Question 10.3: A variable-displacement hydraulic motor supplied with a cons...

The detailed symbolic diagram for this system is shown in Fig. 10.14. The elemental equations are as follows: For the leakage resistor,

Q_{R} =Q_{s}-Q_{t}=\frac{1}{R_{f} } P_{12} .          (10.33)

For the transducer,

T_{t} =C_{1} \psi P_{12} ,          (10.34)

\Omega =\frac{1}{C_{1}\psi } .Q_{t} .          (10.35)

For the inertias and dampers,

T_{t} -T_{1} =\left(J_{m}+J_{1} \right) \frac{d\Omega }{dt} +\left(B_{m}+B_{1} \right)\Omega .          (10.36)

Combining Eqs. (10.33), (10.34), and (10.36) and multiplying all terms by R_{f} C_{1}  yields

T_{t} =C_{1} \psi R_{f } Q_{s} -\left(C_{1} \psi\right) ^{2} R_{f} \Omega .          (10.37)

Combine Eqs. (10.36) and (10.37) to eliminate T_{t}:

C_{1} \psi R_{f } Q_{s} -\left(C_{1} \psi\right) ^{2} R_{f} \Omega -T_{1}=\left(J_{m} +J_{1} \right) \frac{d\Omega }{dt}+ \left(B_{m} +B_{1} \right)\Omega .          (10.38)

Divide all terms by \left(J_{m} +J_{1} \right) and rearrange into state-variable format:

\frac{d\Omega }{dt} =-\frac{B_{m} +B_{1} +\left(C_{1}\psi \right) ^{2}R_{f} }{J_{m} +J_{1}} \Omega +\frac{C_{1}\psi R_{f} }{J_{m} +J_{1}} Q_{s} -\frac{1}{J_{m} +J_{1}} T_{1} .          (10.39)

In this example, the fluid resistor R_{f} is transformed into an equivalent displacement-referenced damper \left(C_{1}\psi \right) ^{2}R_{f}  on the mechanical side of the system.