Inner Pressure Inspection


by Ultrasounds


Introduction

utitled-1024x7661x med hr

In these web pages we have frequently remarked the necessity to answer the fundamental question:

…is it safe, can be sold that bottle?

attacking the basic problem simultaneously by different sides.   Where, sides are the physical principles used by the measurement instruments (the inspections, also named controls) interfering with the object, in our case a container.   We saw with plenty of details elsewhere in this web site:

  Radiator and acoustic transducer of an Inner Pressure Inspection, providing information about cap’s malpositioning and container's pressure, adopting ultrasounds to establish the cap-detector state relation

  • what a role is played by Time during a measurement,
  • in what a way a measurement could negatively affect the measurements’ repeatability,
  • the baffling meaning of the word repeatability.

In the following, we’ll show how an inspection determines by the analyses of the acoustic reaction of a closure stimulated by ultrasounds, that a container has any or else of the following typical defects: 

  • inner pressure too-low, suggesting it’d be a leaking,
  • inner pressure too-high, suggesting it’d be a deformed can,
  • punctured can,
  • cocked crown-cork cap,
  • missing sealing compound,
  • broken thread finish.


Cap-Transducer Correlation 

It is necessary Time to establish a relation between a cap (“Cap”) and the Inner Pressure inspection by ultrasounds (“US”) starting chain of the Inner Pressure Inspection by Ultrasounds, both differentially related with their Environment (“Environment1”, Environment2”).   Time to transform the previous state, in which all possible kinds of correlation coexist, in a following state in which the inspection is “aware” to be correlated to a certain status of the Cap, after having recorded eigenvalues for the eigenfunction ΦiS1 describing a Cap.  Quantomechanical explanation of the measurement process, unaffected by the circularities implicit in the classic explanations.  The vertexes represent interferences

 It is necessary Time to establish a relation between a cap (“Cap”) and the Inner Pressure inspection by ultrasounds (“US”) starting chain of the Inner Pressure Inspection by Ultrasounds, both differentially related with their Environment (“Environment1”, Environment2”).  Time to transform the previous state, in which all possible kinds of correlation coexist, in a following state in which the inspection is correlated to a certain status of the Cap, after having recorded eigenvalues for the eigenfunction ΦiS1 describing a Cap.  Quantomechanical explanation of the measurement process, unaffected by the circularities implicit in the classic explanations.  The vertexes represent interferences













Whatever physical system (micro or macroscopic), caps and lids both included, can be fully represented by its wave function also named, state vector.  The physical meaning of the state vector becomes apparent when making a measurement.  Then the state of the system assumes one of the eigenstates, with probability given by the Born rule, and the result of the measurement is the corresponding eigenvalue.  To definetely perceive a Cap-Ultrasound Transducer state, it is necessary:

  1. Time, to transform the previous state, in which all possible kinds of correlation of the inspection (a measurement instrument also including an acoustic transducer) coexist, in a following state in which the inspection can be considered “aware” to be correlated to a cap, because having recorded eigenvalues for the eigenfunction ΦiS1 describing a cap.  The correlation between the two systems US inspection and Cap, is progressively established during interaction and proportional to the natural logarithm (ln t) of the interaction time t.   An ideal correlation using a single inspection system (a system interfering with the container by mean of a single physical principle), corresponding to a maximised information of the inspection about the cap, can only be reached allowing an infinite time.   This causes the measurements’ fluctuations, a synonimous of the spectrum of the eigenvalues, resulting in the Electronic Inspector’s Inner Pressure inspection’s False Positives (false rejects).   

  At containers’ speed > 2 m/s, the total Time available to establish the correlation between Object and each Detector, following the kind of Detector and Object reduces itself to < 12 ms










  1. Interaction between the systems such that the information in the marginal distribution of the object inspected is never decreased.   In a probability distribution deriving by two random variables, we remember that marginal distribution is where we are only interested in one of them.  Otherwise, we’d have forced a reduction in the sample space of one of the random variables and then, we could not have any more repeatability of the measurements.  As an example, this should be the case if we’d try to detect the position of the cap making use of a beam of high energy neutrons.  Neutrons, rather than low energy photons at the proper frequencies ranging (30 - 300) kHz of the ultrasounds, where the power involved to interact with the closure is reduced to milliwatts and the energy transferred along milliseconds.  The neutrons could modify the molecular structure of the Cap, modifying its eigenstates and then the eigenvalues we expected to derive by the measurement.



Membranes and Closures

Ideal Circular Membranes

drum-large

An initial and basic model ignores properties important for real membranes like the crown-corks, threaded caps or lids, like their bending stiffness, the non-uniform material density, the atmospheric pressure, etc.  Here, an ideal circular membrane is defined by its: 

 Oscillation characteristic values (eigenvalues) are the modes of vibration of the upper surface of a drumhead, each one oscillating its own frequency (  Kreso, 1998-2010)


  • diameter d                     [m] 
  • surface Tension T          [N/m] 
  • superficial density ρ      [kg/m2]

In common with the fixing of cans’ lids and bottles' crown-corks the fact that also the ideal model refers to circular membrane fixed at its circumference.  After receiving energy by a stroke, the membrane vibrates with a superposition of nodal modes of order (m, nwhere the numbers (m, n) designate: 

  • m,         nodal diameters,
  • n,          nodal circles, including the circular boundary.

and whose associated frequencies are νm,n.    On the surface of an ideal membrane, as well as recorded on the surfaces of a drum, can or crown-cork, these nodal lines and circles may be imagined (see figure at right side, above) as defining regions.  Regions are the areas whose vertical (z-axis) displacement for a special mode is minimal.  The term node is here an extension to 2D vibrating membranes of its original use referred to a fixed point in a vibrating 1D string.  Following the combination of nodal diameters and circles, the modes can be divided as:

  1.    m = 0,  n ≥ 1,           circular modes, 
  2.    m ≥ 1,  n = 1,           radial modes,  
  3.    m ≥ 1,  n ≥ 1,           mixed modes,  

At right side three examples of the vibrational modes of a timpano where (m, n) = (1, x) for x = 0, 1, 2.   Visibly, the combination of nodal diameter and circles: 

            (m, n)  =  (1, 0)

is a purely circular mode.  An impulse applied about one-sixth of the diameter away from the edge of the drumhead excites the (m, n) =  (1, 1) vibrational mode.  The frequency ν of the fundamental mode (0,1) is:         





In our special case of a membrane closing a timpano, the ratios existing between the higher frequency modes νm,n and the fundamental mode ν0,1 are in general non-harmonic and named overtones.   



Real Circular Membranes






In the couple of graphics below a visual example of what can be recorded after striking the membrane over a drum.    The upper graphic shows the function y = f(t) of the amplitude versus Time t and the lower the function y = f(ν) of the amplitude y versus Frequency ν.    A frequency spectrum relevant to understand the Inner Pressure Inspection by mean of ultrasounds.    Below, the Amplitude y is measured in [dBFS]: decibel amplitude levels in digital systems having a defined maximum available peak level.     As a matter of fact, we’ll re-encounter a similar damped spectrum in the section: “Vibrations’ damping in association to Leaking Containers”, after a crown-cork or lid have been excited by the energy conveyed by ultrasounds.

 Vibrations’ amplitudes, frequencies and damping well visible in this record of what follows a drum stroke.  The graphics represent the results in the Time and Frequency Domains.  The circular (0,1) mode at a frequency ν0,1 of 140 Hz is marked in the spectrum below (  A. Wagner/2006)


The following video, better than many other kinds of media, illustrates the meaning of the terms we are using: modes of vibrations, normal modes, harmonics, eigenvalues and eigenvectors, resonance, stationary waves, in the practical case of a liquid.



Convolution Operator and Leaking 

Containers

The properties of what is used to strike play an important role in the excitation of the modes. When striking by mean of ultrasounds a circular membrane modelled to be a closure, the electromagnetic energy is transferred from a Generator in the Electronic Inspector, to the entire container.   Entire container, and not just to its closure, lid, threaded cap or crown-cork.  All of the surronding objects, containers, Conveyor belt and atmospheric medium lie in the Generator-driven electromagnetic field.   The form of the function is a complex superposition of several terms, some of them variables.  In the following equations, we’ll approximate temporarily the relation as if it’d be established only between the em field and the membrane-like metal closure.   A meaningful approximation because the closure is the object most strictly related to the ultrasound em field. The energy transfer shall manifest itself as a impulse of force applied by the em field to the closure.  Interaction equivalent to the action of a convolution operator, whose step-by-step definition appears in the figure at right side. Convolution in the Time domain t of the applied force pulse f (t) and the associated response to the impulse of the closure as vibrational system h (t). This results in the vibration s (t) of the closure, where the symbol  identifies the convolution operation:

 Convolution operator or superposition integral.  A system is entirely characterized by its response to an impulse function δ (t), because its forced response to any arbitrary input u (t) may be computed from knowledge of the impulse response alone (  Massachusetts Institute of Technology/2014)






      s (t)  =  f (t)  h (t)    


equivalent to a multiplication in the Frequency ν domain, where the spectrum F(jω) of the force pulse determines the highest modal frequency excited in the container:

membranes irregularities ha med hr


                         S (jω)  =  F (jω) · H (jω)


where ω = 2 π ν.  The vibration of a real membrane like a thin metal closure, implies: 

 The lid of a can resembles a drum-head when looking some of its non-harmonic features. The stay-tab opening mechanism characteristic of most drinking cans, the cuts in the thin Aluminium membrane and its special shape, alter the modal structure (  Marcos Andre/CC BY 2.0 /2006)

  1. bending and shearing of the material.    The elastic deformations resisted until a certain degree due to the thickness and properties of the membrane material.    Resistances which will raise the modal frequencies of the membrane;
  2. altered modal structure, because of irregularities in the material.  As an example, additional rings or layers of mylar on the membrane are used to specifically create irregularities in the membrane structure. These modifications then dampen or shift certain frequency modes and influence the sound of the drum, e.g. make it more harmonic;
  3. air loading.    All bottles or cans are volumes of space closed by the respective crown-corks, threaded caps or lids.  Thus resembling that kind of drum correctly named “timpano” or, eardrum.   It is the closed air volume of the timpani’s kettle what originates a shifting of modes resulting in a nearly harmonic ratio of modes.   This, is true also for the bottles or cans closed by metal resonating membranes: caps or lids; 
  4. damping.    A relevant difference between a timpani’s kettle and our Food and Beverage Bottling containers, is that in our case the equivalent of the timpani’s kettle is mainly occupied by a liquid.  Also, that in all Inner Pressure Inspection systems, the entire container is pressurised.  Reduced volume and gaseous pressurisation both acting as damping factors;
  5. container and liquid vibrations.  Although the motion of the container (can or bottle) shell is extremely small when compared to the displacement of the lid, threaded cap or crown-cork, a strong coupling exists between the closure and the container.  Then, container motion leads to a small however not nil contribution of the container and liquid’s modes, to the final superposed spectrum commonly attributed only to the closure.   

 How a crown-cork resembles a drum-head membrane?  The upper view recalls a circular membrane. Below: the cut of a pry-off crown evidences its plastic sealing gasket. Sealing gasket reducing the mechanical coupling between bottle and closure.  Also, participating to damp the vibrations of the thin (0.22 ± 0.01)mm steel membrane, following the model weighing (175 - 220)mg (abridged by   Vitogiannis Bros. SA/2014)



  The Inner Pressure inspection by Ultrasounds needs the highest possible inner pressures in the containers. That’s why it is typically applied to foaming beverages after the Labeller and Filler Machines. Visible here the Labeller Machine area at the Brewery SAB MIller plc, at Alrode, South Africa



Containers’ Inner Pressure and Ultrasounds


Sequence of the Inner Pressure Inspection by Ultrasounds.  The entrance of a container in the inspection window of an ultrasound transducer, laser-Triggers the consecutive phases of ultrasounds’ emission and following Cap’s or Lid's vibrations. The harmonic content of the waves re-irradiated by Caps in good conditions, applied over bottles whose inner pressure lies in a standard range, also lies in a standard range of energy, frequency and damping domains. Same way,  it may deviate out of those standards due to the superposed effects of changed conditions of the Caps’ or Lids’ and of container’s inner pressure


“The energy transfer shall be manifest as a force applied by the em field to the closure.   Interaction equivalent to the action of a convolution operator  in the Time Domain of the applied force pulse f (t) and the pulse response of the closure as vibrational system h (t)”


The vibrations of the metal closures, after having been radiated by ultrasounds, can be used to verify their status in terms of container’s:

  • excessive pressure;
  • insufficient pressure;
  • not closed.
Leaking metal crown corks. The vibrations of the metal closures, after having been radiated by ultrasounds, can be used to verify their status in terms of container’s:

excessive pressure;
insufficient pressure;
not closed.
Leaking metal crown corks 1280x833@1x
Caps are radiated by an ultrasound oscillator, a transducer electric → electromagnetic,  coupled (a synonimous of the more precise correlated) to a microphone-like transducer responsible for the electromagnetic → electric conversion.  Like in figure below at right side, they'll vibrate as hitted drums, with a tone whose:

amplitude;
harmonic content;
duration;
energy;
damping;
  The Inner Pressure inspection by Ultrasounds needs the highest possible operative inner pressures in the containers. That’s why it is typically applied to foaming beverages after the Labeller and Filler Machines.  Also, it may be applied also to CO2-added mineral water-filled glass bottles

is conditionally related the incoming electromagnetic wave via the intrinsic physical properties of the cap or lid, or threaded cap, following its own:

harmonic content;
duration of the action;
distance of the radiator;
tilt angle existing between cap’s and ultrasound radiator’s axes;
skew angle existing between cap’s and ultrasound radiator’s axes;
and to external conditions, like the:

atmospheric humidity; 
atmospheric pressure;
atmospheric temperature;
other factors whose effects have minor extent;
Correlation cap ultrasound radiator. Containers’ Inner Pressure and Ultrasounds



“The energy transfer shall be manifest as a force applied by the em field to the closure.   

Interaction equivalent to the action of a convolution operator  in the Time Domain of the applied force pulse f (t) and the pulse response of the closure as vibrational system h (t)”

The vibrations of the metal closures, after having been radiated by ultrasounds, can be used to verify their status in terms of container’s:

excessive pressure;
insufficient pressure;
not closed.
Caps are radiated by an ultrasound oscillator, a transducer electric → electromagnetic,  coupled (a synonimous of the more precise correlated) to a microphone-like transducer responsible for the electromagnetic → electric conversion.  Like in figure below at right side, they'll vibrate as hitted drums, with a tone whose:

amplitude;
harmonic content;
duration;
energy;
damping;
is related to properties of the incoming electromagnetic wave, following its own:

intrinsic physical properties of the cap or lid, or threaded cap;
harmonic content;
duration of the action;
distance of the radiator;
tilt angle existing between cap’s and ultrasound radiator’s axes;
skew angle existing between cap’s and ultrasound radiator’s axes;
and to external conditions, like the:

atmospheric humidity; 
atmospheric pressure;
atmospheric temperature.  [The mathematic of the 2D motion of the surface of a drum, metal cap or cans’ lid is relatively complex.   That’s why we suggest to read here (Case Study 3 - Principle of Superposition - Modes of vibration of a piano string) its simpler 1D case, of which the 2D is just a generalisation to a wider space].    A sounded drum or metal cap, results in frequencies that are related in a complex way.   After striking a drum with a mallet, the frequencies of the characteristic modes of vibrations (eigenvalues) are intimately related to the shape of the drum.    For what it refers to the relation between vibration’s amplitude y and vibration's energy E, we know that increasing the amplitude of a sound wave by a factor of two means that it carries four times as much energy.


“The energy transfer shall be manifest as a force applied by the em field to the closure.   

Interaction equivalent to the action of a convolution operator  in the Time Domain of the applied force pulse f (t) and the pulse response of the closure as vibrational system h (t)”

[The mathematic of the 2D motion of the surface of a drum, metal cap or cans’ lid is relatively complex.  That’s why we suggest to read here (Case Study 3 - Principle of Superposition - Modes of vibration of a piano string) its simpler 1D case, of which the 2D is just a generalisation to a wider space].    A sounded drum or metal cap, results in frequencies that are related in a complex way.   After striking a drum with a mallet, the frequencies of the characteristic modes of vibrations (eigenvalues) are intimately related to the shape of the drum.   For what it refers to the relation between vibration’s amplitude y and vibration's energy E, we know that increasing the amplitude of a sound wave by a factor of two means that it carries four times as much energy.

Caps are radiated by an ultrasound oscillator, a transducer electric electromagnetic,  coupled (a synonimous of the more precise correlated) to a microphone-like transducer responsible for the electromagnetic  electric conversion.  Like in figure below at right side, they'll vibrate as hitted drums, with a tone whose:

  1. amplitude;
  2. harmonic content;
  3. duration;
  4. energy;
  5. damping;

  The Inner Pressure inspection by Ultrasounds needs the highest possible operative inner pressures in the containers. That’s why it is typically applied to foaming beverages after the Labeller and Filler Machines.  Also, it may be applied also to CO2-added mineral water-filled glass bottles



is conditionally related the incoming electromagnetic wave via the intrinsic physical properties of the cap or lid, or threaded cap, following its own:

  1. harmonic content;
  2. duration of the action;
  3. distance of the radiator;
  4. tilt angle existing between cap’s and ultrasound radiator’s axes;
  5. skew angle existing between cap’s and ultrasound radiator’s axes;

and to external conditions, like the:

  1. atmospheric humidity; 
  2. atmospheric pressure;
  3. atmospheric temperature;
  4. other factors whose effects have minor extent;

[The mathematic of the 2D motion of the surface of a drum, metal cap or cans’ lid is relatively complex.  That’s why we suggest to read here (Case Study 3 - Principle of Superposition - Modes of vibration of a piano string) its simpler 1D case, of which the 2D is just a generalisation to a wider space]. A sounded drum or metal cap, results in frequencies that are related in a complex way.  

  The Inner Pressure inspection by Ultrasounds’s operative principle in synthesis

After striking a drum with a mallet, the frequencies of the characteristic modes of vibrations (eigenvalues) are intimately related to the shape of the drum.   For what it refers to the relation between vibration’s amplitude y and vibration's energy E, we know that increasing the amplitude of a sound wave by a factor of two means that it carries four times as much energy.


 The area of the Line layout where beer bottles are checked by ultrasounds with maximal Signal-to-Noise ratio, is after the Pasteuriser Machine



Trying to Hear the Shape of a Manifold

The Limits of the Inner Pressure Inspections

Harmonic content of defective and correctly closed containers may be identical. At first sight, a round drum will sound differently from a square or octagonal drum, what is true also for drums made of the same material, and with same surface.   None of these drums will produce a sound with a definite pitch.   Each drum or metal cap shape produces its own “signature” in the form of a well definite ensemble of harmonic frequencies and related amplitudes.   

But, in 1992 it was mathematically demonstrated that two differently shaped drums, also if cubic and cylindrical, can sound exactly the same set of frequencies.





























What provides the physical explanation for the difficulty, the reduced repeatability of whatever electronic inspection trying to discriminate container’s inner pressure or a defective metal cap, in a population of other containers with correct inner pressures or correctly applied, non-leaking caps.   Imagine you hear a vibration arising by a metal can or crown-cork cap stricken.   You hear and record (but cannot see directly what is the shape of) that lid or crown-cork.   What after 1992 is clear is that also the full knowledge of the entire the entire spectrum of the vibrations, does not grant the possibility to infer the shape of what originate them.

The inference remains limited to just:

area
circumference.
The containers’ inner pressure and/or cap/closure/lid inspection by Ultrasounds, results sensible to several macroscopic factors, and between these:

diameter of the cap;
tension of the cap surface.
The last property, being an indirect index of the inner pressure exercised by the fluid existing on the inner side of the cap, into the bottle.

False rejects arising by the Inner Pressure Inspection by Ultrasounds are well known. Conversely, the difficulty to discriminate an actually defective closure mixed to correctly closed containers.  Below at right side a practical example referred to a glass container closed with a common crown-cork, a Production of 50000 bottles-per-hour, inspected at a linear speed ~1.5 m/s.  Here, a container encircled in a red-colour box, shall be rejected only thanks to the fact that it deviated excessively out of the programmed limits in the Frequency Domain.   

domain of frequency amplitu med hr


 The oscillation characteristic values (eigenvalues), or modes of vibration of the upper surface of the membrane of a drum, may be similar.  A container, encircled in a red-colour box, shall be rejected only thanks to the fact that it deviated too much out of the programmed limits in the Frequency domain.  Meaning that the truly defective leaking container was erroneously considered “correct” (Negative) when analysed in the Damping, Energy and Amplitude domains 








Meaning that the truly defective leaking container was erroneously considered “correct” (Negative) when analysed in the Damping-, Energy- and Amplitude-domains.  At first sight, a round drum will sound differently from a square or octagonal, what is true also for drums made of the same material, and with same surface.  None of these drums will produce a sound with a definite pitch.  Each drum or metal cap shape signs with its own signature: a superposition of terms of amplitude y  and non-harmonic overtones of frequency ν.  This, at least, is what until twenty-five years ago was considered true by the Vendors’ Technical Departments developing the Inner Pressure Inspections by Ultrasounds.


Inner Pressure Inspection’s False Rejects

and Convex Sets Theory


Jean-Pierre Serre 1920x1304@1x

  The real problem underlying all equipments trying to infer the status of an object by the exam of the vibration eigenfunctions, like the Inner Pressure inspections by Ultrasounds, started to be understood only after studies of mathematicians published in 1989 (  Dars, et al./2008)

The key question can be rigorously formulated as: 














what Information about a convex compact body (like a can or bottle), closed by a clamped membrane, can be obtained from the sequence of eigenvalues of the eigenvalue problem applied to a Helmholtz wave equation ?    

Yet in 1911 the great German physicist and mathematician Hermann Weyl demonstrated that the area of a vibrating plane membrane is determined by the sequence of its eigenfrequencies.   As a consequence, it was later conjectured that the membrane shape could also determined by this sequence.   In opposition to this conjecture, we could ask ourselves if is it possible to construct two drum heads with different shapes, as an example one cylindric and the other cubic, that share a set of eigenfrequencies.   Around fifty years ago the Polish mathematician Marc Kac published his own comments about the asymptotic behavior of the eigenfrequencies, in the limit of very high frequencies but the subject remained only intuitively defined, based over conjectures.   Readers may think that the point is just academic.   But this is not the case, because all lids and caps presenting themselves to an Inner Pressure Inspection by Ultrasounds, keeping apart exceptions accounting for < 0.001 % of them, have similar area.    But it’s a fact that the defective lids or caps mainly differ by the population because of their shape.   Clearly, if the two drums, lids or caps have an identical set of eigenfrequencies then they would have the same timbre.  Thus transferring the same set of tones to the acoustic transducer of the Inner Pressure Inspection by Ultrasound, even though their shapes are different.   

Convexity


 A set of points in the space is named convex if, whenever it contains two points, it also contains the line segment joining them. The set at right side is not convex because a line segment cannot be traced between all its couples of points (  abridged by Webster/1994)








Drums, seamed cans or crown-cork capped bottles represent examples of the general concept of convex manifold.   To satisfy the requisite described before means that they could exist differently shaped isospectral manifolds.   If this does happen, it means it exists a natural limit preventing us the minimisation of the inspection’s False Rejects.  The question remained unsolved until 1992 when Gordon, Webb, and Wolpert finally constructed two polygons with an identical set of eigenvalues.   Essentially distinct non-convex isospectral membranes.   A visible example of their discovery in the figure below. In the left and right side images, a polygon appears composed of four identical triangles. Identical but coloured in different tones of blue.   

  Understanding the isospectral membranes. Front of you a polygon composed of four identical triangles, all of them coloured in different tones of blue. Around the polygon, three identical triangles red-, grey- and black-coloured. The total area of the polygons at left and right side is the same, the shapes are visibly different.  Against our intuition, the spectrum of the eigenvalues of the polygons at left and right is identical (  abridged by Gruber/2007)





“The pair of isospectral planar regions are not geometrically congruent.   

And, against our intuition, the spectrum of the eigenvalues of the polygons at left and right is identical”





Closures have same area different shape. Isospectral membranes. The key question can be more rigorously formulated as: 

























what Information about a compact body (like a can or bottle) closed by a clamped membrane, can be obtained from the sequence of eigenvalues of the eigenvalue problem applied to a Helmholtz wave equation ?    

Yet the great German physicist and mathematician Hermann Weyl in 1911 demonstrated that the area of a vibrating plane membrane is determined by the sequence of its eigenfrequencies.    As a consequence of this, it was later conjectured that the membrane shape is also determined by this sequence.   In opposition to this conjecture, we could ask ourselves if is it possible to construct two drum heads with different shapes, as an example one cylindric and the other cubic, that share a set of eigenfrequencies.   Around fifty years ago the Polish mathematician Marc Kac made his comments about the asymptotic behavior of the eigenfrequencies in the limit of very high frequencies but the subject remained only intuitively defined and based over conjectures.   The point is not just academic.   All lids and caps presenting themselves to an Inner Pressure Inspection by Ultrasounds, keeping apart exceptions accounting for < 0.001 % of them, have similar area.   But it’s a fact that the defective lids or caps mainly differ by the population because of their shape.   Clearly, if the two drums have an identical set of eigenfrequencies then they would have the same timbre and sound the same tones to the acoustic transdure of the Inner Pressure Inspection by Ultrasound, even though their shapes are different. Drums, seamed cans or crown-cork capped bottles represent examples of the general concept of convex manifold.   To satisfy the requisite described before means that they could exist differently shaped isospectral manifolds.  If this does happen, it means it exists a natural lower limit impeding us to minimise the inspection’s False Rejects.  The question remained unsolved until 1992 when Gordon, Webb, and Wolpert finally constructed two polygons with an identical set of eigenvalues.  Essentially distinct non-convex isospectral membranes.  A visible example of their discovery in the figure below. in the left and right side images, a polygon appears composed of four identical triangles, all of them coloured in different tones of blue.  Around the polygon, three identical triangles red-, grey- and black-coloured.  The total area of the polygons at left and right side is the same but the shapes are visibly different.  Against our intuition, the spectrum of the eigenvalues of the polygons at left and right is identical.  From a mathematical point of view lids and caps, the circular metal membranes used to close a can or a bottle, are external surfaces of convex bodies, themselves a subset of the manifolds.  Manifolds are topologic spaces resembling the Euclidean space near each one of their points.   Lids and caps are homogeneous elastic membranes clamped on smooth closed Jordan curves of given area.   Thus providing also a physical explanation for the reduced repeatability of whatever electronic inspection equipment trying to discriminate particularly low or high container’s inner pressures or a defective metal cap, in a population of other containers whose inner pressures or caps are correct.   Imagine to hear with your ears a vibration arising by a metal can or crown-cork cap stricken.   To hear and record the sounds arising by that closure without be given the possibility to see directly what is the shape.   Thus implying an additional way to establish a correlation between that closure and you.  This is the main limit affecting all Inner Pressure inspections: actually defective closures have the same area of the actually correct, only differing in their shape.  In the meantime, the measurement equipment is trying to discriminate using a technology insensible to the shape and sensible to the area.

Around the polygon, three identical triangles red-, grey- and black-coloured.   The total area of the polygons at left and right side is the same but the shapes are visibly different. The pair of isospectral planar regions visible above are not geometrically congruent.   And, against our intuition, the spectrum of the eigenvalues of the polygons at left and right is identical.  From a mathematical point of view lids and caps, the circular metal membranes used to close a can or a bottle, are external surfaces of convex bodies, themselves a subset of the manifolds.  Manifolds are topologic spaces resembling the Euclidean space near each one of their points.  Lids and caps are homogeneous elastic membranes clamped on smooth closed (Jordan) curves of given area.  What before provides the explanation built on the most solid ground of Physics, for the reduced repeatability of whatever electronic inspection equipment trying to discriminate particularly low or high container’s inner pressures or a defective metal cap, in a population of other containers whose inner pressures or caps are correct.  Reduced repeatability translated in high false reject ratios.  Or, conversely, false negatives, actually defective closures passed to the Market.  Imagine to hear with your ears a vibration arising by a metal can or crown-cork cap stricken.  The main limit affecting all Inner Pressure inspections are actually defective closures having the same area of the actually correct, only differing in their shape.  In the meantime, the measurement equipment trying to discriminate the defective using a technology insensible to the shape and sensible to the area.   



All lids and caps presenting themselves to an Inner Pressure Inspection by Ultrasounds, keeping apart exceptions accounting for < 0.001 % of them, have similar area.   But the defective lids or caps mainly differ by the population because of their shape






How to “Hear” the Shape of a Manifold?
























We are now understanding what really an Inner Pressure Inspection by Ultrasounds is trying to do: to hear the shape of a manifold.    


Instrumental answer

A practical, instrumental answer is that all vibrating objects, be they drumheads, closures, lids, motors or atoms have characteristic vibration frequencies, named eigenfrequencies.  Then, at a first sight, knowing the eigenfrequencies we can really gain information about the vibrating object.  Yet a full understanding of the entire set of relations existing between a vibrat­ing system and its eigenfrequencies has remained elusive.  As an example, an inspection equipment estimates eigenvalues and eigenfrequencies by mean of a finite set of measurements.  Finite set of measurements corresponding to a limited Time available to have a projection in the sample space of an object whose full existence is in the state space.  Comprehensibly, there is no fi­nite set of measurements allowing our equipments to completely captures the shape of a general planar region.  And we remember here the detection condition expressed above following which to definetely perceive a Cap-Ultrasound Transducer state, it is necessary Time.  Time to transform the previous state, in which all possible kinds of correlation of the inspection (a measurement instrument also including an acoustic transducer) coexist, in a following state in which the inspection can be considered “aware” to be correlated to a cap, because having recorded eigenvalues for the eigenfunction ΦiS1 describing a cap.   The correlation between the two systems US inspection and Cap, is progressively established during interaction and proportional to the natural logarithm (ln t) of the interaction Time t.  An ideal correlation using a single inspection system (a system interfering with the container by mean of a single physical principle), corresponding to a maximised information of the inspection about the cap, can only be reached allowing an infinite Time.  This causes the measurements’ fluctuations, a synonimous of the spectrum of the eigenvalues, resulting in the Electronic Inspector’s Inner Pressure inspection’s False Positives (false rejects).  Then, the practical, instrumental answer is negative: we cannot expect a finite sequence of frequencies to contain adequate information about the shape of the region.   

Manifold shape. How to hear the shape of a manifold ?













































We are now understanding what really an Inner Pressure Inspection by Ultrasounds is trying to do: to hear the shape of a manifold.    



Instrumental answer

A practical, instrumental answer is that all vibrating objects, be they drumheads, closures, lids, motors or atoms have characteristic vibration frequencies, named eigenfrequencies.   Then, at a first sight, knowing the eigenfrequencies we can really gain information about the vibrating object.   Yet a full understanding of the entire set of relations existing between a vibrat­ing system and its eigenfrequencies has remained elusive.    As an example, an inspection equipment estimates eigenvalues and eigenfrequencies by mean of a finite set of measurements.  Finite set of measurements corresponding to a limited Time available to have a projection in the sample space of an object whose full existence is in the state space.  Comprehensibly, there is no fi­nite set of measurements allowing our equipments to completely captures the shape of a general planar region.   And we remember here the detection condition expressed above following which to definetely perceive a Cap-Ultrasound Transducer state, it is necessary Time.  Time to transform the previous state, in which all possible kinds of correlation of the inspection (a measurement instrument also including an acoustic transducer) coexist, in a following state in which the inspection can be considered “aware” to be correlated to a cap, because having recorded eigenvalues for the eigenfunction ΦiS1 describing a cap.   The correlation between the two systems US inspection and Cap, is progressively established during interaction and proportional to the natural logarithm (ln t) of the interaction Time t.   An ideal correlation using a single inspection system (a system interfering with the container by mean of a single physical principle), corresponding to a maximised information of the inspection about the cap, can only be reached allowing an infinite Time.   This causes the measurements’ fluctuations, a synonimous of the spectrum of the eigenvalues, resulting in the Electronic Inspector’s Inner Pressure inspection’s False Positives (false rejects).   Then, the practical, instrumental answer is negative: we cannot expect a finite sequence of frequencies to contain adequate information about the shape of the region.   Theoretical answer

The theoretical answer, one based over Differential Geometry and Topology also resulted negative.   As an example, in the figure above is represented a sphere.   Here, the geodesics are the arcs of great circle never deviating from the direction in which they are traveling.  The Equator being a known example.  The list of vibration frequencies of a vibrating manifold is related to the list of lengths of closed geodesics on the manifold.   This because the vibrations themselves that we are trying to hear propagate as waves along geodesics.   Yet in 1964 it was showed that they coexist different manifolds whose list of lengths of closed geodesics is the same.   

 Since 1964 it is known that they coexist manifolds whose shape and size are different but whose sets of eigenfrequencies are identical


Theoretical Answer

The theoretical answer, one based over Differential Geometry and Topology also resulted negative.  As an example, in the figure on side is represented a sphere.  Here, the geodesics are the arcs of great circle never deviating from the direction in which they are traveling. The Equator being a known example.  The list of vibration frequencies of a vibrating manifold is related to the list of lengths of closed geodesics on the manifold.  This because the vibrations themselves that we are trying to hear propagate as waves along geodesics. And, yet in 1964 it was showed that they coexist different manifolds whose list of lengths of closed geodesics is the same.   

 Since 1964 it is known that they coexist manifolds whose shape and size are different but whose sets of eigenfrequencies are identical (  Reis/2014)



Integrating Inductive Controls aside of the 

Ultrasounds' Inner Pressure Inspections














The Inner Pressure Inspection by Ultrasounds is one of those where a redundant inspection based over a different physical principle, reveals itself the most beneficial.    After 1992 it is clear that also acquiring full knowledge of the entire spectrum of the vibrations’ frequencies and amplitudes, does not grant to infer size, shape and mass of what originates them.  The inference remains limited to just:

  • area
  • circumference.

The containers’ inner pressure and/or cap/closure/lid inspection by Ultrasounds, results sensible to several macroscopic factors, and between these:

  • diameter of the cap;
  • tension of the cap surface.

The last property, being an indirect index of the inner pressure exercised by the fluid existing on the inner side of the cap, into the bottle.  What we really want is not merely to know if a crown-cork is cocked or if a lid is perfectly applied on a can, because the: 

  • crown-cork could be in perfect conditions but, in the meantime, the glass bottle itself could be defective (i.e., fractured in its neck or finish);
  • lid could be correctly applied and symmetric but, in the meantime, a micro-hole (typically <0.1 mm) could be affecting the metal can.

 The oscillation characteristic values (eigenvalues) or, modes of vibration, of the upper surface of the lid closing a can may resemble each other.  A deformed lid could be rejected only thanks to the fact that it deviated too much out of the programmed limits in the Frequency domain.  Meaning that an actually defective container was erroneously considered “correct” (Negative) when analysed in the Damping, Energy and Amplitude domains.  This is the rationale for the necessity to integrate in the same Inspector, on side of an Ultrasound Inner Pressure Inspection also an Inductive Inner Pressure Inspection.  It is really the only way to limit to <0.1 % the amount of leaking or deformed lids or caps reaching the Market

The true purposes are wider and include control and rejection of all containers:











  • deformed (cans);
  • whose lids are inclined (cans);
  • whose inner pressure is too-low, suggesting they'd be leaking;
  • whose inner pressure is too-high, suggesting they’d be deformed cans;
  • punctured (cans);
  • whose crown-corks are cocked (bottles);
  • whose metal caps with thread are deformed or uncorrectly applied;
  • missing sealing compound in the crown-corks or thread caps;
  • whose thread in the finish is broken.

Because of the several and different necessities, Quality Assurance really and only means to understand first that to adopt different physical principles is: 

  • the way to pass from the words to the facts;
  • better than making time-consuming measurements adopting a single measurement instrument; then surely basing our conclusions over a single physical principle. 

 

root cause analysis sample  med hr

 Random variables are projections in the Sample Space of objects’ properties having their full existence in the State Space.  The function manifests the relation existing between these two spaces.  On side are shown in red and blue color two different functions.  As an example: red could represent the transfer function of the overtones where it resonates a metal crown-cork and blue the transfer function of the induction of its upper slightly convex surface.  Distinct physical  properties of a single object, whose values appear like two random variables when measured by mean of two measurement devices (e.g., an Inner Pressure Inspection by mean of Ultrasounds and another by mean of an Inductive sensor) in their own Sample Spaces.  Part of the individual points in the Sample Space are simultaneously related to several points in the State Space (  Dong/Hong Kong University/2010)








 

In the framework of Signal Theory, redundancy on the amount of Detectors’ making independent measurements of the same property of the same object, acts allowing more precise estimation of the named true value of the random variable.   Refer to the figure above at right side.   In the same classic framework, to have access to an infinite amount of Detectors is equivalent to observe each one point of the State Space, as seen from an infinite amount of Sample Spaces.  Observation of an infinity of random variables, allowing to record all the fine-details forming a physical Property.   True value that in the classic approximation we always know only approximately because: 

  • registering random variables in a reduced amount of Sample Spaces;
  • having available only a limited time to observe and record random values.


Inductive analog inspection animation. Oscillograms deriving by the analog measurement of the induction of two different lids.  A symmetric one representing an actually closed can. An asymmetric one, representing a leaking can whose inner pressure is low.  Passing one and the same can closed by a lid through one and the same Inner Pressure Inductive Inspection, a statistically significative number of times, i.e. 1000 times, results in 1000 slightly different oscillograms.  From a layman point of view, differences representing the superposed effect of physical properties of the object and of the measurement instrument, changing during the 1000 tests.  Differences making sense of the way IGUSes (for further details about IGUS, see here) observe the state space where the entire description of the object exists.

  Oscillograms deriving by the analog measurement of the induction of two different lids.  A symmetric one representing an actually closed can. An asymmetric one, representing a leaking can whose inner pressure is low.  Passing one and the same can closed by a lid through one and the same Inner Pressure Inductive Inspection, a statistically significative number of times, i.e. 1000 times, results in 1000 slightly different oscillograms.  From a layman point of view, differences representing the superposed effect of physical properties of the object and of the measurement instrument, changing during the 1000 tests. Differences making sense of the way an IGUS (further details about IGUS here) observe the state space where the entire description of the object exists







A bottle, whatever kind of bottle, or a can, have their full existence in the state space and not front of our eyes or front of, as an example, an inspection camera system.   In the case of a complex object like a bottle (PET, glass, metal) or a can, its state space is extremely wide and, i.e., includes all possible: 

  • inclinations of a crown-cork,
  • angles in which that crown-cork inclination may be existing,
  • fractures of a glass bottle’s neck,
  • amounts of missing sealing compound between crown-cork and bottle’s finish,
  • deformations of a metal can,
  • sizes and positions of punctures in a can,
  • inner pressures of a bottle or a can.

A single inspection of that state, measuring i.e. the ternary random variable: 

 (Amplitude,  Energy,  Frequency,  Damping) 


of a crown-cork, corresponds in the random variables’ graphics above to just a few of the many subspaces where the object may be projected as a result of an interaction or, measurement.  And that’s why an over than tenfold reduction in the amount of False Negatives, the actually defective containers sent to the Market, it results useful to complement the system with an Inductive Inspection, like that visible in the figure here at left side. 



 

Other constraints

“...an extremely common crown-cork metal cap resonates over the widest spectrum of its own modes of vibration, under the influence of (typically ceramic) resonators whose diameter is >40 mm”










Additional considerations regard the dimensions, scale and mass relations existing between radiator and the dimensions, scale and mass of what we want to let it vibrate.  Imagine a 100 kg bronze gong.  To let it sounds, would you prefer to strike it with a 5 grams needle or with a hammer having a 2 kg metal mass at one end?  The size and power of the ultrasounds’ radiator is constrained by the: 

  • size, shape, kind of metal and mass of the metal closure or of the cans’ lid.   A common crown-cork metal cap resonates over the widest spectrum of its non-harmonic overtones, when radiated influence of resonators whose diameter is >40 mm.   An ultrasound Generator (transducer electric => electromagnetic) and the device later doing the reverse action, the ultrasounds transduction electromagnetic => electric, cannot simply have whatever size.   On the opposite, their optimal size is constrained by the crown-corks’, threaded caps' or lids’ mechanical Standards.
  • interaction Time.   As seen in the fine-detail before, to establish a relation between a crown-cork, threaded cap or lid and an ultrasounds’ transducer, it is necessary Time.    Now, consider that the interaction Time is inversely proportional by the Production speed in the Conveyor where the Electronic Inspectors is applied. In laymen words, the best practical way to infer the status of a closure by a measurement where the closure receives an impulse by ultrasounds, later transducing and analysing in different domains the vibrational response of the closure, is when keeping the container vertically stand still under the transducer.  [A first hint to the origin of the performances of the Inner Pressure Inspection by Ultrasound, given by several Vendors as Technical Guarantees]



As an example refer to the crown-cork cap depicted before, whose sealing gasket diameter is 22.211 mm.   

  1. In a real application, suppose the containers’ linear speed 900 mm/s (Production being 30000 bottles-per-hour):    
    • to assure measurements’ reproducibility it is necessary to exercise the force impulse and receive the following acoustic transduction inside a box error fluctuating from one cap to another smaller than possible, however <25 % of the closure diameter, meaning a timing whose tolerance is ±2.96 ms,
    • the translation of the diameter defined by the sealing gasket under the ultrasounds Generator and the acoustic transducer happens in <23.7 ms.   
  2. What if their linear speed increseas to 1800 mm/s (Production is 60000 bottles-per-hour) ?      
    • the same measurements’ reproducibility is now requesting us to ameliorate the timing tolerance to just ±1.54 ms,  
    • the translation of the diameter defined by the sealing gasket under the ultrasounds Generator and the acoustic transducer happens now in <12.3 ms. 

  The relative errors expected when applying the Inner Pressure Inspection by Ultrasounds to closures inspected at different linear speed, are different (abridged by   Vitogiannis Bros. SA/2014)




Visibly, increasing the containers’ speed, thus reducing to one-half the interaction Time, results in a proportional:

  • increase of the precision necessary (reduction of the tolerance) to centre the inspection time-window where the acoustis transducer receives the vibrational response of the container’s.   An increase on precision requesting additional tracking (or, triggering) precision, absence of container sliding causes which may affect the movement, constant lubrification of the Conveyor belt, etc. 
  • reduction on the amount of random variables recorded in the sample space where the displacement of each closure are sampled to bel later analysed in the amplitudes, frequencies, dampings and energy domains;

both increasing the relative error of the measurement on which the Inner Pressure Inspection is based, a close relative of the Production losses in the infamous form of False Rejects.   



What Rejects to Expect










Imagine a Full Container Electronic Inspector, operating with sensors in-the-Machine (Labeller or Filler), like frequently happens in the case of the equipments including an Inner Pressure Inspection.   Think the Inspector reduced to a tracking system in-the-Machine and later at-the-Conveyor, some label presence inspection in-the-Labeller Machine and at least an Inner Pressure Inspection with ultrasounds at-the-Conveyor.  Keeping constant: 

  1. container, 
  2. container's inner pressure,
  3. crown-cork, 
  4. sensitivity of the Inner Pressure Inspection,

and knowing the reject ratio at a speed 900 mm/s, what amount to expect for the Total Rejects (Positives  =  True Positives + False Positives), when increasing speed to 1800 mm/s ? 


  Imagine to process a statistically significative sample of bottles by mean of the Inner Pressure Inspection of an Electronic Inspector.  In the meantime, check one-by-one all rejects attribution. On the base of the Sensitivity threshold set, they’ll arise their respective statistical distributions of the hits and of the misses, True Positives, False Positives, True Negative and False Negatives, like visible on side.   If you’d repeat the same identical test increasing two times the containers’ speed, you’d see that the same Sensitivity of the former test is now corresponding to higher rejects TP + FP (abridged by   Jutta234/CC BY-SA 3.0/2006)


































This question, an important one when designing an inspection system and related Conveyors, Rejects' Accumulation Tables, buffers, etc. shall encounter a general answer formulated in another page of this website.    As a matter of fact, all Vendors of Full Containers’ Electronic Inspection systems, tacitly assume independence of the inspection performances by the containers’ speed.  What in general is observed is on the opposite a proportionality, where if we name: 

  •   (0,v0)   rejects observed at speed v0 
  •   (1,v1)   rejects calculated at speed v1 

then:

  •   v0   >  v1     ⇒   0   >  1
  •   v0   <  v1     ⇒   0   <  1

Proportionality existing not only because of the unavoidable change on the relative error of the measurement, due to the change in the amount of random variables sampled in the sample space for a container having full definition of its properties in the state space.   As an example, one and the same container in movement over a horizontal surface, at a certain speed or another two times higher, has in this last case a four times greater kinetic energy.    What means more than the increased power consumption registered by the Frequency Converters regulating the speed of the motors in the Food and Beverage Packaging Lines.   As a first example, different kinetic energy means a different value for its Lagrangian L.   Imagine to know the value of the total rejects 0 for a container’s speed v0  > 0, adopted as a reference, and to desire an estimation for their expected value at another speed v1.    We’ll indicate here, without to provide a demonstration, that the best estimation for the expected total refects R 1 at the new speed v1 shall be:

                             1  ~  R 0 [a (v1 / v0 )2   +  b (v1 / v]

where a, b are adimensional constants, related to the Electronic Inspector configuration:

  1. in-the-Machine or standalone ?
  2. what is the nominal Production speed (mm/s)  speed of the Machine ?
  3. how many locating functions in-the-Machine ?
  4. how many inspections in-the-Machine ?
  5. how many inspections at-the-Conveyor ?
  6. what false reject ratio at the Machine's nominal Production for the inspection(s) in-the-Machine ?
  7. what false reject ratio at the Machine's nominal Production for the inspection(s) at-the-Conveyor ?
  8. …other conditions...

From another standpoint, the first term reflects the change in the Lagrangian counterpart representing the container, and the second term the change in the sampled amount of the random variables. Readers are warned that the total rejects R 1 referred before are the real.  Those which should be truly observed if all of the inspections were truly enabled in the Electronic Inspection system.  It is important to understand that a standalone configured Inspector can be considered as a component of an in-the-Machine configured Inspector.  It controls at the same speed less variables, containers’ identities included. Then, its expected total rejects will be modelled by the approximate relation seen before, but with the important difference that the constants a, b  shall be smaller than in the case of an in-the-Machine configuration. 


Total Defects are also all Falsely Triggered 

Containers




















The real and total defective containers include much more than the containers defective because of a foreign object, underfilled, or without label.   Also, rarely described to the Electronics Maintenance Staff in what menus of the Electronic Inspector they are listed: we are clearly not referring ourselves to the universally-known Production and Rejection Counter menus.    A great share of the additional rejects to be expected when raising the containers velocity from v0   to  v1  is related to their uncertain identity.  The identity of a container has to be considered in the general framework of the Triggering, deeply studied in other pages of this website.  The identity of each one container is its most basic Information: 

  • an Information, which may be expressed in bits, and not a Physical Property.   Something more basic than a physical property (what became clear only after 2000) and which cannot be confused with it.  Some examples of physical properties being the cap's colour, under- or over-filling values, extent of containers' leakage, lid’s inclination, measured weight of a cluster of cans, etc.    
  • allows containers to be arranged in sets and counted,
  • may not split: if a container is broken into two pieces at most one of the pieces has the same identity,   
  • may also be destroyed if the simplest description of the system at a point in Time changes from identifying the object to not identifying it, a typical case of the Empty and Full Container Inspectors, where the containers’ velocity acts blurring the porcentage of containers univoquely identitified,     
  • created at the first point in Time that the simplest model of the system allows to identify it: the first Trigger in a tracking system,
  • all times a container is composed by components (i.e., finish, neck, sidewall, external base, inner base, body-label, neck-label, back-label, etc.), each one of them is an object whose physical properties lie completely within the boundary of the container.  This way, the identity of a container is unique and to this unique information are referred the results of the inspections, mere measurements of physical properties of the container’s components.



Container Identity 

The Most Important Information Prone to 

Ambigous Definition and Loss






















































No container may be processed in a Shifting-Register without an identity.  Identity which may be lost, what happens constantly as an example because of any or more of the following causes:

  1. excessive sliding during the Conveyors’ accelerated ramp-down, controlled by a Labeller, Blowformer, Capper, Closer or Filler Machine, during an emergency stop,
  2. lateral guides installation and fitting flaws, causing sliding of the containers, 
  3. oscillations of the tracking Triggers’ Signals,
  4. dirty lenses and/or mirrors front of the tracking Triggers,
  5. electromagnetic disturbances randomly induced in the tracking Trigger wiring carrying the Signals,
  6. electromagnetic disturbances randomly induced in the tracking Trigger wiring carrying the Signals,
  7. design flaws, as an example, because of a conveyor cross-over whose differences on chains' speed are not accounted by different Encoders, 
  8. design flaws, as an example, because a tracking Trigger wiring is not shielded in an Environment including hundredths of powerful motors,
  9. design flaws, as an example, because a tracking Trigger wiring is shielded to prevent the interferences deriving by an Environment including hundredths of powerful motors, but the shielding was not grounded,
  10. design flaws, as an example, because an Electronic Inspector’s I/O card  is “losing” Trigger Signals because the Chain Cycle is too fast,
  11. design flaws, as an example, because an Electronic Inspector’s CPU processing an I/O is “losing” Trigger Signals because the Chain Cycle is too fast,
  12. design flaws, as an example, because an Electronic Inspector’s software application, running in a CPU processing an I/O carrying a Trigger Signal, is “losing” Signals because the Chain or Machine cycle is too fast (too brief) with respect to the microprocessor’s own cycle time.  A lived example of this, some PET Empty Bottle Inspectors in-the-Blowformer Machine, designed by a known Vendor with just one CPU and Signals moved along tens of meters of Ethernet cables, trying to track PET Empty Bottles literally flying at a speed of 4.2 meter-per-second !    Another lived example of this, too many Empty Can Electronic Inspectors, guaranteed be capable to track cans flying at over 7 meter-per-second, until later discovering this true only after admitting amounts of containers losing their previous identity one-hundred times bigger than the technical Guarantee over the False rejects, 
  13. design, as an example, because a tracking Trigger wiring is shielded to prevent the interferences deriving by an Environment including hundredths of powerful motors; shielding was grounded too, but the Factory final connection to Earth presents an excessive impedance (because the ground is very dry and there are not enough ground rods) toward the high-frequency Noise, jitters and spikes, 
  14. commissioning flaws in the parameterisation of the tracking system of Triggers and Machine Cycle,
  15. (......)

In all these cases the Tracking Trigger encountering the presence of a container deprived of the identity previously attributed it by the precedent tracking Trigger, shall do what who commissioned the Electronic Inspector programmed it has to do.   To reject these unidentified containers or, to attribute them a new identity and later to reject only if the following inspections encounter any defect.   In this last case, meaning that the precedent inspections become oversight like non-existing nor happened.    What precedes to clear that the Readers typically are trained to know just the surface of the true Total Rejects deriving by the operation of their Electronic Inspection systems.   And that the Total Rejects are proportional to the containers’ speed.


Force Does Matter

If the Time available for the measurements is reduced by the Production and related linear speed constraints, there is however an important way-out to measure without any sensible increase of the relative errors of the measurements.   Relative errors otherwise immediately translated in False Rejects or, production losses.   Readers understand that we correctly proceed toward the evaluation of the state space of an object like a metal membrane: 

  • mathematically characterised by an infinity of modes of vibration,   
  • infinity of modes of vibration limited by modern Physics to a limited amount, however an extremely high amount, 

if and only if we interact with it in such a way to excite the greatest possible share of its modes until amplitudes we can detect.    To obtain closures’ vibrations in their wider spectrum of frequencies and amplitudes it is not just the Frequency of the ultrasounds used to act to be relevant.   The energy transferred from the ultrasounds' Generator to the container really does matter.   What is more true, as we’ll see in the following, when considering that the containers are in relatively fast (<2.5 m/s) movement.   The energy is transferred via a Force, countered by the container and, mainly, by its closure.  The figure below shows the relation between Force and Acceleration for a crescendo of forces applied to a membrane where Force 8)  is  33 times bigger than Force 1).   The Acceleration signal shows that this vibration becomes stronger with increasing impact Force.   The net result of this is that to greater Forces shall correspond higher values for the Signal-to-Noise ratio of a measurement system sensible to the vibrations, because of lower relative errors.

 The Force countered by the closure is related to closure Acceleration and vibrations. Graphic of the Force applied to the centre of a membrane versus its Acceleration, for increasing values of the Force.   Force 8) is 33 times bigger than Force 1).  The Acceleration signal shows that this vibration becomes stronger with increasing impact Force.  Net result of this is that to greater Forces correspond higher values for the Signal-to-Noise ratio of a measurement system, because of lower relative errors (  A. Wagner/2006)

Summarising the constraints, all of them conditions out of our control:

  1. Closure's (cap or lid) shape and material                                                             A Standard, impossible to modify
  2. Time available for the interaction Resonator => Closure                                     Inversely proportional to containers’ speed
  3. Time available for the interaction Closure => Ultrasounds’ Transducer              Inversely proportional to containers’ speed









Then, as Food and Beverages’ Bottlers have their possible packagings’ choices forced in a thin band, this is consequently true also for the Inspectors’ Vendors.   With a big difference: whatever Bottler is free to use the same identical lid or crown-cork, with the difference with respect to a Competitor reduced to the logo’s motive and colours.   But this is not true when speaking of the ultrasounds' resonators and related transducers that Vendors have to use: the optimal characteristics could yet had been patented by any of the Competitors.   Imagine that a particular Vendor owning a great know-how in the ultrasounds’ technologies, applications and business, arrives first to patent certain kinds, shapes and sizes of resonators and transducers and….. you’ll be close to fully understand why some frequently observed Closure inspections with ultrasounds are so poorly performing.   And also why some highly performing inspections by a Vendor, corresponding to the positive outcome of its Research and Development, starts to reappear as an inspection by other Vendors ~10 years after the first patent and sale by the Vendor who invested in R&D.   It is simply the time-delay created by the 10 years time-lag started by the patent.   Worse than that, if you’d start to compare the digits provided by the Vendors as Technical Guarantees, with the reality the Vendor’s Service Technician prove to be capable to commission, you’d immediately discover something.   The Technical Guarantees refer to illusory cases and the reality are performances >10 times lower than contractually guaranteed.   What, translated to your real cases of malpositioned caps, means open bottles introduced in the Market under your responsibility (not Vendor’s responsibility).

   Ultrasounds resonator's and transducer’s shape, dimension and mass are related to the optimal way to let crown-corks resonate over the widest possible spectrum of their own modes of vibration.   Resonance damped by the rubber sealing below, by the inner pressure exercised by the fluid into the container, etc.   Beverage Bottlers adopts crown-corks’ and lids’ standardised sizes, thus constraining the optimal resonators' and transducers’ dimensions and masses   

   

To give a further idea of the reality hidden back of too many digits and brochures, we remember some models of inspections using ultrasounds to establish the cap-transducer relation.   The “tac” sound associated to the resonance is so loud that it can be heard in the noisy environment of a 40000 bottles-per-hour running Glass Returnable Bottling Line from a distance of ~15 meters.   In other cases, corresponding to ultrasounds’ inspection systems by other Vendors, to hear the resonance of their ultrasound inspection is necessary to go close-up with ears.   Close to just a few centimeters (<0.3 m) to hear something which “maybe could be that tac” arising by a resonance.   Take out your own conclusions: if you and your own 100 billions of neurons, are nearly not capable to hear a resonance, shall be capable that inspection's transducer (electromagnetic => electric) ?    It’ll be but, ...under what extent of unavoidable relative error of the measurement, a close relative of False Rejects’ ratio, and the strictest parent of the infamous losses on Production.   Why they seems strangely designed to be “False Rejects’ Generators” rather than “inspections”.    Then, take out your own conclusions about the possibilities to detect the entire signature of the many harmonics getting out of a crown-cork metal cap, irradiated by, as an example, a resonator <30 mm.   One more time, the 5 grams needle shows itself not the optimal way to let a 100 kg gong sounds.    But, it’d be possible to do this with a 20 grams microscopic gong…    Transposing the gong paraphrasis to our technological area of interest and reality, there is a true problem, one not having any one solution.    Namely that nearly no one capping application in the worldwide Beverage Bottling adopts caps so small to resonate in their widest spectrum of modes under the effect of such a small ultrasounds' radiator.  




Different Ways to Approach a Single Problem 

“The Technical Guarantees refer to illusory cases and the reality are performances >10 times lower than contractually guaranteed.    

What, translated to the real cases of malpositioned caps, means open bottles introduced in the Market under Bottler’s responsibility”




Consider three possible basic cases of coupling between a container and a closure:

  1. concave shape of the lid/closure;
  2. convex shape of the lid/closure;
  3. flat shape of the lid/closure;

The basic operative principle of the metal cap closure inspection, with ultrasounds implies a Generator of electromagnetic waves in the band of the ultrasounds, whose associated frequency lies in the range (30 - 300) kHz, close to passing-by closures, lids or caps.   These last, resonating like stricken drums.   The non-harmonic content of overtones of the waves re-emitted by each closure, say both series of: 

  • amplitudes y             y0,  y1, y2 , ....,  yN
  • frequencies ν            ν1 , ν2 , ν, …,  νN 

is later separately analysed in the domains of Energy, Frequency, Amplitude and Damping (see graphics below).

Metal cap oscillations damped by the impedance due to the superposed effects of the crown-cork rubber sealing ring (thermal) dissipation, inner pressure of the bottle, environmental humidity, etc.  Consider three possible basic cases, of coupling between a container and a closure:

concave shape of the lid/closure;
convex shape of the lid/closure;
flat shape of the lid/closure;
The basic operative principle of the metal cap closure inspection, with ultrasounds implies a resonator generating electromagnetic waves, whose associated frequency is <300 kHz, close to passing-by closures, lids or caps.   These last, resonating as hitted drums.   

Being known the harmonic content of the emitted wave for bothe the series of: 

amplitudes            y0,  y1, y2 ,...,  yN
frequencies           ν1 , ν2 , ν3 ,…, νN 
the received electromagnetic waves re-emitted by each closure are then separately analysed in the domains of Energy, Time (harmonic content ) and Damping (see graphics below).


  Metal cap vibrations damped by the impedance due to the superposed effects of the crown-cork sealing compound (thermal) dissipation, beverage inner gases, etc.   Visibly, the pink-coloured envelope is an exponential function of the Time





Analysis in the Damping Domain 

Damping expected by an Inner Pressure inspection by Ultrasounds. The dampings associated to correctly closed containers corresponds to a wide band of overtones’ frequencies and related amplitudes.  A reason for the associated uncertainty of the Inner Pressure Inspection with ultrasounds when determining the Leakage state of a container

As visible by the graphics above, the damping law expected is exponential yet for a correctly closed container.   Dissipations of energy due to container’s inner gases, elasticity of the cap's inner sealing compound, elasticity of the can’s thin sidewall, etc. are yet part of a non-leaking correct container.     


The Damping associated to correctly closed containers corresponds to a wide band of overtones’ frequencies and related amplitudes.  A reason for the associated uncertainty of the Inner Pressure Inspection with ultrasounds when determining the Leakage state of a container














To the normally expected exponential function it can be summed the non-systematic effect of a leakage, thus changing the amplitude y and the non-harmonic overtones ν out of the ranges where these amplitudes, energies, frequencies and dampings are expected.   Refer to the graphics above at right side, showing in the white-coloured area the normally expected dampings.  We’ll have a partition of the outcomes where the Dampings very: 

  1. low, shall be translated in a high amplitudes, around the resonance frequency ν0 .    In the figure above represented by the points lying between the white coloured and the green colour line,    
  2. high, shall be translated in low amplitudes, around the resonance frequency ν0 .     In the figure above represented by the points lying between the white coloured and the violet colour line.   

Dampings very low or very high with respect to the distribution of a population referred to a statistically significative sample, are indicators of defects like a leaking container, malpositioned- or cocked-caps, broken finish thread, too-low or too-high inner pressure, etc.     For each one of these domains the measurements are compared with ranges set during commissioning, considering defective those closures deviating out of them.


Analysis in the Frequency Domain










ν is the fundamental resonance frequency of a crown-cork cap or the can’s lid, and ν2 , ν, …, νN  in the case of harmonic overtones its integer multiples.   Metal caps or lids closing bottles or cans are not however harmonic musical instruments.    As an example, a can or a bottle, filled with pressurised (because of CO2 + thermal energy), resembles vaguely a musical instrument named “timpano”.   The completely closed version of drum, having just a single elastic membrane.    But, as you may imagine, surely we do not expect to play a great symphony with such a kind of timpano ...filled with pressurised beer.    In our present context meaning that the overtones of cans or bottles are not harmonic: the majority of them are not integer multiples of the fundamental frequency.  Their analysis in the Frequency Domain: 

  • simplifies many of the complex computations otherwise having to be performed in the Time/Space domain (see figure below);
  • we can see more the Signals’ peculiar features.   



  Analysis in the Frequency Domain of a function sampled in the Time or in the Space Domains.  In the frequency Domain we are plotting the amplitudes as functions of individual frequencies which are terms of the functions Fourier analysis.  In the Time or Space Domains, are examined the amplitudes associated to Times or Distances (   King Fahd University/2008)















Today, however, the analysis in the Frequency Domain is rarely accomplished taking full advantage of its yet known possibilities.   If you’d investigate deeply what yet thirty years ago the Scientific, Aerospatial and Military applications were doing, taking advantage of the Frequency Domain analysis, you’d remain astonished.   Astonished comparing that with what the most modern, expensive Electronic Inspectors offer today.   How comparatively obsolete is in the sad reality what is offered still today to the Beverage Bottlers by the Vendors of Electronic Inspection systems.   As an example, in the Frequency Domain, we could also easily: 

  • reshape a Signal, 
  • suppress undesired parts of it; 
  • enhance some frequencies;
  • isolate a Signal that we wish to consider primary, from the influence of extraneous effects; 
  • many others yet used in other applications Industrial, Aerospatial and Military;

In brief, a container closed with some kind of metal closure, in the end is a single format. Then, one where each one of the physical properties of the non-harmonic overtones of the entire system where a correlation between: 


cap  +  container  +  beverage  + gas  +  conditions (environmental, dynamical)




is established, lies in a certain expected range of amplitudes, occurs with some frequencies, etc.


Line Layout and Inner Pressure Inspection

There are limits to the application of this inspection:











  • it lies typically after the Capper following a Filler Machine, or after the Labeller.   
  • imagine an Inner Pressure Inspection acting after a Filler Machine filling beer.  Here a nozzle sprays a thin jet of hot-water into the bottles’ travelling from the Filler tank to the Closer to let them foam reducing the amount of air in the head space.  Its effect increases the inner pressure of the bottles, increasing the Signal-to-Noise ratio of the Inner Pressure inspection;
  • imagine an Inner Pressure Inspection acting after a Labeller Machine in a beer bottling line.   With minor exceptions, all bottles reach the Labeller from a Pasteurizer Machine where they experience a long lasting overheating, increasing the inner pressure of the CO2-added beverage containing.  Thus increasing the Signal-to-Noise ratio of the Inner Pressure inspection set immediately after the Labeller Machine;
  • the product filled has however to assure bottle’s internal pressure.  I.e., mineral water with standard amounts of CO2 added or beer fit the requirement;
  • caps diameter:  >28 mm;
  • bottle + cap summed height tolerance:  <1.0 mm.






In the following a serie of oscillograms referred to two kinds of acoustic resonators, operating in different conditions: a tuning-fork and a 28 mm crown-cork cap.


Tuning Fork

The entire vibration period can last tens of seconds.   A tuning-fork (or, diapason) oscillating in the air at a frequency of 400 Hz is damped 10% in 12 seconds.   The same forced to vibrate in a vacuum shows a damping <10% after the same 12 seconds.   Say, air is the damping agent.  The oscillogram on side is better interpreted displaying the many states in which a simple mechanical system exists after a disturbance which increased its energy level.  Disturbance propagated to the tuning-fork after an interference (or, interaction) by an external agent.  Being a system nearly undamped, it is possible to observe the extreme density of the overtones of the fundamental mode of vibration.


Crown-cork Cap

  • Cap correctly applied 
  • Non-leaking bottle 

Commonly expected shape of the vibrations arising by a 28 mm crown-cork cap, after ultrasounds' excitation.   Damped much faster than in the tuning-fork case above, with the energy released in <0.2 seconds. 



  • Cap correctly applied
  • Non-leaking bottle
  • Malpositioned cap
  • Leaking bottle

Another expected shape of the vibrations arising by a 28 mm crown-cork cap, after ultrasounds' excitation.   Visibly absent the high amplitude vibrations at the onset of the resonance.   A difference expressing the superposition of many factors and conditions, accelerating the damping: inner pressure of the container relatively low, slightly malpositioned cap, excessive distance between detector and cap, broken base of a glass bottle, slightly inclined cap, broken finish thread, etc.   But which may still be a non-leaking bottle, correctly capped. 



  • Malpositioned cap
  • Cocked cap
  • Low inner pressure
  • Leaking container
  • Broken finish threads
  • Erroneous mechanical setup



  • Malpositioned cap
  • Cocked cap
  • Low inner pressure
  • Leaking container
  • Broken finish threads

Anomalous heavy damping.   Occurrence which may originate by one of the typical defects whose detection is the purpose of the Inner Pressure Inspection by ultrasounds, i.e., a deformed can.  But, between several other possible causes, also a possible effect of an external temporary cause, like a brief contact with another object.



  • Anomalous heavy damping

Anomalous heavy damping, most probably originating by any or more of the following causes:

  • tracking error, because of sliding;
  • tracking error, because of erroneous Trigger setup;
  • defective ultrasound generator;
  • defective acoustic transducer;
  • defective hardware processing the US's signal;
  • erroneous mechanical setup;
  • beverage in the ultrasounds’ transducer;
  • etc. 




Closure Height Compensation 

When:

                           bottle  +  cap height tolerance  >  1.0 mm







the ultrasound inspection of the pressure existing in bottles closed with metal closures, is prone to provoke a false positive, say losses on production (false rejects).   To prevent this, it exists an optional inspection, based on an inductive sensor (see figure below) which allows to:

  • measure before the distance to the cap surface, proportional to the height of the cap;
  • linearize with this information, the evaluation of the following ultrasound inspection of the pressure.

  Heineken at Mons-en-Barouel, France, brews what is named the “Champagne of the North”, devoting special efforst toward the optimisation of its Quality levels. Caring also the ultrasonic metal closures’ inspections


Performances and Respect of Contracts



































The situations hinted before with respect to the Containers’ Inner Pressure Inspection by mean of ultrasounds, have the strictest relation with the illusory guarantees of the High Frequency Fill Level inspections: 

  1. those sold to detect and reject an under- or over-filling of 2.0 mm with a hit ratio >99.9 % and an associated false rejects ratio <0.01 %;    
  2. later discovering that the environmental conditions (ambient humidity and temperature), condition negatively all of the High Frequency Fill level Inspections in the World, whatever their Vendor.   Condition reducing the inspection performances to >5.0 mm under-filled filling detected with a hit ratio >99.9 % and an associated false rejects ratio <0.01 %.   [There are also Vendors who cutted in their brochures 10 times the hit ratio to let it look 99.99 %, 10 times better than a Competitor's 99.9 %…]   
  3. then, you change the position of that identical High Frequency Fill Level inspection, setting it directly in a thermostatic and humidity-controlled clean room.  A typical case of the Aseptic filling; 
  4. and you discover only minimal ameliorations, far from what is written in the Technical Guarantees; 
  5. now, Vendor says that the problem arise directly in the fluctuations of the beverage’s temperature, what is true;
  6. and you invest to upgrade the High Frequency Fill Level inspection, adding a thermal linearisation of the fill level inspection on the base of the beverage's temperature, measured directly in the Filler tank;   
  7. and, one more time, you discover that there is no way to detect and reject a 2.0 mm under- or over-filling with a hit ratio 99.9 % and an associated false rejects ratio <0.01 %;    
  8. now, you are just a phone call far from hearing what no one shall never write you in a Technical Guarantee or in an e-mail (also when filled with Disclaimers’ statements): that Technical Guarantee is not really a Technical Guarantee.  It is fulfilled only if your bottle is (comically) “standing still stopped over the Conveyor, under the High Frequency Fill Level inspection bridge”!     


That 5 % of the Bottling Companies

Finally, if your Company is one in that 5 % of those capable to let their contractual rights be defended, you know you have full right to ask and obtain alternatively:

  1. replacement of that inspection with another of that same Vendor (offering superior performances), and the commissioning of all formats included;
  2. replacement of just that inspection with one from another Vendor, paid completely by the Vendor, installation and commissioning of all formats included, what is feasible and better choice when the Inspector you acquired has digital inputs available to process, count and reject the named External Inputs.  “External Inputs” corresponding to optocoupled, potential-free digital signals (on-off, 0 or 5 volt) 
  3. replacement of that Electronic Inspector with one from another Vendor, paid completely by the Vendor, installation and commissioning of all formats included,
  4. an indemnisation, and remain with the present system,

or, pass that Contract and a list of evidences for all what happened to your Legal Dept. who shall treat the case under the rules of the Civil Law enforced in the country agreed in the Contract imagining unexpected developements like these.


Links to other pages:




This website has no affiliation with, endorsement, sponsorship, or support of Heuft Systemtechnik GmbH, MingJia Packaging Inspection Tech Co., Pressco Technology Inc., miho Inspektionsysteme GmbH, Krones AG, KHS GmbH, Bbull Technology, Industrial Dynamics Co., FT System srl, Cognex Co., ICS Inex Inspection Systems, Mettler-Toledo Inc., Logics & Controls srl, Symplex Vision Systems GmbH, Teledyne Dalsa Inc., Microscan Systems Inc., Andor Technology plc, Newton Research Labs Inc., Basler AG, Datalogic SpA, Sidel AG, Matrox Electronics Systems Ltd. 

                                                                                                                                                                                                                                                                                                                                                                                                                                                         
Webutation
                                                                                                                       © 2013-2015 Graphene.  All rights reserved                                                         DMCA.com Protection Status                    

                                     
                                              
TRUSTe Privacy Policy Privacy Policy
Site protected by 6Scan