Free, Forced and Damped Oscillations — Revision Notes

⚡ 30-Second Revision

Free Oscillation: — No damping, no external force. Oscillates at $omega_0$. $x(t) = A cos(omega_0 t + phi)$. Energy conserved.
Damped Oscillation: — Damping force $F_d = -bv$. Amplitude decays exponentially: $A(t) = A_0 e^{-gamma t}$. $gamma = b/(2m)$. Damped frequency $omega_d = sqrt{omega_0^2 - gamma^2}$.

- Underdamped: $gamma < omega_0$, oscillates with decreasing amplitude. - Critically Damped: $gamma = omega_0$, fastest return to equilibrium without oscillation. - Overdamped: $gamma > omega_0$, slow return to equilibrium without oscillation.

Forced Oscillation: — External driving force $F_{ext} = F_0 cos(omega t)$. System oscillates at driving frequency $omega$.
Resonance: — Occurs when $omega approx omega_0$. Amplitude is maximum. Resonance frequency $omega_r = sqrt{omega_0^2 - 2gamma^2}$. At resonance, phase difference $delta = pi/2$.
Quality Factor (Q): — $Q = omega_0/(2gamma) = momega_0/b$. High Q means sharp resonance, low damping.

2-Minute Revision

Oscillations are repetitive motions. Free oscillations are ideal, undamped motions at a system's natural frequency ($omega_0$), with constant amplitude and conserved energy. In reality, damped oscillations occur due to dissipative forces (like friction), causing the amplitude to decay exponentially over time ($A(t) = A_0 e^{-gamma t}$).

The damping factor $gamma = b/(2m)$ determines the decay rate. Damping also slightly reduces the oscillation frequency to $omega_d = sqrt{omega_0^2 - gamma^2}$. Depending on damping strength, motion can be underdamped (oscillates), critically damped (fastest return to equilibrium without oscillation, e.

g., shock absorbers), or overdamped (slow return without oscillation).

Forced oscillations involve a continuous external periodic force driving the system. The system eventually oscillates at the driving frequency ($omega$). The amplitude of these oscillations depends on $omega$, $omega_0$, and damping.

Resonance is a critical phenomenon in forced oscillations where the amplitude becomes maximum when the driving frequency is close to the natural frequency ($omega_r = sqrt{omega_0^2 - 2gamma^2}$).

At resonance, the phase difference between force and displacement is $90^circ$. The Quality Factor (Q-factor), $Q = omega_0/(2gamma)$, quantifies the sharpness of the resonance peak; high Q means low damping and a very sharp, tall peak.

5-Minute Revision

Let's quickly review the core concepts of oscillations. At its heart, an oscillation is a repetitive motion around an equilibrium point. We categorize them based on external influences and energy dissipation.

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Free Oscillations: — This is the simplest case. A system, once displaced, oscillates under its own internal restoring forces (like a spring's elasticity or gravity for a pendulum). Ideally, there's no energy loss, so the amplitude remains constant, and it oscillates at its unique natural frequency ($omega_0$). For a spring-mass system, $omega_0 = sqrt{k/m}$; for a simple pendulum, $omega_0 = sqrt{g/L}$. Energy is conserved, continuously converting between kinetic and potential forms.

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Damped Oscillations: — Real-world systems always experience energy loss due to dissipative forces (e.g., air resistance, friction). These forces, often proportional to velocity ($F_d = -bv$), cause the amplitude of oscillation to decrease exponentially over time: $A(t) = A_0 e^{-gamma t}$, where $gamma = b/(2m)$ is the damping factor. Damping also slightly reduces the oscillation frequency to $omega_d = sqrt{omega_0^2 - gamma^2}$.

* **Underdamped ($gamma < omega_0$):** The system oscillates with decreasing amplitude (e.g., a swing slowing down). * **Critically Damped ($gamma = omega_0$):** The system returns to equilibrium as quickly as possible without oscillating (e.g., car shock absorbers). * **Overdamped ($gamma > omega_0$):** The system returns to equilibrium slowly without oscillating (e.g., a door closer that moves very sluggishly).

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Forced Oscillations: — Here, an external, periodic driving force ($F_{ext} = F_0 cos(omega t)$) continuously acts on the system. The system eventually settles into oscillating at the driving frequency $omega$. The amplitude of these forced oscillations depends on the driving frequency, the system's natural frequency, and the amount of damping.

The most significant phenomenon here is Resonance. This occurs when the driving frequency ($omega$) is equal or very close to the system's natural frequency ($omega_0$). At this point, the amplitude of oscillation becomes maximum because energy is transferred most efficiently from the driver to the oscillator.

The exact resonance frequency for maximum amplitude in a damped system is $omega_r = sqrt{omega_0^2 - 2gamma^2}$. At resonance, the displacement lags the driving force by $90^circ$ ($pi/2$ radians).

The Quality Factor (Q-factor), $Q = omega_0/(2gamma)$, quantifies the sharpness of the resonance peak. A high Q-factor means low damping, a very sharp and tall resonance peak, and high selectivity (responds strongly only to frequencies very close to $omega_0$). Low Q means high damping, a broad and short peak, and less selectivity.

Worked Mini-Example: A spring-mass system has $m=1,\text{kg}$, $k=100,\text{N/m}$, and damping coefficient $b=2,\text{Ns/m}$. Find its natural frequency, damping factor, and damped frequency.

Solution:

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Natural frequency: — $omega_0 = sqrt{k/m} = sqrt{100/1} = 10,\text{rad/s}$.
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Damping factor: — $gamma = b/(2m) = 2/(2 \times 1) = 1,\text{s}^{-1}$.
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Damped frequency: — $omega_d = sqrt{omega_0^2 - gamma^2} = sqrt{10^2 - 1^2} = sqrt{100 - 1} = sqrt{99} approx 9.95,\text{rad/s}$.

Since $gamma < omega_0$, it's an underdamped system, and $omega_d < omega_0$, as expected.

Prelims Revision Notes

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Free Oscillations:

* No external driving force, no damping. * Oscillates at its natural angular frequency $omega_0$. * For spring-mass: $omega_0 = sqrt{k/m}$. Period $T_0 = 2pisqrt{m/k}$. * For simple pendulum (small angles): $omega_0 = sqrt{g/L}$. Period $T_0 = 2pisqrt{L/g}$. * Amplitude is constant, mechanical energy is conserved.

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Damped Oscillations:

* Presence of dissipative forces (e.g., viscous damping $F_d = -bv$). * Amplitude decays exponentially: $A(t) = A_0 e^{-gamma t}$. * Damping factor $gamma = b/(2m)$. * Damped angular frequency $omega_d = sqrt{omega_0^2 - gamma^2}$.

Note $omega_d < omega_0$. * Types of Damping: * **Underdamped ($gamma < omega_0$):** Oscillates with decreasing amplitude. * **Critically Damped ($gamma = omega_0$):** Returns to equilibrium fastest without oscillation (e.

g., shock absorbers). * **Overdamped ($gamma > omega_0$):** Returns to equilibrium slowly without oscillation. * Mechanical energy is dissipated (converted to heat).

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Forced Oscillations:

* External periodic driving force $F_{ext}(t) = F_0 cos(omega t)$ is applied. * System oscillates at the driving angular frequency $omega$ in steady state. * Amplitude depends on $omega$, $omega_0$, and damping.

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Resonance:

* Occurs in forced oscillations when the driving frequency $omega$ is close to the natural frequency $omega_0$. * Amplitude of oscillation becomes maximum. * Resonance angular frequency $omega_r = sqrt{omega_0^2 - 2gamma^2}$.

For light damping, $omega_r approx omega_0$. * At resonance, the phase difference between driving force and displacement is $delta = pi/2$ (displacement lags force by $90^circ$). * Maximum amplitude at resonance is finite and inversely proportional to damping coefficient $b$.

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Quality Factor (Q-factor):

* $Q = omega_0/(2gamma) = momega_0/b = sqrt{mk}/b$. * Measures the sharpness of the resonance peak. High Q = low damping = sharp peak. * Also, $Q = 2pi \times \frac{\text{Energy stored}}{\text{Energy dissipated per cycle}}$.

Key Graphs:

Damped Oscillation (x vs t): — Exponentially decaying sinusoidal curve.
Forced Oscillation (Amplitude vs $omega$): — Shows a peak at $omega_r$. Higher damping means lower and broader peak.

Vyyuha Quick Recall

For Damped Forced Resonance: Free means no external push, Damped means dying out, Forced means a constant push, Resonance means the right push makes it HUGE!