Worked examples — Forced oscillations — driving frequency

1.6.11 · D3 · Physics › Oscillations & Waves › Forced oscillations — driving frequency

Shuru karne se pehle, yahan sirf do tools hain jo humein chahiye, copy karke likhe hain taaki kuch assume na ho:

Do symbols ko abhi plain words mein samjhana zaroori hai use karne se pehle:

Ek aur quantity Example 2 mein aati hai, toh chaliye usse abhi plain words mein pin down karte hain:

---

Scenario matrix

Is topic ka har question inhi cells mein se kisi ek mein fit hota hai. Neeche har worked example ka tag us cell ke saath hai jo woh fill karta hai.

Cell	Kya vary karta hai	Kya tricky hai	Example
A. Slow drive (${\omega }_d\ll {\omega }_0$)	limiting value	Hooke's law $A\to F_0/k$ tak reduce hona chahiye	Ex 1
B. Exact resonance denominator (${\omega }_d={\omega }_0$)	mismatch term = 0	sirf damping bachti hai; $\varphi =90^{\circ }$	Ex 2
C. True amplitude peak (${\omega }_d={\omega }_{res}$)	peak ${\omega }_0$ se neeche hai	$\sqrt{{\omega }_0^2-2{\gamma }^2}$ use karna padega	Ex 3
D. Phase below resonance (${\omega }_d<{\omega }_0$)	${\omega }_0^2-{\omega }_d^2>0$	$\tan \varphi >0$, $\varphi$ acute	Ex 4
E. Phase above resonance (${\omega }_d>{\omega }_0$)	${\omega }_0^2-{\omega }_d^2<0$	$\tan \varphi <0$ — quadrant fix zaroori	Ex 5
F. Fast drive (${\omega }_d\gg {\omega }_0$)	limiting value	inertia jeet jaata hai, $A\to F_0/(m{\omega }_d^2)$, $\varphi \to 180^{\circ }$	Ex 6
G. Zero damping ($\gamma =0$)	degenerate input	${\omega }_0$ par $A\to \infty$; $\varphi$ $0\to 180^{\circ }$ jump karta hai	Ex 7
H. Real-world word problem	modelling	words ko symbols mein translate karo	Ex 8
I. Exam twist (${\omega }_d$ nikalo diye gaye $A$ se)	inverse problem	root ke andar chhupe unknown ke liye solve karo	Ex 9

Hum number-crunching ke liye ek base system use karte hain taaki results comparable rahein: $m=0.5\text{ kg}$, $k=200\text{ N/m}$, $b=2\text{ kg/s}$, $F_0=10\text{ N}$. Toh ${\omega }_0=\sqrt{200/0.5}=20\text{ rad/s}$ aur $2\gamma =b/m=4\Rightarrow \gamma =2{\text{ s}}^{-1}$.

Figure s01 — Amplitude resonance curve (alt-text). Horizontal axis driving frequency ${\omega }_d$ hai $0$ se $45$ rad/s tak; vertical axis steady-state amplitude $A$ hai $0$ se lagbhag $0.28$ m tak. Ek blue curve static value $0.05$ m ke paas se shuru hoti hai (dashed gray line, labelled $A\to F_0/k$), ek sharp red peak tak $\approx 0.251$ m par ${\omega }_{res}\approx 19.8$ rad/s par uthti hai (${\omega }_0=20$ ko mark karti dotted orange line se thodi left), phir fast drives ke liye zero ki taraf girti hai. Coloured dots mark karte hain kahan har cell curve par hai: green "A slow" left rise par neeche, orange "B ${\omega }_0$" peak ke thoda right mein, red "C peak" bilkul top par, gray "F fast" right tail mein kaafi neeche.

Upar wali figure is base system ke liye amplitude-vs-frequency curve hai, jisme cells A–G mark kiye hain jahan woh curve par hain. Baar-baar isko dekhte raho: har example bas "is curve ka ek point read karo" ya "phase graph ka ek point read karo" hai.

---

Cell A — Slow drive (static limit)

---

Cell B — Exact resonance denominator (${\omega }_d={\omega }_0$)

---

Cell C — True amplitude peak (${\omega }_d={\omega }_{res}$)

---

Cell D — Phase lag resonance ke neeche (${\omega }_d<{\omega }_0$)

---

Cell E — Phase lag resonance ke upar (quadrant trap)

Figure s02 — Phase lag vs frequency (alt-text). Horizontal axis driving frequency ${\omega }_d$ hai $0$ se $45$ rad/s tak; vertical axis phase lag $\varphi$ degrees mein hai $0$ se $180$ tak. Ek blue S-shaped curve slow drives ke liye $0^{\circ }$ ke paas se shuru hoti hai, dashed gray $90^{\circ }$ line se steeply rise karti hai exactly dotted orange ${\omega }_0=20$ par, phir fast drives ke liye $180^{\circ }$ ki taraf level off karti hai. Lower-left band ($90^{\circ }$ se neeche) green tinted hai aur labelled hai "below res: $\varphi$ in (0,90)"; upper band red tinted hai aur labelled hai "above res: $\varphi$ in (90,180)". Ek green dot Ex 4 mark karta hai (${\omega }_d=10$, $\varphi =7.6^{\circ }$); ek red dot Ex 5 mark karta hai (${\omega }_d=30$, $\varphi =166.5^{\circ }$, annotated "add 180!").

---

Cell F — Fast drive (inertia limit)

---

Cell G — Zero damping (degenerate input)

---

Cell H — Real-world word problem

---

Cell I — Exam twist (inverse problem)

---

Recall Matrix ke across quick self-test

$A$ ka slow drive limit kya hai? ::: $F_0/k$ (Hooke's law), Cell A. Amplitude at ${\omega }_d={\omega }_0$? ::: $F_0/(2m\gamma {\omega }_0)=Q\cdot F_0/k$, Cell B. Phase lag at ${\omega }_d={\omega }_0$? ::: $\varphi =90^{\circ }=\frac{\pi }{2}$ rad — force velocity ke in phase, max power, Cell B. True peak location? ::: ${\omega }_{res}=\sqrt{{\omega }_0^2-2{\gamma }^2}$, ${\omega }_0$ se thoda neeche, Cell C. Resonance ke upar $\tan \varphi$ ka sign, aur fix? ::: Negative; $180^{\circ }$ add karo taaki $\varphi \in (90^{\circ },180^{\circ })$, Cell E. ${\omega }_d\to \infty$ par $A$ kya hai? ::: $F_0/(m{\omega }_d^2)\to 0$, Cell F. $\gamma =0$ par ${\omega }_0$ par $A$ ka kya hota hai? ::: Infinity tak diverge karta hai; phase $0\to 180^{\circ }$ jump karta hai, Cell G. Inverse problem: ek below-peak amplitude kitni drive frequencies deta hai? ::: Do — peak ke ek-ek taraf; ${\omega }_d^2$ mein quadratic solve karo, Cell I.

Yeh bhi dekho: Complex Exponential Method $A$ aur $\varphi$ ek saath nikalne ke slick tarike ke liye, Damped Oscillations us transient ke liye jiske upar yeh steady states baithe hain, aur Simple Harmonic Motion $b=0,F_0=0$ ancestor ke liye jo yahan sab ka bunyaad hai.