Forced oscillations — driving frequency — Physics notes

WHAT is a forced oscillation?

WHY does the system end up oscillating at $\omega_d$ ? Two motions coexist: a transient (the system's own damped wobble at $\approx\omega_0$ , which dies out as $e^{-\gamma t}$ ) and a steady state (driven by the force, never dies). After enough time only the steady state survives, and a periodic force of frequency $\omega_d$ can only sustain a response of the same frequency $\omega_d$ . The system is "enslaved" to the driver.

HOW to derive the steady-state amplitude (from scratch)

Step 1 — Newton's second law. $m\ddot{x} = -kx - b\dot{x} + F_0\cos(\omega_d t)$

Why this step? We just add every force acting on the mass. Spring $-kx$ , drag $-b\dot x$ , drive $+F_0\cos\omega_d t$ .

Step 2 — Standardise. Divide by $m$ and define $\omega_0^2=k/m$ , $\;2\gamma=b/m$ : $\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = \frac{F_0}{m}\cos(\omega_d t)$

Why this step? Grouping constants into $\omega_0$ and $\gamma$ keeps the algebra clean and shows the physics: natural frequency vs damping rate.

Step 3 — Guess the steady-state form. Since the drive is periodic at $\omega_d$ , try $x(t)=A\cos(\omega_d t-\phi)$ where $A$ is amplitude and $\phi$ is the phase lag behind the force.

Why this step? A linear equation driven at $\omega_d$ must respond at $\omega_d$ . The lag $\phi$ exists because damping makes the response unable to keep up instantly.

Step 4 — Substitute. Use $\dot{x}=-A\omega_d\sin(\omega_d t-\phi)$ , $\ddot{x}=-A\omega_d^2\cos(\omega_d t-\phi)$ . Plug in and demand it hold for all $t$ . The cleanest route is to balance the cosine and sine components. After collecting terms (or using complex exponentials $e^{i\omega_d t}$ ) you get:

$A = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega_d^2)^2 + (2\gamma\omega_d)^2}}$

$\tan\phi = \frac{2\gamma\omega_d}{\omega_0^2-\omega_d^2}$

Why this step? The denominator is the "impedance" of the oscillator. The term $(\omega_0^2-\omega_d^2)$ is the mismatch between drive and natural frequency; the term $2\gamma\omega_d$ is the damping cost. Big mismatch → small amplitude. Small mismatch → amplitude limited only by damping.

Reading the amplitude curve (Dual Coding)

Figure — Forced oscillations — driving frequency

Intuition Three regimes of

$\omega_d$

$\omega_d \ll \omega_0$ (slow push): $A \to F_0/(m\omega_0^2)=F_0/k$ . Spring-dominated; mass moves in phase with the force ( $\phi\to 0$ ). It just follows the force statically.
$\omega_d \approx \omega_0$ (resonance): denominator is tiny, $A$ peaks. Energy injected every cycle. Phase $\phi\to\pi/2$ : the force leads the displacement by 90°, which means the force is exactly in phase with the velocity ( $\dot x \propto \sin(\omega_d t-\phi)$ ). Since power input is $F\dot x$ , this in-phase condition gives maximum power transfer.
$\omega_d \gg \omega_0$ (fast push): $A \to F_0/(m\omega_d^2)\to 0$ . Mass-inertia dominated; it can't keep up. Phase $\phi\to\pi$ (displacement opposite the force).

Resonance frequency (peak of $A$ ): maximise $A$ ⇒ minimise the denominator. Differentiate $(\omega_0^2-\omega_d^2)^2+(2\gamma\omega_d)^2$ w.r.t. $\omega_d$ : $\omega_{res}=\sqrt{\omega_0^2-2\gamma^2}$ Why? Damping shifts the peak below $\omega_0$ slightly. For light damping ( $\gamma\ll\omega_0$ ), $\omega_{res}\approx\omega_0$ .

Worked examples

Worked example 1 — Amplitude at resonance

A mass $m=0.5$ kg, $k=200$ N/m, $b=2$ kg/s, driven by $F_0=10$ N. Find amplitude at $\omega_d=\omega_0$ .

$\omega_0=\sqrt{200/0.5}=20$ rad/s. Why: $\omega_0=\sqrt{k/m}$ .
$2\gamma=b/m=4 \Rightarrow \gamma=2$ s⁻¹. Why: $2\gamma=b/m$ from standardising.
At $\omega_d=\omega_0$ , the mismatch term vanishes: denominator $=2\gamma\omega_d=4\times20=80$ .
$A=\dfrac{F_0/m}{80}=\dfrac{10/0.5}{80}=\dfrac{20}{80}=0.25$ m. Why: plug into $A$ with $(\omega_0^2-\omega_d^2)=0$ .

Common mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine pushing your friend on a swing. The swing has a natural rhythm — push too fast or too slow and nothing much happens; the swing barely moves and your pushes sometimes fight it. But if you push exactly in time with the swing, each push adds energy and it goes really high. "Driving frequency" is just how often you push. When your push-rhythm matches the swing's natural rhythm, that's resonance — biggest swing for least effort. Friction (air, rusty chain) is what stops it from going infinitely high.

Active recall

What frequency does a steady-state forced oscillator vibrate at?

The driving frequency

\omega_d

, NOT the natural frequency.

Write the steady-state amplitude formula.

A=\dfrac{F_0/m}{\sqrt{(\omega_0^2-\omega_d^2)^2+(2\gamma\omega_d)^2}}

Why does amplitude peak near

\omega_0

The mismatch term

(\omega_0^2-\omega_d^2)

vanishes, so the denominator is minimised (only damping limits

A

At what exact frequency is the amplitude maximum?

\omega_{res}=\sqrt{\omega_0^2-2\gamma^2}

, slightly below

\omega_0

for nonzero damping.

What is the phase lag

\phi

at resonance, and what does it mean physically?

\phi=\pi/2

(90°): force leads the displacement by 90°, so force is in phase with velocity → maximum power input.

What is the amplitude in the static limit

\omega_d\to0

A\to F_0/k

(Hooke's law — pure spring displacement).

What is

A

\omega_d\to\infty

A\to F_0/(m\omega_d^2)\to 0

— inertia dominates, response vanishes, phase

\to\pi

What happens to the transient solution over time?

It decays as

e^{-\gamma t}

, leaving only the steady-state driven motion.

Does

F_0

affect the resonance frequency?

No —

F_0

only scales the amplitude linearly;

\omega_{res}

depends on

\omega_0,\gamma

only.

Connections

Damped Oscillations — provides the $-b\dot x$ term and the decaying transient.
Simple Harmonic Motion — the $F_0=0,\ b=0$ limit giving pure $\omega_0$ .
Resonance and Quality Factor — sharpness of the peak set by $Q=\omega_0/2\gamma$ .
Energy in Oscillations — power input maximal at $\omega_d=\omega_0$ .
Waves and Standing Waves — resonance of strings/air columns is forced oscillation in disguise.
Complex Exponential Method — the clean tool to solve Step 4.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, forced oscillation ka matlab hai ki ek spring-mass system pe hum bahar se ek periodic force laga rahe hain, jaise $F_0\cos(\omega_d t)$ . Yahan teen cheezein ek saath kaam karti hain: spring ka restoring force, damping (friction), aur humara external push. Sabse important baat — final steady state mein system apni natural frequency $\omega_0$ pe nahi, balki driving frequency $\omega_d$ pe hilta hai. System apni marzi bhool jaata hai aur jo frequency hum de rahe hain usi pe naachne lagta hai.

Amplitude ka formula hai $A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega_d^2)^2+(2\gamma\omega_d)^2}}$ . Iska denominator do parts ka hai: ek mismatch $(\omega_0^2-\omega_d^2)$ — yani driver aur natural frequency mein farak — aur ek damping cost $2\gamma\omega_d$ . Jab $\omega_d$ ko $\omega_0$ ke paas le jaate ho, mismatch zero ho jaata hai, denominator chhota ho jaata hai, aur amplitude phat se bahut bada ho jaata hai. Isi ko resonance kehte hain — jhoole wali example yaad rakho, sahi rhythm pe push karo toh jhoola bahut upar jaata hai.

Phase lag $\phi$ bhi samajhna: slow push pe response force ke saath in-phase ( $\phi\approx0$ ). Resonance pe $\phi=90°$ — yani force, displacement se 90° aage hota hai, jiska matlab force aur velocity bilkul in-phase ho jaate hain. Power $P=F\dot x$ hota hai, aur jab force-velocity in-phase hon tabhi maximum power transfer hota hai. Fast push pe $\phi=180°$ yani displacement bilkul ulta. Yaad rakho: resonance pe force velocity ko "lead" nahi karta — woh velocity ke saath same phase mein hota hai.

Do common galtiyan: (1) Mat sochna ki system $\omega_0$ pe hilega — steady state mein hamesha $\omega_d$ pe hilega. (2) Resonance peak bilkul exactly $\omega_0$ pe nahi, thoda neeche $\omega_{res}=\sqrt{\omega_0^2-2\gamma^2}$ pe hota hai jab damping present ho. Aur $F_0$ sirf amplitude badhata hai, peak ki jagah nahi. Ye intuition bridges, buildings, radios sab mein kaam aati hai!

Forced oscillations — driving frequency

WHAT is a forced oscillation?

HOW to derive the steady-state amplitude (from scratch)

Reading the amplitude curve (Dual Coding)

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Oscillations & Waves

Connections