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DOI: 10.14489/hb.2023.01.pp.016-026
Корытов М. С., Щербаков В. С., Кашапова И. Е.
АНАЛИТИЧЕСКОЕ РЕШЕНИЕ ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВЫНУЖДЕННЫХ КОЛЕБАНИЙ ВИБРОЗАЩИТНОЙ СИСТЕМЫ С КУСОЧНО-ЛИНЕЙНОЙ СТАТИЧЕСКОЙ СИЛОВОЙ ХАРАКТЕРИСТИКОЙ ПРИ ГАРМОНИЧЕСКОМ КИНЕМАТИЧЕСКОМ ВОЗБУЖДЕНИИ ПЕРЕМЕЩЕНИЙ ОСНОВАНИЯ
(c. 16-26)
Аннотация. Необходимость защиты от вибрации операторов строительных и дорожных машин обусловливает актуальность разработки математических моделей для исследования динамических процессов вынужденных колебаний виброзащищаемой массы сиденья с оператором. Аналитические решения имеют преимущество в виде наибольшей точности. Для расчетной схемы виброзащитной системы с одной степенью свободы в виде линейного осциллятора с кинематическим возбуждением получено аналитическое решение задачи вынужденных гармонических колебаний при заданных значениях параметров: начальных абсолютных перемещений и скорости виброзащищаемой массы, начальной фазы заданных гармонических колебаний основания сиденья. При этом выражение статической силовой характеристики виброзащитной системы представляло прямую с вертикальным смещением, что позволило разбить переходный процесс на отдельные временны́е промежутки, внутри которых локальная координата, показывающая деформацию виброзащитного механизма, находится в пределах отдельного прямолинейного сегмента кусочно-линейной статической характеристики. Результаты сравнительного анализа показали существенное расхождение между кусочно-аналитическим решением и решением, полученным методом численного интегрирования.
Ключевые слова: вибрация; виброзащита; кинематическое возбуждение; аналитический; гармонический.
Korytov M. S., Shcherbakov V. S., Kashapova I. E.
ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF FORCED OSCILLATIONS OF A VIBRATION PROTECTION SYSTEM WITH A PIECEWISE LINEAR STATIC FORCE CHARACTERISTIC UNDER HARMONIC KINEMATIC EXCITATION OF BASE DISPLACEMENTS
(pp. 16-26)
Abstract. The need to protect operators of mobile machines from vibrations determines the relevance of the development of mathematical models for studying the dynamic processes of forced vibrations of the vibration-protected mass of the seat with the operator. Analytical solutions have the advantage of being the most accurate. For the design scheme of a vibration protection system with one degree of freedom in the form of a linear oscillator with kinematic excitation, an analytical solution to the problem of forced harmonic oscillations was obtained for the given values of the parameters: the initial absolute displacements and speed of the vibration-protected mass, the initial phase of the given harmonic oscillations of the seat base. At the same time, the expression of the static power characteristic of the vibration protection system was a straight line with a vertical displacement, which made it possible to break the transient process into separate time intervals, within which the local coordinate, showing the deformation of the vibration protection mechanism, is within a separate rectilinear segment of the piecewise linear static characteristic. The results of the comparative analysis showed a significant discrepancy between the piecewise analytical solution and the solution obtained by the numerical integration method.
Keywords: Vibrations; Vibration protection; Kinematic excitation; Analytical; Harmonic.
М. С. Корытов, С. Щербаков, И. Е. Кашапова (Сибирский государственный автомобильно-дорожный университет, Омск, Россия) E-mail:
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M. S. Korytov, V. S. Shcherbakov, I. E. Kashapova (The Siberian State Automobile and Highway University, Omsk, Russia) E-mail:
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1. Kamgar R., Gholami F., Sanayei H., Heidar-zadeh H. Modified Tuned Liquid Dampers for Seismic Protection of Buildings Considering Soil-Structure Interaction Effects // Iranian Journal of Science and Technology-Transactions of Civil Engineering. 2020. V. 44, No. 1. P. 339 – 354. DOI: 10.1007/s40996-019-00302-x
2. Optimal Design and Seismic Performance of Multi-Tuned Mass Damper Inerter (MTMDI) Applied to Adjacent High-Rise Buildings / D. De Domenico, H. Qiao, Q. Wang, et al. // Structural Design of Tall and Special Buildings. 2020. V. 29, No. 14. P. e1781. DOI: 10.1002/тал.1781
3. Burian Yu. A., Silkov V. M., Sitnikov V. D. Quasi-Zero Stiffness Vibration Isolation Support with Stiffness Corrector Based on a Rubber-Cord Air Spring // AIP Conference Proceedings. 2020. V. 2285. P. 030007. DOI: 10.1063/5.0027543
4. Bratan S., Kharchenko A., Vladetskaya E., Kharchenko A. Analysis and Synthesis of Vibration Isolation System of A Grinding Machine with Account of the Operational Reliability of its Elements // Metal Working and Material Science. 2019. V. 21, No. 1. P. 35 – 49. DOI: 10.17212/1994-6309-2019-21.1-35-49
5. Seated Postural Loads Caused by Shock-Type Whole-Body Vibration when Driving Over Obstacles / T. Stenlund, R. Lundstrom, O. Lindroos, et al. // International Journal of Forest Engineering. 2020. V. 31, No. 3. P. 184 – 191. DOI: 10.1080/14942119.2020.1761745
6. Zhao H., Wang G., Lv W. Optimization of Hydropneumatic Suspension for Articulated Wheel Loader Based on Kriging Model and Particle Swarm Algorithm // Advances in Mechanical Engineering. 2018. V. 10, No. 11. P. 1687814018810648. DOI: 10.1177/1687814018810648
7. Lynas D., Burgess-Limerick R. Whole-Body Vibration Associated with Dozer Operation at an Australian Surface Coal Mine // Annals of Work Exposures and Health. 2019. V. 63, No. 8. P. 881 – 889. DOI: 10.1093/Annweh/Wxz054
8. Mian J., Shoushi L., Yong G., Jigang W. The Improvement on Vibration Isolation Performance of Hydraulic Excavators Based on the Optimization of Powertrain Mounting System // Advances in Mechanical Engineering. 2019. V. 11, No. 5. DOI: 10.1177/1687814019849988
9. Robinah N., Safiki A., Thomas O., Annette B. Impact of Road Infrastructure Equipment on the Environment and Surroundings // Global Journal of Environmental Science and Management-GJESM. 2022. V. 8, No. 2. P. 251 – 264. DOI: 10.22034/Gjesm.2022.02.08
10. Galdin N. S., Semenova I. A., Galdin V. N. Analysis of the Striker Stroke Impact on the Hydropneumatic Impact Devices Energy Performance // Journal of Physics: Conference Series. 2019. V. 1260, No. 11. P. 112010. DOI: 10.1088/1742-6596/1260/11/112010
11. Tyuremnov I. S., Fyodorova D. V., Morev A. S. Study of Impact of Amount of Shock Absorbers on Parameters of Vibrations of Drum and Frame of Vibrating Roller // Proceedings of the 5th International Conference on Industrial Engineering (ICIE 2019). ICIE 2019. Lecture Notes in Mechanical Engineering. P. 765 – 773. DOI: 10.1007/978-3-030-22063-1_82
12. Jia J., Liu H., Wan Y. Dynamic Characteristics Modelling of the Tamper-Asphalt Mixture Interaction: Application to Predict Asphalt Mat Density // International Journal of Pavement Engineering. 2019. V. 20, No. 5. P. 530 – 543. DOI: 10.1080/10298436.2017.1316642
13. Loprencipe G., Zoccali P. Ride Quality Due to Road Surface Irregularities: Comparison of Different Methods Applied on a Set of Real Road Profiles // Coatings. 2017. V. 7, No. 5. P. 59. DOI: 10.3390/Coatings7050059
14. Adam S., Jalil N., Rezali K., Ng Y. The Effect of Posture and Vibration Magnitude on the Vertical Vibration Transmissibility of Tractor Suspension System // International Journal of Industrial Ergonomics. 2020. V. 80. P. 103014. DOI: 10.1016/J.Ergon.2020.103014
15. Wieckowski J., Rafajlowicz W., Moczko P., Rafajlowicz E. Ata From Vibration Measurement in a Bucket Wheel Excavator Operator’s Cabin with the Aim of Vibrations Damping // Data in Brief. 2021. V. 35. P. 106836. DOI: 10.1016/J.Dib.2021.106836
16. Dhanjee K. C., Sanjay K. P., Vivekanand K., Netai C. K. Whole-Body Vibration Exposure of Heavy Earthmoving Machinery Operators in Surface Coal Mines: a Comparative Assessment of Transport and Non-Transport Earthmoving Equipment Operators // International Journal of Occupational Safety and Ergonomics: JOSE. 2020. P. 1 – 10. DOI: 10.1080/10803548.2020.1785154
17. Burian Y. A., Sitnikov D. V., Silkov M. V., Belkov V. N. The Active System of Vibration Isolation with Digital Twin and Control by Acceleration // Journal of Physics: Conference Series. 2021. V. 1791, No. 1. P. 012007. DOI: 10.1088/1742-6596/1791/1/012007
18. Korytov M. S., Shcherbakov V. S., Titenko V. V., Pochekueva I. E. Study of the Antivibration Suspended Seat Oscillations with Quasi-Zero Stiffness Effect under Sinusoidal Excitation // Journal of Physics: Conference Series. 2021. V. 1901(1). P. 012120. DOI: 10.1088/1742-6596/1901/1/012120
19. A Quasi-Zero-Stiffness Dynamic Vibration Absorber / Y. Chang, J. Zhou, K. Wang et al. // Journal of Sound and Vibration. 2021. V. 494. P. 115859. DOI: 10.1016/J.Jsv.2020.115859
20. Vitorino M. V., Vieira A., Rodrigues M. S. Effect of Sliding Friction in Harmonic Oscillators // Scientific Reports. 2017. V. 7. P. 3726. DOI: 10.1038/S41598-017-03999-W
21. Song H., Hofmann H. Robust, Accurate Systems-Based Power Electronic Circuit Models in Simulink // IEEE 19th Workshop on Control and Modeling for Power Electronics (COMPEL). 2018. P. 1 – 8. DOI: 10.1109/COMPEL.2018.8460049
22. True H., Christiansen A. H., Knutz S. E., Rasmussen L. B. On the Dynamics of a Four-Axle Railway Vehicle with Dry Friction Yaw Damping // International Journal of Heavy Vehicle Systems. 2020. V. 27, No. 5. P. 600 – 621. DOI: 10.1504/IJHVS.2020.111261
1. Kamgar R., Gholami F., Sanayei H., Heidar-zadeh H. (2020). Modified Tuned Liquid Dampers for Seismic Protection of Buildings Considering Soil-Structure Interaction Effects. Iranian Journal of Science and Technology-Transactions of Civil Engineering, Vol. 44, (1), pp. 339 – 354. DOI: 10.1007/s40996-019-00302-x
2. Domenico D. De, Qiao H., Wang Q. et al. (2020). Optimal Design and Seismic Performance of Multi-Tuned Mass Damper Inerter (MTMDI) Applied to Adjacent High-Rise Buildings. Structural Design of Tall and Special Buildings, Vol. 29, 14. DOI: 10.1002/тал.1781
3. Burian Yu. A., Silkov V. M., Sitnikov V. D. (2020). Quasi-Zero Stiffness Vibration Isolation Support with Stiffness Corrector Based on a Rubber-Cord Air Spring. AIP Conference Proceedings, Vol. 2285. DOI: 10.1063/5.0027543
4. Bratan S., Kharchenko A., Vladetskaya E., Kharchenko A. (2019). Analysis and Synthesis of Vibration Isolation System of A Grinding Machine with Account of the Operational Reliability of its Elements. Metal Working and Material Science, Vol. 21, (1), pp. 35 – 49. DOI: 10.17212/1994-6309-2019-21.1-35-49
5. Stenlund T., Lundstrom R., Lindroos O. et al. (2020). Seated Postural Loads Caused by Shock-Type Whole-Body Vibration when Driving Over Obstacles. International Journal of Forest Engineering, Vol. 31, (3), pp. 184 – 191. DOI: 10.1080/14942119.2020.1761745
6. Zhao H., Wang G., Lv W. (2018). Optimization of Hydropneumatic Suspension for Articulated Wheel Loader Based on Kriging Model and Particle Swarm Algorithm. Advances in Mechanical Engineering, Vol. 10, (11). DOI: 10.1177/1687814018810648
7. Lynas D., Burgess-Limerick R. (2019). Whole-Body Vibration Associated with Dozer Operation at an Australian Surface Coal Mine. Annals of Work Exposures and Health, Vol. 63, (8), pp. 881 – 889. DOI: 10.1093/Annweh/Wxz054
8. Mian J., Shoushi L., Yong G., Jigang W. (2019). The Improvement on Vibration Isolation Performance of Hydraulic Excavators Based on the Optimization of Powertrain Mounting System. Advances in Mechanical Engineering, Vol. 11, (5). DOI: 10.1177/1687814019849988
9. Robinah N., Safiki A., Thomas O., Annette B. (2022). Impact of Road Infrastructure Equipment on the Environment and Surroundings. Global Journal of Environmental Science and Management-GJESM, Vol. 8, (2), pp. 251 – 264. DOI: 10.22034/Gjesm.2022.02.08
10. Galdin N. S., Semenova I. A., Galdin V. N. (2019). Analysis of the Striker Stroke Impact on the Hydropneumatic Impact Devices Energy Performance. Journal of Physics: Conference Series, Vol. 1260, (11). DOI: 10.1088/1742-6596/1260/11/112010
11. Tyuremnov I. S., Fyodorova D. V., Morev A. S. (2019). Study of Impact of Amount of Shock Absorbers on Parameters of Vibrations of Drum and Frame of Vibrating Roller. Proceedings of the 5th International Conference on Industrial Engineering (ICIE 2019). ICIE 2019. Lecture Notes in Mechanical Engineering, pp. 765 – 773. DOI: 10.1007/978-3-030-22063-1_82
12. Jia J., Liu H., Wan Y. (2019). Dynamic Characteristics Modelling of the Tamper-Asphalt Mixture Interaction: Application to Predict Asphalt Mat Density. International Journal of Pavement Engineering, Vol. 20, (5), pp. 530 – 543. DOI: 10.1080/10298436.2017.1316642
13. Loprencipe G., Zoccali P. (2017). Ride Quality Due to Road Surface Irregularities: Comparison of Different Methods Applied on a Set of Real Road Profiles. Coatings, Vol. 7, (5). DOI: 10.3390/Coatings7050059
14. Adam S., Jalil N., Rezali K., Ng Y. (2020). The Effect of Posture and Vibration Magnitude on the Vertical Vibration Transmissibility of Tractor Suspension System. International Journal of Industrial Ergonomics, Vol. 80. DOI: 10.1016/J.Ergon.2020.103014
15. Wieckowski J., Rafajlowicz W., Moczko P., Rafajlowicz E. (2021). Ata From Vibration Measurement in a Bucket Wheel Excavator Operator’s Cabin with the Aim of Vibrations Damping. Data in Brief, Vol. 35. DOI: 10.1016/J.Dib.2021.106836
16. Dhanjee K. C., Sanjay K. P., Vivekanand K., Netai C. K. (2020). Whole-Body Vibration Exposure of Heavy Earthmoving Machinery Operators in Surface Coal Mines: a Comparative Assessment of Transport and Non-Transport Earthmoving Equipment Operators. International Journal of Occupational Safety and Ergonomics: JOSE, pp. 1 – 10. DOI: 10.1080/10803548.2020.1785154
17. Burian Y. A., Sitnikov D. V., Silkov M. V., Belkov V. N. (2021). The Active System of Vibration Isolation with Digital Twin and Control by Acceleration. Journal of Physics: Conference Series, Vol. 1791, (1). DOI: 10.1088/1742-6596/1791/1/012007
18. Korytov M. S., Shcherbakov V. S., Titenko V. V., Pochekueva I. E. (2021). Study of the Antivibration Suspended Seat Oscillations with Quasi-Zero Stiffness Effect under Sinusoidal Excitation. Journal of Physics: Conference Series, Vol. 1901, (1). DOI: 10.1088/1742-6596/1901/1/012120
19. Chang Y., Zhou J., Wang K. et al. (2021). A Quasi-Zero-Stiffness Dynamic Vibration Absorber. Journal of Sound and Vibration, Vol. 494. DOI: 10.1016/J.Jsv.2020.115859
20. Vitorino M. V., Vieira A., Rodrigues M. S. (2017). Effect of Sliding Friction in Harmonic Oscillators. Scientific Reports, Vol. 7. DOI: 10.1038/S41598-017-03999-W
21. Song H., Hofmann H. (2018). Robust, Accurate Systems-Based Power Electronic Circuit Models in Simulink. IEEE 19th Workshop on Control and Modeling for Power Electronics (COMPEL), pp. 1 – 8. DOI: 10.1109/COMPEL.2018.8460049
22. True H., Christiansen A. H., Knutz S. E., Rasmussen L. B. (2020). On the Dynamics of a Four-Axle Railway Vehicle with Dry Friction Yaw Damping. International Journal of Heavy Vehicle Systems, Vol. 27, (5), pp. 600 – 621. DOI: 10.1504/IJHVS.2020.111261
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