Two connected cars with an applied input force \(u(t)\) and negligible rolling friction can be represented by a translational mechanical system as shown below.

(a) Draw the free-body diagrams of this mechanical system.

(b) Write down the differential equations that represent the system.

(c) Explain why the vector of state variables should be chosen as \[
\mathbf{x}(t)=\left[\begin{array}{llll}
x_{1} & v_{1} & x_{2} & v_{2}
\end{array}ight]^{T}
\]

(d) Express these differential equations in the State-Variable Matrix model, where the output is the frictional force in \(b_{1}\).

Transcribed Image Text:

X1 X2 u(t) k k ww www Input force m m2 b O b Translational Mechanical System 77