Consider the following mechatronic system consisting of a damping spring-mass system actuated by an electromagnet. The mass is of a magnetic material which is attracted by the magnetic force generated by the electromagnet. This class of mechatronic systems is widely used for door opening systems, magnetic actuators, electromagnetic brakes, solenoids among others. The differential equation that models the dynamics of the system is given by:

Where: Mass Coefficient of viscous friction Spring constant Electromagnetic constant It is a constant that arises from the calculation of the reluctance of the magnetic circuit and the space between the electromagnet and the mass () = Displacement of the mass referred to the coordinate system whose origin is () 0 when (): = 0. Distance between mass and electromagnet when () = 0. , Real constants of distance and current that are empirically determined in a laboratory.

Let the vector of states = (1, 2, 3)^T = (, , )^{T 3}. The system input is the electromagnet voltage. (i.e () = ())

1. Determine the state space representation of the system in the form = (, ). 2. Write a MATLAB program to simulate the system for the following voltage inputs: a) () = 12 b) () = () c) () = d) () = 2 () The initial conditions of the system are: (, , )^T = (1 (0), 2 (0), 3 (0))^T = (0,0,0)^T3 3. Include all the graphs of the states and the voltage applied to the mechatronic system.

GIVEN DATA: =1.54 =0.0659 / =38.94 / =7.957105/ =10 =5 =0.0016 =0.0017 =2 =2

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