Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler–Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned–pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.     

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.     

APPROXIMATE SOLUTIONS FOR TRANSIENT RESPONSE OF CONSTRAINED DAMPING LAMINATED CANTILEVER PLATE

The series composed by beam mode function is used to approximate the displacement function of constrained damping of laminated cantilever plates, and the transverse deformation of the plate on which a concentrated force is acted is calculated using the principle of virtual work.By solving Lagrange＇s equation, the frequencies and model loss factors of free vibration of the plate are obtained, then the transient response of constrained damping of laminated cantilever plate is obtained, when the concentrated force is withdrawn suddenly.The theoretical calculations are compared with the experimental data, the results show：both the frequencies and the response time of theoretical calculation and its variational law with the parameters of the damping layer are identical with experimental results.Also, the response time of steel cantilever plate, unconstrained damping cantilever plate and constrained damping cantilever plate are brought into comparison, which shows that the constrained damping structure can effectively suppress the vibration.     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

三种典型轴向运动结构的振动特性对比

轴向运动结构的横向振动一直是动力学领域的研究热点之一.目前大多数的文献只涉及对一种模型的研究,而针对几种模型的对比分析较少.本文对3种典型轴向运动结构(Euler梁、窄板和对边简支对边自由的板)的振动特性进行了对比分析.针对工程中不同的结构参数,本文为其理论研究中选择更加合理的模型提供了参考.通过复模态方法求解了3种模型的控制方程,给出了其相应的固有频率及模态函数.对于板模型,同时考虑了其自由边界的两种刚体位移以及弯扭耦合振动3种情况.通过数值算例给出了3种模型的前四阶固有频率随轴速和长宽比的变化情况,并应用微分求积法对复模态方法得到的解析解进行验证.特别采用三维图的形式分析了不同的轴速、阻尼、刚度和长宽比等参数混合时对3种模型第一阶固有频率的影响,着重研究了窄板和梁的不同的长宽比和轴速混合时对两者的第一阶固有频率的相对误差的影响.结果表明:随着轴速的增大,3种模型的固有频率逐渐减小. 窄板是板的一种简化模型.在各参数值发生变化时,阻尼对第一阶固有频率的影响最小.长宽比很大,轴速很小或为零时,复杂模型可以简化为简单模型.      

Non-local continuum modeling of carbon nanotubes: Physical interpretation of non-local kernels using atomistic simulations

The longitudinal, transverse and torsional wave dispersion curves in single walled carbon nanotubes (SWCNT) are used to estimate the non-local kernel for use in continuum elasticity models of nanotubes. The dispersion data for an armchair (10,10) SWCNT was obtained using lattice dynamics of SWNTs while accounting for the helical symmetry of the tubes. In our approach, the Fourier transformed kernel of non-local linear elastic theory is directly estimated by matching the atomistic data to the dispersion curves predicted from non-local 1D rod theory and axisymmetric shell theory. We found that gradient models incur significant errors in both the phase and group velocity when compared to the atomistic model. Complementing these studies, we have also performed detailed tests on the effect of length of the nanotube on the axial and shear moduli to gain a better physical insight on the nature of the true non-local kernel. We note that unlike the kernel from gradient theory, the numerically fitted kernel becomes negative at larger distances from the reference point. We postulate and confirm that the fitted kernel changes sign close to the inflection point of the interatomic potential. The numerically computed kernels obtained from this study will aid in the development of improved and efficient continuum models for predicting the mechanical response of CNTs.     

基于Riesz势空间分数阶算子的非局部粘弹性力学元件

将幂函数引入Eringen非局部线粘弹性本构,导出Riesz势形式的应力-应变关系。利用该关系,构造非局部弹簧和非局部阻尼器两类元件;利用元件的串联和并联,建立非局部Kelvin和非局部Maxwell粘弹性模型,推导模型的松弛模量和蠕变柔量。进一步,给出非局部粘弹性模型在生物组织超声波耗散建模中的应用。     

Global and local behaviour based composite damping studies on thin-walled box structure

The local (plate)/global (beam) vibration and damping behavior of composite thin-walled box member subjected to vibratory environment are studied and presented in this paper. This investigation is carried out by the use of 3D beam and thin plate finite elements and the corresponding modal frequencies/damping values of composite box beam are predicted by modal strain energy method. Application examples illustrate the ability of the 3D beam and thin plate element to predict both local and global modal frequencies/damping of hollow box sections. The influence of L/h ratio, b/h ratio and ply angle on the frequency and loss factor of composite box beam is investigated. In addition, an attempt has also been made to investigate the effect of temperature on the composite damping characteristics of rectangular box type section.     

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C⁰ beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.     

Dynamic analysis of a deployable/retractable damped cantilever beam

Deployable/retractable damped cantilever beams are a class of time-varying parametric structures which have attracted considerable research interest due to their many potential applications in the intelligent robot field and aerospace. In the present work, the dynamic characteristics of a deployable/retractable damped cantilever beam are investigated experimentally and theoretically. The time-varying damping, as a function of the beam length, is obtained by both the enveloped fitting method and the period decrement method. Furthermore, the governing equation of the deployable/retractable damped cantilever beam is derived by introducing the time-varying damping parameter,and the corresponding closed-form solution and vibration principles are investigated based on the averaged method. The theoretical predictions for transient dynamic responses are in good agreement with the experimental results. The dynamic mechanism analysis on time-varying damping offers flexible technology in mechanical and aerospace fields.     

基于广义卡尔曼滤波的桥梁结构物理参数识别

基于广义卡尔曼滤波提出了随机荷载作用下桥梁结构物理参数的识别方法。首先,以荷载为观测对象,推导出基于有限元模型的桥梁结构系统的观测方程,以结构待识别的物理参数为状态向量,建立系统状态方程;然后,对该状态方程和观测方程构成的非线性参数系统应用广义卡尔曼滤波,从而识别出结构的物理参数。对一座简支梁桥和一座三跨连续梁桥在不同工况下的物理参数识别进行了数值仿真,结果表明本文方法能够准确地识别桥梁结构全部刚度参数、质量参数和阻尼参数,且具有很强的抗噪性能,从而验证了本文方法的有效性和鲁棒性,可应用于识别大型桥梁结构的物理参数。     

Structural damping identification method using normal FRFs

All structures exhibit some form of damping, but despite a large literature on the damping, it still remains one of the least well-understood aspects of general vibration analysis. The synthesis of damping in structural systems and machines is extremely important if a model is to be used in predicting vibration levels, transient responses, transmissibility, decay times or other characteristics in design and analysis that are dominated by energy dissipation. In this paper, new structural damping identification method using normal frequency response functions (NFRFs) which are obtained experimentally is proposed and tested with the objective that the damped finite element model is able to predict the measured FRFs accurately. The proposed structural damping identification is a direct method. In the proposed method, normal FRFs are estimated from the complex FRFs, which are obtained experimentally of the structure. The estimated normal FRFs are subsequently used for identification of general structural damping. The effectiveness of the proposed structural damping identification method is demonstrated by two numerical simulated examples and one real experimental data. Firstly, a study is performed using a lumped mass system. The lumped mass system study is followed by case involving numerical simulation of fixed–fixed beam. The effect of coordinate incompleteness and robustness of method under presence of noise is investigated. The performance of the proposed structural damping identification method is investigated for cases of light, medium, heavily and non-proportional damped structures. The numerical studies are followed by a case involving actual measured data for the case of a cantilever beam structure. The results have shown that the proposed damping identification method can be used to derive an accurate general structural damping model of the system. This is illustrated by matching the damped identified FRFs with the experimentally obtained FRFs.     

Macroscopic damping model for structural dynamics with random polycrystalline configurations

In this paper the macroscopic damping model for dynamical behavior of the structures with random polycrystalline configurations at micro–nano scales is established.First, the global motion equation of a crystal is decomposed into a set of motion equations with independent single degree of freedom(SDOF) along normal discrete modes, and then damping behavior is introduced into each SDOF motion.Through the interpolation of discrete modes, the continuous representation of damping effects for the crystal is obtained.Second, from energy conservation law the expression of the damping coefficient is derived, and the approximate formula of damping coefficient is given. Next, the continuous damping coefficient for polycrystalline cluster is expressed,the continuous dynamical equation with damping term is obtained, and then the concrete damping coefficients for a polycrystalline Cu sample are shown. Finally, by using statistical two-scale homogenization method, the macroscopic homogenized dynamical equation containing damping term for the structures with random polycrystalline configurations at micro–nano scales is set up.     

FREE VIBRATION ANALYSIS OF SIMPLY SUPPORTED BEAM WITH BREATHING CRACK USING PERTURBATION METHOD

In this paper, a new analytical method for vibration analysis of a cracked simply supported beam is investigated. By considering a nonlinear model for the fatigue crack, the governing equation of motion of the cracked beam is solved using perturbation method. The solution of the governing equation reveals the superhaxmonics of the fundamental frequency due to the nonlinear effects in the dynamic response of the cracked beam. Furthermore, considering such a solution, an explicit expression is also derived for the system damping changes due to the changes in the crack parameters, geometric dimensions and mechanical properties of the cracked beam. The results show that an increase in the crack severity and approaching the crack location to the middle of the beam increase the system damping. In order to validate the results, changes in the fundamental frequency ratios against the fatigue crack severities are compared with those of experimental results available in the literature. Also, a comparison is made between the free response of the cracked beam with a given crack depth and location obtained by the proposed analytical solution and that of the numerical method. The results of the proposed method agree with the experimental and numerical results.     

Approximate analytical solutions and experimental analysis for transient response of constrained damping cantilever beam

Vibration mode of the constrained damping cantilever is built up according to the mode superposition of the elastic cantilever beam. The control equation of the constrained damping cantilever beam is then derived using Lagrange＇s equation. Dynamic response of the constrained damping cantilever beam is obtained according to the principle of virtual work, when the concentrated force is suddenly unloaded. Frequencies and transient response of a series of constrained damping cantilever beams are calculated and tested. Influence of parameters of the damping layer on the response time is analyzed. Analyitcal and experimental approaches are used for verification. The results show that the method is reliable.     

Long-range cohesive interactions of non-local continuum faced by fractional calculus

A non-local continuum model including long-range forces between non-adjacent volume elements has been studied in this paper. The proposed continuum model has been obtained as limit case of two fully equivalent mechanical models: (i) A volume element model including contact forces between adjacent volumes as well as long-range interactions, distance decaying, between non-adjacent elements. (ii) A discrete point-spring model with local springs between adjacent points and non-local springs with distance-decaying stiffness connecting non-adjacent points. Under the assumption of fractional distance-decaying interactions between non-adjacent elements a fractional differential equation involving Marchaud-type fractional derivatives has been obtained for unbounded domains. It is shown that for unbounded domains the two mechanical models revert to Lazopoulos and Eringen model with fractional distance-decaying functions. It has also been shown that for a confined bar, the stress–strain relation is substantially different from that obtained simply using the truncated Marchaud derivatives since a double integral instead of convolution integral appears. Moreover, in the analysis of bounded domains, the governing equations turn out to an integro-differential equation including only the integral part of Marchaud fractional derivatives on finite interval. The mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations, and the constitutive law of the proposed continuum model has been reported by mathematical induction. Several numerical applications have been reported to show, verify and assess the concepts listed in this paper.     

A realistic estimation of the effective breadth of ribbed plates

In this paper a realistic estimation of the effective breadth of a stiffened plate is presented. For the estimation of the effective breadth the adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems, which are nonlinearly coupled, is achieved using the analog equation method. After the solution of the plate––beams system is achieved, the distribution of the axial stresses across the plate, resulting from both the bending and the inplane action of the plate, is obtained. Integrating this distribution across the plate the values of the effective breadth are obtained. The influence of these values from the beam stiffness and their variation along the longitudinal direction of the plate are shown as compared with those obtained from various codes through numerical examples with great practical interest.     

On non-local and non-homogeneous elastic continua

A thermodynamic framework endowed with the concept of non-locality residual is adopted to derive non-local models of integral-type for non-homogeneous linear elastic materials. Two expressions of the free energy are considered: the former yields a one-component non-local stress, the latter leads to a two-component local–non-local stress since the stress is expressed as the sum of the classical local stress and of a non-local component identically vanishing in the case of constant strains. The attenuation effects are accounted for by a symmetric space weight function which guarantees the constant strain requirement as well as the dual constant stress condition everywhere in the body. The non-local and non-homogeneous elastic structural boundary-value problem under quasi-static loads is addressed in a geometrically linear range. The complete set of variational formulations for the structural problem is then provided in a unitary framework. The solution uniqueness of the non-local structural model is proved and the non-local FEM is addressed starting from the non-local counterpart of the total potential energy. Numerical applications are provided with reference to a non-homogeneous bar in tension using the Fredholm integral equation and the non-local FEM. The solutions show no pathological features such as numerical instability and mesh sensitivity for degraded bar conditions.