Harmonic Oscillator
Mechanical Oscillator:
To derive the equation for the displacement of a mechanical harmonic oscillator in the steady state, we can start with the equation of motion for the oscillator. Assuming there is a driving force (F) acting on the oscillator, the equation of motion can be written as:
\[m \frac{{d2 x}}{{dt2}} + r \frac{{dx}}{{dt}} + kx = F(t)\]
When:
\[F(t) = F_0 \cos(\omega t)\]
Where:
- (m) is the mass of the oscillator,
- (r) is the damping coefficient,
- (k) is the spring constant of the oscillator,
- (x) is the displacement of the oscillator as a function of time,
- (F(t)) is the driving force applied to the oscillator, which is (F_0 \cos(\omega t)) in this case.
To find the steady-state solution, we assume that the oscillator has reached a stable condition, where its motion is periodic with the same frequency as the driving force. Therefore, we can write the displacement (x) as:
\[x(t) = A \cos(\omega t + \phi)\]
Where:
- (A) is the amplitude of the steady-state motion,
- (\(\phi\)) is the phase angle.
We can differentiate this equation twice to find the corresponding velocities and accelerations:
\[\dot{x}(t) = -A \omega \sin(\omega t + \phi)\]
\[\ddot{x}(t) = -A \omega^2 \cos(\omega t + \phi)\]
Substituting these expressions into the equation of motion, we get:
\[-m A \omega^2 \cos(\omega t + \phi) - r A \omega \sin(\omega t + \phi) + k A \cos(\omega t + \phi) = F_0 \cos(\omega t)\]
Now, we equate the coefficients of the trigonometric functions on both sides of the equation:
\[\cos(\omega t + \phi) \left(-m \omega2 + k\right) - \sin(\omega t + \phi) (r \omega) = F_0 \cos(\omega t)\]
Since the cosine functions on both sides of the equation must be equal for all values of (t), we can equate the coefficients:
\[-m \omega^2 + k = F_0\]
\[-r \omega = 0\]
From the second equation, we get (r = 0), which implies there is no damping in the system. This assumption is commonly made when discussing steady-state solutions.
Simplifying the first equation, we obtain:
\[\omega^2 = \frac{k}{m} = \omega_0^2\]
Where (\(\omega_0\)) is the natural angular frequency of the oscillator.
\[\omega _0=\sqrt\frac {k}{m}\]
Finally, we can rewrite the equation as:
\[\omega = \sqrt{\omega_0^2 - \omega^2}\]
Solving for (\(\omega\)) gives:
\[\omega^2 = \frac{F_0}{m}\sqrt{\frac{1}{(\omega_0^2 - \omega^2)^2}}\]
Taking the inverse square root of both sides and rearranging, we arrive at:
\[\frac{1}{\omega^2 - \omega_0^2} = \frac{m}{F_0}\sqrt{(\omega^2 - \omega_0^2)^2}\]
Simplifying further, we get:
\[\frac{1}{\omega^2 - \omega_0^2} = \frac{m}{F_0}\left| \omega^ 2 - \omega_0^2\right|\]
Multiplying both sides by \(\omega^2 - \omega_0^2\), we obtain:
\[1 = \frac{m}{F_0}\left| \omega^2 - \omega_0^2\right|(\omega^2 - \omega_0^2)\]
Simplifying and rearranging, we arrive at the following equation for the amplitude ((A) in the steady state:
\[x_0 = \frac{F_0}{m\sqrt{(\omega^2 - \omega_0^2)^2 + \left(\frac{r}{m}\right)^2 \omega^2}}\]
Note that the derivation assumes no damping ((r = 0)), and the amplitude of the driving force (\(F_0\)) is the same as the amplitude of the steady-state motion.
Damping:
Undamped, underdamped, and overdamped are terms used to describe the behavior of a second-order system, such as an RLC circuit or a harmonic oscillator, in response to a disturbance or input. These terms refer to the damping characteristics of the system and how they affect its response.
The behavior of a system can be classified as undamped, underdamped, or overdamped based on the damping factor, which is determined by the ratio of the actual damping to the critical damping. The critical damping represents the minimum amount of damping required to prevent oscillations.