so I got this question which is asking about a bridge undergoing resonance when people walk on it (with no damping). The force from people walking is modeled by the function $A\cos\left(\frac{t}{2\pi}\right)$. The mass of the people and the bridge is $m = 1$.
Its undamped harmonic oscillation is modeled by the differential equation:
$$ y'' + ky = A\cos\left(\frac{t}{2\pi}\right) $$
where
- $y(0) = 0$ and
- $A >0$ and
- $k > 0$
We're asked to find the 'natural frequency' of the bridge (i.e when there are no people on the bridge).
How would I find the natural frequency? When I solve the equation and sub it back in everything ends up canceling out and I can't find a value of $\omega$.
When the bridge is idle we have
$$ y''+ky = 0 $$
which has general solution in the form
$$ y = C_1 \cos(\sqrt k t)+C_2 \sin(\sqrt k t) $$
the natural frequency graphic can be build having in mind the concept of transference function
$$ y''+k y = u\Rightarrow (k-\omega^2)Y(j\omega) = U(j\omega)\Rightarrow G(j\omega) = \frac{Y(j\omega)}{U(j\omega)} = \frac{1}{k-\omega^2}\Rightarrow |G(j\omega)| = \frac{1}{|k-\omega^2|} $$
so the natural resonance angular frequency is $\omega = \sqrt k$
NOTE
Here $Y(s) = \mathcal{L}(y(t))$ is the Laplace transform.