Numerical and experimental investigations of dynamic contact phenomena in jointed structures

Structural dynamic properties like stiffness and damping are strongly influenced by contact phenomena in jointed structures. Therefore numerical and experimental investigations are presented, in order to understand and predict the behaviour of such mechanical devices. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于模式搜索的轮胎压力监测方法研究

提出了一种含有两个参数的轮胎纵向力模型,并通过该模型利用基于模式搜索的最小二乘法对轮胎的纵向刚度和滚动半径进行了动态估计.该算法应用于车辆的实际试验数据,结果证明轮胎的压力与轮胎纵向刚度近似成反比,这为利用轮胎纵向刚度监测轮胎压力变化,提供了可靠的理论依据.     

Sensitivity analysis of weakened beams on elastic foundation with Green's functions

The approach of Sensitivity Analysis with Green's Functions (SAGF) [1,2] was developed to predict changes in deformations, stresses or eigenfrequencies of structures resulting from stiffness modifications or cracking by considering only the weakened or damaged parts of the structures. This approach results in a local analysis instead of a global analysis by recalculating the whole structure. Consequently, it is computationally less time-consuming than the conventional methods based on a global analysis. The key idea of the SAGF approach is based on the comparison between the elastic strain energies of the original and the weakened structures and the substitution of the virtual displacements by the corresponding Green's functions [1, 2]. Furthermore, an approximate approach for the sensitivity analysis was suggested which is described in [1, 3] in details. This approach enables us to predict the changes in the structural responses due to the stiffness weakening in the beam or in the elastic Winkler foundation by considering only the internal forces or the deflections of the original unweakened system. In addition, an iterative method was developed to enhance the accuracy of the SAGF approach. In this paper, the local SAGF method for sensitivity analysis of elastic beams on Winker foundation with stiffness weakening is presented. The accuracy and the efficiency of the proposed method are verified by using a numerical example. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed Micromechanics and Continuum Damage Mechanics Approach to Transverse Cracking in [S, 90n]s Laminates

The stiffness reduction in [S, 90 _n ] _s laminates due to transverse cracking in 90-layers is analyzed using the synergistic continuum damage mechanics (SCDM) and a micromechanics approach. The material constants involved in the SCDM model are determined using the stiffness reduction data for a reference cross-ply laminate. The constraint efficiency factor, which depends on the stiffness and geometry of neighboring layers, is assumed to be proportional to the average crack opening displacement (COD). The COD as a function of the constraint effect of adjacent layers and crack spacing is described by a simple power law. The crack closure technique and Monte Carlo simulations are used to model the damage evolution: the 90-layer is divided into a large number of elements and the critical strain energy rate G _c having the Weibull distribution is randomly assigned to each element. The crack density data for a [0₂/90₄] _s cross-ply laminate are used to determine the Weibull parameters. The simulated crack density curves are combined with the CDM stiffness reduction predictions to obtain the stiffness versus strain. The methodology developed is successfully used to predict the stiffness reduction as a function of crack density in [±/90₄] _s laminates.     

Analytical prediction of Young's modulus of carbon nanotubes using a variational method

Molecular mechanics and solid mechanics are linked to establish, a nanoscale analytical continuum theory for determination of stiffness and Young's modulus of carbon nanotubes. A space-frame structure consisted of representative unit cells has been introduced to describe the mechanical response of carbon nanotubes to the applied loading. According to this assumption a novel unit cell, given the name mechanical unit cell here is introduced to construct a graphene sheet or the wall of the carbon nanotubes. Incorporating the Morse potential function with the strain energy of the mechanical unit cells in a carbon nanotube is the key point of this study. The structural model of the carbon nanotube is solved to obtain its Young's modulus by using the principle of minimum total potential energy. It was found that the Young's modulus of the zigzag and armchair single-walled carbon nanotubes are 1.42 and 1.30 TPa, respectively. The results indicate sensitivity of the stiffness and Young's modulus of carbon nanotubes to chirality but show no dependence on its diameter. The presented analytical investigation provides a very simple approach to predict the Young's modulus of carbon nanotubes and the obtained results are in good agreement with the existing experimental and theoretical data.     

Modeling Matrix Cracking in Composite Laminates Under Thermo‐mechanical Loading

In this paper a theoretical approach is presented for analyzing the ply cracking in general symmetric laminates subjected to any combination of in‐plane mechanical loading and uniform temperature changes. The energy release rate for a ply cracking is explicitly expressed as a function of stiffness reduction parameters of the laminate before the ply cracking and after the ply cracking. Finally, the model is used to predict the thermomechanically cracking stress for the first‐ply‐cracking.     

A constitutive model for continuous fiber reinforced polymer plies

In the present paper a constitutive model is reviewed which can be used to predict the non-linear behavior of continuous fiber reinforced laminates with polymeric matrix materials. The constitutive model considers stiffness degradation and plastic strain accumulation at the length scale of the individual plies (laminae). These effects are modeled via two different phenomenological approaches, however, their interaction is considered when the constitutive equations are solved by an implicit integration scheme. To demonstrate the predictive capabilities of the individual model parts, examples are given where the above mentioned effects are decoupled. This way, their impact on the laminate's response can be studied independently. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a Compliant Mechanism for Positioning in the cm-Range

The kinematics and kinetics of a compliant mechanism are analyzed quasi-statically. The mechanism consists of a parallelkinematic (Biglide type) with the particular feature, that its revolute joints are implemented by flexure hinges. In the considered example the flexure hinges are uniform beams similar to leaf springs. The preliminary considerations with a rigid body system and by linear theory of elasticity result in an initial geometry. In order to predict the large displacement behavior more precisely the theory of Elastica is used. An optimization method is applied to solve the mechanism's state of deformation. It was found that some aspects can be approximated by a pseudo rigid body system, while others, namely the rotation, cannot be rendered. The decreased stiffness in the highly deformed operating state appears as key problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Correlation of experimental flap-lag-torsion damping—a case study

In 1982, carefully done experiments measured the flap-lag-torsion damping of hingeless rotor at various RPM's and for several combinations of flexure geometry and stiffness. The best elastic-blade stability codes of that time were unable to predict this damping and typically gave errors of the order of 100% (as percentage of the largest measured damping, 5 rad/s). Through the past ten years, various unsuccessful efforts have been made to improve those predictions. Now, improved correlation has been performed with errors of 0% to 20% of 5 rad/s. The results show that most of the former discrepancies can be attributed to aerodynamic effects, the largest of which is the dynamics of the three-dimensional wake.     

Fast simplicial finite element algorithms using Bernstein polynomials

Fast algorithms for applying finite element mass and stiffness operators to the B-form of polynomials over d-dimensional simplices are derived. These rely on special properties of the Bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor-product techniques in rectangular domains. First, special structure leading to fast application of mass matrices is developed. Then, by factoring stiffness matrices into products of sparse derivative matrices with mass matrices, fast algorithms are also obtained for stiffness matrices.     

Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.     

Implementation of a cohesive zone model for crack closure analysis of rotors

The presence of a crack in a rotor introduces a local flexibility which affects its dynamic response. Moreover, the crack may open and close during the vibration period. The crack status is a function of time and also depends on the rotational speed and the vibration amplitude of the rotor. This nonlinear case is still a challenging research topic especially in the field of closing crack in the rotating shaft. A cohesive zone model is developed in order to analyze the stiffness of a crack in a rotating shaft. The proposed expression will be compared to three different crack models, namely, a breathing crack model, a switching crack model and an open crack model. Moreover, a cohesive law to predict and to analyse the stress at the crack tip is presented. The numerical model is implemented using a finite element formulation. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix–vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension.     

Transverse cracks in cross-ply laminates 2. Stiffness degradation

From the results of stress analysis between two transverse cracks in cross-ply laminate [1], a model for the stiffness reduction based on generalized plane strain assumptions has been developed. Simple analytical expressions are obtained for the longitudinal modulus and the Poisson's ratio as a function of the transverse crack density. Apart from the crack density, these expressions depend only on the elastic and geometrical properties of constituent laminae and the average crack opening displacement (ACOD) normalized in the proper way. Calculations of the ACOD are performed and analyzed with the FEM and analytical models used for the stress analysis in [1]. The predicting capabilities of approximate models are discussed in comparison with experimental data and FEM results. In order to predict the stiffness degradation for a wide variety of laminates, a simple procedure requiring only one FEM calculation for some average laminate with average crack spacing is proposed and has been proved effective.Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 211–233, March–April, 1998.     

A preconditioning strategy for boundary element Galerkin methods

The Dirichlet and Neumann problems for the Laplacian are reformulated in the usual way as boundary integral equations of the first kind with symmetric kernels. These integral equations are solved using Galerkin's method with piecewise-constant and piecewise-linear boundary elements, respectively. In both cases, the stiffness matrix is symmetric and positive-definite, and has a condition number of order N, the number of degrees of freedom. By contrast, the condition number of the product of the two stiffness matrices is bounded independently of N. Hence, we can use the Neumann stiffness matrix to precondition the Dirichlet stiffness matrix, and vice versa. © 1997 John Wiley & Sons, Inc.     

An analytical dynamic model for single-cracked beams including bending,axial stiffness,rotational inertia,shear deformation and coupling effects

This paper develops an analytical dynamic model for cracked beams including bending, axial stiffness, rotational inertia, shear deformation and the coupling of the last two effects. The damage is modelled using a rotational spring that simulates the crack based on fracture mechanics theory. The developed model is used to predict variations on natural frequencies for several crack sites and damage magnitude along the beam. The importance of this work lies in the development of an analytical model that has no approximation due to discretization of the displacement field. This initial theoretical approach describes the expected behaviour for changes in the natural frequencies for simply-supported and clamped-free beams with the precision that only analytical methods allow. The results provide a useful benchmark to compare with approximate numerical methods that can be used to model and analyse the problem. The model showed similar results for long span beams, but the inclusion of rotational inertia and shear deformation effects rendered improvements in the dynamic behaviour mainly in the case of slender and short span beams when compared with the simplified Euler–Bernoulli model.     

Hopf bifurcation for delay differential equation with application to machine tool chatter

In the machining process, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece, hence the relevant measures must be taken to predict and avoid this phenomenon of instability. In this paper, we propose a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. It is proved that Hopf bifurcation exists when bifurcation parameter equals a critical value, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are given by using the normal form method and center manifold theorem. Numerical simulations show excellent agreement with the theoretical results.     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

Correction of stiffness and mass matrices utilizing simulated measured modal data

Measured and analytical data are unlikely to be equal due to measured noise, model inadequacies and structural damage, etc. It is necessary to update the physical parameters of analytical models for proper simulation and design studies. Starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes, this study presents the equations to update the physical parameters of stiffness and mass matrices simultaneously for analytical modelling by minimizing a cost function in the satisfaction of the dynamic constraints of orthogonality requirement and eigenvalue function. The proposed equations are straightforwardly derived by Moore–Penrose inverse matrix without using any multipliers. The cost function is expressed by the sum of the quadratic forms of both the difference between analytical and updated mass, and stiffness matrices. The results are compared with the updated mass matrix to consider the orthogonality requirement only and the updated stiffness matrix to consider the eigenvalue function only, respectively. Also, they are compared with Wei’s method which updates the mass and stiffness matrices simultaneously. The validity of the proposed method is illustrated in an application to correct the mass and stiffness matrices due to section loss of some members in a simple truss structure.     

Development of generalized Iwan model to simulate frictional contacts with variable normal loads

Most friction models are originally proposed to predict restoring forces in mechanical contacts with constant normal load. In practice the contact interface kinematics may involve normal motion in addition to the tangential displacements, leading to variation of the contact normal load. This phenomenon is observed most strongly in contacts with high lateral vibration amplitudes and is known as slap. The current study establishes a general friction model to account for variation in the normal load and enables one to predict the behavior of a contact more precisely. Iwan model (1966) [5] is a suitable candidate for contact interface modeling and is able to represent the stick-micro/macro slip behavior involved in a friction contact. This physical based model is employed in the current work and its physical parameters are generalized to include the normal load variation effects. The model is characterized by a slippage distribution density function and a linear stiffness at stick state. Both these parameters, defined in presence of constant normal load in the original model, are derived considering normal load variation leading to generalization of the contact model. Conventional models with constant normal loads produce symmetric contact interface hysteresis loops, but the developed generalized Iwan model is capable of generating asymmetric hysteresis loops similar to those frequently seen in experiments. The generalized contact model is employed to simulate the measured behavior of a beam with frictional support observed in an experimental test set-up. The contact slippage distribution function is first identified in a constant normal load condition. Next in low levels of contact preloads where variation of the normal load is significant, the identified distribution function in generalized form is employed to predict the experimental observations.