Question 9.1: Find a set of state-variable equations and develop the input...

The elemental equations are as follows: For the fluid resistor,

P_{12} =R_{f} Q_{R} .          (9.13)

For the inertor,

P_{23} =I\frac{dQ_{R} }{dt} .          (9.14)

For the fluid capacitor,

Q_{R} =C_{f} \frac{dP_{3r} }{dt} .          (9.15)

Continuity is satisfied by use of Q_{R} for Q_{I} and Q_{c}. To satisfy compatibility.

P_{s} =P_{1r} =P_{12}+P_{23}+P_{3r}.          (9.16)

Combining Eqs. (9.13), (9.14), and (9.16) to eliminate P_{12} and P_{23} yields

I\frac{dQ_{R} }{dt} =P_{s} -R_{f} Q_{R}-P_{3r}.          (9.17)

Rearranging Eq. (9.17) yields the first state-variable equation:

\frac{dQ_{R} }{dt} =-\frac{R_{f} }{I} Q_{R}-\frac{1}{I} P_{3r} +\frac{1}{I} P_{s}.          (9.18)

Rearranging Eq. (9.15) yields the second state-variable equation:

\frac{dP_{3r} }{dt} =\frac{1}{C_{f} } Q_{R}.          (9.19)

Combining Eqs. (9.18) and (9.19) to eliminate Q_{R} and multiplying all terms by I yields the input–output system differential equation:

C_{f} I\frac{d^{2}P_{3r} }{dt^{2} } +R_{f}C_{f}\frac{dP_{3r} }{dt} +P_{3r}=P_{s}.          (9.20)