Expression for Steady state of Forced vibration [duplicate]

Question

In my book under the topic Steady state of the forced oscillator, they started with the equation: $$\frac{d^2x}{dt^2}+γ\frac{dx}{dt}+ω_0^2x=fe^{jωt}$$

I know the equation for damped oscillation but it has no imaginary part in the equation we begin with.

Please explain how this equation is formed.

Also specifically in this equation they did not explain the expression for γ and $ω_0$( Like for damped oscillation $γ=\frac{p}{m}$ and $ω_0=\sqrt{k/m}$ where p is damping force per unit velocity, k is restoring force per unit displacement and m is the mass of the oscillator.)

Answer 1

They are using a standard mathematical tool, where complex numbers can aid the solution of a problem which concerns real numbers.

Let $s$ be the position of a driven oscillator. If the driving force is periodic then we can write the equation of motion $$ \frac{d^2s}{dt^2}+γ\frac{ds}{dt}+ω_0^2s=f \cos(ωt). \tag{1} $$ Now introduce a complex number $x$, such that $s$ is the real part of $x$, and consider the equation $$ \frac{d^2x}{dt^2}+γ\frac{dx}{dt}+ω_0^2x=f e^{jωt}. \tag{2} $$ You can confirm that equation (1) is the real part of equation (2).

Now whenever we have an equation involving complex numbers, both the real part and the imaginary part of the equation have to be satisfied. It follows that the real part of the solution to equation (2) will be a solution to equation (1).

So the method is to first solve equation (2), and then you have a complex number $x(t)$. Take the real part of this complex number and you have $s(t)$ which is what you wanted to know.

(This issue has been covered in previous Q/A on this site but I decided to write an answer anyway.)