Forced Harmonic Oscillator — Obtaining the Solution
A forced harmonic oscillator is a system that oscillates under the influence of both
a restoring force and an external periodic driving force.
Typical examples include an electric circuit driven by an AC source,
or a pendulum driven by a periodic torque.
1. Equation of Motion
Consider a mass \( m \) attached to a spring of stiffness \( k \),
subjected to a damping force proportional to velocity, and driven
by an external periodic force \( F_0 \cos(\Omega t) \).
The net force is given by:
\[
F_{\text{net}} = -kx - c\dot{x} + F_0 \cos(\Omega t)
\]
Applying Newton’s second law:
\[
m\ddot{x} + c\dot{x} + kx = F_0 \cos(\Omega t)
\]
Dividing through by \( m \):
\[
\ddot{x} + 2\beta\dot{x} + \omega_0^2 x = f_0 \cos(\Omega t)
\]
where:
\[
\omega_0 = \sqrt{\frac{k}{m}}, \quad 2\beta = \frac{c}{m}, \quad f_0 = \frac{F_0}{m}
\]
2. General Solution
The total solution of this differential equation consists of two parts:
\[
x(t) = x_h(t) + x_p(t)
\]
where:
- \( x_h(t) \): solution of the homogeneous (transient) equation
- \( x_p(t) \): particular (steady-state) solution due to the driving force
3. Homogeneous Solution (Transient Part)
The homogeneous equation is:
\[
\ddot{x} + 2\beta\dot{x} + \omega_0^2 x = 0
\]
which is the same as the damped oscillator equation.
Its solution depends on the damping condition:
\[
x_h(t) = A e^{-\beta t} \cos(\omega_d t + \phi)
\]
where \( \omega_d = \sqrt{\omega_0^2 - \beta^2} \).
This term decays with time due to the exponential factor \( e^{-\beta t} \)
and eventually vanishes, leaving only the steady-state solution.
4. Particular Solution (Steady-State Response)
To find the steady-state response, assume a trial solution of the form:
\[
x_p(t) = X \cos(\Omega t - \delta)
\]
where:
- \( X \): amplitude of forced oscillation
- \( \delta \): phase difference between the driving force and displacement
Substitute this trial solution into the equation of motion:
\[
-\Omega^2 X \cos(\Omega t - \delta)
+ 2\beta\Omega X \sin(\Omega t - \delta)
+ \omega_0^2 X \cos(\Omega t - \delta)
= f_0 \cos(\Omega t)
\]
Using trigonometric identities and comparing coefficients of \( \cos(\Omega t) \)
and \( \sin(\Omega t) \), we obtain two equations:
\[
X(\omega_0^2 - \Omega^2) = f_0 \cos\delta
\quad \text{and} \quad
2\beta\Omega X = f_0 \sin\delta
\]
5. Amplitude of Steady-State Oscillation
Squaring and adding the two equations:
\[
X^2 \big[ (\omega_0^2 - \Omega^2)^2 + (2\beta\Omega)^2 \big] = f_0^2
\]
Hence,
\[
\boxed{X = \frac{f_0}{\sqrt{(\omega_0^2 - \Omega^2)^2 + (2\beta\Omega)^2}}}
\]
This shows that \( X \) depends on both the driving frequency \( \Omega \)
and the damping constant \( \beta \).
6. Phase Difference
The phase lag \( \delta \) is obtained from:
\[
\tan\delta = \frac{2\beta\Omega}{\omega_0^2 - \Omega^2}
\]
- For \( \Omega \ll \omega_0 \): \( \delta \approx 0 \) → displacement in phase with force.
- For \( \Omega = \omega_0 \): \( \delta = \pi/2 \).
- For \( \Omega \gg \omega_0 \): \( \delta \approx \pi \) → displacement out of phase with force.
7. Resonance
The amplitude \( X \) reaches a maximum when the denominator is minimum,
i.e. when:
\[
\frac{dX}{d\Omega} = 0
\]
This occurs approximately at:
\[
\boxed{\Omega_r = \sqrt{\omega_0^2 - 2\beta^2}}
\]
known as the resonance frequency.
For weak damping (\( \beta \ll \omega_0 \)), \( \Omega_r \approx \omega_0 \).
8. Velocity and Energy of Forced Oscillator
The velocity of oscillation is:
\[
v(t) = -\Omega X \sin(\Omega t - \delta)
\]
and the average power absorbed from the driving force is:
\[
\langle P \rangle = \frac{1}{2} F_0 \Omega X \sin\delta
= \frac{F_0^2}{2m} \frac{2\beta\Omega^2}
{(\omega_0^2 - \Omega^2)^2 + (2\beta\Omega)^2}
\]
The power absorbed is maximum when \( \Omega = \omega_0 \),
that is, at resonance.
9. Summary Table
| Quantity |
Expression |
| Equation of motion |
\( \ddot{x} + 2\beta\dot{x} + \omega_0^2 x = f_0 \cos(\Omega t) \) |
| Steady-state amplitude |
\( X = \dfrac{f_0}{\sqrt{(\omega_0^2 - \Omega^2)^2 + (2\beta\Omega)^2}} \) |
| Phase lag |
\( \tan\delta = \dfrac{2\beta\Omega}{\omega_0^2 - \Omega^2} \) |
| Resonance frequency |
\( \Omega_r = \sqrt{\omega_0^2 - 2\beta^2} \) |
| Average power absorbed |
\( \langle P \rangle = \dfrac{F_0^2}{2m} \dfrac{2\beta\Omega^2}
{(\omega_0^2 - \Omega^2)^2 + (2\beta\Omega)^2} \) |
10. Physical Interpretation
- At low frequencies, the mass follows the driving force almost exactly.
- Near resonance, the system absorbs maximum energy.
- At high frequencies, inertia dominates and displacement is nearly opposite in phase.
- Damping reduces both the peak amplitude and sharpness of resonance.
Few questions like question 5 is out of Syllabus.
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