Stability Analysis of Milling Via the Differential Quadrature Method

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Ye Ding ◽  
LiMin Zhu ◽  
XiaoJian Zhang ◽  
Han Ding
Keyword(s):  
Stability Analysis ◽  
Analytical Solution ◽  
Free Vibration ◽  
Forced Vibration ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Algebraic Equations ◽  
Sampling Grid ◽  
Grid Points

This paper presents a time-domain semi-analytical method for stability analysis of milling in the framework of the differential quadrature method. The governing equation of milling processes taking into account the regenerative effect is formulated as a linear periodic delayed differential equation (DDE) in state space form. The tooth passing period is first separated as the free vibration duration and the forced vibration duration. As for the free vibration duration, the analytical solution is available. As for the forced vibration duration, this time interval is discretized by sampling grid points. Then, the differential quadrature method is employed to approximate the time derivative of the state function at a sampling grid point within the forced vibration duration by a weighted linear sum of the function values over the whole sampling grid points. The Lagrange polynomial based algorithm (LPBA) and trigonometric functions based algorithm (TFBA) are employed to obtain the weight coefficients. Thereafter, the DDE on the forced vibration duration is discretized as a series of algebraic equations. By combining the analytical solution of the free vibration duration and the algebraic equations of the forced vibration duration, Floquet transition matrix can be constructed to determine the milling stability according to Floquet theory. Simulation results and experimentally validated examples are utilized to demonstrate the effectiveness and accuracy of the proposed approach.

Harmonic Differential Quadrature Method for Surface Location Error Prediction and Machining Parameter Optimization in Milling

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Ye Ding ◽  
XiaoJian Zhang ◽  
Han Ding
Keyword(s):  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Machining Parameters ◽  
Quadrature Method ◽  
Location Error ◽  
Sampling Grid ◽  
Surface Location Error ◽  
Harmonic Differential Quadrature Method ◽  
Surface Location ◽  
Grid Points

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.

scholarly journals A Mixed Layerwise-Differential Quadrature Method for 3-D Vibration and Buckling Analyses of Orthogonally Stiffened Composite Conical Shell

2019 ◽  
Vol 24 (2) ◽  
pp. 217-227
Author(s):  
Mostafa Talebitooti
Keyword(s):  
Boundary Conditions ◽  
Conical Shell ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Algebraic Equations ◽  
Discrete Elements ◽  
Arbitrary Boundary ◽  
Stiffened Shells

A layerwise-differential quadrature method (LW-DQM) is developed for the vibration analysis of a stiffened laminated conical shell. The circumferential stiffeners (rings) and meridional stiffeners (stringers) are treated as discrete elements. The motion equations are derived by applying the Hamilton’s principle. In order to accurately account for the thickness effects and the displacement field of stiffeners, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations applying the DQM in the meridional direction. The advantage of the proposed model is its applicability to thin and thick unstiffened and stiffened shells with arbitrary boundary conditions. In addition, the axial load and external pressure is applied to the shell as a ratio of the global buckling load and pressure. This study demonstrates the accuracy, stability, and the fast rate of convergence of the present method, for the buckling and vibration analyses of stiffened conical shells. The presented results are compared with those of other shell theories and a special case where the angle of conical shell approaches zero, i.e. a cylindrical shell, and excellent agreements are achieved.

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

2019 ◽  
Vol 55 (1-2) ◽  
pp. 42-52
Author(s):  
Milad Ranjbaran ◽  
Rahman Seifi
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

scholarly journals A Coupled Pseudospectral-Differential Quadrature Method for a Class of Hyperbolic Telegraph Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Fangzong Wang ◽  
Yong Wang
Keyword(s):  
Differential Quadrature Method ◽  
Pseudospectral Method ◽  
Engineering Calculation ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Quadrature Methods ◽  
Telegraph Equations ◽  
Lagrange Interpolation Polynomials ◽  
Grid Points ◽  
Interpolation Polynomials

Pseudospectral methods and differential quadrature methods are two kinds of important meshless methods, both of which have been widely used in scientific and engineering calculation. The Lagrange interpolation polynomials are used as the trial function of the two methods, and the same distribution of grid points is used. This paper points out that the differential quadrature method is a special form of the pseudospectral method. On the basis of the above, a coupled pseudospectral-differential quadrature method (PSDQM) is proposed to solve a class of hyperbolic telegraph equations. Theoretical analysis and numerical tests show that the new method has spectral precision convergence in spatial domain and has A-stability in time domain. And it is suitable for solving multidimensional telegraph equations.

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