Question 10.4.1: Figure 10.4.3 shows a hydraulic implementation of proportion......
Figure 10.4.3 shows a hydraulic implementation of proportional action to control the angle of an aircraft rudder, elevator, or aileron (see [Cannon, 1967]). The input motion y is produced by the motion of the pilot’s control stick acting through cables. Analyze its motion assuming that the rudder inertia is small, and show that the system gives proportional action.
When the input motion y occurs, the beam pivots about its lower end. For small motions x = L_3θ and the beam geometry is such that
z=\frac{L_1 + L_2}{L_2} y-\frac{L_1}{L_2} x (1)
The proof of this equation for small angular motions is as follows. First, suppose that x is fixed to be 0. Then from similar triangles,
\frac{z}{L_1 + L_2}=\frac{y}{L_2} \quad \text { or } \quad z=\left(L_1+L_2\right) \frac{y}{L_2}Now suppose that y is fixed to be 0. From similar triangles,
\frac{z}{L_1}=-\frac{x}{L_2} \quad \text { or } \quad z=-L_1 \frac{x}{L_2}Since the general motion is the superposition of the two individual motions, adding these two relations gives equation (1).
For the servomotor, as shown in Example 7.4.7, if the rudder inertia is small,
These equations give the following transfer function.
\frac{\Theta(s)}{Y(s)}=\frac{L_1 + L_2}{L_1 L_3} \frac{1}{\tau s + 1}where
\tau=\frac{L_2 A}{L_1 C_1}If the servomotor gain C_1/A is large, then τ ≈ 0, and the transfer function becomes
\frac{\Theta(s)}{Y(s)}=\frac{L_1 + L_2}{L_1 L_3} \equiv K_PThus the system implements proportional control with proportional gain K_P if the motion y represents the error signal and θ represents the controller output.