Mechanics of Regenerative Vibration in Drilling: Analytical and Experimental Study of the Formation of Lobed Holes

2001 ◽  
Author(s):  
Philip V. Bayly ◽  
Michael T. Lamar ◽  
Sean G. Calvert
Keyword(s):  
Transition Matrix ◽  
State Transition ◽  
Rotation Frequency ◽  
Static Model ◽  
Simplified Model ◽  
Eigenvalues And Eigenvectors ◽  
Current Position ◽  
Discrete State ◽  
Large Amplitude Vibrations ◽  
Regenerative Vibration

Abstract Large-amplitude vibrations in drilling often occur near multiples of the rotation frequency, even when these frequencies are much lower than the system’s first natural frequency. These vibrations are responsible for out-of-round, “lobed” holes. A simplified model of the mechanics of this phenomenon is presented in this paper. The model includes cutting and “rubbing” forces on the drill, but inertia and damping of the tool are neglected at low speeds. This quasi-static model remains dynamic because of the regenerative nature of cutting; the force on each cutting element depends on both the tool’s current position and its position at the time of the previous tooth passage. Characteristic solutions, including unstable retrograde “whirling” modes, are found in terms of eigenvalues and eigenvectors of a discrete state-transition matrix. These unstable modes correspond closely to behavior observed in drilling tests.

Low-Frequency Regenerative Vibration and the Formation of Lobed Holes in Drilling

2002 ◽  
Vol 124 (2) ◽  
pp. 275-285 ◽  
Author(s):  
Philip V. Bayly ◽  
Michael T. Lamar ◽  
Sean G. Calvert
Keyword(s):  
Transition Matrix ◽  
State Transition ◽  
Low Frequency ◽  
Rotation Frequency ◽  
Static Model ◽  
Eigenvalues And Eigenvectors ◽  
Current Position ◽  
Discrete State ◽  
Large Amplitude Vibrations ◽  
Regenerative Vibration

Large-amplitude vibrations in drilling often occur at frequencies near multiples of the rotation frequency, even when these are much lower than the system’s first natural frequency. These vibrations are responsible for out-of-round, “lobed” holes. A simplified model of the mechanics of this phenomenon is presented in this paper. The model includes cutting and “rubbing” forces on the drill, but inertia and damping of the tool are neglected at low speeds. This quasi-static model remains dynamic because of the regenerative nature of cutting; the force on each cutting element depends on both the tool’s current position and its position at the time of the previous tooth passage. Characteristic solutions, including unstable retrograde “whirling” modes, are found in terms of eigenvalues and eigenvectors of a discrete state-transition matrix. These unstable modes correspond closely to behavior observed in drilling tests.

scholarly journals Mathematical Model for Tapping Simulation to Predict Radial Pitch Diameter Difference of threads

2021 ◽  
Author(s):  
Jie Ren ◽  
Xianguo Yan
Keyword(s):  
State Transition ◽  
Great Influence ◽  
Spindle Speed ◽  
Quality Inspection ◽  
Eigenvalues And Eigenvectors ◽  
Current Position ◽  
Damping Force ◽  
Discrete State ◽  
Main Thread

Abstract As the most important component of parts, thread has a great influence on the mechanical properties and service performance of parts. In order to ensure the quality of the thread, the thread quality inspection standard involves 11 main thread characteristics, of which the surface roughness of the thread is the most studied, and the research on Radial Pitch Diameter Difference (RPDD) is still blank. In this paper, a quasi-static model of the tapping process is developed based on the roundness error mechanism of the hole, which includes cutting force and cutting damping force. Due to the regenerative nature of cutting, the force on each cutting edge depends on both the tool’s current position and previous position. According to the eigenvalues and eigenvectors of the discrete state-transition matrix, RPDD is finally determined, and the influence of the chamfer length and the spindle speed on RPDD is simulated by this model. The results demonstrate that the chamfer length and spindle speed will affect RPDD, and the RPDD is the smallest when the chamfer length is 2 threads and the spindle speed is 1400 rev/min. The development of this model not only provides a cheap and effective method for the study of RPDD, but also lays a foundation for further experimental research.

scholarly journals Neural Representation of Spatial Topology in the Rodent Hippocampus

2014 ◽  
Vol 26 (1) ◽  
pp. 1-39 ◽  
Author(s):  
Zhe Chen ◽  
Stephen N. Gomperts ◽  
Jun Yamamoto ◽  
Matthew A. Wilson
Keyword(s):  
State Space ◽  
Transition Matrix ◽  
State Transition ◽  
State Transition Matrix ◽  
Neural Representation ◽  
Two Dimensional ◽  
Discrete State ◽  
Spatial Topology ◽  
Population Codes ◽  
Spiking Activity

Pyramidal cells in the rodent hippocampus often exhibit clear spatial tuning in navigation. Although it has been long suggested that pyramidal cell activity may underlie a topological code rather than a topographic code, it remains unclear whether an abstract spatial topology can be encoded in the ensemble spiking activity of hippocampal place cells. Using a statistical approach developed previously, we investigate this question and related issues in greater detail. We recorded ensembles of hippocampal neurons as rodents freely foraged in one- and two-dimensional spatial environments and used a “decode-to-uncover” strategy to examine the temporally structured patterns embedded in the ensemble spiking activity in the absence of observed spatial correlates during periods of rodent navigation or awake immobility. Specifically, the spatial environment was represented by a finite discrete state space. Trajectories across spatial locations (“states”) were associated with consistent hippocampal ensemble spiking patterns, which were characterized by a state transition matrix. From this state transition matrix, we inferred a topology graph that defined the connectivity in the state space. In both one- and two-dimensional environments, the extracted behavior patterns from the rodent hippocampal population codes were compared against randomly shuffled spike data. In contrast to a topographic code, our results support the efficiency of topological coding in the presence of sparse sample size and fuzzy space mapping. This computational approach allows us to quantify the variability of ensemble spiking activity, examine hippocampal population codes during off-line states, and quantify the topological complexity of the environment.

scholarly journals GRACETOOLS—GRACE Gravity Field Recovery Tools

Geosciences ◽  
2018 ◽  
Vol 8 (9) ◽  
pp. 350 ◽  
Author(s):  
Neda Darbeheshti ◽  
Florian Wöske ◽  
Matthias Weigelt ◽  
Christopher Mccullough ◽  
Hu Wu
Keyword(s):  
Open Source ◽  
Gravity Field ◽  
Transition Matrix ◽  
State Transition ◽  
Satellite Observations ◽  
Gravity Field Recovery ◽  
Grace Data ◽  
Range Rate ◽  
Analysis Platform ◽  
Recovery Tools

This paper introduces GRACETOOLS, the first open source gravity field recovery tool using GRACE type satellite observations. Our aim is to initiate an open source GRACE data analysis platform, where the existing algorithms and codes for working with GRACE data are shared and improved. We describe the first release of GRACETOOLS that includes solving variational equations for gravity field recovery using GRACE range rate observations. All mathematical models are presented in a matrix format, with emphasis on state transition matrix, followed by details of the batch least squares algorithm. At the end, we demonstrate how GRACETOOLS works with simulated GRACE type observations. The first release of GRACETOOLS consist of all MATLAB M-files and is publicly available at Supplementary Materials.

Numerical Algebra Solution: A New Algorithm for the State Transition Matrix

2017 ◽  
Vol 60 (12) ◽  
pp. 2620-2629 ◽  
Author(s):  
Wenfeng Nie ◽  
Tianhe Xu ◽  
Yujun Du ◽  
Fan Gao ◽  
Guochang Xu
Keyword(s):  
Transition Matrix ◽  
State Transition ◽  
The State ◽  
State Transition Matrix ◽  
Numerical Algebra
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