Governors

Governor vs flywheel

Terms

• Sensitiveness: A governor is sensitive if it responds to a small change of speed. It is measured as

$$\text{Sensitiveness} = \frac{N_2 - N_1}{N} = \frac{2(N_2 - N_1)}{N_1 + N_2}$$

where $N_1, N_2$ are the minimum and maximum equilibrium speeds and $N$ is the mean speed.

• Stability: A governor is stable if, for every speed in its working range, there is only one radius of the balls at which the sleeve is in equilibrium, and the ball radius increases with speed.
• Isochronism: The limiting case of a governor whose equilibrium speed is the same for all radii of rotation of the balls. Its range of speed is zero, so it is infinitely sensitive, but it is unstable (the balls move to an extreme position).
• Hunting: Continuous fluctuation of the engine speed above and below the mean caused by an over-sensitive governor: it overcorrects, the sleeve moves up and down, and the speed keeps oscillating.
• Effort: The mean force exerted at the sleeve for a given fractional change of speed (e.g. 1%). It is the force that can be used to move the control mechanism.
• Power: The work done at the sleeve for a given percentage change of speed; power $=$ (mean effort) $\times$ (lift of the sleeve).

Data: $l = 0.3$ m (arms and links equal, hung from the axis), $m = 5$ kg, $M = 25$ kg, $r_1 = 0.15$ m, $r_2 = 0.2$ m.

Equilibrium relation

For a Porter governor with arms and links of equal length meeting the axis ($\alpha = \beta$, so $q = \tan\beta/\tan\alpha = 1$):

$$\omega^2 = \frac{(m+M)\,g}{m\,h}, \qquad h = \sqrt{l^2 - r^2}$$

         O (pivot on axis)
        /|\
       / | \   arms l
      /  |h \
     B   |   B   balls m at radius r
      \  |  /   links l
       \ | /
        [M]  sleeve load

Lowest position ($r_1 = 0.15$ m)

$$h_1 = \sqrt{0.3^2 - 0.15^2} = 0.2598\ \text{m}$$ $$\omega_1^2 = \frac{(5+25)\times 9.81}{5 \times 0.2598} = 226.55 \Rightarrow \omega_1 = 15.05\ \text{rad/s},\quad N_1 = 143.7\ \text{rpm}$$

Highest position ($r_2 = 0.2$ m)

$$h_2 = \sqrt{0.3^2 - 0.2^2} = 0.2236\ \text{m}$$ $$\omega_2^2 = \frac{30 \times 9.81}{5 \times 0.2236} = 263.23 \Rightarrow \omega_2 = 16.22\ \text{rad/s},\quad N_2 = 154.9\ \text{rpm}$$

Range, lift and sensitiveness

$$\text{Range} = N_2 - N_1 = 154.9 - 143.7 = 11.2\ \text{rpm}$$ $$\text{Lift of sleeve} = 2(h_1 - h_2) = 2(0.2598 - 0.2236) = 0.0724\ \text{m} = 72.4\ \text{mm}$$ $$\text{Sensitiveness} = \frac{N_2 - N_1}{N_{mean}} = \frac{11.2}{149.3} = 0.0750$$

Answer: Speed range $= 143.7$ to $154.9$ rpm (11.2 rpm); sleeve lift $= 72.4$ mm; sensitiveness $= 0.0750$.

Data: $m = 2.5$ kg, ball arm $x = 120$ mm, sleeve arm $y = 100$ mm, $r_1 = 0.1$ m, $r_2 = 0.14$ m, $N_1 = 300$ rpm, $N_2 = 330$ rpm.

        ball (m)        x = 120
          O------+  fulcrum
                 |  y = 100
                 +--> sleeve --> spring force S

Centrifugal force on each ball

$$\omega_1 = \frac{2\pi \times 300}{60} = 31.42\ \text{rad/s}, \qquad \omega_2 = \frac{2\pi \times 330}{60} = 34.56\ \text{rad/s}$$ $$F_{c1} = m\omega_1^2 r_1 = 2.5 \times (31.42)^2 \times 0.1 = 246.7\ \text{N}$$ $$F_{c2} = m\omega_2^2 r_2 = 2.5 \times (34.56)^2 \times 0.14 = 418.0\ \text{N}$$

Spring force

Taking moments about the fulcrum of the bell crank (per ball, the sleeve force is $S/2$ for each of the two levers):

$$F_c \times x = \frac{S}{2}\times y \Rightarrow S = \frac{2F_c\,x}{y}$$ $$S_1 = \frac{2 \times 246.7 \times 120}{100} = 592\ \text{N}, \qquad S_2 = \frac{2 \times 418.0 \times 120}{100} = 1003\ \text{N}$$

Sleeve lift and stiffness

The sleeve moves in proportion to the ball movement:

$$\text{Lift} = (r_2 - r_1)\frac{y}{x} = (0.14 - 0.10)\times\frac{100}{120} = 0.0333\ \text{m} = 33.3\ \text{mm}$$ $$s = \frac{S_2 - S_1}{\text{lift}} = \frac{1003 - 592}{0.0333} = 12329\ \text{N/m} = 12.33\ \text{N/mm}$$

Initial compression

$$\delta_0 = \frac{S_1}{s} = \frac{592}{12329} = 0.0480\ \text{m} = 48.0\ \text{mm}$$

Answer: $S_1 = 592$ N; $S_2 = 1003$ N; spring stiffness $= 12.33$ N/mm; initial compression $= 48.0$ mm.