Determine the principal coordinates for the system of Chapter Problem 6.9. Write the differential equations which the principal coordinates satisfy. if \(I=0.03 \mathrm{~kg} \cdot \mathrm{m}^{2}, c=0 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}, m=3 \mathrm{~kg}, a=0.3 \mathrm{~m}, b=0.4 \mathrm{~m}\) and \(k=3000 \mathrm{~N} / \mathrm{m}\).

Data From Chapter Problem 6.9:

Derive the differential equation of the two degree-of-freedom model of the machine tool of Chapter Problem 6.8 using \(x\) and \(\theta\) as generalized coordinates.

Data From Chapter Problem 6.8:

A two degree-of-freedom model of a machine tool is illustrated in Figure P6.8. Using \(x_{1}\) and \(x_{2}\) as generalized coordinates, derive the differential equations governing the motion of the system.

k m, I G X x1 FIGURE P6.8. k