Find the natural frequency?

Question

For the system u"+3u=4sin(ωt), find the natural frequency, the value of ω that produces resonance and a value of ω that produces beats.

natural frequency=sqrt(3)

How do I find the beats?

Answer 1

The general solution of this equation is

                           u = A sin (ω₀t) + B cos(ω₀t) + p sin(ωt)

where psin(ωt)  is a particular solution of the inhomogeneous equation. You can make sure that

                                                        p = 4/(3 - ω²)

while 3 = ω₀² - square of natural frequency.

The frequaency of beats is the difference between natural frequency and  the frequaency of external force (ω). Thus it is equal to

                                                             Δω = √3 - ω.