(Solved): \( 2.28 \) A closed loop hydraulic system consists of a variable-displacement ...

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\( 2.28 \) A closed loop hydraulic system consists of a variable-displacement pump driving a fixed displacement motor as shown diagrammatically in figure A.24. (a) Show that \[ \frac{\Omega_{\mathrm{m}}(s)}{Y(s)}=\frac{K_{\mathrm{p}}}{d_{\mathrm{m}}}\left(\frac{1}{1+\tau s}\right) \] where \( s \) is the Laplace transform function, \( K_{\mathrm{p}} \) is the pump flow constant for a given speed \( \Omega_{\mathrm{p}}, d_{\mathrm{na}} \) is the motor displacement, and \( \tau \) is the time constant. (Neglect the fluid compressibility.) The combined leakage coefficient for the pump and motor is \( \lambda \). (b) Determine from first principles the response of the system to a unit step input after a time equal to \( 3 \pi \). Calculate the value of \( \tau \). In a particular system the values are Load inertia, \( I=100 \mathrm{Nm} \mathrm{s}^{2} \) Leakage coefficient, \( \lambda=12 \times 10^{-3} \mathrm{l} / \mathrm{min} / \mathrm{bar} \) Motor displacement, \( d_{\mathrm{m}}=25 \mathrm{ml} \) per radian Pump flow constant, \( K_{n}=5 \times 10^{-3} \mathrm{~m}^{2} \mathrm{~s}^{-1} \) (c) Draw a polar plot for the frequency response of the system to an input \( \sin \omega t \) for values of \( \omega \) between 1 and 10 radians per second.