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∙s^{2}∙m^{–3}, the first proper frequency changes within 500 to 750 s^{-1} (80 to 119 Hz) at soil elastic deformation coefficient 2∙10^{5} 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. |
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Copyright © The Author(s), 2021. Published by Science Publishing Group |
Root Crop, Longitudinal Vibrations, Amplitude, Damper
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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
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
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
@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} }
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 -