spectral-Tchebychev Technique Research Articles

Spectral-Tchebychev technique for the free and stochastic vibration analysis of functionally graded plates with piezoelectric patches

This paper presents a Spectral-Tchebychev (S-T) dynamic model to examine the free and random vibration properties of functionally gradient plates with piezoelectric patches (FGPPP) under random excitation. Firstly, the plate is divided into several sub-plates according to the position of the piezoelectric patches through the multi-partitioning strategy. Then, the dynamical equations for each sub-plate are derived based on the first-order shear deformation theory (FSDT) and Hamilton's variational principle. The boundary conditions and the coupling relationships of the adjacent plates are handled using the artificial spring technique. Finally, the free and stochastic vibration characteristics of FGPPP are obtained using the Spectral-Tchebychev technique, where the stationary random load is introduced by the pseudo-excitation method (PEM). The rationality and validity of the S-T model in predicting the vibration characteristics of FGPPP are verified by the comparison with the results from literature and FEM. The influence mechanism of some key parameters including the thickness ratio, area ratio, patching position, patching type, power law index, etc., on the vibration characteristics are systematically investigated.

Tool-tip dynamics in micromachining with arbitrary tool geometries and the effect of spindle speed

Mechanical micromachining has become a leading approach to fabricating complex three-dimensional microscale features and miniature devices on a broad range of materials. To satisfy the accuracy and productivity demands of various micromachining applications, the tool-tip dynamics, i.e., the dynamic behavior of the tool-ultra high-speed spindle assembly as reflected at the cutting edges of a microtool, should be well-understood. However, existing techniques for predicting tool-tip dynamics pose strict limitations in frequency bandwidth and do not capture the effect of the spindle speed on tool-tip dynamics. In addition, those techniques cannot be applied broadly to predict tool tip dynamics for a myriad of microtool geometries. This paper presents a systematic approach to predicting the tool-tip dynamics accurately in micromachining when using ultra-high-speed (UHS) spindles and for arbitrary microtool geometries. The speed-dependent dynamics of the UHS spindle are obtained using an experimental approach. The dynamics of microtools are obtained analytically using the spectral Tchebychev technique, such that any microtool geometry can be modeled accurately and does not require new testing. The tool-tip dynamics are then predicted by combining (coupling) the spindle and microtool dynamics using a novel modal-Tchebychev domain coupling technique. This technique enabled accurate coupling/decoupling of substructure dynamics within a broad frequency bandwidth (up to 15 kHz) and at different spindle speeds (up to 120,000 rpm). Furthermore, an empirical model for the mode-splitting effect is derived to capture the effect of spindle speeds on tool-tip dynamics. The overall approach is demonstrated and experimentally validated on a UHS spindle with microtool blanks and micro endmills at operational speeds. We conclude that the presented methodology can be used to determine the tool-tip dynamics accurately.

Free vibration analysis of coupled structures of laminated composite conical, cylindrical and spherical shells based on the spectral-Tchebychev technique

In this article, a numerical spectral-Tchebyshev (ST) technique is applied to solve the free vibration strong solution of the coupled structures of laminated composite conical, cylindrical and spherical shells under various boundary conditions. The artificial spring technique is introduced to realize the coupling connection of substructures and simulate the arbitrary boundary conditions. Considering the first-order shear deformation shell theory (FSDST), and using Hamilton variational analysis under its framework to derive the governing equations of motions of three coupled structures: coupled conical-cylindrical shells, coupled spherical-cylindrical shells and coupled spherical-cylindrical-conical shells. The solution of the governing equations of motions is obtained using the spectral-Tchebyshev technique. The variables along the generatrix direction are discretized based on the Gauss-Lobatto nodes, while the Tchebyshev polynomials are used for spectral approximation. The variables along the circumferential direction are approximated by Fourier series. The calculated free vibration characteristics of the coupled structures are compared with the numerical results based on previous references and the finite element method, which completely reflects the excellent convergence and calculation accuracy of the spectral-Tchebyshev technique. On this basis, the effect of the material properties and geometric dimensions of the substructures (conical, cylindrical and spherical shells) on the free vibration frequencies of the coupled structure is further studied.

Spectral-Tchebychev technique for the free vibration analysis of composite laminated stepped and stiffened cylindrical shells with arbitrary boundary conditions

By combining the first-order shear deformation theory (FSDT) with the spectral-Tchebychev (ST) technique, including the arbitrary boundary conditions, this paper analyzes the free vibration characteristics of composite laminated stepped and stiffened cylindrical shells. Based on artificial spring technique, the boundary conditions at both ends of the structure and the coupling continuity conditions between the shell segments are simulated. Then the Hamilton’s principle is applied to obtain the governing differential equations of each shell segments, the governing equations of the boundary conditions at both ends of the structure and the elastic coupling governing equations between adjacent shell segments. The displacement components of the composite laminated stepped and stiffened cylindrical shells are spectrally expanded into the finite terms of Tchebyshev polynomials, and then discretized according to Gauss-Lobatto points to obtain the matrix form characteristic equations of the structure. The frequency of free vibration of the structure and the corresponding mode shape can be obtained by solving the characteristic equation. In the numerical example, the convergence analysis of the model established by the technique in this paper is carried out, and then the frequency results of composite laminated stepped and stiffened cylindrical shells under arbitrary boundary conditions calculated in this paper are compared with the calculation results of related literature and finite element software. It is fully verified that the proposed spectral-Tchebychev technique not only has the characteristics of rapid convergence, but also can accurately analyze the free vibration behaviors of composite laminated stepped and stiffened cylindrical shells under arbitrary boundary conditions. After proving its effectiveness and accuracy, the parameter studies are finally carried out to analyze the influence of material parameters and rib geometric parameters on the free vibration behaviors of composite laminated stiffened cylindrical shells.

Three-dimensional vibration behavior of bi-directional functionally graded curved parallelepipeds using spectral Tchebychev approach

This paper presents the application of the three-dimensional spectral-Tchebychev technique to accurately predict the vibration behavior of bi-directional functionally graded material curved parallelepipeds including geometries such as beams, thin/thick plates, and solids. In this study, the material distribution within the domain of the structure is obtained using bi-directional Mori-Tanaka method. To derive the boundary value problem governing the dynamics of functionally graded curved parallelepipeds, three-dimensional elasticity equations are used together with extended Hamilton’s principle. Numerical solution of the integral boundary value problem is performed using the three-dimensional spectral Tchebychev approach. To validate and assess the performance of the presented solution approach, a number of case studies are conducted. In each case study, the non-dimensional natural frequencies and mode shapes are calculated and compared to those found using a finite element solution approach. Furthermore, computational time of the simulation is measured in each case. It is shown that the presented solution technique enables accurate prediction of vibration behavior of bi-directional functionally graded curved parallelepipeds as precise as a finite elements method, but for a fraction of the computational cost.

A Model for Bending, Torsional, and Axial Vibrations of Micro- and Macro-Drills Including Actual Drill Geometry—Part I: Model Development and Numerical Solution

Part I of this work presents a combined one-dimensional/three-dimensional approach for obtaining a unified model for the dynamics of micro- and macro-drills. To increase the numerical efficiency of the model, portions of the drill with circular cross-section (shank, extension, and tapered sections) are modeled using one-dimensional beam models without compromising model accuracy. A three-dimensional model is used for an accurate representation of the fluted section, considering the actual geometry with the pretwisted shape and axially varying (nonaxisymmetric) cross-section. The actual cross-section of the drills is incorporated to the model through a polynomial mapping while the pretwist effect is captured by defining a rotating reference frame. The boundary-value problem for both one- and three-dimensional models are derived using a variational approach, based on the extended Hamilton’s principle, and are subsequently solved by applying the spectral-Tchebychev technique. A component-mode synthesis is used for connecting the individual sections to obtain the dynamic model for the entire drill. Convergence of the model is studied by varying the number of polynomials for each section. The experimental validation of the model is included in Part II for both macro- and micro-drills. Also included in Part II is an analysis of drill dynamics for varying drill-geometry parameters and axial (thrust) force.

A spectral-Tchebychev technique for solving linear and nonlinear beam equations

This paper presents a spectral-Tchebychev technique for solving linear and nonlinear beam problems. The technique uses Tchebychev polynomials as spatial basis functions, and applies Galerkin's method to obtain the spatially discretized equations of motion. Unlike alternative techniques that require different admissible functions for each different set of boundary conditions, the spectral-Tchebychev technique incorporates the boundary conditions into the derivation, and thereby enables the utilization of the solution for any linear boundary conditions without re-derivation. Furthermore, the proposed technique produces symmetric system matrices for self-adjoint problems. In this work, the spectral-Tchebychev solutions for Euler–Bernoulli and Timoshenko beams are derived. The convergence and accuracy characteristics of the spectral-Tchebychev technique is studied by solving eigenvalue problems with different boundary conditions. It is found that the convergence is exponential, and a small number of polynomials is sufficient to obtain machine-precision accuracy. The application of the technique is demonstrated by solving: (1) eigenvalue problems for tapered Timoshenko beams with different boundary conditions, taper ratios, and beam lengths; (2) an Euler–Bernoulli beam problem with spatially and temporally varying forcing, elastic boundary, and damping; (3) large-deflection (nonlinear) Euler–Bernoulli beam problems with different boundary conditions; and (4) a micro-beam problem with nonlinear electrostatic excitation. The results obtained from the spectral-Tchebychev solutions are seen to be in excellent agreement with those presented in the literature.

An Analytical Model for Micro-Endmill Dynamics

This article presents an analytical model of the transverse vibration of rotating micro-endmills in the presence of three-dimensional tilt and rotary axis misalignment. To accurately capture the rotary inertia and shear deformations of non-slender micro-endmills, the Timoshenko beam model is used. The boundary-value problem is derived using the extended Hamilton's principle. The numerical solution of the problem is obtained through a spectral Tchebychev technique. Non-smooth diameter variations along the length are handled by considering the micro-endmill as a sectioned beam and using a component mode synthesis method. The mode shapes and natural frequencies from the model are compared to those from a commercial finite element solver. The effectiveness of the model is illustrated by applying it to analyze (1) the effect of the rotational speed on the micro-endmill dynamics, (2) the effect of geometry on mode shapes and natural frequencies, (3) the influence of three-dimensional tilt and eccentricity on vibrations, and (4) the dynamic response of the micro-endmills to harmonic forcing. It was shown that the derived model and associated spectral Tchebychev technique effectively and efficiently capture the dynamic behavior of rotating micro-endmills. The model can be used for designing micro-tools with specified dynamic characteristics, as well as for modeling and stability analysis of the micromilling process.

Microendmill Dynamics Including the Actual Fluted Geometry and Setup Errors—Part I: Model Development and Numerical Solution

This paper presents an analytical model for the microendmill dynamics, including the setup errors, the axial force, the sectioned tool geometry, and the actual crosssection of the twisted fluted section. Different tool geometries and tool-grinding errors can also be incorporated in the presented model. To include the shear deformations and rotary inertia effects arising from the stubby nature of the microendmill, the Timoshenko beam equations are used in the derivation. The boundary-value problem is derived for each of the shank, taper, and fluted sections of the microendmill, and a component mode synthesis technique is used to combine the individual sections. A new spectral-Tchebychev technique is utilized to obtain the numerical solution of the boundary-value problem without resorting to the finite-elements technique. As a result, a simple yet accurate description of the microendmill dynamics is obtained. The solution is experimentally validated in Part II of the paper.

An improved tool–holder model for RCSA tool-point frequency response prediction

Abstract In addition to the precise kinematic motions of the machine tools and spindles, machining accurate parts necessitates controlling the dynamic behavior of the tool tip with respect to the workpiece. High-fidelity models of tool-tip dynamics can be used to select operating parameters that improve the accuracy by reducing the effect of vibrations. To effectively model the tool-tip dynamics for arbitrary tool-and-holder combinations using the receptance coupling substructure analysis (RCSA) technique, highly accurate and numerically efficient models of the tool–holder dynamics are needed. In this paper, we present a tool–holder model that incorporates a spectral-Tchebychev technique with the Timoshenko beam equation to obtain a completely parameterized solution. Comparison of the tool–holder model to a three-dimensional finite elements solution shows that the dynamic behavior is captured with sufficient accuracy. The tool–holder model is then coupled with the experimentally determined spindle–machine dynamics through RCSA to realize a model of the tool-tip dynamics. The coupled model is validated through experiments for three different tool overhang lengths. The presented technique can be used to predict the tool-tip dynamics for different tool-and-holder combinations and for optimization studies without the need for extensive experimentation.