# Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility

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3 Theta(rad) L(m) Theta(rad) L(m) B. Friction Model To derive appropriate friction model from physical laws alone is impossible. Empirical approach can be taken to model friction in a system by reproducing effects observed in experiments. Among different interpretations of friction the Lu-Gre friction model is found to capture most of the friction behaviour that has been, in general, observed experimentally in control systems []. The friction occurring in the linear actuator and in the trolley motor has a very significant effect on the system behaviour. The hoist motor friction is mainly due to the gear. The efficiency of the hoist motor with the attached gear box is 5 % for the laboratory scale gantry crane model in []. Thus it is assumed that 5 % of the torque is lost because of friction, therefore T fric = -.5T load for the hoist motor. The friction in the trolley motor and in the linear actuator is modelled using the Lu-Gre friction model. The standard parameterization of the Lu-Gre model is given by The PID controlled output without friction is obtained, as in Fig. The trolley displacement reached a magnitude instantaneously due to lightly damped nature of crane and absence of friction. The cable length magnitude initially oscillates due to the combined effect of gravity and the controller. It finally settles to the reference value, in about seconds, due to the control action. It is observed that angular swing is zero for zero friction when trolley velocity is zero. Also it is noticed that angular swing is not affected by cable length variation in the absence of friction. Fig 3 shows the effect of friction on the output. Due to friction the trolley displacement and cable length are found to settle at values below their corresponding reference values. The swing angle is reaching up to about 5 radians initially and also all three outputs are taking much time to settle. Therefore a more suitable controller is required Hoist length.5 (7) Where, α i and v are the static parameters and σ i are the dynamic ones. The state variable z is related to the bristle interpretation of friction and is the average bristle deflection. z is not measurable. The friction torque is given by F, which is a function of the trolley speed v. The friction torques calculated are subtracted from the corresponding inputs Hoist Length Angular swing Fig. D model output without friction.5 Fig 3. D model output with friction (3.6) Swing angle C. d Non-Linear Model With 3d Angular Swing Fig 4 shows the swing motion of the load caused by trolley movement of a 4DOF (x displacement of trolley, y displacement of trolley, hoisting cable length and angular swing) gantry crane system. The trolley and the load are considered as point masses and are assumed to move in a 3D plane. The trolley can move along the girder in Y direction and the girder moves along the X direction. Here, θ x is angular swing component in x direction (rad), θ y is angular swing component in y direction (rad), x is displacement of the girder (m), y is displacement of trolley along girder(m), l is hoist cable length in meters, f x is control force applied to the trolley in the X-direction (N), f y is control force applied to the ISSN : 39 38, Volume-, Issue-, 3

4 l(m) l(m) trolley in the Y-direction (N), f l is control force applied to the payload in the l -direction (N), m p is payload mass(kg) and I is mass moment of inertia of the payload (kgm ) Hoist length 3 Theta y Angular swing component in y direction Angular swing component in x direction Theta x Fig 4. Trolley-cart model of 3D overhead crane [5] The gantry crane system considered in this paper is of 3 degree of freedom having no x displacement. When the girder is in rest and trolley is in motion, the three dimensional crane model resembles the two-dimensional crane model [6].Thus the girder displacement x and its derivatives are zero. But forces in X direction can be present in the form of disturbances acting directly on payload and is taken as F x. Therefore we get, (8) Theta y Theta x Fig 5. Model output without disturbances in X direction Hoist length Angular swing component in y direction Angular swing component in x direction For the given system, the generalised coordinate vector q and force vector F are given by, Deriving the Lagrangian equations using the above data, the dynamic equations of the system is obtained. (8) (9) Fig 6. Model output with disturbances in X direction The PID controlled output without payload disturbances (Fig 5) are obtained by giving F x =. Comparing with Fig, the new model output without friction (Fig 5) shows initial oscillations for trolley displacement and also presence of angular swing of (9) load. From Fig 5 and Fig 6 it can be seen that angular swing component in X direction, θ x, is present only when disturbance forces F x is acting directly on the payload. But angular swing is present in the Y direction due to trolley motion in Y direction. Here, F x has been given as a single pulse at time 3 seconds. Angular swing component in Y direction is seen to settle faster than that in X direction since it is influenced by the y displacement which is settling whereas x displacement is absent. Thus swing output can be controlled using trolley displacement. ISSN : 39 38, Volume-, Issue-, 3

5 l(m) Adding friction to the new D model, the trolley displacement and hoist cable length does not reach the reference values as observed previously. But, compared to new model output without friction (Fig 5), output with friction (Fig 7) shows that oscillations are taking longer time to settle due to the combined effect of friction and control action v(l,t) can be found using the solution of Euler- Bernoulli beam equation using assumed modes method. In this method, deflection is expressed as a sum of two functions: a function of displacement along the length of the cable (Shape function i x ) and a function of time (generalized co-ordinates δ i (t )) [7]. From the solution of Euler-Bernoulli beam equation [8], deflection v at a distance x along the cable is,.5 Theta y Angular swing component in y direction Theta x.5.5 Hoist Length Angular swing component in x direction Where, () () () Fig 7. Model output with 3D angular swing and friction A. d Non-Linear Model With Load Hoisting And Flexible Cable Fig 8 shows the load swing in a trolley-pendulum model of a D 3DOF gantry crane system with a flexible cable. v(x,t) is the deflection of a point on the cable at a distance x from the trolley end of cable in meters. All the rest of the notations are the same as that in Fig.Here, v is a function of x as well as time t. Then, deflection of the cable tip is v(l,t) where l is the length of the cable. for a loaded beam [3]. The coordinates of the payload is given by, Kinetic and potential energies of the system is given by, (4) If the cable is assumed to be mass-less the kinetic energy of the cable K cable can be neglected. For the given system, the generalised coordinate vector q and force vector F are given by, q = [y l θ δ] T F=[F y F l ] T (5) Fig 8. D gantry crane with load hoisting and flexible rod Using the above equations, the dynamic equations of the system are obtained using Lagrange s equation of motion 3 ISSN : 39 38, Volume-, Issue-, 3

6 Theta(rad) L(m) Length of Hoist Swing angle Fig 9. D model open loop response without friction and cable flexibility for 6s This difference in the actual angle will be having important implications on controller performance. E. 3d Non-Linear Model With Load Hoisting And Flexible Cable It has been observed previously that the transients are better captured by the model including 3D angular swing dynamics. Therefore, the same derivation is attempted here including the dynamics of cable flexibility. The 3D gantry crane model used before is again considered with having a flexible cable (Fig ). The two components of the additional deflection due to the flexibility of the cable v(x,t) are given by v(l y,t) and v(l x,t). Where, Fig. 3D gantry crane with load hoisting and flexible cable Fig. D model open loop response with friction and cable flexibility for 6s (t =.3n) Comparing Fig 9 and Fig, the effect of friction is observed to decrease the rate of increase of trolley displacement and hoist cable length. The effect of flexibility is seen in angular swing of payload. The swing angle is seen to decrease significantly compared to the model output without considering cable flexibility. A possible explanation to this phenomenon can be explained with Fig. The coordinates of the payload is given by, (6) (7) The kinetic and potential energies of the system will be (8) Fig. Deflection of a flexible rope 4 ISSN : 39 38, Volume-, Issue-, 3

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