Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance

The aim of the study described in this paper is to investigate the forced dynamics of an axially moving viscoelastic beam. The governing equation of motion is obtained via Newton's second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin–Voigt, a two-parameter rheological model, energy dissipation mechanism, in which material, not partial, time derivative is employed in the viscoelastic constitutive relation. The dimensionless partial differential equation of motion is discretized using Galerkin's scheme with hinged–hinged beam eigenfunctions as the basis functions. The resulting set of nonlinear ordinary differential equations is then solved using the pseudo-arclength continuation technique and a direct time integration. For the system with the axial speed in the sub-critical regime, the response of the system is examined when possessing an internal resonance and when not. By employing a direct time integration, it is shown how the bifurcation diagrams of the system are modified by the presence of the dissipation terms—i.e. by both the time-dependant and steady (due the simultaneous presence of the axial speed and the energy dissipation mechanism) energy dissipation terms. Moreover, the amplitude–frequency responses and bifurcation diagrams of Poincaré maps are presented for several values of the system parameters.     

A new approach for the analysis solution of dynamic systems containing fractional derivative

Fractional derivative models, which are used to describe the viscoelastic behavior of material, have received considerable attention. Thus it is necessary to put forward the analysis solutions of dynamic systems containing a fractional derivative. Although previously reported such kind of fractional calculus-based constitutive models, it only handles the particularity of rational number in part, has great limitation by reason of only handling with particular rational number field. Simultaneously, the former study has great unreliability by reason of using the complementary error function which can’t ensure uniform real number. In this paper, a new approach is proposed for an analytical scheme for dynamic system of a spring-mass-damper system of single-degree of freedom under general forcing conditions, whose damping is described by a fractional derivative of the order 0< α< 1 which can be both irrational number and rational number. The new approach combines the fractional Green’s function and Laplace transform of fractional derivative. Analytical examples of dynamic system under general forcing conditions obtained by means of this approach verify the feasibility very well with much higher reliability and universality.     

MULTI-LAYER PIEZOELECTRIC ACTUATOR AND ITS APPLICATION IN CONTROLLABLE CONSTRAINED DAMPING TREATMENT

A kind of novel multi-layer piezoelectric actuator is proposed and integrated with control- lable constrained damping treatment to perform hybrid vibration control.The governing equation of the system is derived based on the constitutive equations of elastic,viscoelastic and piezoelectric materials,which shows that the magnitude of control force exerted by multi-layer piezoelectric actua- tor is the quadratic function of the number of piezoelectric laminates used but in direct proportion to control voltage.This means that the multi-layer actuator can produce greater actuating force than that by piezoelectric laminate actuator with the same area under the identical control voltage.The optimal location placement of the multi-layer piezoelectric actuator is also discussed.As an example,the hybrid vibration control of a cantilever rectangular thin-plate is numerically simulated and carried out experimentally.The simulated and experimental results validate the power of multi-layer piezoelectric actuator and indicate that the present hybrid damping technique can effectively suppress the low fre- quency modal vibration of the experimental thin-plate structure.     

含增粘剂润滑油非线性本构方程的探讨

本文针对含增粘剂润滑油的流变特性，将线性多组元粘弹性模型在特殊的共转坐标系下推广，得到了能同时反映含增粘剂润滑油剪切稀化及其粘弹性的本构方程。将该本构关系给出的表现粘度与实验结果进行了对比，证明该本构关系能够较全面含增粘剂润滑油的流变特性且形式简单便于工程应用。     

Modelling the plane strain compression test to obtain constitutive equations of aluminium alloys

Hot plane strain compression tests on 1050, 1198, 3003 and 3004 aluminium alloys have been conducted. Based on these experiments and on a set of internal type constitutive equations for hot working, the values of the parameters in the constitutive functions are determined. The constitutive equations proposed here, with the constitutive functions and material parameters associated, accurately reproduce the basic tests. The procedure used to fit the material parameters is improved, in comparison with classical slip line analysis, by using a finite element modelling of the plane strain compression test. It is demonstrated that accurate plane strain or three-dimensional large strain finite element analysis can be used to correct the friction and lateral spread effects. Furthermore, it is demonstrated from comparison with the experimental observations that microstructural parameters can be accurately determined from numerical modelling. The constitutive equations and finite element procedure proposed here can be useful for obtaining an improved analysis of hot rolling of aluminium alloys.     

温度对橡胶隔振器刚度阻尼特性的影响

在不同温度( 20~80 ℃ )下对橡胶材料进行了单轴静态拉伸试验与动态频率扫描试验,分析了应力应变关系、储能模量及损耗模量随温度的变化规律,拟合获得了不同温度下表征橡胶力学性能的超弹性和粘弹性本构模型参数。对某倒 V 形橡胶隔振器进行了不同温度本构参数下的静态与动态仿真,结果表明,温度对橡胶隔振器力学性能有较大影响,研究结果可为橡胶隔振器的工程应用提供参考。     

Die compaction of copper powder designed for material parameter identification

This article presents uniaxial compaction experiments of a fine copper powder in a cylindrical die. The compaction process consists of monotonic loading and of loading paths with inserted unloading and reloading cycles. An experimental setup that has been developed for determining the axial and radial stresses during the compaction is described and the calibration of the new device using highly accurate p-finite element simulations of the dies response to internal pressure is shown. The experimental results were subsequently used for the identification of the material parameters of a constitutive model for granular materials recently proposed by Bier and Hartmann [A finite strain constitutive model for metal powder compaction using a unique and convex single surface yield function. accepted for publication by European Journal of Mechanics, Series A/Solids 2006.]. The identification of the elasticity parameters was treated with special attention.     

Modeling the powder compaction process using the finite element method and inverse optimization

This paper focuses on studying and adapting modeling techniques using the finite element method to simulate the rigid die compaction of metal powders. First, it presents the implementation of the cap constitutive model into ABAQUS FE software using the closest point projection algorithm. Then, an inverse modeling procedure was proposed to alleviate the problems raised by the interpretation of the experimental tests and to more accurately determine the material parameters. The objective function is formed, based on the discrepancy in density data between the numerical model prediction and the experiment. Minimization of the objective function with respect to the material parameters was performed using an in-house optimization software shell built on a modified Levenberg?CMarquardt method. Thus, an integrated simulation module consisting of an inverse optimization method and a finite element method was developed for modeling the powder compaction process as a whole. The simulation and identification module developed was applied to simulate the compaction of some industrial parts. The results reveal that the maximum absolute error between densities is 2.3%. It corresponds to the precision of the experimental method.     

聚合物注塑内应力计算的蠕变模型

从聚合物粘弹性理论出发,构建了注塑制品内应力计算的蠕变模型。结合蠕变试验结果,分析聚合物注塑粘弹性行为的非线性响应,并提出在应力、温度作用下聚合物弹性模量和粘度的计算公式。在此基础上,利用有限元法对工程塑料ABS(Acrylonitrile butadiene styrene)平板注塑制件脱模前的内应力进行了模拟计算并与相关试验数据及模拟结果进行比较。结果表明该模型计算精度较高,且计算过程简便,为进一步准确预测注塑制件内应力提供了新的途径。     

A viscoelastic analysis of the tensile weakening of deep femoral head articular cartilage

Articular cartilage from below the surface of the femoral head of the hip joint shows a profound age-dependent weakening in its tensile mechanical properties. This ageing is also associated with a reduced viscoelastic response in the older tissue. A constitutive model of the viscoelastic behaviour of deep articular cartilage (as discussed by Egan in 1988) is used to generate a graphical pattern which represents the mechanical behaviour. This constitutive approach suggests that the tensile weakening of the older cartilage is due to an age-related reduction in the recruitment of load-carrying structures as the tissue is deformed. The viscoelastic constitutive model also predicts a reduction in the tensile strength of deep articular cartilage with rate of deformation. This prediction is supported by experimental fracture stress data. A weakening of the tensile integrity of the microstructure of articular cartilage could make the tissue less able to sustain normal compressive physiological loading without damage and thus make the tissue more susceptible to osteoarthritic degeneration. The constitutive approach indicates that the weakening of the older tissue may be related to changes within the microstructure which determine how applied mechanical energy is stored and dissipated.     

Biomechanical behaviour of heel pad tissue: experimental testing, constitutive formulation, and numerical modelling

This paper deals with the constitutive formulation of heel pad tissue and presents a procedure for identifying constitutive parameters using experimental data, with the aim of developing a computational approach for investigating the actual biomechanical response. The preliminary definition of constitutive parameters was developed using a visco-hyperelastic formulation, considering experimental data from in vitro compression tests on specimens of fat pad tissue and data from in vivo tests to identify the actual trend of tissue stiffness. The discrepancy between model results and experimental data was evaluated on the basis of a specific cost function, adopting a stochastic/deterministic procedure. The parameter evaluation was upgraded by considering experimental tests performed on the fat pad tissues of a cadaveric foot using in situ indentation tests at 0.01 and 350 mm/s strain rates. The constitutive formulation was implemented in a numerical model. The comparison of data from in situ tests and numerical results led to an optimal domain of parameters based on an admissible discrepancy criterion. Numerical results evaluated for different sets of parameters inside the domain are reported and compared with experimental data for a reliability evaluation of the proposed procedure.     

经验应力松弛模量与Maxwell模量的转换计算方法

基于MATLAB软件,利用单纯形法搜索最小误差范数,提出了一种针对粘弹材料的经验松弛函数KohlrauschWilliamWatts(KWW)方程与Maxwell模量转换的计算方法,实现了对粘弹材料的广义Maxwell模型的拟合。并以1050℃的粘弹光纤玻璃为例,给出了一组计算实例。     

五参量结构阻尼模型及其在弹性机构动力学中的应用

将阻尼合金视为粘弹性材料,利用五参量本构关系来描述阻尼合金材料的应力应变关系。在试验的基础上,利用优化算法拟合出本构关系式中的五个参量。导出了以五参量表示阻尼和刚度特性的单元运动微分方程。为便于计算,将包含卷积运算的微分方程转换成一个四阶微分方程,进而装配出含有阻尼合金构件的弹性连杆机构的系统动力学方程。利用状态空间法对导出的高阶时变微分方程组进行了数值求解。计算实例结果表明所提模型是正确、有效的。     

A study on formulation of objective functions for determining material models

Optimisation techniques have been developed and used to determine material constants arising in unified creep/viscoplastic constitutive equations based on experimental data. Objective functions (OF) have been formulated as pointers to the quality of fit between the equations and experimental data and a set of criteria is presented to assess the suitability of the objective functions. Convergence features of two existing objective functions are analysed. The problems of using these objective functions are studied. To overcome difficulties arising in the existence of different scales of individual sub-objectives, a novel error definition is introduced, which has a natural unitless form and can provide a measure for “true” error. A novel weighting technique is introduced, which can also be chosen automatically to compensate the loss of credits in individual data points and curves. Using these techniques, a novel objective function is formulated, which meets the set criteria. The objective function, together with an evolutionary programming (EP) solver, are employed to determine material constants in three sets of unified constitutive equations which are formulated to match experimental data. The convergent features of the objective functions are compared and analysed.     

基于Eyring本构模型的磁流变液阻尼器设计原理与试验研究

根据试验得出的Bingham塑性流体模型的模型参数, 建立了磁流变液切应力的误差函数,利用多参数优化理论和数据拟合方法对Eyring本构模型的参数进行辨识。建立了基于Eyring模型的环状混合准稳态流动方程,得出了磁流变液在环形通道中流动的速度分布函数。在给定活塞速度和环形通道的几何尺寸条件下,对混合工作模式的汽车磁流变液阻尼器产生的阻尼力进行理论预测研究。按照长安之星微型汽车前悬架的技术要求,设计和制作了微型汽车磁流变液阻尼器,并对此进行了试验测试,试验结果表明：应用所提出的理论分析方法预测磁流变液阻尼器的特性是可行的。     

Numerical derivation of constitutive models for unbonded flexible risers

In this paper a new constitutive model for flexible risers is proposed and a procedure for the identification of the related input parameters is developed using a multi-scale approach. The constitutive model is formulated in the framework of an Euler-Bernoulli beam model, with the addition of suitable pressure terms to the generalized stresses to account for the internal and external pressures, and therefore can be efficiently used for large-scale analyses. The developed non-linear relationship between generalized stresses and strains in the beam is based on the analogy between frictional slipping between different layers of a flexible riser and frictional slipping between micro-planes of a continuum medium in non-associative elasto-plasticity. Hence, a linear elastic relationship is used for the initial response in which no-slip occurs; an onset-slip function is introduced to define the ‘no-slip’ domain, i.e. the set of generalized stresses for which no slip occurs; a non-associative rule with linear kinematic hardening is used to model the full-slip phase. The results of several numerical simulations for a riser of small-length, obtained with a very detailed (small-scale) non-linear finite-element model, are used to identify the parameters of the constitutive law, bridging in this way the small scale of the detailed finite-element simulations with the large scale of the beam model. The effectiveness of the proposed method is validated by the satisfactory agreement between the results of various detailed finite-element simulations for a short riser, subject to internal and external uniform pressure and uniform cyclic bending loading, with those given by the proposed constitutive law.     

Constitutive model of tendon responses to multiple cyclic demands (II)

The hereditary integral form of a quasi-linear viscoelastic law has been employed. Four new concepts have been employed: 1. a reduced relaxation function with a non-linear exponential function of time, 2. an inverse method to determine the scale factor of the elastic response, 3. an instant elastic recovery strain during unloading, and 4. the results of a constitutive model for cyclic tests may be a function of the Heavyside class. These concepts have been supported by agreement between measured and predicted responses of soft connective tissue to three types of multiple cyclic tests which include rest periods of no extension and alternations between different strain levels. Such agreement has not been attained in the previous studies. Chun and Hubbard (2001) is our companion experimental analysis paper.     

大视场双目立体视觉柔性标定

为了实现双目立体视觉系统大范围高精度三维测量,提出了一种大视场双目立体视觉系统柔性标定方法,该方法将系统中各相机内部参数标定与相机间的姿态标定进行分离,标定内部参数时,只需要令标靶相对于相机任意摆放至少三个姿态,对标靶上的编码标志点进行识别,根据标靶上编码标志点信息,建立各姿态下视图的对应关系,粗略计算标志点的初始三维坐标;建立多姿态下逆向投影误差最小的目标函数,采用非线性最小二乘优化获取精确的相机内部参数和标志点三维坐标;最后,建立基于双相机逆向投影误差最小的目标函数,优化得到精确的相机间姿态的外部参数。实验结果表明:当测量空间为1 200mm×1 000mm×1 000mm时,立体视觉系统的测量精度优于0.1mm,满足大范围双目立体视觉系统的高精度测量需求。     

Nonlinear dynamic analysis of cantilever tube conveying fluid with system identification

The vibration of a flexible cantilever tube with nonlinear constraints when it is subjected to flow internally with fluids is examined by experimental and theoretical analysis. These kinds of studies have been performed to find the existence of chaotic motion. In this paper, the important parameters of the system leading to such a chaotic motion such as Young’s modulus and the coefficient of viscoelastic damping are discussed. The parameters are investigated by means of system identification so that comparisons are made between numerical analysis using the design parameters and the experimental results. The chaotic region led by several period-doubling bifurcations beyond the Hopf bifurcation is also re-established with phase portraits, bifurcation diagram and Lyapunov exponent so that one can define optimal parameters for system design.     

Dimensionless correlation for sand erosion of families of polymers

A single correlation for sand erosion of a “family” of polyurethanes is presented. By “family” is meant a group of chemically similar compounds. For the first time, an implication of Truesdell’s [C. Truesdell, The natural time of a viscoelastic fluid: its significance and measurement, Phys. Fluids 7 (8) (1964) 1134–1142] observations regarding the number of dimensional parameters in viscoelastic constitutive equations has been used together with the Pi theorem to prepare a single dimensionless correlation for a group of materials with similar but different forms of constitutive equations, different forms of stress-deformation behavior. To this point, such correlations have been for materials having the same form of constitutive equation, such as Hooke’s law or Newton’s law of viscosity.