Final answer:
To find the free response of the given system model with initial conditions, identify the initial parameters and select the appropriate equation to solve for Xreaction. Apply the initial conditions to the chosen method for solving the system's dynamics.
Step-by-step explanation:
To obtain the free response of the given system models, we must first identify the initial conditions and the variables of interest. Since ū (the initial conditions) are known and we take xo-reaction to be 0, the variable we want to find is Xreaction. The best equation to use in this case would typically be one derived from the system's differential equation, which characterizes the system's dynamics. This solution could involve solving the characteristic equation of the system to find the natural frequencies and modes of response, applying Laplace transforms, or utilizing state-space representation if the model is given in that form.
Once the proper equation or method is chosen, it's a matter of applying the initial conditions and proceeding with the mathematical solution to find Xreaction. The resolution may involve calculating eigenvalues, eigenvectors, or employing inversion of Laplace transforms, according to the nature of the system described by the model.