For an assignment in one of my maths units at uni, I've been asked to derive and solve the differential equation of motion for a forced harmonic oscillator, with the forcing function having the form $F_0sin(\omega t)$, with some given properties (mass = 1kg, spring constant = 400N/m, amplitude of the sinusoidal driving force = 20N).
I am reasonably confident of the solution I found;
$$x(t) = -0.0245\cos(20t) + \frac{\omega}{\omega^2 - 400}\sin(20t) + \frac{20}{400 - \omega^2}\sin(wt) + 0.0245$$
Next the question tells us that the spring will fail if its extension exceeds 1 metre, and asks us what forcing frequencies will allow for safe oscillations.
I formulated this condition as;
$$1 \geq |-0.0245\cos(20t) + \frac{\omega}{\omega^2 - 400}\sin(20t) + \frac{20}{400 - \omega^2}\sin(wt)+0.0245|$$
Obviously this condition needs to hold for all t, so it seems to me that I need only look at the sum of the amplitudes, however $\cos(20t)$ and $\sin(20t)$ will never superimpose entirely constructively, and I cannot find an adequate way of expressing them as one trig function.
How should I approach this?
