12 Damped harmonic oscillator

Let $t$ be time and $y$ displacement from rest of a mass on a spring. The velocity is $y^{\prime }$ and acceleration $y^{\prime \prime }$, the mass is driven by an external driving force $u(t)$. The equation is

$$y^{\prime \prime }+\beta y^{\prime }+\gamma y=u$$

where $\beta$ and $\gamma$ are constants. This models the suspension of a car moving along a road. Here $u(t)$ represents the force imparted on the car by the road; the suspension involves a spring which gives a restoring force $-\gamma y$, while the shock absorbers give a damping force $-\beta y^{\prime }$. We can write this as a feedback system, using the formula

$$y=-\beta \int y-\gamma \int \int y+\int \int u.$$

We often choose input $u(t)=e^{i\omega t}=\cos (\omega t)+i\sin (\omega t)$, with angular frequency $\omega$.