Root locus of K/[s(s²+10s+24)] at ζ = 0.7

Result: at $\zeta=0.7$, $K \approx 28.4$, with $\omega_n \approx 1.98$ rad/s and $\omega_d \approx 1.42$ rad/s.

The root locus is sketched with the standard rules. The $\zeta = 0.7$ line is then drawn from the origin, and the magnitude condition at its intersection with the locus gives $K$.

$$G(s)H(s) = \frac{K}{s(s^2+10s+24)} = \frac{K}{s(s+4)(s+6)}$$

Step 1: Poles and zeros

Poles: $0,\ -4,\ -6$ ($n=3$). No zeros ($m=0$), so 3 branches go to infinity.

Step 2: Real-axis segments

Between $0$ and $-4$, and from $-6$ to $-\infty$.

Step 3: Asymptotes

$$\theta = \frac{(2q+1)180^\circ}{3} = 60^\circ,\ 180^\circ,\ 300^\circ, \qquad \sigma_A = \frac{0-4-6}{3} = -3.33$$

Step 4: Breakaway point

$$\begin{aligned} K &= -(s^3+10s^2+24s) \\ \frac{dK}{ds} &= -(3s^2+20s+24) = 0 \\ s &= \frac{-20\pm\sqrt{400-288}}{6} = -1.57 \ \text{or}\ -5.10 \end{aligned}$$

$s=-5.10$ is not on the locus. So the breakaway point is $s=-1.57$, where $K = |s||s+4||s+6| = 16.9$.

Step 5: Imaginary-axis crossing

Characteristic equation: $s^3+10s^2+24s+K=0$

$K = 240$ makes the $s^1$ row zero, and $10s^2+240=0$ gives $s = \pm j4.90$. The system is stable for $0<K<240$.

Step 6: Point with $\zeta = 0.7$

The constant-$\zeta$ line makes angle $\theta = \cos^{-1}0.7 = 45.57^\circ$ with the negative real axis. Points on it are $s = -0.7\omega_n + j0.714\omega_n$. Using the angle condition $\angle s + \angle(s+4) + \angle(s+6) = 180^\circ$ (solved by trial):

$$\omega_n = 1.984\ \text{rad/s}, \qquad s = -1.389 \pm j1.417$$

Magnitude condition:

$$K = |s|\,|s+4|\,|s+6| = 1.984 \times 2.971 \times 4.824 = 28.4$$

The third closed-loop pole is then at $s \approx -7.22$, far to the left, so the complex pair is dominant.

Sketch

 Upper half shown; lower half is its mirror image.

                            jw      /
                             |     / 60 deg asymptote
                             *    /  (from -3.33)
                            /| j4.90 (K=240)
                           / |
                          /  |
     zeta=0.7 point -->  *   |  -1.39+j1.42 (K=28.4)
                         |.  |
                         | . |   . = zeta line
                         |  .|
  <====x------x==========o===x----> sigma
      -6     -4       -1.57  0
                     (K=16.9)

Answer: at $\zeta = 0.7$: $K \approx 28.4$, natural frequency $\omega_n \approx 1.98$ rad/s (damped frequency $\omega_d = \omega_n\sqrt{1-\zeta^2} \approx 1.42$ rad/s). Closed-loop dominant poles are at $-1.39 \pm j1.42$.