Mathematical Model of Root Crop Digging with Longitudinal Vibrations

Volodymyr Bulgakov; Aivars Aboltins; Hristo Beloev; Ivan Holovach; Valerii Adamchuk; Semjons Ivanovs; Yevhen Ihnatiev

doi:doi:10.11648/j.engmath.20210502.13

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Mathematical Model of Root Crop Digging with Longitudinal Vibrations

Volodymyr Bulgakov, Aivars Aboltins, Hristo Beloev, Ivan Holovach, Valerii Adamchuk, Semjons Ivanovs, Yevhen Ihnatiev

Published in Engineering Mathematics (Volume 5, Issue 2)

Received: 10 June 2021 Accepted: 22 June 2021 Published: 9 August 2021

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Abstract

The problem how to reduce damage to tubers when they are dug up is urgent. For the new design of a vibrating digging working body for root crops the mathematical model of longitudinal vibrations of a root crop in the soil is developed as an elastic body in an elastically damped medium. The Ostrogradsky-Hamilton variational principle is applied for the analytical description of the process. The Ritz method was applied to find the frequencies of natural vibrations, the amplitudes of forced vibrations of a root crop as a solid elastic body when it is captured by a vibrating digging body. The frequency equation for the discussed vibrational process was obtained. The values of the first proper frequency of longitudinal vibrations of the considered elastic body of the root crop with specific geometric physical parameters are found. Graphs of the dependence of the first natural frequency upon the elastic deformation coefficient, the damping coefficient of the soil as an elastic damping medium are obtained. When the soil damping coefficient changes within 0 to 10 N∙s²∙m^–3, the first proper frequency changes within 500 to 750 s^-1 (80 to 119 Hz) at soil elastic deformation coefficient 2∙10⁵ N∙m^–3. Dependence of the elastic body forced vibration amplitude upon the change in the amplitude of the disturbing force have been obtained. When the amplitude of the disturbing force changes within 100 to 600 N, the amplitude of forced vibrations of the root crop body changes within 0.30 to 0.68 mm.

Published in	Engineering Mathematics (Volume 5, Issue 2)
DOI	10.11648/j.engmath.20210502.13
Page(s)	25-38
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Root Crop, Longitudinal Vibrations, Amplitude, Damper

References

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APA Style

Volodymyr Bulgakov, Aivars Aboltins, Hristo Beloev, Ivan Holovach, Valerii Adamchuk, et al. (2021). Mathematical Model of Root Crop Digging with Longitudinal Vibrations. Engineering Mathematics, 5(2), 25-38. https://doi.org/10.11648/j.engmath.20210502.13

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ACS Style

Volodymyr Bulgakov; Aivars Aboltins; Hristo Beloev; Ivan Holovach; Valerii Adamchuk, et al. Mathematical Model of Root Crop Digging with Longitudinal Vibrations. Eng. Math. 2021, 5(2), 25-38. doi: 10.11648/j.engmath.20210502.13

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AMA Style

Volodymyr Bulgakov, Aivars Aboltins, Hristo Beloev, Ivan Holovach, Valerii Adamchuk, et al. Mathematical Model of Root Crop Digging with Longitudinal Vibrations. Eng Math. 2021;5(2):25-38. doi: 10.11648/j.engmath.20210502.13

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@article{10.11648/j.engmath.20210502.13,
  author = {Volodymyr Bulgakov and Aivars Aboltins and Hristo Beloev and Ivan Holovach and Valerii Adamchuk and Semjons Ivanovs and Yevhen Ihnatiev},
  title = {Mathematical Model of Root Crop Digging with Longitudinal Vibrations},
  journal = {Engineering Mathematics},
  volume = {5},
  number = {2},
  pages = {25-38},
  doi = {10.11648/j.engmath.20210502.13},
  url = {https://doi.org/10.11648/j.engmath.20210502.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20210502.13},
  abstract = {The problem how to reduce damage to tubers when they are dug up is urgent. For the new design of a vibrating digging working body for root crops the mathematical model of longitudinal vibrations of a root crop in the soil is developed as an elastic body in an elastically damped medium. The Ostrogradsky-Hamilton variational principle is applied for the analytical description of the process. The Ritz method was applied to find the frequencies of natural vibrations, the amplitudes of forced vibrations of a root crop as a solid elastic body when it is captured by a vibrating digging body. The frequency equation for the discussed vibrational process was obtained. The values of the first proper frequency of longitudinal vibrations of the considered elastic body of the root crop with specific geometric physical parameters are found. Graphs of the dependence of the first natural frequency upon the elastic deformation coefficient, the damping coefficient of the soil as an elastic damping medium are obtained. When the soil damping coefficient changes within 0 to 10 N∙s2∙m–3, the first proper frequency changes within 500 to 750 s-1 (80 to 119 Hz) at soil elastic deformation coefficient 2∙105 N∙m–3. Dependence of the elastic body forced vibration amplitude upon the change in the amplitude of the disturbing force have been obtained. When the amplitude of the disturbing force changes within 100 to 600 N, the amplitude of forced vibrations of the root crop body changes within 0.30 to 0.68 mm.},
 year = {2021}
}

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TY  - JOUR
T1  - Mathematical Model of Root Crop Digging with Longitudinal Vibrations
AU  - Volodymyr Bulgakov
AU  - Aivars Aboltins
AU  - Hristo Beloev
AU  - Ivan Holovach
AU  - Valerii Adamchuk
AU  - Semjons Ivanovs
AU  - Yevhen Ihnatiev
Y1  - 2021/08/09
PY  - 2021
N1  - https://doi.org/10.11648/j.engmath.20210502.13
DO  - 10.11648/j.engmath.20210502.13
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 25
EP  - 38
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20210502.13
AB  - The problem how to reduce damage to tubers when they are dug up is urgent. For the new design of a vibrating digging working body for root crops the mathematical model of longitudinal vibrations of a root crop in the soil is developed as an elastic body in an elastically damped medium. The Ostrogradsky-Hamilton variational principle is applied for the analytical description of the process. The Ritz method was applied to find the frequencies of natural vibrations, the amplitudes of forced vibrations of a root crop as a solid elastic body when it is captured by a vibrating digging body. The frequency equation for the discussed vibrational process was obtained. The values of the first proper frequency of longitudinal vibrations of the considered elastic body of the root crop with specific geometric physical parameters are found. Graphs of the dependence of the first natural frequency upon the elastic deformation coefficient, the damping coefficient of the soil as an elastic damping medium are obtained. When the soil damping coefficient changes within 0 to 10 N∙s2∙m–3, the first proper frequency changes within 500 to 750 s-1 (80 to 119 Hz) at soil elastic deformation coefficient 2∙105 N∙m–3. Dependence of the elastic body forced vibration amplitude upon the change in the amplitude of the disturbing force have been obtained. When the amplitude of the disturbing force changes within 100 to 600 N, the amplitude of forced vibrations of the root crop body changes within 0.30 to 0.68 mm.
VL  - 5
IS  - 2
ER  -

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Author Information

Volodymyr Bulgakov

Department of Mechanics, Faculty of Construction Design, National University of Life Environmental Sciences of Ukraine, UA Kyiv, Ukraine
Aivars Aboltins

Institute of Agricultural Machinery, Faculty of Engineering, Latvia University of Life Sciences and Technologies, Jelgava, LV, Latvia
Hristo Beloev

Department of Agricultural Machinery, “Angel Kanchev” University of Ruse, Ruse, Bulgaria
Ivan Holovach

Department of Mechanics, Faculty of Construction Design, National University of Life Environmental Sciences of Ukraine, UA Kyiv, Ukraine
Valerii Adamchuk

National Scientific Centre, Institute of Mechanization Electrification of Agriculture, Kyiv, Ukraine
Semjons Ivanovs

Institute of Agricultural Machinery, Faculty of Engineering, Latvia University of Life Sciences and Technologies, Jelgava, LV, Latvia
Yevhen Ihnatiev

Department of Machine-Using in Agriculture, Dmytro Motornyi Tavria State Agrotechnological University, Melitopol, Ukraine

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APA Style

Volodymyr Bulgakov, Aivars Aboltins, Hristo Beloev, Ivan Holovach, Valerii Adamchuk, et al. (2021). Mathematical Model of Root Crop Digging with Longitudinal Vibrations. Engineering Mathematics, 5(2), 25-38. https://doi.org/10.11648/j.engmath.20210502.13

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ACS Style

Volodymyr Bulgakov; Aivars Aboltins; Hristo Beloev; Ivan Holovach; Valerii Adamchuk, et al. Mathematical Model of Root Crop Digging with Longitudinal Vibrations. Eng. Math. 2021, 5(2), 25-38. doi: 10.11648/j.engmath.20210502.13

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AMA Style

Volodymyr Bulgakov, Aivars Aboltins, Hristo Beloev, Ivan Holovach, Valerii Adamchuk, et al. Mathematical Model of Root Crop Digging with Longitudinal Vibrations. Eng Math. 2021;5(2):25-38. doi: 10.11648/j.engmath.20210502.13

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@article{10.11648/j.engmath.20210502.13,
  author = {Volodymyr Bulgakov and Aivars Aboltins and Hristo Beloev and Ivan Holovach and Valerii Adamchuk and Semjons Ivanovs and Yevhen Ihnatiev},
  title = {Mathematical Model of Root Crop Digging with Longitudinal Vibrations},
  journal = {Engineering Mathematics},
  volume = {5},
  number = {2},
  pages = {25-38},
  doi = {10.11648/j.engmath.20210502.13},
  url = {https://doi.org/10.11648/j.engmath.20210502.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20210502.13},
  abstract = {The problem how to reduce damage to tubers when they are dug up is urgent. For the new design of a vibrating digging working body for root crops the mathematical model of longitudinal vibrations of a root crop in the soil is developed as an elastic body in an elastically damped medium. The Ostrogradsky-Hamilton variational principle is applied for the analytical description of the process. The Ritz method was applied to find the frequencies of natural vibrations, the amplitudes of forced vibrations of a root crop as a solid elastic body when it is captured by a vibrating digging body. The frequency equation for the discussed vibrational process was obtained. The values of the first proper frequency of longitudinal vibrations of the considered elastic body of the root crop with specific geometric physical parameters are found. Graphs of the dependence of the first natural frequency upon the elastic deformation coefficient, the damping coefficient of the soil as an elastic damping medium are obtained. When the soil damping coefficient changes within 0 to 10 N∙s2∙m–3, the first proper frequency changes within 500 to 750 s-1 (80 to 119 Hz) at soil elastic deformation coefficient 2∙105 N∙m–3. Dependence of the elastic body forced vibration amplitude upon the change in the amplitude of the disturbing force have been obtained. When the amplitude of the disturbing force changes within 100 to 600 N, the amplitude of forced vibrations of the root crop body changes within 0.30 to 0.68 mm.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Model of Root Crop Digging with Longitudinal Vibrations
AU  - Volodymyr Bulgakov
AU  - Aivars Aboltins
AU  - Hristo Beloev
AU  - Ivan Holovach
AU  - Valerii Adamchuk
AU  - Semjons Ivanovs
AU  - Yevhen Ihnatiev
Y1  - 2021/08/09
PY  - 2021
N1  - https://doi.org/10.11648/j.engmath.20210502.13
DO  - 10.11648/j.engmath.20210502.13
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 25
EP  - 38
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20210502.13
AB  - The problem how to reduce damage to tubers when they are dug up is urgent. For the new design of a vibrating digging working body for root crops the mathematical model of longitudinal vibrations of a root crop in the soil is developed as an elastic body in an elastically damped medium. The Ostrogradsky-Hamilton variational principle is applied for the analytical description of the process. The Ritz method was applied to find the frequencies of natural vibrations, the amplitudes of forced vibrations of a root crop as a solid elastic body when it is captured by a vibrating digging body. The frequency equation for the discussed vibrational process was obtained. The values of the first proper frequency of longitudinal vibrations of the considered elastic body of the root crop with specific geometric physical parameters are found. Graphs of the dependence of the first natural frequency upon the elastic deformation coefficient, the damping coefficient of the soil as an elastic damping medium are obtained. When the soil damping coefficient changes within 0 to 10 N∙s2∙m–3, the first proper frequency changes within 500 to 750 s-1 (80 to 119 Hz) at soil elastic deformation coefficient 2∙105 N∙m–3. Dependence of the elastic body forced vibration amplitude upon the change in the amplitude of the disturbing force have been obtained. When the amplitude of the disturbing force changes within 100 to 600 N, the amplitude of forced vibrations of the root crop body changes within 0.30 to 0.68 mm.
VL  - 5
IS  - 2
ER  -

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