American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 1214-1223. doi: 10.3934/proc.2011.2011.1214

Physical statistical modelling of bending vibrations

Nils Raabe 1, and  Claus Weihs 1,

1. 

TU Dortmund University, Faculty of Statistics, 44221 Dortmund, Germany, Germany

Received  July 2010 Revised  April 2011 Published  October 2011

One serious problem in deep-hole drilling is the formation of a dynamic disturbance called spiralling which causes holes with several lobes. One explanation for the occurrence of spiralling is the intersection of time varying bending eigenfrequencies with multiples of the rotational frequency of the boring bar leading to a regenerative e ect. This e ect results from the periodical tilt of the drillhead cutting in each lobe after each revolution and continues in a self exciting manner even when the original causing eigenfrequency keeps changing. We propose a physical-statistical model consisting of a system of coupled di erential equations and allowing the explicit Maximum Likelihood estimation of the modal parameters and by this the implicit estimation of the bending eigenfrequency courses. An extensive simulation for the evaluation of the properties of these estimators and tted courses has now been conducted. It is shown that the results of the model can be improved by tting polynomial local regressions frequency band wise. With the tted eigenfrequency courses it is possible to set up the machining parameters in a way that intersections of speci c eigenfrequencies with multiples of the rotational frequency and spiralling correspondingly get unlikely.
Keywords: Statistical Modelling, Maximum Likelihood Estimation, Physical Modelling.
Mathematics Subject Classification: Primary: 37N15, 60K35.
Citation: Nils Raabe, Claus Weihs. Physical statistical modelling of bending vibrations. Conference Publications, 2011, 2011 (Special) : 1214-1223. doi: 10.3934/proc.2011.2011.1214
[1]

Johnathan M. Bardsley. A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Problems & Imaging, 2010, 4 (1) : 11-17. doi: 10.3934/ipi.2010.4.11
[2]

Jie Huang, Xiaoping Yang, Yunmei Chen. A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation. Inverse Problems & Imaging, 2011, 5 (3) : 645-657. doi: 10.3934/ipi.2011.5.645
[3]

Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems & Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167
[4]

Juan Pablo Aparicio, Carlos Castillo-Chávez. Mathematical modelling of tuberculosis epidemics. Mathematical Biosciences & Engineering, 2009, 6 (2) : 209-237. doi: 10.3934/mbe.2009.6.209
[5]

Oren Barnea, Rami Yaari, Guy Katriel, Lewi Stone. Modelling seasonal influenza in Israel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 561-573. doi: 10.3934/mbe.2011.8.561
[6]

Jacques Demongeot, Jean Gaudart, Julie Mintsa, Mustapha Rachdi. Demography in epidemics modelling. Communications on Pure & Applied Analysis, 2012, 11 (1) : 61-82. doi: 10.3934/cpaa.2012.11.61
[7]

Deborah C. Markham, Ruth E. Baker, Philip K. Maini. Modelling collective cell behaviour. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123
[8]

Tiffany A. Jones, Lou Caccetta, Volker Rehbock. Optimisation modelling of cancer growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 115-123. doi: 10.3934/dcdsb.2017006
[9]

Ingenuin Gasser, Marcus Kraft. Modelling and simulation of fires in tunnel networks. Networks & Heterogeneous Media, 2008, 3 (4) : 691-707. doi: 10.3934/nhm.2008.3.691
[10]

Peng Feng, Menaka Navaratna. Modelling periodic oscillations during somitogenesis. Mathematical Biosciences & Engineering, 2007, 4 (4) : 661-673. doi: 10.3934/mbe.2007.4.661
[11]

Geoffrey Beck, Sebastien Imperiale, Patrick Joly. Mathematical modelling of multi conductor cables. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 521-546. doi: 10.3934/dcdss.2015.8.521
[12]

Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794
[13]

Philippe Destuynder, Caroline Fabre. A modelling of springing, whipping and slamming for ships. Communications on Pure & Applied Analysis, 2009, 8 (1) : 209-235. doi: 10.3934/cpaa.2009.8.209
[14]

Ingenuin Gasser. Modelling and simulation of a solar updraft tower. Kinetic & Related Models, 2009, 2 (1) : 191-204. doi: 10.3934/krm.2009.2.191
[15]

Nirav Dalal, David Greenhalgh, Xuerong Mao. Mathematical modelling of internal HIV dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 305-321. doi: 10.3934/dcdsb.2009.12.305
[16]

Oliver Penrose, John W. Cahn. On the mathematical modelling of cellular (discontinuous) precipitation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 963-982. doi: 10.3934/dcds.2017040
[17]

Liumei Wu, Baojun Song, Wen Du, Jie Lou. Mathematical modelling and control of echinococcus in Qinghai province, China. Mathematical Biosciences & Engineering, 2013, 10 (2) : 425-444. doi: 10.3934/mbe.2013.10.425
[18]

Sara D. Cardell, Amparo Fúster-Sabater. Modelling the shrinking generator in terms of linear CA. Advances in Mathematics of Communications, 2016, 10 (4) : 797-809. doi: 10.3934/amc.2016041
[19]

Gilles Pijaudier-Cabot, David Grégoire. A review of non local continuum damage: Modelling of failure?. Networks & Heterogeneous Media, 2014, 9 (4) : 575-597. doi: 10.3934/nhm.2014.9.575
[20]

Koichi Osaki, Hirotoshi Satoh, Shigetoshi Yazaki. Towards modelling spiral motion of open plane curves. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1009-1022. doi: 10.3934/dcdss.2015.8.1009

 Impact Factor: 

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences