Lecture 2: Frequency & Time Domains presented by David Shires Packaging Dynamics Lecture 2: Frequency & Time Domains presented by David Shires Editor-in-Chief, Packaging Technology & Science Chief Consultant, Pira International

Time Domain This may seem obvious but: When we record shock or vibration we record it as a function of time We record / observe the progression of an event as time passes With digital recording our sequence of samples is at regular time intervals

Time Domain We record the data over time Easy to show that data graphically Easy to understand the data in general terms It relates to our experience of the world

Time Domain We can measure the event: When it happened How long it happened for What happened before / after Was it periodic – if so what frequency

Time Domain Sine wave, 0.6G peak to peak, 22.5Hz Easy to define Magnitude Shape Frequency

Time Domain How can we describe this waveform? Is it more, or less damaging than a sine wave at 22.5 Hz?

Time Domain

Frequency Domain Frequency √(G2) Frequency √(G2)

Time Domain Frequency Domain √(G2) See wave shape Identify periodicity / phase Could add absolute time or place detail Difficult to determine frequency content / complexity or simplicity See frequency content See acceleration as function of frequency Could add absolute time or place detail Very difficult to visualise wave shape and no periodicity or phase information

Why are we interested in frequency? Single degree of freedom system

Why are we interested in frequency? Multiple Degrees of Freedom If our input has high energy at 17 Hz our product will be excited If our input has high energy at 26Hz our product will hardly respond

Why are we interested in frequency? When we unitize packages we build columns of similar damped springs The columns have a natural frequency – typically between 10Hz and 25Hz Around the natural frequency the top layers move the most Most vibration damage to packages occurs in the top layers

Why are we interested in frequency? The product inside the packs may have sensitive components with natural frequencies If fn product >> fn pack the packs will filter out vibration at damaging frequencies If fn product << fn pack the packs will transfer the input vibration without change If fn product ≈ fn pack the packs will amplify the input vibration at damaging frequencies

What about shock ? If the pallet is dropped…. Will the packs: transfer the shock to the product filter it out amplify it? This will depend on the frequencies in the shock pulse and the frequency response of the collated goods

Fourier Series French mathematician Fourier showed that any periodic waveform can be defined as a sum of simple waves (harmonics) f(x) = a0/2 + a1 cos (x) + a2 cos (2x) + a3 cos (3x) + . . . + an cos (nx) + b1 sin (x) + b2 sin (2x) + b3 sin (3x) + . . . + bn sin (nx)

f(x) = sin(2x) + 0.7sin(5x) + 0.3sin(14x) Fourier Series f(x) = sin(2x) + 0.7sin(5x) + 0.3sin(14x)

Fourier Transform A Fourier Transform (FT) is a special case of the Fourier series for non-periodic and discreet functions. The function is considered to be a very short portion of a waveform of infinite periodicity The Fourier Transform transforms a waveform from the time domain to the frequency domain.

Fourier Transforms A reverse Fourier Transform transforms a waveform from the frequency domain into the time domain A fast Fourier Transform (FFT) is an algorithm to allow the efficient computation of FT’s.

Random Waveforms Statistically unpredictable Non-periodic Contain all frequencies within a measurement band Transport vibration is similar to random vibration

Power Spectral Density Imagine listening to white noise through a filter White noise is a random wave and contains equal power at all frequencies The narrower the filter’s bandwidth the quieter the noise The wider the filter’s bandwidth the louder the noise We can only quantify volume as a function of bandwidth Even if we don’t filter the noise our hearing, or the speakers, have a bandwidth.

Power Spectral Density In transport vibration we are interested in acceleration and power power is the equivalent of volume in sound power relates to the work done on a package Power ά (acceleration)2 We can only quantify power in terms of bandwidth We are interested in which frequencies of the vibration have high power - we chose a narrow bandwidth – 1Hz Our unit of power spectral density is g2/Hz

Power Density Spectrum (psd plot) Power Density g2/Hz Frequency Hz

The lowest frequency we can determine: = 1/t Hz We can do a FT to discover the frequency content of our random vibration We can do it over a short period, a longer period or the whole record

Shock in Time Domain Examples Half sine 50G shock pulse 3.2 ms duration Magnitude Shape Duration

Shock in Time Domain How do we describe the white shock pulse? Which pulse is most damaging?

Shock in Frequency Domain We can do a FT on the shock pulse: Show us the power spectral density We need also to understand the frequency response of the product or product and pack We need to combine the FT with the frequency response

Shock Response Spectrum 10 20 30 40 50 60 1 2 3 4

Shock Response Spectrum 10 20 30 40 50 60 1 2 3 4

Shock Response Spectrum Specify complex shocks by their SRS Synthesise a (simpler) shock giving the same SRS

Shock Response Spectrum Depending on the SRS of the product: The peak acceleration of the product might be higher or lower than that of the cushion Shock inside product depends on SRS of pulse and natural frequencies of product. It is often higher than at cushion

Summary of Lecture 2 We measure data in the time domain but gain increased understanding of it in the frequency domain Fourier transform is a mathematical tool to analyse data in the frequency domain Products have natural frequencies which increase the risk of damage We can understand shock events by using the shock response spectrum