Chapter 5, Problem 32P

The be vibrations of the body of a helicopter due to the periodic force applied by the rotation of the rotor can be modeled by a frictionless spring-mass-damper system subjected to an external periodic force. The position x (t) of the mass is given by the equation:

$x(t)=\frac{2f_0}{{\omega }_n^2-{\omega }^2}\sin \left(\frac{{\omega }_n-\omega }{2}t\right)\sin \left(\frac{{\omega }_n-\omega }{2}t\right)$ where $F\left(t\right)=F_0\sin \omega t$ , and f₀ = F₀/m, $\omega$ is the frequency of the applied force, and ${\omega }_n$ , is the natural frequency of the helicopter. When the value of $\omega$ is close to the value of ${\omega }_n$ , the vibration consists of fast oscillation with slowly changing amplitude called beat. Use F₀/m = 12 N/kg, ${\omega }_n=10\text{rad/s}$ , and $\omega =12\text{rad/s}$ to plot x (t) as a function of t for 0 = t = 10 s.