As I posted, it is easier to work from the generic equation ##\theta=A\cos(\omega t+\phi)##. You can calculate ω from the coefficient in the differential equation ##(\ddot \theta+\omega^2\theta=0)##, and read off the amplitude and φ from the graph.

Is the coefficient of the θ variable always = ω^{2}?

If so, can't I just get ω^{2} = √27.513 (the coefficient in front of θ) = 5.245?

Is that the natural frequency value?

How do I go about reading the amplitude and φ from the graph anyway? A seems to be about 3.9 (?) but how do I read φ ? The sin wave seems to be shifted 5.5 rad to the right so is θ = 3.9sin(ω(0)-5.5)? (⇒ θ = 2.75 rad?

You could measure with a ruler, but by eye I would put the amplitude about 0.037 or 0.038.
For the phase, the first peak is at about ωt=0.8 or 0.9 radians. What value of φ would result in that?

Ah they cancel, that's cool! But does that only work for this particular example? With the θ' coefficient being = 0?

Basically, I'm asking if that's a way to calculate ω in every question, not just this one? Because most past exam questions ask for ω_{n}, so if I find the equation of motion, then will ω_{n} always be the coefficient of the "x" term (θ in this problem)?

On a different topic, I'm able to do most of the questions now, but I have just 2 last queries for you;

1. In Question 6(a) I derived the formula fine, but what are the units for ω_{n} in terms of E,I,L and m? I did it as ω_{n} = √(Gpa*mm^{4})/(mm^{3}*kg) = √(Gpa*mm)/kg....does that results in the correct units for the natural frequency? If not, what does?

2.In Question 6(c) how does the factor of safety come into play here? The "hint" with λ in it actually made me think I have no idea how to answer this correctly because none of my lecture notes have that info on modes or λ(c). Any ideas on how to approach this?

Thanks so much to you and @BvU for all your help so far, you really helped me take the edge off this difficult subject.

Yes, but don't forget the square root. If ##\ddot x + kx=0## then the angular frequency is ##\sqrt k##.
(If k is negative it is not SHM, since the system will accelerate away from the neutral position, like a pencil balanced on its point).

Careful with signs. Substitute that in the equation with ωt=0.9 to check.

It certainly produces the right dimension (1/time). Do you know dimensional analysis? But you need to check whether the units are s^{-1}.

This gets beyond my competence. Knowing the natural frequency etc. and the applied frequency and power, there must be some formula that tells you the maximum stress. Look up forced oscillations.

All this is saying is that the mode is the simplest possible: at any instant, the displacement at offset x is c(x)cos(ωt), where c(x) gives the shape of the beam under a static deflection.