## What is this article about?

### Vibration resonance: fascinating in technology, problematic in the courtroom

In the world of machines and technology, vibrations are omnipresent. For vibration experts they are even “fascinating”. The underlying physics and mathematics are particularly exciting. But in court, vibrations and resonance can become a real problem. In this article, I will explain why vibrations and resonance are a “special issue” in litigation and why they are often problematic for the party presenting evidence.

### Trials and the "time trap" of securing evidence

Court proceedings are often lengthy. But in the case of machine vibrations , it cannot wait until a forensic expert assesses the situation on site. Often the problems are eliminated during repairs, which at the same time destroys the evidence. This is destruction of evidence , but not intentional, but rather a systematic occurrence in the case of damage caused by vibrations (resonance). In such cases, the party with the burden of proof faces a significant problem.

### Burden of proof and fact finding: challenges in court

In addition to the “time trap” of securing evidence, there are other aspects that make vibration phenomena in court proceedings (vibration resonance) a challenge. The burden of proof lies with the party bearing the burden of proof. This means that she must explain and prove to the court that the vibrations are the cause of the damage.

Added to this is the complex task of establishing facts by court experts. Vibration phenomena are often difficult to measure and assess. This requires expertise and experience on the part of the expert, but in any case

*a still existing vibration system*

### Fascination and litigation: The two sides of the vibration coins

The world of machine oscillations, resonances and vibrations in mechanical engineering is undoubtedly fascinating. But for companies and private individuals involved in legal disputes due to machine vibrations, this issue can quickly become a nightmare. The complex questions of evidence and the challenges in establishing facts make it a “special issue” that requires considerable expertise and strategic action even before the lawsuit is filed.

## Short introduction to machine vibrations (vibration resonance)

An extremely short introduction is necessary in order to give even technical laypeople a very, very small impression of the underlying physics and mathematics and thus the complexity of the topic of ” *vibrations* “.

Figure 1 shows the result of the simulation of a nonlinear mathematical model, which depicts the essential physical properties of a car windshield wiper.

Court proceedings are often lengthy. But in the case of machine vibrations, it cannot wait until a forensic expert assesses the situation on site. Often the problems are eliminated during repairs, which at the same time destroys the evidence. This is destruction of evidence, but not intentional, but rather a systematic occurrence in the case of damage caused by vibrations and resonance. In such cases, the party with the burden of proof faces a significant problem.

Figure 1 shows, even for laypeople, an oscillation curve over time, where “repetitions” can clearly be seen. So this is a good example of a simple definition of a vibration.

A vibration is a process in which…

*the variable of interest (distance, speed) changes over time in such a way that certain features recur. *

### Vibration-capable systems

Systems capable of oscillation are

*physically always structured the same way*

and can generally always be described mathematically

- Dimensions,
- rigidity,
- Damping and if necessary
- the stimulation of the system,

if it is not a free system.

Equation 1 now describes a forced oscillation for a damped 1-mass oscillator. It’s an extremely simple example. The natural frequency of this system can be easily determined from

These include:

*m:*

*d:*

*c:*

*f(t):*

*x**,x*,x:*

(vibrating) mass;

Damping;

rigidity;

excitation of the vibration system;

Acceleration, speed, distance.

### Resonance

Figure 2 shows different magnification functions (resonance curves) purely as examples.

In an oscillation system, the oscillation response increases the closer the excitation frequency approaches the natural frequency of the oscillation system. This connection is shown – purely as an example – in Figure 2.

In the “undamped” case, when the oscillation system is excited at the natural frequency, the oscillation response becomes “infinitely” large. In “damped” systems, the vibration response also increases significantly. Purely mathematically, it reaches “finite” values. The greater the damping of the system, the smaller the increase in the resonance peak (also called resonance peak).

In general, this is understood as:

*Resonance is the correspondence between natural frequency and interference frequency.*

At this point it must be mentioned that in the damped system the resonance frequency does not coincide with the resonance frequency of the undamped oscillator.

### Real vibration systems

The main difference between theoretical models with few degrees of freedom and reality is that real oscillation systems have a

*infinite number of degrees of freedom*

feature. This also results in

*infinitely many natural frequencies and natural shapes that always overlap in the vibration.*

#### Oscillation systems have infinite degrees of freedom

Vibration systems are usually reduced to “only” *n* degrees of freedom. Equation 2 shows how

*real systems result in exorbitantly large systems of equations.*

Equation 2 can generally be formulated in the following form for *n* degrees of freedom:

At this point it should also be clear to the vibration technology layman that

*Such real systems cannot be solved “just like that” by looking at them or using technical “feeling”.*

Any “quick” hypothesis put forward by “experts” and sometimes “creative” lawyers, which is used as a “quick explanation” for a vibration-related cause of damage, is required

*of course the technical, experimental and/or mathematical-physical verification.*

### Summary for machine vibrations and vibrations

Now that the very complex relationships in real vibration systems have been explained – hopefully simply and understandably – in very few words, it is clear that vibration technology is difficult and anything but understandable ” *at first glance* “.

#### Important:

Each vibration problem must be viewed individually. A solution is not possible by “just looking at it”, neither when it comes to vibrations in court proceedings nor when testing vibrations or machine vibrations on behalf of the party.

## Differences between statics and dynamics

Engineers specializing in structural dynamics often joke that statics is merely the “by-product” of structural dynamics – and there is actually some truth in that.

This provocative statement underlines the fascinating contrast between the two disciplines. While statics deals with the unchangeable loads on a component or machine, structural dynamics delves deep into the exciting world of movements and vibrations. Every change, every vibration, every resonance can have an effect depending on time.

Through the eyes of a structural dynamicist, it becomes clear how lively and complex the world of mechanics really is. Instead of rigid structures, the focus here is on the play of forces and movements. Even for the layman this will (hopefully) become clear when looking at equation 2.

### Statics

So let us briefly turn to the “waste product”.

If we delete the accelerations and velocities in equation 2 and equation 3, we obtain the greatly simplified general

This is the proverbial “waste”.

#### Tensions

A stress is an auxiliary quantity which, in a __one-dimensional__ case, results quite simply from the quotient between the acting force and the underlying area. Thus, one can simplify the following relationship:

In the static design of components, structures and machines, the stresses in the structure (“stresses”) determined from the external load (e.g. “force”) are usually compared with the mechanical material properties present in the material, taking into account so-called safety factors.

If the maximum permissible voltage is exceeded, damage will occur. This is

*The type and manner of stress is of particular importance.*

In general, a distinction is made between static and dynamic stress. The situation is that in the

*In the dynamic case, considerably lower maximum stress values can be tolerated than in the static load case.*

#### When does structural damage occur in steel?

In the tensile test, steels exhibit a stress-strain diagram, which can in principle be derived from the general example in Figure 3.

It starts with the linear-elastic region “1”. In tensile tests, the so-called Hooke’s law applies. There is a linear relationship between stress and strain in the form:

The elastic modulus *E* , a material constant, can be defined in the tensile test via the inclination of the straight line in the linear range “1”.

The *yield strength Re* characterizes the end of the linear-elastic material state and thus describes the

*Failure due to flow,*

i.e. by the onset of plastic deformation, see Figure 3.

In many structural steels, the linear elastic region is followed by a *pronounced yield region* with a strain of 1-3% (region IIa in Figure 3), which is characterized by the fact that the bar initially elongates without any further increase in stress. For these materials, the yield strength relevant for the strength calculation is determined by the stress *ReH (upper yield point)* at the end point of the Hooke’s line before the first significant load drop. The smallest value that occurs during pronounced flow (without transient phenomena) is called *the lower yield strength ReL* .

It is essential that there is damage in the statics when the linear elastic range is left and the stresses are in the range “2”.

After a discharge, the structure

*permanently deformed*

and thus

*damaged.*

### Structural dynamics

However, the majority of technical components are subject to

*temporally changing load.*

Vibration stress, which is due to mechanical and thermal operating loads, represents a particular challenge with regard to the strength design of components. These loads, which often overlap with a static base load, can have serious consequences.

Vibration stresses arise in various application areas, for example through the bending of shafts or the start-up and shut-down processes of machines. The repetitive load cycles, whether frequent or less frequent,

*can*

lead to progressive material damage, even

*Material fatigue*

called. Such effects can lead to cracks and ultimately to the breakage of the component.

#### Wöhler diagram

Figure 4 shows a purely schematic Wöhler diagram. These diagrams are used to describe the fatigue strength behavior of components as a function of the number of fatigue cycles. The number of cycles is plotted logarithmically.

There are essentially two areas:

- an inclined line and
- a horizontal.

If this diagram is based on a Cartesian coordinate system, the stresses are taken into account on the ordinate and the vibration cycles on the abscissa.

Figure 4 must now be read as follows:

If the stresses occurring in a cross-section to be assessed as a result of vibration loading are less than or equal to the value for the fatigue strength (here A _{D} ), then in the ideal case we are at point N _{D} and the structure is fatigue-resistant for this load in the technical sense.

If the stresses occurring due to a stress amplitude are smaller – here the point S1 – than the fatigue strength, then we are exclusively in the area or below the horizontal line. Regardless of the number of cycles, the structure is fatigue-resistant for such stress. This means that a fatigue fracture will not occur at this stress S1 within the lifetime.

If the stresses occurring in the cross-section are greater than the permissible value A _{D} , here for example point S2, the structure is no longer fatigue-resistant. Such a condition is reached when the ideal structure per se is stressed too highly or, if it is “theoretically borderline”, but there are manufacturing influences that lead to a (significant) increase in the stresses determined for the “borderline” case.

In the example of a stress S2 occurring in the cross section, the number of cycles can be read accordingly at point N2. If the voltages continue to rise, the ordinate, the voltage amplitude, moves “upwards”. The tolerable number of cycles N is reduced to the same extent. You “wander” along the abscissa to lower values.

The theoretical limit here is that the crack occurs for a fatigue life of zero. Then in the static case the tensile strength would be exceeded. This is the theoretical limiting case in the transition from dynamics to statics.

## Examples of damage caused by vibrations

There are countless fascinating examples of vibrations that I could present. I would just like to discuss two impressive cases that illustrate the challenges and peculiarities of machine vibrations.

### Vibration fracture of a roller

A roller broke during operation.

If you look more closely, you can see that the fracture cross-section has different areas. This is shown in Figure 3.

The roller is destroyed. However, even with the naked eye, two different areas can be seen in the fracture cross-section when viewed macroscopically in Figure 3.

The area marked “1” in Figure 3 shows an obviously different area than the much smaller area marked “2” in Figure 3.

The area marked with “2” was

*Damaged by continuous stress (“material fatigue”).*

The area marked with “1” is the

*Residual violence.*

From the size ratio of the areas “1”/“2” it can be immediately seen that the nominal stresses in the cross section were high. With respect to the S-N diagram in Figure 4, the stresses in the cross section were far to the left in the yellow area.

The stresses in the cross-section therefore tended to be high for the mechanical properties of the material under dynamic loading and there was pre-damage, so that the cross-sectional area became smaller over time. If the area becomes smaller under the same external load, the stress in the cross-section increases and will further damage an already damaged cross-section.

This is a classic example of a

*Damage caused by dynamic stress.*

In this case, the structure is destroyed. For example, it is possible to explain this case “backwards” in the analysis, so to speak, based on the known dimensions of the roller and the determination of the material with its mechanical properties and the stress on the roller during operation.

It is possible to determine which external load was necessary to produce this damage pattern. Once this has been done, one can, for example, discuss in a legal dispute under which conditions the necessary burdens that caused this damage could have arisen. Such cases can also be resolved retrospectively within the framework of dynamics.

### Vibration fracture of a shaft

In Figure 4 one can also clearly see two different areas in the fracture surface. The failure indicates a tough bending-force fracture.

In Figure 4 one can also clearly see two different areas in the fracture surface. The failure indicates a tough bending-force fracture.

Zone 1 runs transverse to the axial direction, is low-deformation, flat and matt.

Zone 2 runs diagonally to the structure and is visibly deformed, flat and partially shiny.

This is the drive shaft of the carousel of a large industrial milling machine. It was stretched over eight straps. Of course, one can make the argument that the belt tension was too high. However, if the shaft fails, the system is completely destroyed.

In the present case, one could also try to recreate the damage pattern through analyses in order to then also “backwards” find technical indications that the belt tension was too high. The fact is, however, that with this case of damage, the situation with the belts can no longer be determined exactly because the system is destroyed. We then have:

*Destruction of evidence by the vibration system or the dynamic system.*

### Vibrations as a cause of fire

The previous examples have shown that the cross-sections of broken structures usually also contain clues to the past.

In reality, however, there are numerous cases of damage that are attributed to vibration resonance and can no longer be proven.

In the example shown in Figure 5, a pipe connection had come loose. There was an oil leak and ultimately a fire. Vibrations were suggested as the cause.

If the vibration system is destroyed, the problem can no longer be detected retrospectively. Then measurements are just as impossible as analyses, for example using the finite element method.

Furthermore, if cross-sectional information is no longer available or can no longer be found, it is technically impossible to draw conclusions about the past.

Such matters go in a dispute

*always to the detriment of the party who has the burden of proof.*

Why is the vibration system so important? Clues can be obtained simply by looking at the exemplary system in the section “ *Vibration systems have an infinite number of degrees of freedom* ” as well as the following section. This section goes into more detail about the problems of the vibration system.

## Why is the vibration system so important?

It is of fundamental importance

*the natural frequencies and**the natural vibration forms*

to know or determine the vibration system in machine vibrations. An expert must therefore understand the vibration system in order to enable substantiated findings regarding vibrations in court proceedings.

Regardless of the absolute magnitude of the system’s excitations (“ *excitations* ”) and dampings, knowledge of the essential natural frequencies and mode shapes is very helpful in order to be able to evaluate the dynamic behavior.

There are measurement methods that can be used to represent the natural vibration patterns of real structures.

If you have the design data available and compare it with the structure implemented in reality, you can examine real structures relatively easily – albeit with a lot of time – using suitable software.

Depending on the discretization of the structure, the analysis results, for example using the finite element method (FEM), are very precise with regard to the natural vibration forms. The natural frequencies then determined are:

*there are always sometimes clear deviations from reality.*

However, knowledge of the natural vibration forms is essential because you then know which “shape” you need to shift towards the desired frequency ranges through “which” interventions in the structure in order to avoid undesirable effects.

The following must be taken into account:

*No expert can do something like this just by looking at it*

and especially not if

- the damage has already happened and
- the structures dismantled or
- are destroyed.

In all of these cases, essential technical information was lost. This is with vibrations in court proceedings

*always associated with a disadvantage for the party responsible for providing evidence.*

The following simple example will illustrate why the vibration system is so important.

### Influence of boundary conditions on vibrations using a simple example

In Figure 6 you can see a top view of two simple rectangular plates that were depicted in a so-called shell model in an FEM analysis.

With the label “1”, “left” in Figure 6, the two light blue edges are completely blocked (“boundary condition 1” in Table 1). Shifts are excluded along these blue edges.

With the label “2”, “right” in Figure 6, only the red edges are completely blocked. Shifts are possible “bottom right” (“boundary condition 2” in Table 1).

#### Natural vibration forms (“modes”)

Figure 7 shows the comparison of the first mode shape. In case “1” with both completely blocked edges, the classic antinode results, which runs exactly in the middle if there is symmetry.

In case “2” only the exposed edge area vibrates. The associated frequency is a full 2 Hz lower than in case “1” with clamping over the entire edge.

##### Nodal lines for oscillations, vibrations, resonances

Figure 8 also shows significant differences in the natural vibration shape and of course in the natural frequency. In both cases you can see the so-called *nodal lines* .

In two-dimensional structures, nodal lines mean that natural vibration forms along these lines

*are at rest “despite vibration” and remain so.*

You can see in the left part of Figure 8 (“Boundary condition 1”) that the expected symmetry is present here. In the case with the edge only partially clamped, a natural vibration form results, which is a combination of the vibration antinode present in the case of symmetry according to Figure 7 (“left”) and the vibration behavior according to Figure 7 (“right”).

Mode | “1” | “2” |
---|---|---|

1 | 3,47 Hz | 1,40 Hz |

2 | 4,11 Hz | 3,41 Hz |

3 | 6,73 Hz | 4,45 Hz |

4 | 9,58 Hz | 5,6 Hz |

Table 1: Influence of boundary conditions.

If you now look at the determined natural frequencies 3 and 4 according to Table 1, you can also see clear differences here.

This example is intended to make it clear that boundary conditions such as:

- connections,
- storage,
- movement options, etc.

*significantly influence the vibration behavior of structures.*

This change refers to the

*Natural frequencies and the natural vibration forms (“modes”) associated with these natural frequencies.*

This is intended to make it clear again for the layperson at this point that one must know

- what you are looking for and most importantly
- at what frequency,

so that technical conclusions can be drawn about events and – ideally – avoided in the future. It follows:

*Vibration problems such as machine vibrations are therefore a technically very demanding area of responsibility for specialists.*

## Technical recommendations

The explanations provide a brief insight into the fascinating world of vibration technology (vibration resonance) and the associated challenges in mechanical engineering.

*Technical expertise is essential.*

In order to identify potential vibration problems and assess their relevance in a litigation, in-depth technical understanding is essential. Parties involved should therefore obtain comprehensive information or consult experts in this field.

*Proof: Not as simple as it seems.*

Initiating an independent evidentiary proceeding or a lawsuit due to vibration problems requires careful consideration. Evidence must be able to be presented in a way that is admissible in court. Although an expert report is a common form of evidence, it loses its value if the vibration system has been destroyed or altered.

*Precise question for the evidence decision*

If a technical analysis shows that vibration problems can be proven in court, the precise formulation of the questions of evidence is crucial for the decision on the evidence. This task should not be left to lawyers, but should be carried out in cooperation with technically experienced experts.

*Cooperation between technology and law*

This recommendation does not in any way arise from distrust of lawyers. On the contrary, working with legal experts in complex technical cases is always enriching. However, the area of expertise of lawyers lies in law, while vibration technology is an extremely complex technical subject. Therefore, the involvement of excellent technical expertise in vibration issues in litigation is essential.

*Carefully weigh up litigation risk*

Before initiating litigation over vibration problems, the litigation risk should be carefully weighed. Since vibrations are often difficult to determine retrospectively in court, there are often unresolved aspects that can benefit the party not required to provide evidence.

Vibration technology in mechanical engineering is a fascinating and at the same time demanding field that requires both technical know-how and legal tact. Cooperation

*of experts in both fields is the key to successfully dealing with vibration problems in legal disputes.*

### Litigation risk: The dangers of overestimating the ability of court experts in the field of machine vibrations

In legal disputes, there is a risk that court experts will comment on technical issues that are beyond their expertise. This is especially true for the complex area of machine vibrations, where in-depth knowledge of vibration engineering, vibrations, resonances, structural mechanics and structural dynamics is required.

*The pitfalls of “seemingly simple” questions*

Even seemingly simple questions about machine vibrations can become a challenge for experts without specialist knowledge. As the general representation of a system of differential equations (equation 2) shows, it is impossible to predict the influence of parameter changes “just like that”.

*Beware of experience and overestimation*

Statements such as “ *from experience I can tell you that…”* should therefore be viewed with caution when referring to machine vibrations. Such assessments are often based on experience and subjective judgments rather than on sound specialist knowledge. In the complex interplay of vibration phenomena, even the smallest changes can have serious consequences that cannot be predicted without in-depth analysis.

*Avoid expensive wrong decisions*

Incorrect assessments by experts can lead to fatal wrong decisions. In the worst case, they become a party to a legal dispute because they believe they can definitely win the legal dispute based on the assessment of an expert or even a lawyer. That’s why I recommend involving experts in advance and, of course, for an appropriate fee in order to minimize the risk.

It is therefore essential to work with specialized experts and specialists to make informed decisions in legal disputes involving vibration issues. Only experts with in-depth specialist knowledge and many years of experience in vibration technology can assess the complex interrelationships. I therefore recommend:

- If you have vibration problems in legal disputes, rely on specialized experts with proven expertise in vibration technology.
- Don’t be blinded by general empirical values or seemingly simple explanations.

## Conclusion and final thoughts: Vibrations Resonance

The world of vibrations is fascinating, but also fraught with dangers. In technology, vibrations can lead to damage, product defects and even health problems. In legal disputes over responsibility, a Gordian knot often arises:

*the vibration system is usually destroyed.*

### Burden of proof: A serious problem also due to destruction of evidence

At the heart of legal disputes involving vibration issues is the burden of proof . In reality, destruction of evidence often means that it is no longer possible to make any significant technical findings in court proceedings regarding vibrations (vibration resonance), because:

- the oscillating system was changed,
- the machine was destroyed by the damage and
- relevant information was destroyed.

Although damaged components can sometimes provide information about the damage process, a clear clarification is often not possible.

*In these cases, the party with the burden of proof bears the risk.*

While “statics” is generally widely recognized, vibration engineering , the far more complex dynamics behind it, is often underestimated. The statics are merely a “by-product” of vibration engineering.

In legal disputes, a glaring deficit often becomes apparent:

*Clear technical requirements and a comprehensive understanding of the vibration problem are missing.*

The result is uninformed promises and the naive belief that vibration problems can be easily solved.

Vibration engineering is usually extremely complex and time-consuming.

*Lawyers understandably often underestimate this complexity.*

Publicly appointed and sworn experts can act as technical experts and develop solutions for complex vibration problems. However, this only happens on behalf of the party and against an appropriate fee.

The court expert in the vibration procedure can only clarify where errors were made, who bears the technical responsibility and what effects these errors had on the agreed performance. He cannot offer a technical solution in the court’s order.

### Conclusion: Specialists are indispensable

Vibration engineering problems require in-depth specialist knowledge. Quick solutions often lead to further technical and legal problems. In legal disputes involving vibration issues, it is therefore essential to involve specialized experts. Only experts can assess the complex relationships, prepare well-founded reports and represent your interests effectively.

*Contact specialized experts early on to protect your interests in vibration procedures.*