# Damped Forced Oscillator with initial conditions

The equation of motion of a damped forced oscillator is;

$$\ddot{x}(t)+\gamma\dot{x}(t)+\omega_0^2x(t)=F(t),$$

$$F(t) = F_0 \cos(\omega_dt);$$ also for the purpose of this problem we may set $$\omega_d=\omega_0=\omega$$, just not inside cos and sine arguments.

and $$x(t)=A(\omega_d)\cos(\omega_dt-\delta),$$ which is a steady-state solution. From my understanding, it is a particular solution only.

We need to find a full sol. ($$x(t)=x_\text{characteristic} + x_\text{particular}$$) to a damped forced oscillator with initial conditions, $$x(0)=x'(0)=0$$.

Here's my result;

$$𝑥(t)=\frac{F_0}{r\omega}𝑒^{−\gamma 𝑡/2}\sin(\omega 𝑡) + \frac{F_0}{r\omega}\cos\left(\omega _d 𝑡 + \frac{\pi}{2}\right)$$

Now, I want to calculate the total power on the system and integrate the result for further interpretation.

I know that power = force $$\times$$ velocity

Power due to external/driving force: $$P = F(t) \cdot \frac{dx}{dt}$$

Power due to damping force: $$P = −\gamma \left(\frac{dx}{dt}\right)^2$$

But which $$x(t)$$ sol. do I use for $$\frac{dx}{dt}$$?

• What's the problem? You have $x(t)$, so $P(t)=x'(t)\times F(t)$. – Gert Oct 13 at 23:17
• Yes, but do I use x(t) - the steady-state solution? Or should I use the solution I obtained through applying initial conditions? – Laura Oct 13 at 23:59
• There is only one $x(t)$ and that's the one to use. Your 4th equation. – Gert Oct 14 at 2:29
• What about the steady-state solution, i.e. the third equation? Would I use that in the absence of initial conditions to calculate the power? – Laura Oct 14 at 12:10
• If you want to know $P(t)$ at steady state only, then yes. – Gert Oct 14 at 15:27