Forced oscillations — driving frequency

1.6.11 · Physics › Oscillations & Waves

Forced oscillation KYA hoti hai?

System ${\omega }_d$ par oscillate kyun karne lagta hai? Do motions saath mein exist karti hain: ek transient (system ki khud ki damped wobble $\approx {\omega }_0$ par, jo $e^{-\gamma t}$ ki tarah khatam ho jaati hai) aur ek steady state (force dwara driven, kabhi khatam nahin hoti). Kaafi time ke baad sirf steady state bachti hai, aur frequency ${\omega }_d$ ki periodic force sirf usi frequency ${\omega }_d$ par response sustain kar sakti hai. System driver ka "gulam" ban jaata hai.

Steady-state amplitude derive kaise karein (scratch se)

Step 1 — Newton ka doosra niyam. $m\ddot{x}=-kx-b\dot{x}+F_0\cos ({\omega }_{d}t)$

Yeh step kyun? Hum sirf mass par act karne wali har force ko add karte hain. Spring $-kx$, drag $-b\dot{x}$, drive $+F_0\cos {\omega }_{d}t$.

Step 2 — Standardise karo. $m$ se divide karo aur ${\omega }_0^2=k/m$, $2\gamma =b/m$ define karo: $\ddot{x}+2\gamma \dot{x}+{\omega }_0^{2}x=\frac{F_0}{m}\cos ({\omega }_{d}t)$

Yeh step kyun? Constants ko ${\omega }_0$ aur $\gamma$ mein group karna algebra ko clean rakhta hai aur physics dikhata hai: natural frequency vs damping rate.

Step 3 — Steady-state form guess karo. Kyunki drive ${\omega }_d$ par periodic hai, try karo $x(t)=A\cos ({\omega }_{d}t-\varphi )$ jahan $A$ amplitude hai aur $\varphi$ force ke peechhe phase lag hai.

Yeh step kyun? ${\omega }_d$ par driven ek linear equation ka response ${\omega }_d$ par hi hona chahiye. Lag $\varphi$ isliye exist karta hai kyunki damping response ko instantly keep up karne se rokti hai.

Step 4 — Substitute karo. $\dot{x}=-A{\omega }_d\sin ({\omega }_{d}t-\varphi )$, $\ddot{x}=-A{\omega }_d^2\cos ({\omega }_{d}t-\varphi )$ use karo. Plug in karo aur demand karo ki yeh sab $t$ ke liye hold kare. Sabse clean route cosine aur sine components ko balance karna hai. Terms collect karne ke baad (ya complex exponentials $e^{i{\omega }_{d}t}$ use karke) milta hai:

$A=\frac{F_0/m}{\sqrt{({\omega }_0^2-{\omega }_d^2{)}^2+(2\gamma {\omega }_d{)}^2}}$

$\tan \varphi =\frac{2\gamma {\omega }_d}{{\omega }_0^2-{\omega }_d^2}$

Yeh step kyun? Denominator oscillator ki "impedance" hai. Term $({\omega }_0^2-{\omega }_d^2)$ drive aur natural frequency ke beech ka mismatch hai; term $2\gamma {\omega }_d$ damping cost hai. Bada mismatch → chhoti amplitude. Chhota mismatch → amplitude sirf damping se limited.

Amplitude curve padhna (Dual Coding)

Resonance frequency ($A$ ka peak): $A$ maximize karo ⇒ denominator minimize karo. $({\omega }_0^2-{\omega }_d^2{)}^2+(2\gamma {\omega }_d{)}^2$ ko ${\omega }_d$ ke w.r.t. differentiate karo: ${\omega }_{res}=\sqrt{{\omega }_0^2-2{\gamma }^2}$ Kyun? Damping peak ko ${\omega }_0$ se thoda neeche shift kar deta hai. Light damping ke liye ($\gamma \ll {\omega }_0$), ${\omega }_{res}\approx {\omega }_0$.

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

• Damped Oscillations — $-b\dot{x}$ term aur decaying transient provide karta hai.
• Simple Harmonic Motion — $F_0=0,b=0$ limit jo pure ${\omega }_0$ deta hai.
• Resonance and Quality Factor — peak ki sharpness $Q={\omega }_0/2\gamma$ se set hoti hai.
• Energy in Oscillations — power input ${\omega }_d={\omega }_0$ par maximum hota hai.
• Waves and Standing Waves — strings/air columns ki resonance disguise mein forced oscillation hai.
• Complex Exponential Method — Step 4 solve karne ka clean tool.

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