# Complex Vibrations and Waveform Analysis

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### Complex vibration

the sum of two or more simple vibrations

### Frequency components or partials

the simple vibrations that make up a complex vibration

### Fourier Theorem

states that any complex vibration is the sum of various sinusoidal motions of varying amplitude, frequency, and phase.

### Aperiodic vibration

A vibrations without a repeating pattern in time

### Periodic motion

a motion that repeats itself in regular intervals until it is stopped by external action

### Period

the time it takes to complete a certain nubmber of cycles
measured in seconds
T = t/cycles

### Frequency

the number of cycles per second
measured in Hz
F = cycles/t

### Waveform Synthesis

The process of combining several individual sinusoidal motions into a complex waveform

### Harmonics

Frequency components of a complex waveform that are whole-number multiples of its fundamental frequency

### Greatest Common Factor

The fundamental frequency of a complex waveform; how we calculate fundamental frequency

### Missing fundamental

A complex periodic wave in which no component is equal to the fundamental frequency; there is no energy at the fundamental frequency

### Waveform Analysis

the process of breaking down a complex waveform and determine its components

shows amplitude and frequency

### Spectrum

a graphical representation of a complex waveform showing the waveform energy (amplitudes) of the individual components (y axis) arranged in order of frequency (x axis)

### Spectrum components

Individual components arranged in order of frequency

### Continuous spectrum

When thousands of spectral components are in a spectrum they do not appear as individual lines, but because they are so close together, they cannot be differentiated from each other

### Octave

The doubling of frequency

### Impedance

The opposition to the flow of energy through a system

For a given force (F) applied to a mechanic system, the complex mechanic characteristics of the system affect its velocity (v)
F = Z x v

Measured in ohms
relates the velocity of a system to the force acting on the system

### Compliance (C)

The inverse of stiffness (K)

C = 1/K

1. Periodic
2. Aperiodic

### Aperiodic

- Sloppy, cannot make out one period from the next
- Random
- Noise

### Periodic

Repeating pattern

### Frequency domain

the domain for analysis signals with respect to frequency, rather than time.

### Periodic Motion

motion that repeates iteself in regular intervals until it is stopped by external action

### Simple Sinusoidal Motions are graphed

as a function of time
the x axis indicates time
the y axis is the magnitude of a quantity

Example: