1. 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

2. 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

3. 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

4. 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

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

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

7. Time Domain

8. Frequency Domain
Frequency
√(G2)
Frequency
√(G2)

9. 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

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

11. 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

12. 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

13. 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

14. 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

15. 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)

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

17. 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.

18. 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.

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

20. 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.

21. 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

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

23. 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

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

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

26. 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

27. Shock Response Spectrum10 20 30 40 50 60 1 2 3 4

28. Shock Response Spectrum10 20 30 40 50 60 1 2 3 4

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

30. 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

31. 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