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  3. the system a model the dynamics of the system completely...

Question: the system a model the dynamics of the system completely...

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The system a) Model the dynamics of the system completely, ie., derive the mathematical equatio motion for the system. Hint: Draw the free body diagrams and write the equations for balance of forces for bot trolley and the load. For the load, write the equation for balance of torques (since the represents a rotational system) A crane system which consists of a trolley, rope and payload is shown in the figure below The notation refers to the following parameters m-mass of the trolley mi- mass of the load, x, - horizontal displacement of the trolley xi-horizontal displacement of the load, p- angular displacement of the rope L -length of the rope b-coefficient of friction between the trolley and the rail, bi- coefficient of friction on the load (caused by pivot joint friction and air friction), F-force applied to the trolley c) Obtain the Laplace transfer function relating the angular displacement of the load ф to the horizontal displacement of the trolleyxr. Hint: Note that the notion of transfer function applies to linear systems, and the model of the crane system is nonlinear. Therefore, first linearize the system in the vicinity of the stable equilibrium point ф-0 and ф-0: hence, assuming small angular displacements ф m the neighborhood of the equilibrium point, we can substitute sinφ ~ cos: 1, and φ,0. Using these approximations, a linearized model of the crane system will be obtained, and afterwards the transfer function will be derived. Tt br d) Obtain the Laplace transfer function relating the angular displacement of the load ф to the horizontal velocity of the trolley v. Hnt: First linearize the model equations, similarly to the solution in part c). mi al

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