kinetics in the bi-functional NC materials. This NC Model for the multicomponent mass transfer in the bi-functional NC matrix includes into the consideration the proposed key conception—two co-existing routes: I—chemical reactions onto the active NC centers-sites, and II—diffusion mass transfer inside the bi-functional NC matrix. All the results are presented in the terms of the additional key concept: propagating multicomponent concentration waves (W+) in the NC matrix. The used W+ concept for the description of the multicomponent NC mass transfer kinetics give the clear interpretation of the computerized results. The mass transfer process in the NC matrix has been described theoretically by computerized simulation. The results of the calculations are new and illustrated by author’s animations showing visually the propagation of the multicomponent concentration waves (W) inside the various NC matrixes: r-beads, cylindrical ro-fibers, or planar L-membranes. Two variants of modeling for mass transfer diffusion kinetics in the bi-functional NC matrixes with one (Variant 1), or two (Variant 2) dissociation-association reactions at the active nano-sites (R0) are considered theoretically.

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.

Parametrization of Transfer Matrix:for One-Dimensional Anderson Model with Diagonal Disorder

In this paper, we developed a new parametrization method to calculate the localization length in one-dimensionalAnderson model with diagonal disorder.This method can avoid the divergence difficulty encountered in theconventional methods, and significantly save computing time as well.