K for sustained oscillation of k/[s(s+3)(s²+s+1)]

Sustained oscillation occurs at the marginal value of $k$, where the $s^1$ row of the Routh array is zero. The auxiliary equation from the $s^2$ row then gives the frequency.

Characteristic equation

$$\begin{aligned} s(s+3)(s^2+s+1) + k &= 0 \\ s^4 + 4s^3 + 4s^2 + 3s + k &= 0 \end{aligned}$$

Routh array

For stability: $3-\frac{16k}{13}>0$ and $k>0$, so $0 < k < \frac{39}{16} = 2.4375$.

Value of k for sustained oscillation

$$3 - \frac{16k}{13} = 0 \Rightarrow k = \frac{39}{16} = 2.4375$$

Oscillation frequency

Auxiliary equation:

$$\begin{aligned} \frac{13}{4}s^2 + k &= 0 \\ \frac{13}{4}s^2 + \frac{39}{16} &= 0 \\ s^2 &= -\frac{3}{4} \Rightarrow s = \pm j0.866 \end{aligned}$$

Check: at $k=2.4375$ the roots are $\pm j0.866$, $-2.866$ and $-1.134$.

Answer: $k = 2.4375$ gives sustained oscillation at $\omega = 0.866$ rad/s.