DYNAMIC SIMULATION OF A HIGH ALTITUDE GANTRY CRANE WITH CABLE HOISTING. PART ONE. 2D MODEL Текст научной статьи по специальности «Строительство и архитектура»

ИНЖЕНЕРНЫЕ СИСТЕМЫ В СТРОИТЕЛЬСТВЕ

УДК 621.873:004.9 DOI: 10.22227/1997-0935.2021.5.615-622

Dynamic simulation of a high altitude gantry crane with cable

hoisting. Part one. 2D model

Mostafa Jafari, Evgeniy M. Kudryavtsev

Moscow State University of Civil Engineering (National Research University) (MGSU);

Moscow, Russian Federation

ABSTRACT

Introduction. Simulation of the 2D dynamic motion of a high altitude wide span gantry crane with a rope hoisting mechanism is addressed. Such large gantry cranes with a height of more than 50 meters, have been unstudied very well so far. A small swing angle of the payload, it's fast hoisting, and the fast motion of the trolley are critical for these cranes and, hence, they need to be analyzed in detail.

Materials and methods. The generalized formulation of the two-dimensional crane dynamics is efficiently performed and simulated in Mathcad. This is a single mass model that has a non-elastic cable. The formulation is derived using the Lagrange method, and differential equations are correctly solved using the Runge-Kutta method in Mathcad. In this model the crane is fixed, and all the subsystems are considered as rigid bodies without any deflection in terms of the trolley and the payload. Results. The results are verified using MSC ADAMS (Academic) that indicates satisfactory convergence. The considerable influence of the payload oscillation on the trolley motion is visible in both Mathcad and ADAMS models. The implemented Mathcad code can be useful for students and researchers.

Conclusions. The maximum speed of the trolley is 1.716 m/s to prevent the payload swinging angle from exceeding 0.5 deg. The calculated velocity of the trolley is reasonable for such a large crane if limitations like wind effects and resonance are ignored. e "

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KEYWORDS: gantry crane, hoisting mechanism, trolley, Mathcad, MSC ADAMS, Lagrange method, oscillation 3 i

FOR CITATION: Jafari M., Kudryavtsev E.M. Dynamic simulation of a high altitude gantry crane with cable hoisting. Part one. S _ 2D model. Vestnik MGSU [Monthly Journal on Construction and Architecture]. 2021; 16(5):615-622. DOI: 10.22227/1997- S r 0935.2021.5.615-622 (rus.). U O

Динамическое моделирование высокого козлового крана

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Мостафа Джафари, Е.М. Кудрявцев О &

Национальный исследовательский Московский государственный строительный университет

(НИУ МГСУ); г. Москва, Россия

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Введение. Рассмотрено моделирование динамического движения высокого и широкого козлового крана с канатным ш 0

подъемным механизмом. Такие большие козловые краны высотой более 50 м недостаточно хорошо изучены. Умень- о 6

шение угла поворота полезной нагрузки при быстром подъеме и быстром движении тележки является критическим С ®

для этих кранов и нуждается в дополнительном анализе. i о

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Материалы и методы. Обобщенная формулировка двумерной динамики крана эффективно выполняется и моде- | § лируется в Mathcad. Модель изучалась как монокристаллическая масса с неупругим кабелем. Формулировка полу- r § чена путем реализации метода Лагранжа, а дифференциальные уравнения корректно решаются с помощью реша- • теля Рунге-Кутты в системе Mathcad. В этой модели кран останавливается, и все подсистемы рассматриваются как < Т

твердые тела без какого-либо прогиба, как единая точка массы для тележки и полезной нагрузки. 1 °

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Результаты. Результаты верифицируются с помощью MSC ADAMS (Academic), что указывает на удовлетворитель- 3 i ный итог. Значительное влияние колебаний полезной нагрузки на движение тележки видно как в моделях Mathcad, ® . так и в моделях ADAMS. Реализованный код Mathcad может быть полезен студентам и исследователям. Выводы. Максимальная скорость тележки составляет 1,716 м/с, чтобы избежать колебаний полезной нагрузки бо ^ _ лее чем на 0,5 град. Рассчитанная скорость тележки является разумной для такого большого крана, игнорируя при s у этом такие ограничения, как влияние ветра и резонанс. ф к

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КЛЮЧЕВЫЕ СЛОВА: козловой кран, механизм подъема, тележка, Mathcad, MSC ADAMS, метод Лагранжа, колебания

ДЛЯ ЦИТИРОВАНИЯ: Джафари М., Кудрявцев Е.М. Dynamic simulation of a high altitude gantry crane with cable hoisting. Part one. 2D model // Вестник МГСУ. 2021. Т. 16. Вып. 5. С. 615-622. DOI: 10.22227/1997-0935.2021.5.615-622

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© Mostafa Jafari, Evgeniy M. Kudryavtsev, 2021

Распространяется на основании Creative Commons Attribution Non-Commercial (CC BY-NC)

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INTRODUCTION

The value of the global crane market reached USD 31.15 billion in 2018 and is expected to increase by about 4.26 % during the forecast period (2019-2024)1. Cranes are increasingly used in transportation and construction industries. They are also becoming larger, faster, and higher, thus, necessitating efficient controllers to guarantee little turnover time with account taken of the safety requirements. Over the last 60 years, we have evidenced a growing interest in research on the modeling and control of cranes2 of all types, including tower, overhead, gantry cranes, etc.

Gantry cranes are very popular construction site machines and research items. Methods of control of this type of mobile cranes are very interesting for scientists (over 105 papers from 2007 till 2020 indexed in Scopus), their dynamic models are also extensively covered in literature [1-3]. Other topics in this area include optimization methods [4, 5], structural design [6], analysis of failures [7, 8], emergency behavior [9], vibration reduction and stability [10-13], robotics and automatization [14, 15], energy saving [16], virtual reality [17], market analysis3, crane scheduling and assignment problems [18, 19], etc.

Large gantry cranes are more widely used, and many companies use them every year. However, despite much research on cranes, such super cranes have not been sufficiently studied [20, 21], although many of their problems [21-23] require new solutions and advanced technologies.

In this paper, a three-degrees-of-freedom dynamic analysis of a high altitude wide span gantry crane is performed with regard for the effect of a single rope hoisting mechanism on the payload swing angle. A generalized formulation of the two-dimensional crane behaviour is simulated in Mathcad [24] and verified using MSC ADAMS [25] (Academic) that shows good convergence.

MATERIALS AND METHODS

System modeling

Fig. 1 shows a dynamical model. All the subsystems are considered as rigid bodies without any deflection, and the crane is fixed. The mechanism selected for the hoisting system has a single rope having a single

y(t)

F(t)

mt, trolley

Fig. 1. Illustration of the 2D gantry crane having a single rope hoisting system

mass point for the hook and the payload. The 2D crane model has two masses that are the trolley and the pay-load, mt, m, respectively. The system has three degrees of freedom, y, and z for the position of the payload and the trolley and ^(t) for the oscillation of the payload (sway angle) on the x-z axis.

According to Fig. 1, the trolley moves along the j axis with the help of Ft(t) as a controlling force. Besides, the payload goes up and down along the k axis and oscillates in the y-z plane using the hoisting system control force Fh(t). The hoisting system takes advantage of the rope length changing mechanism, the acceleration of the trolley, and its velocity will be discussed in the next section (Fig. 1).

Since the lumped model is used in this paper, the rope is non-elastic. The Lagrange method is employed to identify kinetic and potential energies (T and V)4 5:

T = 0.5mby(t)2 + 0.5m[y(t)2 +l(tf +l(tftytf

(1)

+2 y(t)i(t) sin <|>(i) + 2j>(?)/(0<K0 COS <K0);

V = -mgl cos (|) (t). (2)

Here are the three-degrees-of-freedom equations of motion of the hoisting payload and the travelling trolley, given Fh(t) control forces and Ft(t):

1 Crane market — growth, trends, COVID-19 impact, and forecast (2021-2026). URL: https://www.mordorintelligence. com/industry-reports/crane-market

2 Abdel-Rahman E.M. Dynamics and control of cranes: A review // Journal of Vibration and Control. 2003. No. 7. Pp. 863-908.

3 Global portable gantry crane market size 2021 — Industry analysis and forecast 2027 available at Absolute Reports. URL: https://www.marketwatch.com/press-release/global-portable-gantry-crane-market-size-2021---industry-analysis-and-forecast-2027-available-at-absolute-reports-2021-04-05

/(04»(0 + 2/(0<j>(0 + g sin <K0 + K0 cos <K0 = 0; (3) (mt + m)y(t) + ml(t)(§{t) cos <\>(t) - <j>(i)2 sin <|>(f)) + +mi {t) sin <)>0) + 2mi(t)§(t) cos <)>(/) = Ft ;

nil it) - ml(t)§(t)2 - mg cos <|>(/) + +my(t) sin <Ki) = ~Fh.

(4)

(5)

4 Kim Yong-Seok, Hong Keum-Shik, Sul Seung-Ki. Anti-sway control of container cranes: Inclinometer, observer, and state Feedback // International Journal of Control, Automation and Systems. 2004. Vol. 2. No 4. Pp. 435-449.

5 Lee H. Modeling and control of a three-dimensional overhead crane // Journal of Dynamic Systems Measurement and Control. 1998. Vol. 120. No. 4. Pp. 471-476. DOI: 10.1115/1.2801488

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Gantry Crane 2D model in Mathcad besides, the results are displayed in Fig. 6 to 10.

Modeling of the gantry crane dynamics in Mathcad will find y" and using equations (3), the Mathcad software [26] is shown in Fig. 2 and 5; (4) [27].

Fig. 2. Modeling gantry crane dynamics in the Mathcad software

According to Fig. 3, a suitable quadratic trolley control force of Ft(t) is projected for the crane span of 80 meters having a 30-sec payload hoisted up. Also, this control force is used in ADAMS to find a suitable cable hoisting rate by modeling the cable system (Fig. 4).

Differential equations are solved in Mathcad using the Runge-Kutta method with a variable step that modifies the integral step size based on the prediction of an integration error. The solution procedure is provided below (Fig. 5).

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Fig. 3. Mathcad model: automatic derivation ofy" and 4>"

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Fig. 4. Cable hoisting profile projection (using ADAMS software): a — cable length; b — cable hoisting acceleration

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The solution of the differential equations of the pay load motion with Math cad:

Trollev mass ooirU: Pavlod mass point:

Length of the pendulum: Final time for integration: Gravitational acceleration;

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mt > 5.35T429J66SE-H)Î m > 5 .¿3 62-16312E-HW

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Trolley Action Force Ft(t)

Payload Action Force Fh(t)

Trolley displacement: (y. y1) Xq

Angel: s, ™ « tend

N1 := 200 t:=0.-„tend tt:=t

N1

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Ft(t) - L^tJ-m-Bii^Sjl + gm-co^s^-sii^:^1 m — m-co^sil1 + mt

Ft(t)-co^s11 + 2-L'(t)-m-H3 4- 2-L'(t)-mt-t- gm-sii^X} | + g-mt-sii^Xjl 4- j-Eu^x^ — L"(t)-m-co^x1 j-sinfs^ - 2 - L'( t) - m - - c o ^ h j ] 2

10 := 0

L< t) -m - L(É) -m-co J2 + L{ t) -mt tl := tend N:= N1 Z := RIcadapt (x,tO,tl D)

JÙ „<2> „<3> Ji

t-Z" Xfl := Z " xi := Z " X2 := Z " X3 := Z

Fig. 5. The solution to the differential equations describing gantry crane dynamics in the Mathcad software RESULTS OF THE RESEARCH

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Fig. 6, 7 and 8 illustrate the results of the simulation. Fig. 7 shows the swing angle of the payload that remains below 0.5 deg because of the appropriate trolley and payload control forces. Fig. 4 shows trol-

ley displacement and velocity. The maximum speed of the trolley is 1.716 m/s to prevent the payload swinging angle from exceeding 0.5 deg. The trolley velocity is reasonable for this type of cranes, if such limitations as wind effects are ignored.

Fig. 6. x0: Trolley Displacement (y); x2: Trolley velocity (y') in Mathcad

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Fig. 7. xr Swing angle ao of payload (4); x3: Payload angular velocity (40 in Mathcad

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-Trolley Ax (MSC ADAMS) - - Ft(t)/mt -Trolley Ax (Mathcad)

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Fig. 8. Trolley acceleration in MSC ADAMS and Mathcad

100.0

150.0

The effect of payload oscillation on the trolley dynamics in Mathcad and MSC ADAMS models is shown in Fig. 8. The difference between input control force F(t) using both ADAMS and Mathcad models is visible. Fig. 9 compares payload and trolley velocities. Payload oscillation is clear in Fig. 7 to 9. Fig. 10 shows hoisting animation and the payload trajectory using MSC ADAMS.

CONCLUSION AND DISCUSSION

In this study, the dynamic analysis of a gantry crane having a single rope hoisting mechanism was performed. The dynamic model was considered as

a lumped mass having a non-elastic cable. All the subsystems were considered as rigid bodies without any deflection. The formulation performed using the Lagrange method and the Mathcad solution were exhaustively described that can be useful for researchers.

The results were verified using MSC ADAMS, and they demonstrate excellent convergence. The effect of payload oscillation on the trolley movement was visible in both Mathcad and ADAMS models. The maximum velocity of the trolley is 1.716 m/sec to avoid the payload swing angle to exceed 0.5 deg. The trolley velocity is reasonable for this type of cranes, if such limitations as wind effects and resonance are ignored.

2.0

&

0.0 - 0.25

// tj to \ s \ s — Payload Velocity (ADAMS) — ■ Trolley velocity (mathcad)

// Vv. V.

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50.0

100.0

150.0

Time, s

Fig. 9. Comparison of the Trolley and the Payload Velocity in MSC ADAMS and Mathcad

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Fig. 10. Hoisting animation using MSC ADAMS (a); payload hoisting trajectory (b)

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Bionotes: Mostafa Jafari — Postgraduate student of the Department of Construction Mechanization; Moscow State University of Civil Engineering (National Research University) (MGSU); 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; mostafa.jafari@mail.ru;

Evgeniy M. Kudryavtsev — Doctor of Technical Sciences, Professor of the Department of Construction Mechanization; Moscow State University of Civil Engineering (National Research University) (MGSU); 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; ID RISC: 691899; sdm@mgsu.ru.

ЛИТЕРАТУРА

1. Shehu M.A., Li A. A novel smooth super-twisting control method for perturbed nonlinear double-pendulum-type overhead cranes // Arabian Journal for Science and Engineering. 2021. DOI: 10.1007/s13369-021-05340-0

2. Kudryavtsev E., Jafari M. Simulation of internal forces of a cantilevered beam with a moving load // IOP Conference Series: Materials Science and Engineering. 2021. Vol. 1103. No. 1. P. 012007. DOI: 10.1088/1757-899X/1103/1/012007

3. Кудрявцев Е.М., Гавриленко А.В., Джафа-ри М. Компьютерное моделирование динамических нагрузок на грузовую балку козлового крана // Качество и жизнь. 2020. № 4 (28). С. 23-28. DOI: 10.34214/2312-5209-2020-28-4-23-28

4. Qu X., Xu G., Fan X., Bi X. Intelligent optimization methods for the design of an overhead travelling crane // Chinese Journal of Mechanical Engineering. 2015. Vol. 28. No. 1. Pp. 187-196. DOI: 10.3901/cjme.2014.1008.157

5. Ku L.P., Lee L.H., Chew E.P., Tan K.C. An optimisation framework for yard planning in a container terminal: Case with automated rail-mounted gantry cranes // OR Spectrum. 2010. Vol. 32. No. 3. Pp. 519541. DOI: 10.1007/s00291-010-0200-9

6. Fan X.N., Zhi B. Design for a crane metallic structure based on imperialist competitive algorithm and inverse reliability strategy // Chinese Journal of Mechanical Engineering. 2017. Vol. 30. Pp. 900-912. DOI: 10.1007/s10033-017-0139-8

7. Domazet T., Luksa F., Bugarin M. Failure of two overhead crane shafts // Engineering Failure Analysis. 2014. Vol. 44. Pp. 125-135. DOI: 10.1016/j. engfailanal.2014.05.001

8. Alam M.R., Hassan S.F., Amin M.A., Arif-Uz-Zaman K., Karim M.A. Failure Analysis of a Mobile Crane: A Case Study // Journal of Failure Analysis and Prevention. 2018. Vol. 18. No. 3. Pp. 545-553. DOI: 10.1007/s11668-018-0437-1

9. Ma B., Fang Y., Zhang Y. Switching-based emergency braking control for an overhead crane system // IET Control Theory & Applications. 2010. Vol. 4. No. 9. Pp. 1739-1747. DOI: 10.1049/iet-cta.2009.0277

10. Shin J.H., Lee D.H., Kwak M.K. Vibration suppression of cart-pendulum system by combining the input-shaping control and the position-input positionoutput feedback control // Journal of Mechanical

Science and Technology. 2019. Vol. 33. No. 12. Pp. 5761-5768. DOI: 10.1007/s12206-019-1120-5

11. Ursavas E. Crane allocation with stability considerations // Maritime Economics & Logistics.

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Поступила в редакцию 9 января 2021 г. Принята в доработанном виде 17 мая 2021 г. Одобрена для публикации 17 мая 2021 г.

Об авторах: Мостафа Джафари — аспирант кафедры механизации строительства; Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ); 129337, г. Москва, Ярославское шоссе, д. 26; mostafa.jafari@mail.ru;

Евгений Михайлович Кудрявцев — доктор технических наук, профессор кафедры механизации строительства; Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ); 129337, г. Москва, Ярославское шоссе, д. 26; РИНЦ ГО: 691899; sdm@mgsu.ru.

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