Vibration Foundations

Lesson 1: SHM & Phase Dynamics

Sequence: Longitudinal Wave Mechanics

Unit: 01 // Foundations

The Energy Balance Hook

Consider a standard mass-spring system in SHM. We know energy cycles between Kinetic and Potential.

Challenge:

At exactly what displacement \( x \) is the kinetic energy of the system precisely equal to its potential energy?

KE = PE?

Solving for \( x \) in terms of Amplitude \( A \)...

Defining the Restoring Force

A system executes SHM if the restoring force is directly proportional to displacement:

\[ F = -kx \]

Where \( k \) is the stiffness constant and \( x \) is displacement from equilibrium.

Applying Newton's Second Law (\( F=ma \)):

\[ m\frac{d^2x}{dt^2} + kx = 0 \] \[ \frac{d^2x}{dt^2} + \omega^2x = 0 \]

Fundamental Equation of Motion: \( \omega = \sqrt{k/m} \)

General Solutions & Phase

The Solution

The position of the particle at any time \( t \) is:

\( x(t) = A\cos(\omega t + \phi) \)

\( A \): Amplitude (Max displacement)
\( \omega \): Angular Frequency
\( \phi \): Phase Constant

Velocity and Acceleration

Velocity \( v(t) \)

\( -A\omega\sin(\omega t + \phi) \)

Acceleration \( a(t) \)

\( -A\omega^2\cos(\omega t + \phi) \)

The Phase Relationship

Note that Velocity leads Displacement by \( \pi/2 \) radians (90°), and Acceleration leads Velocity by another \( \pi/2 \) radians.

Total Mechanical Energy

Potential Energy (U)

\( U = \frac{1}{2}kx^2 = \frac{1}{2}kA^2\cos^2(\omega t + \phi) \)

Kinetic Energy (K)

\( K = \frac{1}{2}mv^2 = \frac{1}{2}kA^2\sin^2(\omega t + \phi) \)

Conservation of Energy

The sum of Potential and Kinetic energy is constant:

\( E_{total} = \frac{1}{2}kA^2 \)

Since \( \sin^2\theta + \cos^2\theta = 1 \), the energy transfer is seamless and lossless in an ideal system.

Solving the Energy Challenge

We want to find \( x \) such that \( K = U \).

Statement: \( \frac{1}{2}kx^2 = \frac{1}{2}mv^2 \)

Total Energy: \( U + K = E_{total} \implies 2U = E_{total} \)

Substitute: \( 2(\frac{1}{2}kx^2) = \frac{1}{2}kA^2 \)

Result: \( x = \frac{A}{\sqrt{2}} \approx 0.707A \)

This occurs at exactly 45 degrees (\( \pi/4 \)) into the oscillation phase.

Longitudinal Wave Mechanics

Vibration Foundations

Wave Equation Derivation

Pressure Phase Dynamics

Wave Energy Mechanics

Superposition Mathematics

Vibration Foundations Slides

Vibration Analysis Worksheet

Vibration Analysis Answer Key

Deriving Wave Equation Slides

Wave Equation Derivation Lab

Wave Equation Derivation Solutions

Pressure Phase Slides

Pressure Analysis Worksheet

Pressure Analysis Answer Key

Wave Energy Slides

Intensity Mechanics Worksheet

Intensity Mechanics Answer Key

Superposition Mathematics Slides

Interference Showdown Worksheet

Interference Showdown Answer Key

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