IntermediateTopic 66 of 104Engineering Mathematics
Damping and resonance
Under-, critically and over-damped responses; resonance occurs when forcing matches natural frequency.
In depth
With damping ratio ζ=c/(2√(km)), motion is under-damped for ζ<1 (oscillating decay), critical at ζ=1 and over-damped for ζ>1. Forced vibration amplitude peaks near ωₙ, and lower damping gives sharper and taller peaks.
Key points
- Damped frequency ω_d=ωₙ√(1-ζ²)
- Quality factor Q≈1/(2ζ)
- Car suspensions target ζ≈0.3–0.7
ExampleA bridge with little damping may be driven into dangerous resonance by wind.
What is Damping and resonance?
Under-, critically and over-damped responses; resonance occurs when forcing matches natural frequency.
Why is it important to understand damping and resonance?
Damping controls how a structure or circuit settles after a disturbance, and resonance explains why small periodic forces can cause large vibrations.
What is critical damping?
The damping ratio equals one, the fastest possible return to equilibrium without overshoot. Many suspensions are designed close to this value.
Why is resonance dangerous?
Near the natural frequency, the response amplitude can grow large with little forcing, and fatigue or failure can follow even when the force seems small.
How is Damping and resonance used in practice?
Typical applications include choosing a shock absorber that avoids overshoot in a vehicle, avoiding resonance in a pump mounted on a floor and tuning a radio receiver to a station frequency.
What should I learn before Damping and resonance?
Study "Variation of parameters" first (topic 65 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Damping and resonance?
Damped frequency ω_d=ωₙ√(1-ζ²). Quality factor Q≈1/(2ζ). Car suspensions target ζ≈0.3–0.7.
Can you give an example of Damping and resonance?
A bridge with little damping may be driven into dangerous resonance by wind.
What comes after Damping and resonance?
The next topic is "Laplace transform", which builds directly on this one.
How difficult is Damping and resonance?
It is classed as Intermediate in Engineering Mathematics (topic 66 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.