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.

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.

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.

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.

Optimal design of 6-DOF vibration isolation platform based on transfer matrix method for multibody systems

The structure parameters of 6-degree of freedom(DOF)vibration isolation platform have a significant effect on its performance.To make the designed vibration isolation platform perform well,non-dominanted sorting genetic algorithm version II(NSGA-II)was applied to optimize its structure based on the transfer matrix method for multibody systems.Firstly,the Jacobian matrix of 6-DOF vibration isolation platform was solved based on kinematics.Secondly,the transfer equation of 6-DOF vibration isolation system was established by the linear transfer matrix method for multibody systems.And the formula of its natural frequency was derived according to the boundary conditions of the system.Thirdly,the manipulability index was constructed based on a dimensionless Jacobian matrix.And a new performance index function was established considering the influence of dynamic isotropic and legs mass.Fourthly,genetic algorithm(GA)and NSGA-II were used to optimize the structure of the 6-DOF vibration isolation platform under the same conditions,respectively.It showed that NSGA-II had higher optimization efficiency,better calculation accuracy and shorter optimization time than that of GA.Finally,NSGA-II was adopted for multi-objective optimization design of 6-DOF vibration isolation platform based on the constraint conditions.Optimal Pareto solutions were obtained,which provides structural parameters for subsequent design work.Therefore,the proposed optimization method and the performance index in this paper provide a theoretical basis for the optimal design of relevant vibration isolation mechanism.

A COMBINED FINITE ELEMENT----RICCATI TRANSFER MATRIX METHOD FOR FREE VIBRATION OF STRUCTURES

A combination of Riccati transfer matrix method and finite element method is proposed for obtaining vibration frequencies of structures. This method reduces the propagation of round-off errors produced in the standard transfer matrix method and finds out the values of the frquency by Newton-raphson method. As the application of this method, a numerical example is given for the case of vibration of a cantilever plate. By the result obtained, the method is shown to be more effective.

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.

Reduced multibody system transfer matrix method using decoupled hinge equations

In the multibody system transfer matrix method(MSTMM),the transfer matrix of body elements may be directly obtained from kinematic and kinetic equations.However,regarding the transfer matrices of hinge elements,typically information of their outboard body is involved complicating modeling and even resulting in combinatorial problems w.r.t.various types of outboard body's output links.This problem may be resolved by formulating decoupled hinge equations and introducing the Riccati transformation in the new version of MSTMM called the reduced multibody system transfer matrix method in this paper.Systematic procedures for chain,tree,closed-loop,and arbitrary general systems are defined,respectively,to generate the overall system equations satisfying the boundary conditions of the system during the entire computational process.As a result,accumulation errors are avoided and computational stability is guaranteed even for huge systems with long chains as demonstrated by examples and comparison with commercial software automatic dynamic analysis of the mechanical system.

FREE VIBRATION ANALYSIS AND PHYSICAL PARAMETER IDENTIFICATION OF NON-UNIFORM BEAM CARRYING SPRING-MASS SYSTEMS

To analyze a multibody system composed of non-uniform beam and spring-mass subsystems, the model discretization is carried on by utilizing the finite element method（FEM）, the dynamic model of non-uniform beam is developed by using the transfer matrix method of multibody system（MS-TMM）, the transfer matrix of non-u- niform beam is derived, and the natural frequencies are computed. Compared with the numerical assembly method （NAM）, the results by MS-TMM have good agreement with the results by FEM, and are better than the results by NAM. When using the high precision method, the global dynamic equations of the complex multibody system are not needed and the orders of involved system matrices are decreased greatly. For the investigation on the re- verse problem of the physical parameter identification of multibody system, MS-TMM and the optimization tech- nology based on genetic algorithms（GAs） are combined and extended. The identification problem is exchanged for an optimization problem, and it is formulated as a global minimum solution of the objective function with respect to natural frequencies of multibody system. At last, the numerical example of non-uniform beam with attach- ments is discussed, and the identification results indicate the feasibility and the effectivity of the proposed aop- proach.

Eigenvalue analysis of planar linear multibody system under conservative force based on the transfer matrix method

The linear multibody system transfer matrix method(LMSTMM)provides a powerful tool for analyzing the vibration characteristics of a mechanical system.However,the original LMSTMM cannot resolve the eigenvalues of the systems with ideal hinges(i.e.,revolute hinge,sliding hinge,spherical hinge,cylindrical hinge,etc.)or bodies under conservative forces due to the lack of the corresponding transfer matrices.This paper enables the LMSTMM to solve the eigenvalues of the planar multibody systems with ideal hinges or rigid bodies under conservative forces.For a rigid body,the transfer matrix can now consider coupling terms between forces and kinematic state perturbations.Also,conservative forces that contribute to the eigenvalues can be considered.Meanwhile,ideal hinges are introduced to LMSTMM,which enables the treatment of eigenvalues of general multibody systems using LMSTMM.Finally,the comparative analysis with ADAMS software and analytical solutions verifies the effectiveness of the proposed approach in this paper.

Transfer matrix method for multibody systems(Rui method) and its applications

The transfer matrix method for multibody systems, namely the "Rui method", is a new method for studying multibody system dynamics, which avoids the global dynamics equations of the system, keeps high computational speed, and allows highly formalized programming. It has been widely applied to scientific research and key engineering of lots of complex mechanical systems in 52 research directions. The following aspects regarding the transfer matrix method for multibody systems are reviewed systematically in this paper: history, basic principles, formulas, algorithm, automatic deduction theorem of overall transfer equation, visualized simulation and design software, highlights, tendency, and applications in 52 research directions in over 100 key engineering products.

Multibody system transfer matrix method:The past, the present, and the future

The multibody system transfer matrix method(MSTMM),a novel dynamics approach developed during the past three decades,has several advantages compared to conventional dynamics methods.Some of these advantages include avoiding global dynamics equations with a system inertia matrix,utilizing low‐order matrices independent of system degree of freedom,high computational speed,and simplicity of computer implementation.MSTMM has been widely used in computer modeling,simulations,and performance evaluation of approximately 150 different complex mechanical systems.In this paper,the following aspects regarding MSTMM are reviewed:basic theory,algorithms,simulation and design software,and applications.Future research directions and generalization to more applications in various fields of science,technology,and engineering are discussed.

Dynamic modeling of ultra-precision fly cutting machine tool and the effect of ambient vibration on its tool tip response

The dynamic performances of an ultra-precision fly cutting machine tool(UFCMT)has a dramatic impact on the quality of ultra-precision machining.In this study,the dynamic model of an UFCMT was established based on the transfer matrix method for multibody systems.In particular,the large-span scale flow field mesh model was created;and the variation in linear and angular stiffness of journal and thrust bearings with respect to film thickness was investigated by adopting the dynamic mesh technique.The dynamic model was proven to be valid by comparing the dynamic characteristics of the machine tool obtained by numerical simulation with the experimental results.In addition,the power spectrum density estimation method was adopted to simulate the statistical ambient vibration excitation by processing the ambient vibration signal measured over a long period of time.Applying it to the dynamic model,the dynamic response of the tool tip under ambient vibration was investigated.The results elucidated that the tool tip response was significantly affected by ambient vibration,and the isolation foundation had a good effect on vibration isolation.

Impact vibration properties of locally resonant fluid-conveying pipes

Fluid-conveying pipe systems are widely used in various equipments to transport matter and energy.Due to the fluid–structure interaction effect,the fluid acting on the pipe wall is easy to produce strong vibration and noise,which have a serious influence on the safety and concealment of the equipment.Based on the theory of phononic crystals,this paper studies the vibration transfer properties of a locally resonant(LR)pipe under the condition of fluid–structure interaction.The band structure and the vibration transfer properties of a finite periodic pipe are obtained by the transfer matrix method.Further,the different impact excitation and fluid–structure interaction effect on the frequency range of vibration attenuation properties of the LR pipe are mainly considered and calculated by the finite element model.The results show that the existence of a low-frequency vibration bandgap in the LR pipe can effectively suppress the vibration propagation under external impact and fluid impact excitation,and the vibration reduction frequency range is near the bandgap under the fluid–structure interaction effect.Finally,the pipe impact experiment was performed to verify the effective attenuation of the LR structure to the impact excitation,and to validate the finite element model.The research results provide a technical reference for the vibration control of the fluid-conveying pipe systems that need to consider blast load and fluid impact.

Torsional Vibration of Roller Oscillating Tooth Gear Drive

Torsional vibration of roller oscillating tooth gear drive (ROTGD) is studied in this paper. On the basis of conservation law for kinetic energy and potential energy, the mathematical expressions are developed which describe transformation of moment of inertia of inertial components into input shaft. Also, the formula is derived which expresses transformation of contact stiffness of elastic components into input shaft torsional stiffness. Besides, torsional vibration model of ROTGD is presented by using the transfer matrix method, and natural frequencies and vibration mode shapes are determined. Eventually, an example is given.

Discrete time transfer matrix method for dynamics analysis of complex weapon systems

Efficient, precise dynamic modeling and analysis for complex weapon systems have become more and more important in their dynamic design and performance optimizing. As a new method developed in recent years, the discrete time transfer matrix method of multibody system is highly efficient for multibody system dynamics. In this paper, taking a shipboard gun system as an example, by deducing some new transfer equations of elements, the discrete time transfer matrix method of multibody sys- tem is used to solve the dynamics problems of complex rigid-flexible coupling weapon systems successfully. This method does not need the global dynamic equations of system and has the low order of system matrix, high computational efficiency. The proposed method has advantages for dynamic design of complex weapon systems, and can be carried over straightforwardly to other complex mechanical systems.

Reduced-order modeling and vibration transfer analysis of a fluid-delivering branch pipeline that consider fluid–solid interactions

The efficient dynamic modeling and vibration transfer analysis of a fluid-delivering branch pipeline(FDBP)are essential for analyzing vibration coupling effects and implementing vibration reduction optimization.Therefore,this study proposes a reduced-order dynamic modeling method suitable for FDBPs and then analyzes the vibration transfer characteristics.For the modeling method,the finite element method and absorbing transfer matrix method(ATMM)are integrated,considering the fluid–structure coupling effect and fluid disturbances.The dual-domain dynamic substructure method is developed to perform the reduced-order modeling of FDBP,and ATMM is adopted to reduce the matrix order when solving fluid disturbances.Furthermore,the modeling method is validated by experiments on an H-shaped branch pipeline.Finally,transient and steady-state vibration transfer analyses of FDBP are performed,and the effects of branch locations on natural characteristics and vibration transfer behavior are analyzed.Results show that transient vibration transfer represents the transfer and conversion of the kinematic,strain,and damping energies,while steady-state vibration transfer characteristics are related to the vibration mode.In addition,multiple-order mode exchanges are triggered when branch locations vary in frequency-shift regions,and the mode-exchange regions are also the transformation ones for vibration transfer patterns.

Hybrid multibody system method for the dynamic analysis of an ultra‐precision fly‐cutting machine tool

The dynamics of an ultra‐precision machine tool determines the precision of the machined surface.This study aims to propose an effective method to model and analyze the dynamics of an ultra‐precision fly‐cutting machine tool.First,the dynamic model of the machine tool considering the deformations of the cutter head and the lathe head is developed.Then,the mechanical elements are classified into M subsystems and F subsystems according to their properties and connections.The M‐subsystem equations are formulated using the transfer matrix method for multibody systems(MSTMM),and the F‐subsystem equations are analyzed using the finite element method and the Craig-Bampton reduction method.Furthermore,all the subsystems are assembled by combining the restriction equations at connection points among the subsystems to obtain the overall transfer equation of the machine tool system.Finally,the vibration characteristics of the machine tool are evaluated numerically and are validated experimentally.The proposed modeling and analysis method preserves the advantages of the MSTMM,such as high computational efficiency,low computational load,systematic reduction of the overall transfer equation,and generalization of its computational capability to general flexible‐body elements.In addition,this study provides theoretical insights and guidance for the design of ultra‐precision machine tools.

Spatial Vibration and Its Numerical Analytical Method of Four-high Rolling Mills

The rolls in contemporary four-high mills cannot be maintained parallel during the rolling process. There- fore, four-high rolling mill vibrations take place in six degree of freedom （DOF） leading to spatial behaviors invol- ving vertical, horizontal, axial, torsional, cross and swinging vibration modes resulting in complex relative motions between the rolls. Two numerical methods, modified Riccati-transfer matrix method （Riccati-TMM） and finite ele- ment method （FEM）, are presented to analyze a spatial vibration characteristic of two four-high rolling mills with different stability. The natural frequency and mode shape of four-high rolling mills are obtained, and the clearance has a great effect on natural frequency and mode shape. In addition, field testing experiment is also conducted to measure natural frequency by power spectrum analysis of rolling mill vibration. Experimental results basically agree with those calculated by Riccati-TMM and FEM, which means that the Riccati-TMM and the FEM can be used for analysis of spatial vibration of four-high rolling mill. Meanwhile, the spatial vibration shows more compound vibra- tion behaviors and the negative effect of horizontal, vertical, cross and swinging vibration modes are effectively con- trolled after redesign of rolling mill. These advantages have a great significance for the rolling mill to be operated with a much higher rolling speed and improved yield of products.

Dynamic modeling and H_∞ independent modal space vibration control of laminate plates

In this paper, combining the transfer matrix method and the finite element method, the modified finite element transfer matrix method is presented for high efficient dynamic modeling of laminated plates. Then, by constructing the modal filter and the disturbance force observer, and using the feedback and feedforward approaches, the H ∞ independent modal space control strategy is designed for active vibration control of laminate plates subjected to arbitrary, immeasurable disturbance forces. Compared with ordinary dynamic modeling and control methods of laminated plate structures, the proposed method has the low memory requirement, high computational efficiency and robust control performance. Formulations as well as some numerical examples are given to validate the method and the control performance.