Damped oscillations — underdamped, critically damped, overdamped

1.6.9 · Physics › Oscillations & Waves

HUM KYA DESCRIBE KAR RAHE HAIN?

Velocity-proportional kyun? Kyunki viscous drag (ek body jo slowly fluid/air mein move karti hai) sach mein speed ke saath scale karta hai — honey mein dheema motion jitna tez push karo utna zyada resist karta hai. Yeh sabse simple aur physically common case hai, isliye hum yahan se shuru karte hain.

EQUATION OF MOTION DERIVE KAISE KAREIN

Newton ka second law: $m\ddot{x}=\sum F$. Forces hain spring $-kx$ aur damping $-b\dot{x}$:

$m\ddot{x}=-kx-b\dot{x}$

Sab kuch ek side le jao aur $m$ se divide karo:

$\ddot{x}+\frac{b}{m}\dot{x}+\frac{k}{m}x=0$

Yeh step kyun? $m$ se divide karne par do physically meaningful frequencies saamne aati hain. Hum define karte hain:

$2\gamma$ mein factor 2 kyun? Pure convenience — yeh baad ke square roots ko clean banata hai ($\sqrt{{\gamma }^2-{\omega }_0^2}$ mein koi extra factor nahi).

Exponentials guess karke solve karo

Ek linear constant-coefficient ODE ko $x=e^{\lambda t}$ se solve kiya jata hai. Yeh guess kyun? Kyunki differentiation sirf $e^{\lambda t}$ ko $\lambda$ se multiply kar deta hai, calculus ko algebra mein badal deta hai. Substitute karo:

${\lambda }^2+2\gamma \lambda +{\omega }_0^2=0\text{(characteristic equation)}$

Quadratic formula:

$\lambda =-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}$

Sab kuch ${\gamma }^2-{\omega }_0^2$ ke sign par depend karta hai — yaani discriminant negative, zero, ya positive hai ya nahi. Woh ek sign teen regimes create karta hai.

Teen regimes

Oscillation kyun? $\lambda$ ka imaginary part ek oscillation hai (Euler: $e^{i{\omega }_{d}t}$ = sines/cosines); real part $-\gamma$ decay hai. Spring abhi bhi "wobble" ki ladai jeet raha hai, lekin har swing chhoti hoti jati hai.

Dheera kyun? Slower root ${\lambda }_1=-\gamma +\sqrt{{\gamma }^2-{\omega }_0^2}$ magnitude mein chhota hai (0 ke close), isliye woh term kaafi der tak rehta hai. Heavy damping system ko sluggish bana deta hai, jaise ek door closer jo bahut tight set ho.

Extra $t$ kyun? Repeated root ke saath, $e^{\lambda t}$ akele sirf EK solution deta hai, lekin ek 2nd-order ODE ko DO chahiye. $t$ se multiply karne par genuinely independent doosra solution milta hai (tum verify kar sakte ho ki yeh ODE satisfy karta hai).

Critical damping special/useful kyun hai? Yeh borderline hai. Thoda bhi kam damping ⇒ overshoot/wobble; thoda bhi zyada ⇒ sluggish. Isliye yeh sabse tez non-oscillatory settling hai — exactly wahi jo car suspensions, analog meter needles, aur door closers ke liye chahiye.

Energy & "kitna damped hai?" — Quality factor

Underdamped oscillator mein energy decay hoti hai kyunki amplitude $\propto e^{-\gamma t}$ aur energy $\propto$ amplitude²:

$E(t)=E_0{e}^{-2\gamma t}$

2 kyun? Energy $\sim \frac{1}{2}k{A}^2$, aur $A^2\to (e^{-\gamma t}{)}^2=e^{-2\gamma t}$.

Worked examples

Common mistakes (steel-manned)

Flashcards

Connections

• Simple Harmonic Motion — $\gamma =0$ limit (${\omega }_d\to {\omega }_0$, koi decay nahi).
• Forced Oscillations & Resonance — ek driving force add karo; damping resonance sharpness set karta hai ($Q$).
• Second-order linear ODEs — yahan use ki gayi characteristic-root method.
• Quality factor & bandwidth — $Q$ kaise damping ko resonance width se link karta hai.
• RLC circuits — electrical twin: $L\ddot{q}+R\dot{q}+q/C=0$, same teen regimes.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps