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The distribution cycle of any good carries a series of risks to the load’s integrity, which includes the vibrations. Measuring the vibration parameters allows for the characterization of this phenomenon in a way that allows companies to simulate the damages caused by this motion during the package’s design and create an optimized packaging that protects the load against this threat.

Everybody has at least one natural resonance frequency. If vibrations are caused on the body at this frequency, it will resonate, vibrating at that frequency. If the addition of energy is maintained by an external source and it exceeds the breaking point, the body will break just as sopranos manage to break cups by using their voice.

An example of the resonance phenomenon is the use of children’s swings. Simplifying the system, the weight of the child and the length of the swing makes it so that they have a resonance frequency that children can intuitively find. In order to make the swing oscillate, it needs to be pushed at the same frequency as its natural frequency. If the chain or rope that it hangs from is not resistant enough, it will break, either because the force reaches the material’s breaking point (which can be linked to fragility) or because of the accumulated stress resulting from the repeated oscillations.

Here it is important to mention two types of failures caused by resonance. The first one is as a result of reaching the body’s fragility by attaining such an intensity that it can no longer resist – such as in the previous voice example – or due to continuous stress. The classic example is the Tacoma Narrows bridge in the United States, which collapsed after having been subject to transversal vibrations caused by the wind.

Vibration parameters: what are they

Vibration is the oscillation of an object around a state of equilibrium. There are many types of vibration: mechanical with linear motions, angular or rotational motions, electromagnetic, gravitational… Not only are we surrounded by them, but we are also immersed in them. Body temperature is nothing but an indication of the level of excitement of electrons as they vibrate. Below we will limit ourselves to discussing mechanical, linear and angular vibrations.

Oscillations are usually classified in many ways and in the case of transportation one needs to define whether one is referring to a fixed or variable frequency vibration, whether it is a single or multiple frequencies, or whether it is periodic or random since the body that experiences the vibrations is subjected to oscillations across the entirety or part of the body. As a result, these oscillations experienced by a body translate into vibration forces.

When performing a vibration analysis using a vibration sensor, an accelerometer is commonly used for linear measurements and gyroscopes for angular measurements.

The following parameters are taken into account for a simple frequency oscillation:

 The frequency

The frequency (f) is the number of times per unit of time that something repeats. The unit used in the International System (S.I.) is the Hertz (Hz), or number of cycles per second.

The period (T) is also used in relation to the frequency – to put it in simpler terms, how often the phenomenon repeats. T=1/f, and angular frequency ω=2πf.

 The acceleration

The acceleration is one of the vibration parameters and is defined as the speed variation per unit of time (the unit used in the S.I. is the m/sec2 ). However, it is quite common to measure the acceleration in g or G, which is the acceleration of gravity, since it is a more intuitive value. 1g=9.81m/s2. Along with the frequency, these are the most meaningful parameters when it comes to measuring vibrations.

The formula for speed is: V(t)= dx/dt = vocos (2πf t) = vosin ( 2πf t+ π/2)

vo: maximum speed or peak value

 The speed

A measurable parameter as part of a vibration motion is the speed at which the particle moves. In this case, the unit of measurement in the S.I. is m/s

The formula for acceleration is: a(t)= dv/dt= d2x/dt2= -aosin (2πf t)

ao: maximum acceleration, or peak value

 The displacement

In addition, the displacement or amplitude of a vibrating particle is defined as the distance between the position of the vibrating particle and its position at rest. In this case, the maximum amplitude of specific vibrations is usually measured, and the unit used is the meter.

The formula for displacement is: x(t)= xosin (2πf t)

x(t):instantaneous displacement
xo: maximum displacement or peak value
f: vibration frequency
ω: angular frequency (2πf)

Wave equations can be complicated by including other vibration parameters that are necessary to characterize the motion of this type, including the frequency of the system itself, the resonance or the damping.

If sea waves are idealized, they can be considered to be the main wave defined by the wave equation u0(x,t), where a (much smaller) ripple caused by the wind is superimposed, defined by the wave equation u1(x,t). The principle of superposition allows us to define complex vibration motions through the addition of wave equations. U(x,t)= u0(x,t)+u1(x,t). In this same way, it could be broadened to infinite generation sources, such as the effect of tides and other wave effects U(x,t)= u0(x,t)+u1(x,t)+….(to learn more you can take a look at the studies of d’Alembert, Euler, Bernoulli, or Lagrange.)

By using the previous equations all the deterministic phenomena could be defined. This idealization is a good approach when, for example, a study is performed on a sea where conditions are always the same, but in reality, the wind can change intensity and direction, changing the system in a non-controlled manner. Even the effect of variations in gravity can affect it. The question is, in which way could the vibration response of something that is not predictable be characterized? The answer: by performing a statistical study of all the vibrations that take place. The vibration parameter in random vibration tests is the power spectral density or PSD.

Power Spectral Density (PSD)

Given that random vibrations are neither predictable nor do they maintain a constant cadence, they can be characterized by the amount of energy that they add for each frequency. In addition, those vibrations are particular to a specific transport – it is like the vibrational DNA of that transport. Therefore, since they are transport-specific, they are unrelated to the material being transported. For this reason, they are defined as a density so that they are applicable regardless of the amount of material being transported. Below we show you the vibration parameters that characterize the PSD of random vibrations:


The calculation of the PSD is related to the Fourier transform and allows for a transition from the temporal domain to spectral-domain properties. From a theoretical standpoint, vibrations take place from 0Hz (a constant value) toward infinity. From a practical standpoint, there are studies that show that they only involve values that are up to 3 orders of magnitude lower than the maximum value. The frequency band where it holds true that intensities are among the 3 highest-intensity orders of magnitude is called the bandwidth. The bandwidth is characterized by the lower cutoff frequency and the upper cutoff frequency, which are simply the minimum and maximum value of the bandwidth. Depending on the transport, the bandwidth can vary, it is not the same for a freight ship than for a delivery van or a plane, but it typically oscillates between 0.5Hz and 200Hz.

RMS Acceleration

This is a numeric value that is obtained by calculating the square root after adding the squares of the intensities for the entire bandwidth. This yields an intuitive value that equates vibrations with a constant acceleration. It is very important to be clear on the fact that it is used to compare intensities in a way that they can be simplistically compared. One same RMS acceleration for one same bandwidth can yield completely different results if the shape of the applied profile is different.


The PSD is defined once the intensities for each point on the bandwidth are known. The different profile shapes characterize the means of transportation.

In a first approximation, once the PSD is known, it can be simulated in a laboratory, considering that the random response is Gaussian. This considers that the statistical distribution of the vibrations matches those of a Gaussian distribution. As a result, it normalizes the vibrations in such a way that the peak of the bell coincides with the RMS value and the ends, for a 3-sigma clip, range between ⅓ of the RMS value and 3 times that value. This results in flattened vibrations when compared to the records. The traditional random vibration analysis treats them as if it were a Gaussian phenomenon, but when the statistics of the vibrations that take place on the road are studied, measures do not match a Gaussian response.

Probability density function (PDF)

When studying how many times the same RMS acceleration values take place, one can find that that distribution is not Gaussian. The intensities accumulate at the lowest values and flatten where the intensity is higher. If simplified, and if the values are treated as Gaussian, this results in a compliant mean value, but the higher-intensity vibrations – which are the most harmful – do not occur.

Therefore, it is also important to perform a study of the probability density function, which studies the way in which the intensities are distributed during the trip. A lot more can be said about the subject, but to keep it simple, the most important parameters are the intensities and duration in time for each package being studied.

Vibration analysis: why you should perform it

Our bodies convey information to us regarding the environment around us. When we are in transport, it allows us to feel many things, including the motions and vibrations that take place. We can hear these vibrations, as well as feel them inside us. If the vibrations or noise become much stronger, they annoy us and we instinctively lower the speed or change trajectories so that it changes. Those vibrations are so hazardous that they are taken into account in occupational safety. And noise is also regulated. Vibration motions cause a number of hazardous effects. In the case of the human body, vibrations between 1 to 400Hz are considered to be the cause of anything from sickness to chronic nervous system disorders.

Just like there are people who listen to music at full volume and others that cannot stand this, there are transported products that are more sensitive to vibrations than others. The only way to know the fragility of a product and the implemented packaging solution is by performing a vibration analysis that allows you to know which frequencies the product is more sensitive to and whether the packing system can dampen those frequencies.

In the case of freight transportation, vibrations can cause damages to the load if the packaging has not been optimized so as to protect it.

Here is where the importance of transportation simulation comes into play as a way to design an optimized product-package system that can protect the goods during the distribution cycle.

It is calculated that logistics companies face losses of 50 billion euros every year worldwide due to damages caused during transportation. Investing in transport simulation and vibration analysis is, therefore, a vital step toward avoiding that situation.

The first step consists of collecting data on the vehicles through recording devices such as innRecord. It is the vehicle the one that will generate the vibrations on the product. It is also important to record those that have the highest intensity, since those will be the worst-case scenario. There are areas in the vehicle where vibrations are amplified and others where they get dampened. There are articles that prove that, in road transportation, the highest-intensity vibrations take place on top of the rear axle, on the side that is closer to the edge of the road – the right side for most countries, the left side for countries such as Australia, UK or Japan where traffic drives on the opposite side. Needless to say, the device must be configured correctly in order to capture data as required.

The amount of data collected makes it unfeasible to perform a manual calculation – or even using spreadsheets. It is for that reason that a specific analysis program must be used, such the Drone Software by Safe Load Testing Technologies, where you can see the collected data and appropriately select it.

Automated processing of all the data is not enough. If this were the case, all records would be averaged out, which would lead to the time that the vehicle was stopped smoothing the results. In addition, some transportation processes use several means of transportation. If all data is processed as one when vehicles change, this would result in a mixed vibration profile, which would not be ideal.

When processing the data, the time that the vehicle remains stationary must be eliminated. It is necessary to identify irregularities, such as shocks, which need to be processed separately, since they increase the intensity, but since they are transient, if they are included as vibrations, they never repeat. Some specific programs, such as the Drone Software of the Data Recorder, offer tools to help with this processing.

Once the data has been cleansed you obtain the PSD and PDF, which can be good results for analysis, but are not usable for the simulation. First, because it is necessary to identify the bandwidth, and second, because the PSD can have hundreds or even thousands of points. The PSDs defined in the thousands of points are not directly applicable. Most simulation applications only allow for a handful of points. Others admit a few tens of points. In the case of the Safe Load Testing Technologies’ control software, it can use hundreds. But the PSDs usually have thousands. This implies the need to adapt the PSD profile to the machine. In most situations, users must perform this adaptation themselves. In the case of the DR Software, it offers a tool that allows the user to easily define this profile.

How to measure the vibrations

There are different ways to measure vibrations depending on their nature. For vibrations along with the Cartesian coordinates, accelerometers are used, and for angular coordinates, gyroscopic systems are mostly used.

Measuring the vibration parameters for transportation requires special equipment that can be shipped, which consists on installing a measuring device and vibration recorder that can perform a vibration analysis an measure the intensity of this hazard along a specific route.

By accessing this information, it is possible to perform a transport simulation using equipment specially designed for vibration testing.

How to simulate the vibrations that take place during transportation

Once the information has been recorded and analyzed, it can be applied by means of two alternatives:

  • Vertical vibration table
  • Machine designed to simulate the vertical vibration motions produced by several shipping & handling means of transportation.
  • Pitch&Roll Module

Safe Load patented solution that includes the pitch and roll motions that take place during transportation when measuring vibrations. It is possible to attach this module to any vibration table in order to obtain results that are closer to real-life conditions.

This allows for the simulation of a multi-axis vibration, which will soon be included among ISTA protocols and other international transportation standards

Do you want to know more about vibration parameters and about preventing the vibrations that take place during transportation from affecting your products? At Safe Load TT, our track-record of over two decades in the transport simulation sector and packaging industry are at your service. Contact us and let’s talk about how we can help you.

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