Question 7.1.3: Figure 7.1.4a shows a cylinder and piston connected to a loa......
Figure 7.1.4a shows a cylinder and piston connected to a load mass m, which slides on a frictionless surface. Part (b) of the figure shows the piston rod connected to a rack-and-pinion gear. The pressures p_{1} and p_{2} are applied to each side of the piston by two pumps. Assume the piston rod diameter is small compared to the piston area, so the effective piston area A is the same on both sides of the piston. Assume also that the piston and rod mass have been lumped into m and that any friction is negligible.(a) Develop a model of the motion of the displacement x of the mass in part (a) of the figure, assuming that p_{1} and p_{2} are given functions of time. Also, obtain the expression for the mass flow rate that must be delivered or absorbed by the two pumps. (b) Develop a model of the displacement x in part (b) of the figure. The inertia of the pinion and the load connected to the pinion is I.
a. Assuming that p_{1} > p_{2}, the net force acting on the piston and mass m is (p_{1} − p_{2})A, and thus from Newton’s law,
m{\ddot{x}}=(p_{1}-p_{2})ABecause p_{1} and p_{2} are given functions of time, we can integrate this equation once to obtain the velocity:
{\dot{x}}(t)={\dot{x}}(0)+{\frac{A}{m}}\int_{0}^{t}[p_{1}(u)-p_{2}(u)]\,d uThe rate at which fluid volume is swept out by the piston is A{\dot{x}}, and thus if {\dot{x}} > 0, the pump providing pressure p_{1} must supply fluid at the mass rate \rho A\dot{x}., and the pump providing pressure p_{2} must absorb fluid at the same mass rate.
b. Because we want an expression for the displacement x, we obtain an expression for the equivalent mass of the rack, pinion, and load. The kinetic energy of the system is
\mathrm{KE}={\frac{1}{2}}m\dot{x}^{2}+{\frac{1}{2}}{I}\dot{\theta}^{2}={\frac{1}{2}}\left(m+{\frac{I}{R^{2}}}\right)\dot{x}^{2}because R{\dot{\theta}}={\dot{x}}.
Thus the equivalent mass is
m_{e}=m+\frac{I}{R^{2}}The required model can now be obtained by replacing m with me in the model developed in part (a).