Driven Oscillations & Forced Vibrations

Oscillations are everywhere: the swaying of a suspension bridge, the ringing of a guitar string, the flicker of an electrical circuit.
Many of these motions do not happen in isolation. Instead, they are driven—sustained or influenced by external forces.
Such motion is described as driven oscillation or forced vibration, and it is one of the most important topics in classical physics and engineering.

This in-depth guide explores the theory, mathematics, and wide-ranging applications of driven oscillations and forced vibrations. We will also see how resonance—nature’s dramatic amplifier—emerges when driving conditions match a system’s natural rhythm.

1. Background: Free vs. Driven Oscillations

Before diving into forced vibrations, let’s distinguish two fundamental types of motion:

1.1 Free Oscillations

Occur when a system oscillates on its own after an initial disturbance.
Examples: A pendulum swinging after being released, a plucked guitar string, or an LC electrical circuit discharging.
Frequency equals the system’s natural frequency f0f_0f0.
Amplitude decreases over time if damping is present.

1.2 Driven or Forced Oscillations

Occur when a continuous external force drives the system.
The system responds not only to its natural frequency but also to the driving frequency fdf_dfd.
Energy is continually supplied, allowing oscillations to persist even in the presence of damping.

The key difference is the presence of a sustained external input that keeps the system moving.

2. Real-World Intuition

Imagine pushing a child on a swing. Each push is an external force.

If you push randomly, the swing moves irregularly.
If you time each push to match the swing’s natural rhythm, the motion grows dramatically.

This simple playground scenario captures the essence of driven oscillations and resonance.

3. The Physics of a Driven Harmonic Oscillator

The classic model is a mass–spring–damper system subjected to a periodic external force: md2xdt2+cdxdt+kx=F0cos⁡(ωt)m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F_0 \cos(\omega t)mdt2d2x+cdtdx+kx=F0cos(ωt)

Where:

mmm = mass
ccc = damping coefficient
kkk = spring constant
F0F_0F0 = driving force amplitude
ω\omegaω = driving angular frequency

This second-order differential equation describes a wide range of systems—from mechanical oscillators to electrical RLC circuits.

4. Solution Components: Transient and Steady State

The complete solution has two parts:

Transient Solution:
- Depends on initial conditions.
- Represents free oscillations that decay due to damping.
- Eventually dies out.
Steady-State Solution:
- Oscillation at the driving frequency ω\omegaω.
- Persists as long as the driving force continues.
- Amplitude and phase depend on ω\omegaω, ccc, and kkk.

Over time, only the steady-state motion remains. This is the signature of a driven oscillator.

5. Amplitude Response and Resonance

The steady-state amplitude is: A(ω)=F0/m(ω02−ω2)2+(2ζω0ω)2A(\omega) = \frac{F_0 / m}{\sqrt{(\omega_0^2 – \omega^2)^2 + (2 \zeta \omega_0 \omega)^2}}A(ω)=(ω02−ω2)2+(2ζω0ω)2F0/m

where

ω0=k/m\omega_0 = \sqrt{k/m}ω0=k/m = natural angular frequency
ζ=c/(2mk)\zeta = c/(2\sqrt{mk})ζ=c/(2mk) = damping ratio

5.1 Resonance

When the driving frequency ω\omegaω approaches the natural frequency ω0\omega_0ω0, the denominator is minimized, and the amplitude peaks—this is resonance.

Low Damping: Very large amplitude; can cause structural failure.
High Damping: Broader, lower peak; safer but less responsive.

Resonance explains why a singer can shatter a wine glass and why bridges need special design to avoid catastrophic vibrations.

6. Phase Difference

In forced vibrations, displacement does not always match the driving force in time. The phase difference ϕ\phiϕ varies with driving frequency:

At low frequencies, displacement is almost in phase with the driving force.
At resonance, the phase lag is 90°.
At high frequencies, displacement lags by nearly 180°.

This phase behavior is crucial in engineering control systems and signal processing.

7. Energy Flow

Unlike free oscillations, a driven oscillator continually receives energy from the external force.

At resonance, energy transfer is most efficient.
Power input equals energy dissipated through damping in steady state.

This balance keeps amplitude constant despite energy losses.

8. Everyday Examples

Driven oscillations and forced vibrations appear in countless contexts:

8.1 Musical Instruments

A violin string vibrates because the bow continuously drives it.
A guitar body resonates with the string’s frequency, amplifying sound.

8.2 Bridges and Buildings

Wind or traffic imposes periodic forces.
The infamous 1940 Tacoma Narrows Bridge collapse was partly due to aeroelastic resonance.

8.3 Electrical Circuits

An AC source driving an RLC circuit shows the same math.
Voltage and current resonate at the circuit’s natural frequency.

8.4 Household Devices

Washing machines, loudspeakers, and engines all experience forced vibrations.

8.5 Human Body

The vocal cords are a driven oscillator; airflow from the lungs drives tissue vibrations.
The middle ear bones exhibit forced vibrations to transmit sound.

9. Controlling Forced Vibrations

Engineers often need to manage or exploit forced vibrations.

9.1 Reducing Harmful Effects

Damping Systems: Shock absorbers, tuned mass dampers in skyscrapers.
Isolation: Rubber mounts, flexible couplings to prevent vibration transmission.

9.2 Harnessing Resonance

Musical Instruments: Careful design of cavities to enhance sound.
Sensors: Quartz crystal resonators in watches rely on stable resonance frequencies.

10. Quality Factor (Q)

The Q-factor measures how “sharp” a resonance peak is: Q=ω0ΔωQ = \frac{\omega_0}{\Delta \omega}Q=Δωω0

High Q: Low damping, narrow frequency response (tuning forks, lasers).
Low Q: High damping, broad response (car suspensions).

Controlling Q is essential in filters, oscillators, and many mechanical systems.

11. Nonlinear and Complex Systems

Real systems may deviate from the simple linear model:

Nonlinear Springs: Amplitude-dependent stiffness leads to frequency shifts.
Multiple Degrees of Freedom: Coupled oscillators show complex resonance patterns.
Parametric Excitation: Driving not by a force but by changing parameters (e.g., swing pumping).

Understanding these cases requires advanced mathematics but follows the same core principle: external influence sustains motion.

12. Experimental Demonstrations

Educators often illustrate driven oscillations with accessible setups:

Driven Pendulum: A motorized arm provides periodic torque, showing resonance at certain speeds.
RLC Circuit Labs: Students observe resonance peaks in voltage response.
Tuning Fork and Speaker: A speaker sweeps frequencies; amplitude spikes at the fork’s natural frequency.

Such experiments reveal the beauty and power of resonance firsthand.

13. Forced Vibrations in Nature

Nature itself is full of driven oscillations:

Ocean Tides: Gravitational forcing from the Moon and Sun.
Climate Systems: Seasonal solar forcing leads to oscillatory temperature patterns.
Biological Rhythms: Circadian cycles are driven by external light cues.

These examples show how universal the phenomenon is, from atomic scales to planetary motion.

14. Safety and Engineering Lessons

History offers stark reminders of resonance dangers:

Tacoma Narrows Bridge (1940): Aerodynamic forces matched the bridge’s natural frequency, causing catastrophic collapse.
Millennium Bridge (2000): Pedestrians inadvertently synchronized footsteps, exciting lateral oscillations until dampers were added.
Aviation Flutter: Aircraft wings and control surfaces require careful analysis to avoid resonant vibrations.

Modern engineering standards mandate vibration analysis to prevent such failures.

15. Summary Table: Key Parameters

Parameter	Symbol	Significance
Natural frequency	ω0\omega_0ω0	Frequency of free oscillation
Driving frequency	ω\omegaω	External force frequency
Damping ratio	ζ\zetaζ	Energy loss factor
Amplitude	A(ω)A(\omega)A(ω)	Response magnitude at given drive frequency
Phase difference	ϕ\phiϕ	Lag between force and displacement
Quality factor	QQQ	Sharpness of resonance peak

This compact view highlights the interplay of factors that define driven motion.

16. The Broader Significance

Driven oscillations and forced vibrations are not just textbook physics; they are the foundation of:

Modern Electronics: From radio tuners to quartz clocks.
Mechanical Design: Cars, buildings, and machinery rely on controlled damping.
Medical Technology: Ultrasound transducers and MRI machines use resonance principles.