This paper presents dynamic modelling of a double-pendulum gantry crane system based on closed-form equations of motion. 

A dynamic model of the system incorporating payload is developed and the effects of payload on the response of the system are discussed. Extensive results that validate the theoretical derivation are presented in the time and frequency domains

Mohd Anwar, Zawawi

Mohd Syakirin, Ramli

Raja Mohd Taufika, Raja Ismail

UMP Institutional Repository

Authors:R. M. T. Raja Ismail, M. A. Ahmad, M. S. Ramli, R. Ishak and M. A. ZawawiFaculty of Electrical & Electronics Engineering,Universiti Malaysia Pahang, MalaysiaABSTRACT This paper presents dynamic modelling of a double-pendulum gantry crane system based on closed-form equations of motion.  A dynamic model of the system incorporating payload is developed and the effects of payload on the response of the system are discussed.  Extensive results that validate the theoretical derivation are presented in the time and frequency domains.PROBLEM STATEMENT Purpose of controlling a gantry crane: To transport the load at short period of time (fast) without causing any excessive swing at the final position. Problems that arise: Gantry crane results in a swing motion when the payload stops suddenly after a fast rope movement. It requires more time (larger settling time) to minimize the swing motion (swing angle). The needs for skillful operators to manually control and stop the swing at the right position** Failure to control the crane might cause accident and may harm people and surrounding.OBJECTIVES To study the dynamic modelling of a double-pendulum gantry crane system based on closed-form equations of motion. To investigate the effects of payload on the dynamic behaviour of a double pendulum gantry crane system.BRIEFING ON Gantry Crane Sys :Model structure• The double-pendulum gantry crane system with its hook and load considered in this work is shown below. • Where x is the cart position, mc is the cart mass, mh is the hook mass and mp is the payload mass. • Meanwhile, θ1 is the hook swing angle, θ2 is the load swing angle, l1 and l2 are the cable length of the hook and load, respectively, and F is the cart drive force.BRIEFING ON Gantry Crane Sys :System parameter valuesSymbol Parameter Valuemc Cart mass 5 kgmh Hook mass 2 kgmp Payload mass 1-10 kgl1 Hook pendulum length 2 ml2 Load pendulum length 1 mg Gravity acceleration 9.8 m-s–2F Bang-bang input 10 N (amplitude) / 1 s (width)BRIEFING ON Gantry Crane Sys :System’s variable concernedSymbol Variable The importancex (m) Cart position To achieve steady state position with minimum errorθ1 (rad) Hook swing angle To avoid excessive swing at hook θ2 (rad) Load swing angle To avoid excessive swing at loadPSD of  θ1 (dB)Power spectral density of the hook swing angleTo minimize the vibration at hook due to rope movementPSD of  θ2 (dB)Power spectral density of the load swing angleTo minimize the vibration at load due to rope movementBRIEFING ON Gantry Crane Sys :Other parameters assumption1) Cart friction force is ignored.2) The tension force that may cause the hook and load cables elongate is also ignored.3) The cart (translational) and the payload (rotational) are assumed to move in two dimensional only (2D –movements)BRIEFING ON Gantry Crane Sys :Mathematical model1) The dynamic model of the double-pendulum gantry crane system is expressed as :Where:Fqqqqqq  )(G),(C)(M )cos(cos)(cos)(cos)()(M212122211111llmlmlmmlmmlmmmmmppphphphphcq22212122)cos(coslmllmlmppp12121111)sin(000sin)(0),(Cllmlmmpphqq0)sin(sin12121222llmlmpp2211sinsin)(0)(GglmglmmpphqInertia Centrifugal coriolisGravity SIMULATION RESULTS …(1)0.40.60.811.21.41.61.82Cart position (m)mp = 1 kgmp = 5 kgmp = 10 kg• It is noted that the average final position of the cart decreases and the chattering of the final position increases with increasing payloads.0 2 4 6 8 10 12 14 16 18 2000.2Time (s)Fig 1: Response of the cart position (mp = 1, 3, & 5 kg)SIMULATION RESULTS …(2)-0.6-0.4-0.200.20.40.60.8Hook swing angle (rad)mp = 1 kgmp = 5 kgmp = 10 kg-0.6-0.4-0.200.20.40.60.81Load swing angle (rad)mp = 1 kgmp = 5 kgmp = 10 kg• It is shown that, the hook swing angle and load swing angle responses for various payloads requires more than 20 sec. to settle down. • Besides that, it can be seen the oscillations of the hook swing angle and the load swing angle decrease with increasing payloads.0 2 4 6 8 10 12 14 16 18 20-0.8Time (s)0 2 4 6 8 10 12 14 16 18 20-1-0.8Time (s)Fig 2(a): Response of the hook swing angle (mp = 1, 3, & 5 kg) Fig 2(b): Response of the load swing angle (mp = 1, 3, & 5 kg)SIMULATION RESULTS …(3)-50-40-30-20-100102030PSD of the hook swing angle (dB)mp = 1 kgmp = 5 kgmp = 10 kg-50-40-30-20-100102030PSD of the load swing angle (dB)mp = 1 kgmp = 5 kgmp = 10 kg• Fig. 3 (a),(b) demonstrates that the resonance modes of vibration of the system shift to higher frequencies with increasing payloads.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-80-70-60Frequency (Hz)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-80-70-60Frequency (Hz)Fig 3(a): PSD of the hook swing angle (mp = 1, 3, & 5 kg) Fig 3(b): PSD of the load swing angle (mp = 1, 3, & 5 kg)ANALYSIS AND DISCUSSIONTable 1: Payload vs. Cart position responses Table 2: Payload vs. Hook & load swing anglesPayload(kg)Average cartposition (m)Oscillation(m)0 - -1 1.9920 ±0.36302 1.9151 ±0.38313 1.9145 ±0.42284 1.8521 ±0.49295 1.8419 ±0.5049Payload(kg)Hook swingangle (°)Load swingangle (°)0 - -1 ±0.4132 ±0.88262 ±0.4063 ±0.74183 ±0.3770 ±0.61404 ±0.3493 ±0.53195 ±0.3080 ±0.3982• From table 1, the average cart position decreases but the oscillation itself increases for heavier loads.• Meanwhile, both hook and load swing angle decrease with the load increments (Refer table 2)6 1.7219 ±0.50577 1.6940 ±0.50758 1.6063 ±0.50919 1.5472 ±0.510010 1.5154 ±0.51026 ±0.3049 ±0.37917 ±0.2919 ±0.34318 ±0.2813 ±0.32449 ±0.2333 ±0.300710 ±0.2305 ±0.2902ANALYSIS AND DISCUSSIONTable 3: Payload vs. Hook & load swing angles resonanse freq. (Hz)Payload(kg)Resonance frequency (Hz)Hook swing angle Load swing angleMode 1 Mode 2 Mode 1 Mode 20 - - - -1 0.3662 1.343 0.3662 1.0992 0.3662 1.587 0.3662 1.2213 0.3662 1.709 0.3662 1.2214 0.3662 1.709 0.3662 1.2215 0.4883 1.099 0.4883 1.221• From table 3, it shows that both hook and load swing angles have the same resonance frequencies of mode 1. • It is due to the sway of the payload is always follow the oscillation of the hook.6 0.4883 1.221 0.4883 1.2217 0.4883 1.343 0.4883 1.3438 0.4883 1.343 0.4883 1.3439 0.4883 1.465 0.4883 1.46510 0.4883 1.465 0.4883 1.465• Besides, the system has the same resonance frequencies of mode 1 that is 0.3662 Hz, when the payload is varied from 1 kg to 4 kg and has the same frequency of 0.4883 Hz when the payload is varied from 5 kg to 10 kg. • This shows that, in order to decrease the oscillation of ANALYSIS AND DISCUSSIONthe system, a same control design can be used for several systems although they have different payloads.• Besides, the hook and the load swing angles have different resonance frequencies of mode 2. However, these resonance frequencies do not affect much on the system since the mode 1 frequency is the dominant mode to the systemFUTURE RECOMMENDATION Comparative studies on the cart position, hook & load swing angle as well as their respective PSD for various rope length (l1 & l2) and input forces (F). Implementation of experimental studies by using a different type of crane, (e.g. rotary crane).CONCLUSION Investigation into the development of a dynamic model ofa double-pendulum gantry crane system incorporatingpayload has been presented The dynamic model has been simulated with bang-bangforce input. The cart position, hook swing angle and load swing angleresponses of the gantry system have been obtained andanalysed in time and frequency domains. Moreover, the effects of payload on the dynamiccharacteristic of the system have been studied anddiscussed.REFERENCES[1] D. T. Liu, W. P. Guo and J. Q. Yi and D. B. Zhao, Double-pendulum-type Overhead Crane Dynamics and its Adaptive Sliding Mode Fuzzy Control, Proc. Of the Third International Conference on Machine Learning and Cybernetics, 2004, pp. 423-428.[2] D. T. Liu, W. P. Guo and J. Q. Yi, GA-based Composite Sliding Mode Fuzzy Control for Double-pendulum-type Overhead Crane, Lecture Notes in Computer Science, Springer Berlin, 2005, pp. 792-801.[3] D. T. Liu, W. P. Guo and J. Q. Yi, Dynamics and GA-based Stable Control for a Class of Underactuated Mechanical Systems, International Journal of Control, Automation, and System, Vol. 6, No. 1, 2008, pp. 35-43.[4] D. T. Liu, W. P. Guo and J. Q. Yi, Dynamics and Stable Control for a Class of Underactuated Mechanical Systems, Acta Automatica Sinica, Vol. 32, No. 3, 2006, pp. 422-427.[5] D. Kim and W. Singhose, Reduction of Double-Pendulum Bridge Crane Oscillations, The 8th International Conference on Motion and Vibration Control, 2006, pp. 300-305.REFERENCES (cont…)[6] M. Kenison and W. Singhose, Input Shaper Design for Double-pendulum Planar Gantry Cranes, 1999 IEEE Conference on Control Applications, 1999.[7] M. A. Ahmad, Z. Mohamed and N. Hambali, Dynamic Modelling of a Two-link Flexible Manipulator System Incorporating Payload, 3rd IEEE Conference on Industrial Electronics and Applications, 2008, pp. 96-101.[8] M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control. New Jersey: John Wiley, 2006.[9] M. W. Spong, Underactuated Mechanical Systems, Control Problems in Robotics and Automation. London: SpringerThankyou

Dynamic Modelling of a Double‐Pendulum Gantry Crane System Incorporating Payload

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Dynamic Modelling of a Double‐Pendulum Gantry Crane System Incorporating Payload

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