Forced oscillations — driving frequency: Foundations

1.6.11 · D1 · Physics › Oscillations & Waves › Forced oscillations — driving frequency

Parent note ki ek bhi line padhne se pehle, aapko usmein aane wale har symbol ko khud se samajhna hoga. Yeh page har ek symbol ko zero se build karta hai — pehle plain words mein, phir picture mein, phir yeh batata hai ki topic uske bina kyon nahi chal sakta. Upar se neeche padho; har block upar wale par lean karta hai.

0. Stage: ek spring par mass

Sab kuch ek picture se shuru hota hai — ek block jo left aur right slide kar sakta hai, ek spring se wall se juda hua.

Figure — Forced oscillations — driving frequency

1. Rates of change — dots $\overset{x}{˙}$ aur $\overset{x}{¨}$

Parent $\overset{x}{˙}$ aur $\overset{x}{¨}$ likhta hai bina dots explain kiye. Yahan se aate hain yeh.

2. Teen forces aur unke symbols

Parent teen forces add karta hai. Har ek mein har letter define karte hain. Saare forces newtons (N) mein measure hote hain.

3. Angular frequency $ω$ — topic ka dil

Symbol $ω$ (Greek "omega") har jagah aata hai: $ω_{d}$ , $ω_{0}$ , $ω_{r es}$ . Ise yahan pakad lo.

Definition Angular frequency

$ω$ $ω$ batata hai rhythm kitni tez cycle karta hai, radians per second mein measure kiya gaya.

Picture: oscillation ko ek circle par ghoomte dot ki tarah imagine karo. Ek full lap = ek full aage-peeche. $ω$ yeh hai ki woh dot har second mein us circle ke kitne radians sweep karta hai. (Ek full lap $2 π$ radians hai.)
Units: radians per second (rad/s).
" $cos (ω t)$ " kyun? Jab circle-dot ghoomta hai, ek axis par uski shadow exactly ek cosine wave trace karti hai. Isliye $cos (ω_{d} t)$ ek push hai jo har $2 π / ω_{d}$ seconds mein repeat hoti hai. Bada $ω$ → tezi se wiggling.
Hum $ω$ kyun use karte hain, na ki ordinary "cycles per second": $cos$ function radians "khaata" hai, isliye rate ko radians/second mein package karna $cos (ω t)$ ko koi extra $2 π$ clutter ke bina clean banata hai.

Recall

$ω_{0} = k / m$ kyun hai? Plain spring-mass ke liye, Newton deta hai $m \overset{x}{¨} = - k x$ , yaani $\overset{x}{¨} = - (k / m) x$ . Jo bhi $\overset{x}{¨} = - ω^{2} x$ follow karta hai woh rate $ω$ par wiggle karta hai, isliye $ω^{2} = k / m$ match karne par $ω_{0} = k / m$ milta hai. Stiffer ya lighter → tezi se natural rhythm.

4. Damping rate $γ$

5. Amplitude $A$ aur phase lag $ϕ$

Woh do answers jo topic compute karta hai.

Common mistake "Bas

$ϕ = arctan (2 γ ω_{d} / (ω_{0}^{2} - ω_{d}^{2}))$ lo aur kaam khatam." Kyun sahi lagta hai: parent ek single $tan ϕ$ formula quote karta hai, isliye ek $arctan$ button kaafi lagta hai. Fix: plain $arctan$ sirf $- 90°$ aur $+ 90°$ ke beech angles return karta hai, lekin physical lag $ϕ$ poore $0$ se $180°$ tak jaata hai jab aap driver ko speed up karte ho. Pakad yeh hai ki adjacent side $(ω_{0}^{2} - ω_{d}^{2})$ ka sign kya hai:

Resonance se neeche ( $ω_{d} < ω_{0}$ ): mismatch positive hai, damping cost positive → dono triangle sides positive → $ϕ$ ek chhota acute angle hai, $0 < ϕ < 90°$ . Push ke saath almost sync mein.
Resonance par ( $ω_{d} = ω_{0}$ ): mismatch zero hai, isliye triangle purely vertical hai → $ϕ = 90°$ exactly. (Formula ka denominator $0$ ho jaata hai; ghabrao mat — angle bas ek right angle hai.)
Resonance se upar ( $ω_{d} > ω_{0}$ ): mismatch negative ho jaata hai (adjacent side backwards point karti hai) → $ϕ$ obtuse hai, $90° < ϕ < 180°$ . Response ab push ki opposite side par hai. Safe recipe hai two-argument arctangent: lo $ϕ = atan2 (2 γ ω_{d}, ω_{0}^{2} - ω_{d}^{2})$ , jo dono signs use karta hai taaki $ϕ$ intended range $0 < ϕ < π$ mein aaye. Ek plain $arctan$ resonance ke upar galat se ek negative lag report karega.

6. Zero-damping edge case

Agar friction completely gayab ho jaaye toh upar ki sab cheezein kya ho jaati hain? Yahi woh case hai jo mnemonics quietly skip kar dete hain.

Intuition

$γ \to 0$ : undamped resonance blow up ho jaati hai Amplitude denominator $(ω_{0}^{2} - ω_{d}^{2})^{2} + (2 γ ω_{d})^{2}$ dekho. Doosra term $(2 γ ω_{d})^{2}$ wahi ek cheez hai jo denominator ko zero se door rakhta hai jab aap exactly $ω_{d} = ω_{0}$ par drive karte ho.

Damping ke saath ( $γ > 0$ ): $ω_{d} = ω_{0}$ par pehla term $0$ hai lekin damping term survive karta hai, isliye $A$ bada hai lekin finite — ek rounded peak.
No damping ke saath ( $γ = 0$ ): $ω_{d} = ω_{0}$ par dono terms zero hain, denominator $0$ hai, aur $A \to \infty$ . Push har single cycle mein energy add karta hai bina kuch drain kiye, isliye amplitude without bound grow karta hai (ek real spring pehle toot jaayegi).
Is limit mein Phase: $ϕ$ abruptly $0$ (neeche) se seedha $180°$ (upar) jump kar jaata hai, $90°$ value sirf exactly $ω_{0}$ par touch hoti hai — Section 5 ka smooth $atan2$ transition ek sudden flip mein collapse ho jaata hai. Isliye topic ko damping $b$ chahiye: yahi woh hai jo resonance ko tall-but-finite peak banata hai blow-up ki jagah. Dekho Resonance and Quality Factor.

Prerequisite map

Is map ko topic ki assembly order ki tarah padho: sabse baayein boxes raw ingredients hain jo aapke paas pehle se hone chahiye; arrows dikhate hain kaun sa symbol feed karta hai kaun se mein. Notice karo ki Newton's second law funnel hai — mass $m$ , spring $k$ , damping $b$ aur drive sab usmein pour hote hain, aur doosri taraf se dono headline answers $A$ aur $ϕ$ nikalte hain. Agar koi bhi upstream box fuzzy hai, toh jo box woh point karta hai woh bhi fuzzy hoga, isliye exactly woh foundation dhundhne ke liye yeh use karo jise revisit karna hai.

Substitution step painlessly karne ke liye jo shortcut sablog use karte hain, dekho Complex Exponential Method. Yeh dekhne ke liye ki yeh sab kahan le jaata hai jab oscillations space mein travel karti hain, dekho Waves and Standing Waves. Poora topic yahan hai the parent note.

Equipment checklist

Khud test karo — reveal karne se pehle answer zor se bolo.

m

ka kya matlab hai aur uski units kya hain?

Block ki mass (uski sluggishness); units kilograms (kg).

x

ka ek phrase mein kya matlab hai, aur uski units?

Block ki equilibrium se distance; units metres (m).

Ek dot (

\overset{x}{˙}

) ka kya matlab hai? Do dots (

\overset{x}{¨}

) ka?

Ek dot = velocity in m/s; do dots = acceleration in m/s².

Restoring force

- k x

minus sign kyun carry karta hai, aur

k

ki units kya hain?

Yeh hamesha ghar ki taraf point karta hai, displacement ke opposite;

k

N/m mein hai.

Drag

- b \overset{x}{˙}

kyun likha jaata hai, aur

b

ki units kya hain?

Tez motion zyada drag laata hai aur minus velocity oppose karta hai;

b

N·s/m = kg/s mein hai.

F_{0} cos (ω_{d} t)

ke teen pieces kya hain?

F_{0}

= peak push strength (N);

cos

= smooth aage-peeche shape;

ω_{d}

= aap kitni tez push karte ho (rad/s).

ω

physically kya hai, uski units, aur radians/second kyun?

Rhythm kitni tez cycle karta hai, rad/s mein; radians isliye ki

cos

radians "khaata" hai,

cos (ω t)

ko clean rakhta hai.

ω_{d}

aur

ω_{0}

mein farq?

ω_{d}

driver ki rhythm hai (aap choose karte ho);

ω_{0} = k / m

system ki khud ki preferred rhythm hai.

γ

b

se kya relation hai, aur uski units kya hain?

γ = b / (2 m)

; units 1/s,

ω

jaisi hi taaki directly compare ho sakein.

A

aur

ϕ

dono kya answer karte hain?

A

= steady swing kitna bada hai (m);

ϕ

= motion push se kitna peeche hai (angle ke roop mein).

tan ϕ

ϕ

lete waqt plain

arctan

kyun use nahi kar sakte?

Plain

arctan

sirf

- 90°

+ 90°

span karta hai;

atan2

use karo taaki

(ω_{0}^{2} - ω_{d}^{2})

ka sign

ϕ

ko correctly

0

180°

mein rakhe.

Zero-damping limit

γ \to 0

mein

ω_{d} = ω_{0}

par kya hota hai?

Denominator zero ho jaata hai aur amplitude

A

infinity tak diverge karta hai — undamped resonance blow up ho jaati hai.