---
title: "Forced vibration amplitude and resonance"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:forced-oscillation-amplitude-resonance"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-oscillation-amplitude-resonance
site: BCT Engineering
---

# Forced vibration amplitude and resonance

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2070 Asar, 5 marks; 2070 Chaitra, 5 marks (asked 2 times)
- Video: [Forced Vibration: Amplitude and Velocity, Resonance, Sharpness of Resonance, LCR Circuit](https://www.youtube.com/watch?v=4ZK0s5RNDOA) — Mywbut
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-oscillation-amplitude-resonance
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Distinguish between free and forced vibrations. Give the necessary theory of forced vibration (differential equation), determine the amplitude of oscillation and deduce the condition for resonance amplitude. Explain sharpness of resonance.

## Short answer

Free vibration occurs at natural frequency without external force; forced vibration is driven by periodic force at driving frequency. Amplitude A = (F0/m)/√((ω0²-p²)²+4γ²p²) peaks at resonance p_r = √(ω0²−2γ²) with A_max = F0/(2mγ√(ω0²−γ²)). Sharpness is measured by quality factor Q = ω0/(2γ).

## Answer

#### Free and forced vibrations
| Free vibration | Forced vibration |
|---|---|
| Body vibrates at its natural frequency after an initial disturbance | Body vibrates at the frequency of an external periodic force |
| No external force after start | Periodic force acts continuously |
| Amplitude falls if damped | Amplitude steady (after transients) |
| Example: pluck a string | Example: tuning fork on a table |

#### Theory (differential equation)
Mass m, spring constant k, damping force −b v, driving force F0 cos pt:
```text
m x'' + b x' + k x = F0 cos pt
x'' + 2γ x' + ω0² x = f0 cos pt
```
with 2γ = b/m, ω0² = k/m, f0 = F0/m.

#### Amplitude
Steady state: x = A cos(pt − φ). Substituting and comparing the coefficients of cos(pt − φ) and sin(pt − φ):
```text
A (ω0² − p²) = f0 cosφ
A (2γ p)     = f0 sinφ
```
```text
A = f0 / √[(ω0² − p²)² + 4γ²p²] ,   tanφ = 2γp/(ω0² − p²)
```

#### Amplitude resonance
A is maximum when D = (ω0² − p²)² + 4γ²p² is minimum. dD/d(p²) = −2(ω0² − p²) + 4γ² = 0:
```text
p_r² = ω0² − 2γ²      (p_r < ω0)
A_max = f0 / (2γ √(ω0² − γ²))
```
Resonance condition: the driving frequency equals √(ω0² − 2γ²), very nearly the natural frequency if damping is small. For γ → 0, A becomes very large at p = ω0.

#### Sharpness of resonance
It is the rate at which the amplitude falls as p departs from p_r.
```text
 A |      /\  small damping (sharp)
   |     /  \
   |    / __ \___ large damping (flat)
   |___/_/____\\___ p
        p_r
```
- Small γ gives a tall, narrow peak (sharp); large γ gives a low, broad peak.
- It is measured by the quality factor Q = ω0/2γ = ω0/Δω, where Δω is the half-power bandwidth.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A forced vibration occurs at which frequency?
   A. The frequency of the external periodic force
   B. The natural frequency
   C. Twice the natural frequency
   D. Zero frequency
   Answer: A. After transients die, the body follows the driver. (Key point: Free vs forced)

2. The steady-state amplitude is?
   A. A = f0 / (w0^2 + p^2)
   B. A = f0 / sqrt((w0^2 - p^2)^2 + 4 gamma^2 p^2)
   C. A = f0 sqrt((w0^2 - p^2)^2 + 4 gamma^2 p^2)
   D. A = f0 / (2 gamma p)
   Answer: B. Squaring and adding the coefficient equations. (Key point: Amplitude)

3. The driving frequency for maximum amplitude is?
   A. p = w0 exactly always
   B. p^2 = w0^2 + 2 gamma^2
   C. p^2 = w0^2 - 2 gamma^2
   D. p^2 = 2 gamma^2
   Answer: C. Minimising the denominator gives p_r < w0. (Key point: Amplitude resonance)

4. What gives a sharp resonance?
   A. Large damping gives a tall peak
   B. Sharpness is independent of damping
   C. Large damping gives narrow peak
   D. Small damping gives a tall, narrow peak
   Answer: D. Q = w0/2gamma; small gamma means high Q. (Key point: Sharpness)

## Related in Engineering Physics

- [Sharpness of resonance, forced e-m oscillation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-em-sharpness-current-amplitude) — Engineering Physics, chapter 1
- [Forced LCR oscillation and resonance](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-lcr-resonance-frequency) — Engineering Physics, chapter 1, asked 2×
- [LC oscillation differential equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Alc-oscillation-differential-equation) — Engineering Physics, chapter 1
- [Damped vs forced LCR oscillation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-lcr-differential-compare) — Engineering Physics, chapter 1
- [Sharpness of resonance and quality factor](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asharpness-resonance-quality-factor) — Engineering Physics, chapter 1

## Related in other subjects

- [Free damped vibration: three cases](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Adamped-free-vibration-cases-derivation) — Theory of Mechanism and Machine II, chapter 6