scholarly journals Free vibration and chatter stability of a rotating thin-walled composite bar

2018 ◽  
Vol 10 (9) ◽  
pp. 168781401879826 ◽  
Author(s):  
Jingmin Ma ◽  
Yongsheng Ren
Keyword(s):  
Time Delay ◽  
Free Vibration ◽  
Galerkin Method ◽  
Lagrange Equation ◽  
Chatter Stability ◽  
Milling Force ◽  
Thin Walled ◽  
The Stability ◽  
The Time Domain ◽  
The Galerkin Method

The free vibration and chatter stability of a rotating thin-walled composite bar under the action of regenerative milling force were investigated in this article. The free vibration equations of the rotating thin-walled composite bar were presented and solved by the Galerkin method according to the derived Lagrange equation for the rotating thin-walled composite bar, and the influences of the taper ratio, ply angle, and mode number on the first natural frequency were analyzed. In order to study the chatter stability of the rotating thin-walled composite bar, the time-delay motion equation of the system was derived with consideration of the effects of regenerative milling force and internal/external damping. The two-mode Galerkin method was used to discretize the time-delay free vibration equations, and the semi-discrete method in the time domain was employed to predict the stability lobes. Finally, the correctness of the method was validated, and the stability analysis of the rotating thin-walled composite bar was conducted. Emphatically, the influences of the ply angle, taper ratio, and internal/external damping on the chatter stability of the rotating thin-walled composite bar were analyzed.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

On the Buckling of Circular Cylindrical Shells Under Pure Bending

1961 ◽  
Vol 28 (1) ◽  
pp. 112-116 ◽  
Author(s):  
Paul Seide ◽  
V. I. Weingarten
Keyword(s):  
Galerkin Method ◽  
Compressive Stress ◽  
Cylindrical Shells ◽  
Bending Stress ◽  
Pure Bending ◽  
Circular Cylindrical Shells ◽  
The Stability ◽  
The Galerkin Method

The stability of circular cylindrical shells under pure bending is investigated by means of Batdorf’s modified Donnell’s equation and the Galerkin method. The results of this investigation have shown that, contrary to the commonly accepted value, the maximum critical bending stress is for all practical purposes equal to the critical compressive stress.

The Galerkin method applied to convective instability problems

1968 ◽  
Vol 33 (1) ◽  
pp. 201-208 ◽  
Author(s):  
Bruce A. Finlayson
Keyword(s):  
Exact Solution ◽  
Time Dependence ◽  
Galerkin Method ◽  
Convective Instability ◽  
Fluid Layer ◽  
Oscillatory Instability ◽  
Rotating Fluid ◽  
Exponential Time ◽  
The Stability ◽  
The Galerkin Method

The Galerkin method is applied in a new way to problems of stationary and oscillatory convective instability. By retaining the time derivatives in the equations rather than assuming an exponential time-dependence, the exact solution is approximated by the solution to a set of ordinary differential equations in time. Computations are simplified because the stability of this set of equations can be determined without finding the detailed solution. Furthermore, both stationary and oscillatory instability can be studied by means of the same trial functions. Previous studies which have treated only stationary instability by the Galerkin method can now be extended easily to include oscillatory instability. The method is illustrated for convective instability of a rotating fluid layer transferring heat.

scholarly journals Dynamic Modeling and Cutting Stability of Rotating Tapered Composite Cutter Bar considering Material Damping

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yuhuan Zhang ◽  
Ren Yongsheng ◽  
Bole Ma ◽  
Jinfeng Zhang
Keyword(s):  
Constitutive Relations ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Chatter Stability ◽  
Differential Motion ◽  
Chatter Suppression ◽  
Fixed Type ◽  
The Stability ◽  
The Galerkin Method ◽  
Cutting System

Traditional milling cutter bars are generally made up of metals and exhibit poor capacity of chatter suppression. This study proposes an anisotropic composites tapered cutter bar for increasing natural frequency and damping and finally achieves the goal of enhancing chatter stability. Based on Hamilton principle and Euler–Bernoulli beam theory, the partial differential motion equations of the cutting system with a 3D rotating tapered composite cutter bar are established. Next, using the Galerkin method, the equations of motion are discretized so as to derive ordinary differential equations. In the model, damping modeling of the composite cutter bar is achieved theoretically by using damping dissipation constitutive relations for viscoelastic composites. Moreover, by introducing the rotating effect of the 3D cutter bar in the 2-DOF analytical model of stability analysis first proposed for a fixed-type cutter bar, an improved prediction model is developed and used to solve the stability lobes of the cutting system in the frequency domain analytically. Furthermore, the influences of the gyroscopic effect, material, ply angle, stacking sequence, and taper ratio on chatter stability are also discussed.

Stability Analysis of Rectangular Thin Walled Structure under Nonlinear Distributed Stress

2014 ◽  
Vol 1028 ◽  
pp. 117-121
Author(s):  
De Hui Yu ◽  
Yong Fa Hou ◽  
Jing Sun
Keyword(s):  
Stability Analysis ◽  
Galerkin Method ◽  
Material Properties ◽  
Critical Thickness ◽  
Wall Structure ◽  
Thin Wall ◽  
Nonlinear Stress ◽  
Thin Walled ◽  
The Stability ◽  
Internal Relationship

The stability of rectangular thin wall with 3 simple supported sides and 1 free side, subjected to nonlinear distributed stress, is analyzed. Galerkin method is applied in order to study the internal relationship between the critical thickness and nonlinear stress. Under the same stress circumstance, as the a/b ratio gets larger, the critical thickness for the thin wall structure getting buckled grows too. The influence of different material properties on the critical thickness is also researched. The method proposed in this paper is applicable for more complicated distributed stress and also can provide some reference criterion for design of thin wall structure.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Delays

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
The Stability ◽  
The Galerkin Method ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

In this paper, we develop Galerkin approximations for determining the stability of delay differential equations (DDEs) with time periodic coefficients and time periodic delays. Using a transformation, we convert the DDE into a partial differential equation (PDE) along with a boundary condition (BC). The PDE and BC we obtain have time periodic coefficients. The PDE is discretized into a system of ordinary differential equations (ODEs) using the Galerkin method with Legendre polynomials as the basis functions. The BC is imposed using the tau method. The resulting ODEs are time periodic in nature; thus, we resort to Floquet theory to determine the stability of the ODEs. We show through several numerical examples that the stability charts obtained from the Galerkin method agree closely with those obtained from direct numerical simulations.

Analytical Stability Prediction and Design of Variable Pitch Cutters

1999 ◽  
Vol 121 (2) ◽  
pp. 173-178 ◽  
Author(s):  
Y. Altıntas¸ ◽  
S. Engin ◽  
E. Budak
Keyword(s):  
Time Delay ◽  
Depth Of Cut ◽  
Analytical Prediction ◽  
Time Varying ◽  
Chatter Stability ◽  
Variable Pitch ◽  
Analytical Stability ◽  
The Stability ◽  
Time Invariant ◽  
Regenerative Phase

An analytical prediction of stability lobes for milling cutters with variable pitch angles is presented. The method requires cutting constants, number of teeth, and transfer function of cutter mounted on the machine tool as inputs to a chatter stability expression. The stability is formulated by transforming time varying directional cutting constants into time invariant constants. Constant regenerative time delay in uniform cutters is transformed into nonuniform multiple regenerative time delay for variable pitch cutters. The chatter free axial depth of cut is solved from the eigenvalues of stability expression, whereas the spindle speed is identified from regenerative phase delays. The proposed technique has been verified with extensive cutting tests and time domain simulations. The practical use of the analytical solution is demonstrated by an optimal tooth spacing design application which increases the chatter free depth of cuts significantly.

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