Damping and Resonance

Overview

Most real oscillating systems do not behave as ideal simple harmonic oscillators forever. Energy is usually lost through friction, air resistance, internal deformation, electrical resistance, or sound emission. This causes oscillation amplitude to decrease with time, a process known as damping.

If an external periodic force continuously supplies energy to the system, the motion becomes a forced oscillation. Under suitable conditions, the response can become very large. This phenomenon is called resonance.

Definition

Damping is the gradual loss of mechanical energy from an oscillating system due to resistive or dissipative forces.

A forced oscillation occurs when a system is driven by an external periodic force.

Resonance occurs when a periodic driving force has frequency equal or close to the natural frequency of the system, producing maximum response amplitude.

Why It Matters

Damping and resonance are central to engineering design, structural safety, musical acoustics, electronics, and many natural systems. The same ideas explain vehicle suspensions, musical instruments, bridge vibrations, radio tuning, and unwanted machinery vibration.

Key Representations

Damping

Common causes of damping include:

• air resistance;
• friction;
• fluid drag;
• internal material deformation;
• electrical resistance;
• sound radiation.

As energy decreases, amplitude decreases. Since energy is proportional to amplitude squared in many oscillators:

$$E\propto A^2$$

a gradual reduction in amplitude means continuous energy loss.

Damped Oscillation Graph

Typical displacement-time behaviour:

• oscillatory motion continues;
• successive peaks become smaller;
• equilibrium position remains unchanged.

The envelope often decays approximately exponentially:

$$A(t)=A_0{e}^{-kt}$$

This exact model is beyond the basic syllabus, but the graph shape is useful.

Types of Damping

Light damping, or underdamping, allows oscillations to continue while amplitude gradually decreases.

Critical damping returns the system to equilibrium in the shortest possible time without oscillating. This is often desirable in engineering systems such as car suspensions and analog meter needles.

Heavy damping, or overdamping, returns the system to equilibrium without oscillating, but more slowly than critical damping.

Type	Oscillates?	Speed of Return
Light damping	Yes	Slow to settle
Critical damping	No	Fastest without overshoot
Heavy damping	No	Slower than critical

Forced Oscillations

In a one-dimensional model, the driving force is usually written as a signed component:

$$F(t)=F_0\cos {\omega }_{d}t$$

where $F_0$ is the driving-force amplitude and ${\omega }_d$ is the driving angular frequency.

After initial transients die away, the system oscillates mainly at the driving frequency, not necessarily its natural frequency. The amplitude depends on driving frequency, damping, and strength of forcing.

Natural Frequency

Every oscillating system has one or more natural frequencies determined by its physical properties.

For a spring-mass oscillator:

$${\omega }_0=\sqrt{\frac{k}{m}}$$

For a simple pendulum:

$${\omega }_0=\sqrt{\frac{g}{l}}$$

Resonance

Resonance occurs when:

$$f_{\text{driving}}\approx f_0$$

or:

$${\omega }_d\approx {\omega }_0$$

At resonance, energy transfer each cycle is efficient, so amplitude becomes large.

Frequency Response

A graph of amplitude against driving frequency is called a frequency response curve. It has:

• small amplitude far from resonance;
• a peak near natural frequency;
• lower response again beyond resonance.

Greater damping causes:

• lower resonance peak;
• broader peak;
• less sharp resonance;
• lower maximum amplitude.

Less damping gives a taller and sharper peak.

Phase in Forced Motion

As driving frequency increases:

• at low frequency, the oscillator is nearly in phase with the driver;
• near resonance, phase lag increases;
• at high frequency, the oscillator approaches antiphase.

This is useful enrichment for interpreting resonance curves.

Applications and Risks

Useful resonance examples include musical instruments, radio tuning, MRI/NMR, filters, and sensors.

Dangerous resonance examples include bridges, buildings during earthquakes, rotating machinery, and repeated vibration of mechanical supports.

Soldiers break step on bridges because marching in step applies periodic forces. If this matches a bridge’s natural frequency, resonance may occur.

Car suspension combines a spring and a damper. Too little damping causes repeated bouncing. Too much damping gives a harsh, sluggish response. Near critical damping is often preferred.

Mathematical Model

A damped driven oscillator can be modelled by:

$$m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=F_0\cos {\omega }_{d}t$$

where $m$ is mass, $c$ is damping constant, and $k$ is stiffness. This equation explains resonance curves and phase lag, but detailed solution is beyond JC core requirements.

Common Mistakes

• Thinking damping always stops motion immediately.
• Confusing natural frequency with driving frequency.
• Assuming resonance is always destructive.
• Forgetting damping reduces the resonance peak.
• Assuming resonance only occurs in mechanical systems.

Links

• Main topic: Oscillations and Simple Harmonic Motion
• Related concept: Simple Harmonic Motion
• Related concept: Pendulum Motion
• Related concept: Phase Difference