Sinusoidal Vibration Basics

This tutorial is about the basic quantities that characterize sinusoidal vibration: amplitude, frequency and phase. Also, some interesting properties of sinusoidal vibrations are explored such as amplitude units conversion and polar representation.

Motivation

Sinusoidal vibration is an idealization. There are few machines that will vibrate in a pure sinusoidal fashion (although a notable example are machines that exhibit pure mass unbalance); most real vibration waveforms are much more complex. However, understanding sinusoidal vibration is useful for a number of reasons:

  • For sinusoidal waveforms it is easy to convert overall amplitude values between peak, peak to peak and rms. It is also easy to convert between acceleration, velocity and displacement.

  • Sinusoidal vibration allows to introduce the concept of phase, which is used in some advanced diagnostic techniques and the basic concept used in rotor balancing: if you want to balance a rotor —and understand that is happening— you must definitely understand sinusoidal vibration and phase.

  • Any vibration waveform, no matter how complex, can be decomposed into sinusoidal components. This fact is the base of frequency analysis, perhaps the most known tool for vibration diagnostics.

Sinusoidal vibration

Sinusoidal vibration is the simplest form of vibration, in which a body moves around an equilibrium position in a periodic and smooth way. Perhaps the best known example of sinusoidal motion is the motion of a mass attached to an ideal spring and subject to no friction.

In a formal sense, a vibration is said to be sinusoidal if it corresponds to a sinusoidal function of time, and it can be described with the following equation:

x(t) = X·cos(2πft − φ)

Such a function looks like this:


Sinusoidal vibration waveform

A sinusoidal waveform is completely determined by three parameters:

  • X, the amplitude.
  • f, the frequency.
  • φ, the phase.

In the following sections we will describe more in detail these quantities and their properties.

Notation

We will use the following notation throughout this tutorial:

Sinusoidal displacement waveform: x(t) = X·cos(2πft − φx)
Sinusoidal velocity waveform: v(t) = V·cos(2πft − φv)
Sinusoidal acceleration waveform: a(t) = A·cos(2πft − φa)

The properties of sinusoidal vibration apply regardless of the physical quantity measured (displacement, velocity or acceleration). For simplicity, in most formulas we will use displacement vibration, x(t); you can replace "x" by "v" or "a" as needed.

Tutorial Contents