Chatter stability analysis of high speed machining by means of spectral decomposition modeling

A technique to study the stability of high speed milling machining process and the chatter vibration is proposed. The equation of motion of a one degree of freedom (dof) model of the milling machine-tool system is considered. The system is described by means of a linear, ordinary, time dependent parameter, Periodic Delay Differential Equation (PDDE). A spectral decomposition is then applied to the equation, by means of a generalized harmonic balance technique, so that a linear, ordinary, constant parameter, PDDE is attained. A non standard eigenvalue problem can then be solved in order to study the stability of the solution, making it possible to predict the occurrence of chatter vibration during milling. A numerical example is presented in order to test the proposed approach. The well known Semi-Discretization (SD) method is considered as reference for selecting stable and unstable configurations to be tested. The time solution obtained by means of direct integration of the original equation is used to show that the proposed model preserve the stability characteristics of the original model. Strengths and limits of the proposed approach are critically discussed and compared with the SD method

PB-spline hybrid surface fitting technique

This work considers the fitting of data points organized in a rectangular array to parametric spline surfaces. Point Based (PB) splines, a generalization of tensor product splines, are adopted. The basic idea of this paper is to fit large scale data with a tensorial B-spline surface and to refine the surface until a specified tolerance is met. Since some isolated domains exceeding tolerance may result, detail features on these domains are modeled by a tensorial B-spline basis with a finer resolution, superimposed by employing the PB-spline approach. The present method leads to an efficient model of free form surfaces, since both large scale data and local geometrical details can be efficiently fitted. Two application examples are presented. The first one concerns the fitting of a set of data points sampled from an interior car trim with a central geometrical detail. The second one refers to the modification of the tensorial B-spline surface representation of a mould in order to create a local adjustment. Considerations regarding strengths and limits of the approach then follow

PB-spline finite element shell modeling and refinement technique

This paper presents a Point Based (PB) spline degenerate shell finite element model to analyze the behavior of thin and moderately thick-walled structures. Complex shapes are modeled with several B-spline patches assembled as in conventional finite element technique. The refinement of the solution is carried out by superimposing a tensorial set of B-spline functions on a patch and employing the PB-spline generalization. The domains for the numerical integration are defined by making use of the retained tensorial framework. Some numerical examples are presented. Considerations regarding strengths and limits of the approach then follow

Application of an automatic refinement technique for B2-spline finite element modeling of thin-walled mechanical components

This paper presents a technique for the automatic refinement of a B-spline degenerate shell finite element model for the vibration analysis of curved thin and moderately thick walled structures. A B2-spline finite element shell is defined as a generalization of the B-spline shell element. The proposed element makes it possible the finite element solution on a subdomain inside a selected element to be locally refined without affecting the discretization of the connected elements. A degrees of freedom constraint condition is imposed so that the C0 continuity of the displacement field is restored on the boundaries of the refined subdomains. The choice of the elements to be refined, the position and the extension of the refining subdomains are carried out automatically by means of an iterative procedure. The adaptive technique adopts a pointwise error functional based on the system total potential energy density and a two-step process. The subdomains to be refined are identified by means of the functional value. The number of shape functions on a subdomain is iteratively increased until the difference of the total potential energy, calculated between two consecutive iterations, is below a user defined tolerance. A numerical example is presented in order to test the proposed approach. Strengths and limits of the approach are critically discusse

Analysis of stability in high speed milling machining by means of spectral decomposition

A method to study the chatter stability of a high speed milling machining process is presented. Identification of the chatter-free machining parameters maximizing the metal removal rate is also studied in order to improve the milling process productivity. Starting from the equations of motion of a general machine-tool-workpiece system, resulting in a set of linear, ordinary, time dependent parameter, Periodic Delay Differential Equations (PDDEs), a spectral decomposition technique is applied. A new set of linear, ordinary, constant parameter, PDDEs is derived by applying a spectral decomposition and a generalized harmonic balance technique. The stability of the solution can be studied in advance by solving a non standard eigenvalue problem, making it possible to predict the occurrence of chatter vibration during milling. The proposed approach makes it possible to obtain a straightforward formula for the sensitivity of the real component of eigenvalues with respect to the variation of a technological parameter. A comparison with the well known Semi-Discretization (SD) method is carried out. The SD method is revisited so that a closed formula for the sensitivity of eigenvalues with respect to the variation of a technological parameter is introduced. A numerical example is presented in order to test the proposed approach. Finally strengths and limits of the proposed approach are critically discussed and a comparison with the SD method is carried out

B-spline finite element formulation for laminated composite shells

This paper presents a finite element formulation for the dynamical analysis of general double curvature laminated composite shell components, commonly used in many engineering applications. The Equivalent Single Layer theory (ESL) was successfully used to predict the dynamical response of composite laminate plates and shells. It is well known that the classic shell theory may not be effective to predict the deformational behavior with sufficient accuracy when dealing with composite shells. The effect of transverse shear deformation should be taken into account. In this paper a first order shear deformation ESL laminated shell model, adopting B-spline functions as approximation functions, is proposed and discussed. The geometry of the shell is described by means of the tensor product of B-spline functions. The displacement field is described by means of tensor product of B-spline shape functions with a different order and number of degrees of freedom with respect to the same formulation used in geometry description, resulting in a non-isoparametric formulation. A solution refinement method, making it possible to increase the order of the displacement shape functions without using the well known B-spline \u201cdegree elevation\u201d algorithm, is also proposed. The locking effect was reduced by employing a low-order integration technique. To test the performance of the approach, the static solution of a single curvature shell and the eigensolutions of composite plates were obtained by numerical simulation and are then compared with known solutions. Discussion follows

Dynamical Analysis of Shell Structures by Means of B-Spline Shape Functions

This paper presents a free vibration analysis of double curvature free form shaped shell structures using the B-spline shape functions approximation method. It is based on the Ritz method. The shell formulation is developed following the well known Ahmad degenerate approach including the effect of shear deformation. The assumed displacement field is described through non-uniform B-spline functions of any degree. The effect of locking is investigated and both reduced and modified quadrature integration rules are considered with the purpose of increasing the solution accuracy and diminishing the computational cost. Numerical simulation is reported for the evaluation of the eigensolution of plates, and of single and double curvature shells to test the effectiveness and the efficiency of the approach. The presence of spurious zero energy modes both at local and global level was investigated. The solutions are compared with other available analytical and numerical solutions, and discussed in detai

Free vibration analysis of Double Curvature Thin Walled Structures by a B-Spline Finite Element Approach

This paper presents a free vibration analysis of general double curvature shell structures using B-spline shape functions and a refinement technique. The shell formulation is developed following the well known Ahmad degenerate approach including the effect of the shear deformation. The formulation is not isoparametric, as a consequence the assumed displacement field is described through non-uniform B-spline functions of any degree. A solution refinement technique is considered by means of a high continuity p-method approach. The eigensolution of a plate, and of single and double curvature shells are obtained by numerical simulation to test the performance of the approach. Solutions are compared with other available analytical and numerical solutions, and discussion follows

Thin shell laminate B spline FE model updating by means of experimental FRFs

Fiber-reinforced composites are currently adopted as an alternative to conventional materials. Nevertheless, many parameters are needed to obtain an accurate dynamical model of a system made of fiber-reinforced composite material. In order to use such a model for design, diagnostics or other industrial tasks it is necessary to take into account of model uncertainties, such as unknown material properties, constraint and joint characteristics. In this paper, a technique to update the parameters of a finite element (FE) model for thin-walled structures made of composite material, by means of nondestructive testing is proposed. The updating procedure adopts experimental estimates of frequency response functions (FRFs) as input data. The FE model adopts B-spline basis functions for modeling both the geometry and the displacement field. The effect of damping is considered by adopting a real damping assumption (real eigen-modes), where mode damping ratios are assumed to vary with respect to frequency and modeled using non-parametric B-spline functions. The updating approach is based on the least squares minimization of an objective function dealing with residues, defined as the difference between the model based response and the experimental measured response, at the same frequency. The objective function is iteratively minimized by adopting a sensitivity approach, where a variable transformation is employed to constrain the updated parameters to lie in a compact domain without using additional variables. The parameters defining the material behavior and the damping model are taken into account in the updating process. Some examples are reported using measured data on a carbon-fiber reinforced tubolar structure. Results are shown and critically discussed

B-spline Finite element updating of a railway bridge deck

An updating procedure based on measured Frequency Response Function (FRF) data is proposed to correct the numerical parameters of a continuous B-spline shell dynamical model of a bridge deck. Since standard FRF based parametric updating techniques may fail in practical applications because of resonance frequency shift between measured and analytical simulations, a Frequency Domain Assurance Criterion (FDAC) correlation technique was considered and included in the formulation of the objective function. Updating parameters are normalized and constrained in a compact domain by means of a proper variable transformation choice. An example, concerning the model updating of a real railway bridge, is illustrated and discussed