Section 20.11

Mechanical Vibrations

Finally, we apply our heavy machinery. Mass, springs, and dashpots create the perfect playground for Second Order DEs.

Introduction

By Newton's Second Law (), a mass attached to a spring (constant ) and a damper (constant ) obeys:

Here is displacement.
is acceleration (Inertia).
is velocity (Drag/Damping).
is displacement (Spring Force).

Free Vibrations ($F=0$)

Classification

Undamped ($\gamma = 0$): Simple Harmonic Motion. . Frequency .
Underdamped ($\gamma^2 < 4mk$): Oscillates with decaying amplitude. "Quasi-frequency" .
Critically Damped ($\gamma^2 = 4mk$): Repeated roots. Returns to equilibrium as fast as possible without oscillating.
Overdamped ($\gamma^2 > 4mk$): Real distinct roots. Slowly returns to equilibrium (molasses).

Forced Vibrations

Resonance

If we force the system at its natural frequency (and there is no damping), the amplitude grows linearly without bound!

.
This is the physical manifestation of the "Repeated Roots" case in Nonhomogeneous equations.

Visual: Damping Control

Interactive: The Shock Absorber

Low damping allows oscillation (Underdamped). High damping kills it essentially immediately (Overdamped).

Worked Examples

Example 1: Setting up the Equation

A 2 kg mass stretches a spring 0.5 m (to equilibrium). It is then pulled down 0.2 m and released. Find (Assume ).

1. Find k:

2. Set up IVP:

(down is positive).

(released from rest).

3. Solve:

Example 2: Interpretation

A system has solution . What is the quasi-period?

1. Identify Parameters:

The oscillation part is .

So the quasi-frequency is rad/s.

2. Calculate Quasi-Period:

seconds.

This is the time between zero-crossings (though not exactly peak-to-peak due to damping drift).

Example 3: Resonance

Find the resonant frequency for .

1. Natural Frequency:

2. Condition:

Resonance occurs when forcing frequency equals natural frequency .

So .

At this frequency, amplitude grows linearly ().