Mathematical Modeling of Nonlinear Dynamic. System of the Truck Crane

Transcription

1 Contemporary Engineering Sciences, Vol. 9, 16, no. 1, HIKAI Ltd, Mathematical Modeling of Nonlinear Dynamic System of the Truck Crane Sergei Evgenievich Ivanov ITMO National esearch University (ITMO University) Department of Information Systems 19711, Saint Petersburg, 49 Kronverksky Pr., ussian Federation Copyright 16 Sergei Evgenievich Ivanov. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The mathematical model of the mechanical system of truck crane with three degrees of freedom is considered. The truck crane includes the mechanisms of bridge and electric hoist, which are modeled by translational kinematic pairs. For the automation of handling operations are determined the position of cargo, speed and acceleration of cargo in the initial coordinates system. The device for soft start-stop of the truck crane is simulated by nonlinear polynomial with generalized coordinates. By modified method of polynomial transformations we obtain the analytical solution of nonlinear mathematical model of truck crane. For control the accuracy of analytical results we carried out calculation of mathematical model by the method unge-kutta fourth order. The power of devices necessary for lift and move the cargo is defined. Keywords: truck crane, mathematical modeling, nonlinear dynamical system, the method of polynomial transformations 1 Introduction The truck crane applied for moving heavy cargos in the warehouse, for installing large units of equipment and for the movement of goods between the technological areas. The automation of the truck crane allows increasing the performance.

2 488 Sergei Evgenievich Ivanov The truck crane (figure 1) includes: support, beams, bridge and hoist [1, ]. Fig.1 The truck crane The truck crane includes the mechanism for soft start-stop of crane, the brake mechanism of truck crane, the brake mechanism for lifting, the brake mechanism for load carriage [, 4]. The truck crane controlled automatically or remotely. The electric hoist is fixed to the trolley that moves along the monorail path (figure ). Along the I-beam the electric hoist is moved. The beam is connected to the two beams on which are located the running wheels. Fig. The electric hoist The technical specifications the crane are determined by the following main parameters: the load, length of flight, lifting height, the speed of bridge, the speed of lift truck, the speed of lifting. The truck crane is applied for spans from 4.5 to 8 m. and cargo capacity from.5 to 1 tons. Mathematical modeling of the mechanical system of bridge and electric hoist The mechanical system of bridge and electric hoist has three degrees of freedom. Figure shows the kinematic scheme of truck crane.

3 Mathematical modeling of nonlinear dynamic system 489 x z O1 y z O y x z x O y z O y x Fig. The kinematic scheme of truck crane The mechanical system of the truck crane has geometric characteristics: the length and the width of the span - l1, l and height of lifting - h. We introduce the relative coordinate system O1, O, O associated with bridge of crane, with the lifting truck and with the cargo. The initial coordinate system we denote O and associate with supports of crane. The coordinates the cargo we choose for generalized coordinates q1, q, q The transition from coordinate system O to O 1 occurs through displacement along z-axis by h and displacement along axis x by q 1. The transition from coordinate system O 1 to O occurs through rotation around z-axis by / and displacement along axis x by q. The transition from the coordinate system O to O occurs through rotation around z-axis by / and displacement along axis z by q. We applied the matrix method and Lagrange equations in matrix form to produce the equations of motion. For radius vector of a point in the coordinate system i, we define the column: x y z. i i i i 1 T

4 49 Sergei Evgenievich Ivanov The communication of radius vectors in coordinate systems i-1 and i by the transition matrix A i with formula: i1 A i i The transition matrix from O too 1, the transition matrix fromo 1 too, the transition matrix from O too are equal: 1 q1 1 A1, 1 h q A 1 1, A q 1 The transition matrix from O too 1, the transition matrix fromo too, the transition matrix from O too are equal: A 1 1 q1 1, 1 h 1 A 1 q1 1 q, 1 h 1 A 1 q1 1 q 1 h q 1 We obtain the kinetic energy for bridge of crane, lift trucks and lifted load. For calculate the kinetic energy we apply matrix form with transition matrix: T T tr(a H A ) /, i i i i where A i - derivative of the transition matrix, H i - matrix of inertia. The weights bridge of crane, lift trucks and lifted load are equal: m1, m, m The total kinetic energy is equal the sum of the kinetic energy of the three units: T.5 m m m q.5 m m q.5mq 1 1 In matrix form, the potential energy equals: T where G g i P m G A, T i i i i - the matrix-row for the acceleration of free fall The total potential energy equal to: P mg( h q) We apply the Lagrange equations in matrix form for the equations of motion. d T T P Q i, where Q1, Q, Q - generalized forces of units. dt q q q i i i The nonlinear device for soft start-stop of the bridge and electric hoists is simulated by cubic polynomial. The generalized forces for soft start-stop of the bridge, hoist and lift represented in the form: f q f q f q f 1

5 Mathematical modeling of nonlinear dynamic system 491 By substituting kinetic energy, potential energy and the generalized forces in the Lagrange equation, we obtain the system of equations with three degrees of freedom. m1 m m q1 b1q 1 bq1 bq1 b, m m q c1q cq cq c, mq d1q dq dq d gm For the solution of nonlinear differential equations is applied various analytical methods [5-1]: the harmonic balance method, van der Pol method, the small parameter method, the averaging method, Krylov-Bogolyubov method, the Poincare perturbation method and the polynomial transformations method. The exact solution of the nonlinear system of equations is obtained numerically by the method of unge-kutta fourth order with the following parameters: m 5, m 1, m 1, h 1, g 9.81, l 1, l b.5, b., b.1, c.4, c., c.1, d., d., d The analytical solution is obtained by the modified method of polynomial transformations [11-1]. We computed the generalized coordinates and the speed of the bridge for distance l / / 6b 6 1b1 b b / b q1 asin t w / cos t w 8cos t w 57 /, q a w 15sin t w 5sin t w sin t w / 15, 1 where a 1 1 / / / 4/ 6b 1 b b 1b b b 18 b 1b b w / 1b m m m 1 54b 7b b b 84.75b l l 5b l 6b 8b b l l 5bl 6b 8b1 54b 7b1 bb 4b1 b 9b We computed the generalized coordinates and the speed of lifting truck along the beam of bridge for distance l q a t w a t w a t w q a w sin t w a w sin t w / a w sin t w /15 / / where a 6c 18 c 6 c1c / c cos 1 cos 1 / 8 cos 1 /15,,

6 49 Sergei Evgenievich Ivanov w / 1 6 c / 1 / / 4/ / c c c c c c m m c c c c c m m / / 4 / 84.75c l l 5c l 6c 8c 54c 7c c c c l l 5cl 6c 8c1 54c 7c1c c 4c1c 9c We computed the generalized coordinates and lifting speed for height h / / 6d 6 1d1d d / d q 1. 9 asin t w / asin t w / cos t w / 4asin t w / cos t w / 15 q a w sin t w a w sin t w / a w sin t w /15, where a 6d 1 d d 1d d d 18 d 1d d w / 1d m / / / 4/ d 7d d d 84.75d 8hd 6h d 5h d 16gm d d1d d d1dd d hd1 h d h d gm The figure 4 presents generalized coordinates and the speed of lifting electric hoist for height of 1 meters obtained by analytical method of polynomial transformation (dashed line) and by the numerical method of unge-kutta (solid line). Fig.4 the generalized coordinates and cargo lifting speed We computed the power of devices that needed for lift and move the cargo. The maximum power required for lift on height is equal:

7 Mathematical modeling of nonlinear dynamic system 49 1/ d / / 1/ 1/ 5dh 6dh 8d1h 16gm 6 d 6 d 1d1d 1 N 96 d 6 d 1 d d 1d d 18 d 1d d 1/ 4/ / / 1/ 1/ / 1 1 / d m The figure 5 shows graphs of the generalized forces for move to the distance l of the beam (solid line) and for lifting truck (dashed line). Conclusion Fig.5 the generalized forces to move to distance l The simulation of mechanical systems of the truck crane with a nonlinear device for soft start-stop is carried out. For solve the problem of automation for handling operations is defined the position, speed and acceleration of cargo in the initial coordinate system. By modified method of polynomial transformations we obtained the analytical solution of nonlinear mathematical model of truck crane. For control the accuracy of analytical results is carried out numerical solution of nonlinear model by method of unge-kutta fourth order. The power of devices necessary for lift and move cargo is defined. Acknowledgements. The research was carried out with the financial support of the ussian Foundation for Basic esearch eferences [1]. Tang, J. Huang, Control of bridge cranes with distributed-mass payloads under windy conditions, Mechanical Systems and Signal Processing, 7-7 (16),

8 494 Sergei Evgenievich Ivanov [] W. He, S. S. Ge, Cooperative control of a nonuniform gantry crane with constrained tension, Automatica, 66 (16), [] M. Marzouk, A. Abubakr, Decision support for tower crane selection with building information models and genetic algorithms, Automation in Construction, 61 (16), [4] D. Briskorn, P. Angeloudis, Scheduling co-operating stacking cranes with predetermined container sequences, Discrete Applied Mathematics, 1 (16), [5] E. P. Kolpak, L. S. Maltseva, S. E. Ivanov, On the stability of compressed plate, Contemporary Engineering Sciences, 8 (15), no., [6] S. A. Kabrits, E. P. Kolpak, Numerical Study of Convergence of Nonlinear Models of the Theory of Shells with Thickness Decrease, AIP Conference Proceedings, 1648 (15). [7] S. E. Ivanov, V. G. Melnikov, On the equation of fourth order with quadratic nonlinearity, International Journal of Mathematical Analysis, 9 (15), no. 54, [8] S. E. Ivanov, E. P. Kolpak, Mathematical modeling of the guidance system of satellite antenna, Contemporary Engineering Sciences, 9 (16), no. 9, [9] S. E. Ivanov, V. G. Melnikov, On the two-dimensional nonlinear Korteweg-de Vries equation with cubic stream function, Advanced Studies in Theoretical Physics, 1 (16), no. 4, [1] E. P. Kolpak, S. E. Ivanov, On the three-dimensional Klein-Gordon equation with a cubic nonlinearity, International Journal of Mathematical Analysis, 1 (16), no. 1, [11] S. E. Ivanov, V. G. Melnikov, Mathematical modeling vibration protection system for the motor of the boat, Applied Mathematical Sciences, 9 (15), no. 119, [1] E. P. Kolpak, S. E. Ivanov, Mathematical modeling of the system of drilling rig, Contemporary Engineering Sciences, 8 (15), no. 16,

9 Mathematical modeling of nonlinear dynamic system 495 [1] E. P. Kolpak, S. E. Ivanov, Mathematical and computer modeling vibration protection system with damper, Applied Mathematical Sciences, 9 (15), no. 78, eceived: March 15, 16; Published: May 15, 16