Final answer:
To determine the free response of the given models with initial conditions x(0)=0 and v(0)=1, we need to solve their corresponding differential equations. Model (a) has a complex solution, model (b) has two real and distinct solutions, and model (c) has two real and equal solutions.
Step-by-step explanation:
The free response of the given models can be determined by solving the corresponding differential equations.
For model (a), the equation is ςx˨+4x·+8x=0. The characteristic equation is r²+4r·+8=0. The roots of this equation are complex numbers, which means the solution is of the form x(t)=Ae^(-2t)cos(2t)+Be^(-2t)sin(2t).
For model (b), the equation is x˨+8x·+12x=0. The characteristic equation is r²+8r·+12=0, which has two real and distinct roots. The solution is of the form x(t)=Ae^(-4t)+Be^(-3t).
For model (c), the equation is 2x˨+8x·+8x=0. The characteristic equation is 2r²+8r·+8=0, which has two real and equal roots. The solution is of the form x(t)=Ae^(-2t)+Bte^(-2t).