Section 6.7: Driven Oscillations and Resonance

When an external periodic force is applied to an oscillator, the system is said to undergo driven oscillations. The steady-state motion depends on the driving frequency relative to the natural frequency of the system.

        Equation of Motion with Driving Force:
        \[
        m\ddot{x} + b\dot{x} + kx = F_0 \cos(\omega t)
        \]
        \( F_0 \): amplitude of the driving force
\( \omega \): driving angular frequency
Solution has both a transient (damped) and a steady-state part

      

Resonance:

When the driving frequency \( \omega \) is close to the natural frequency \( \omega_0 \), the amplitude of oscillations becomes very large. This phenomenon is called resonance.

The maximum amplitude occurs near:

\[ \omega \approx \sqrt{\omega_0^2 - 2\beta^2} \]

Example 1

A driven oscillator has \( m = 1 \,\text{kg}, k = 25 \,\text{N/m}, b = 1.0 \,\text{kg/s}, F_0 = 5 \,\text{N} \). Find the natural frequency and discuss when resonance occurs.

\(\omega_0 = \sqrt{\tfrac{k}{m}} = \sqrt{25} = 5 \,\text{rad/s}\)

Damping coefficient: \(\beta = \tfrac{b}{2m} = \tfrac{1}{2} = 0.5 \,\text{s}^{-1}\)

Resonance occurs when \( \omega \approx \sqrt{\omega_0^2 - 2\beta^2} = \sqrt{25 - 0.5} \approx 4.95 \,\text{rad/s} \).

Practice Problems

What distinguishes a driven oscillator from a free oscillator?
For \(m=0.5\) kg, \(k=20\) N/m, and \(b=0.2\) kg/s, find the resonance frequency.
Explain why resonance can be dangerous in buildings and bridges.
Show that the maximum amplitude of forced oscillations depends inversely on damping.
Why is damping necessary in real-world resonance systems like musical instruments and car suspensions?

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