To establish a relationship between a system-specific process or state variable and potential measurands methodically, a catalog system consisting of a physical effect matrix and an associated effect catalog is proposed. In order to develop a suitable catalog system, the basic idea of the established catalog systems by Koller (1998) and Roth (2000) was taken up and combined with the basics of multipole-based modeling. According to the intended purpose, a cause-effect perspective is consistently applied. This results in a physically and logically justified structure of the developed catalog system, which is complete against the background of the underlying modeling.
The overall objective of the developed catalog system, is to establish a relationship between a systemspecific process or state variable and potential measurands. This is achieved by a concatenation of physical effects, socalled physical effect chain. Generalized relationships between a cause and a resulting effect are systematically built and mapped in the twodimensional effect matrix. The information about the individual physical effects that establish these relationships is stored in a onedimensional effect catalog.
4.1 Effect Matrix
The two-dimensional overview catalog is structured in the form of a matrix, schematically shown in Fig. 7, the socalled effect matrix. This matrix lists causes in columns and the resulting effects in rows. Since the effect matrix serves as an overview catalog, the approach below and the resulting structure of the effect matrix represent the elementary result of this section.
4.1.1 Fundamental Approach for Structuring the Effect Matrix
The causes and effects are divided on basis of the main domains of classical physics in mechanics, electricity and magnetism as well as thermodynamics [cf. e.g. Hering et al. (2016)]. The field of periodic changes in state is considered in the context of the domains mechanics (matterbound waves, e.g. fluid sound) and electricity and magnetism (nonmatterbound waves, e.g. light). The physical quantities listed in the outline section are derived from the eight quantities of physics that can be balanced. The general balancing of energy forms the basis for this. For the differentiated classification, the other seven balanceable quantities of classical physics are used: momentum \(p\), angular momentum \(L\), (heavy) mass \({m}_{S}\), volume \(V\), electric charge \(Q\), entropy \(S\) and amount of substance \(n\) [cf. e.g. Maurer (2015)]. According to the basics of system variables introduced in section 2.4, the seven balanceable quantities are used as primary variables \(X\). The respective flow and effort variables \({I}_{M}\) and \(Y\) as well as the extensum \({E}_{X}\) are then derived. In this way, the outline section is physically and logically justified. An overview of the approach to structure the effect matrix is shown in Fig. 8. Figure 10 illustrates an extract of the resulting outline section of the effect matrix using the example of the translational momentum p and the electric charge Q.
The resulting structure and compatibility with multipole-based modeling exhibit further significant advantages:
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On the basis of the differentiation it is possible to draw a direct conclusion on the metrological properties of the quantities. According to section 2.4, the four system variables within a physical domain are differentiated into P-variables, socalled singlepointquantities, and Tvariables, socalled twopointquantities.
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Since the energy exchange between the network elements within a multipole-based model can always be described by the flow and effort variable of a domain, energy flows and thus the changes and transformations of a function variable occurring in the system can be modeled [Janschek (2012)]. This is particularly relevant against the background of the transfer of energies or signals in a technical system.
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In this way, it is possible to structure the system step by step along nodes, e.g., on the basis of flow variables, and to model and view it sequentially [cf. e.g. Vorwerk-Handing et al. (2018)]. The term „node“ goes back to the consideration of the electric current in electric networks according to Kirchhoff‘s node rule (1st Kirchhoff’s law) and can be transferred analogously to other flow variables [cf. e.g. Meschede (2015); MacFarlane (1964)]. In mechanics, e.g., this corresponds to a balancing of forces on a freecut element. Figure 9 illustrates an exemplary visualization of the mentioned relationships.
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Via the respective effort variables, relationships between different discretely modeled substitute elements in a system can be mapped along meshes [MacFarlane (1964)]. In electric networks (cf. Figure 9) this corresponds, e.g., to the consideration of all partial voltages in the circuit of a mesh according to Kirchhoff’s mesh rule (2nd Kirchhoff’s law) [cf. e.g. Meschede (2015)].
In multipole-based modeling, the relationship between the four system variables is established either via design parameters or a temporal relationship [Roth (2000)]. This fact is considered by including the corresponding design parameters and the time \(t\) in the effect matrix. Since the design parameters and the time \(t\) establish the relationship between the system variables, these quantities cannot occur as a cause in a physical effect and are therefore not listed in the input outline section of the effect matrix. However, since design parameters are influenced by system variables, the design parameters as well as the time \(t\) are listed in the line-by-line structure of the effect matrix (cf. Figure 11).
In order to be able to build up and model relationships between function variables of different physical subdomains, the described structure of the output in the outline section of the effect matrix is of fundamental significance for two reasons:
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Design parameters not only establish a relationship between system variables within a subdomain, but also across subdomains. In the latter case, energy is exchanged between the subdomains involved. An example of the establishment of a relationship between system variables of different subdomains is Coulomb’s law. It describes the force \(F\) between two electric charges \(Q\) and \(Q’\), which are idealized as point charges (cf. Figure 12).
Consequently, a relationship between the electric primary variable of the charge\(Q\) (cause) and the flow variable\(F\) (effect) from the domain of mechanics is described. This relationship is established by the design parameter of the length[2], the influence constant \({\epsilon }_{0}\) and the circular number \(\pi\)as shown in Eq. 4.
\(F=\frac{Q\bullet Q{\prime }}{4 \pi \bullet {\epsilon }_{0}\bullet {r}^{2}}\) (4)
In this context, \(F\) is the force emanating from the electric charges \(Q\) and \(Q‘\)and \(r\) is the distance between these two charges, which are assumed to be pointshaped.
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Furthermore, design parameters of a subdomain can be influenced by system variables or function variables of another subdomains. In this case, there is approximately no energy exchange between the subdomains involved. By influencing a design parameter, system variables or function variables of one subdomain indirectly influence another subdomain. These relationships can be used metrologically, e.g. in measuring resistors. In the application case of the measuring resistor, the design parameter of the electric resistance \(R\) (subdomain of electricity) is influenced by the effect of a thermal function variable in the form of the temperature \(T\) (subdomain of thermodynamics). Indirect temperature measurement is realized by measuring the voltage drop across the temperaturedependent electrical resistance \(R\) through which a constant measuring current flow.
In addition, this systematic consideration of design parameters allows a clear distinction between system variables and design parameters, which is, e.g., not present in the matrix of Koller (1998).
In order to meet the level of abstraction chosen by the user when considering the described relationships, the superordinate design parameters are further differentiated into the independent material and geometric properties (cf. Figure 11). The user sees, for example, a relationship between the force \(F\) and the displacement \(s\), which is described by Hooke's law. Depending on the needs and knowledge of the user, this relationship is established on different observation levels. On the one hand, it is possible to establish the relationship via the elasticity or spring constant k of the component under consideration, which can be directly determined in experiments. On the other hand, in a one-dimensional case, the relationship can be established via a differentiated consideration of the surface area \(A\) perpendicular to the acting force \(F\) and the initial length \({l}_{0}\) (geometric properties) and the elastic modulus \(E\) of the material (material property) (cf. Figure 13).
4.1.2 Extensions
The application and the comparison of an effect matrix with an outline section built up according to the explanations with the assignment matrix according to Koller (1998) show that it is reasonable and necessary to include also derived variables. An essential cause for this is a direct metrological relevance of derived quantities.
The basis for the extension is formed by two essential observations: First, the underlying approaches of systems physics and multipole-based modeling cannot represent some areas of physics that are both physically and especially technically meaningful and necessary. Second, physical quantities derived from the listed process or state variables sometimes have a wide practical use. Both aspects will be explained in more detail in the following and, based on this, an extension of the approach for structuring the effect matrix will be presented.
The derived variables include related variables that are derived from the process and state variables already listed via
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a reference to a design parameter (e.g. mass \(m\), length \(l\), surface area \(A\) or volume \(V\)) or
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a derivation according to the time \(t\), an integration over the time \(t\), respectively.
In addition to classical mechanical quantities such as the stress \(б\) or \(\tau\) or the strain \(\epsilon\), electric and magnetic flow quantities are also included in the approach in this way. As an example, the extensions of the basic structure of the effect matrix resulting from this aspect are exemplary shown in Fig. 14 for the quantities of the translational momentum \(p\) and the electric charge \(Q\).
Since the systems physics approach [Maurer (2015)] and the multipolebased modeling based on the quantities of classical physics are not suitable for acquire and model magnetic effects, an extension of the approach is sought. The reason for the described limitation is that the existence of a magnetic monopole or magnetic charge as a counterpart to the electric charge Q is practically not proven [Meschede (2015)]. The necessity of an extension results from the fact that such a magnetic charge would have to be used as a primary variable in the basic system in order to derive the three other system variables. Since magnetic effects are important from a technical and, in particular, a metrological point of view and must be considered in the effect matrix, the magnetic quantities and relationships shown in Fig. 15 are included in the structure part. In order to achieve a representation that is as uniform as possible, the representation of the relationships in Fig. 15 is based on the overview representation used according to section 2.4.[3] However, as described, it represents a deliberately defined exception to the basic approach starting from a primary variable according to Fig. 8.
Aspects of wave theory (matter-bound and non-matter-bound waves) have to be considered in a differentiated manner in the structure part of the effect matrix. Matter-bound waves, e.g. fluid sound, are considered in the domain of mechanics and non-matter-bound waves, e.g. light, in the domain of electricity and magnetism.
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Depending on the propagation medium (solid or fluid), matterbound waves as periodic changes of state can be represented as structureborne sound by function variables of the translatory momentum or as fluid sound by function variables of the balanceable quantity of the volume. This differentiation is based on the fact that in fluids only longitudinal waves occur and propagate in the form of pressure and density fluctuations. In most cases, a dynamic field variable, the sound pressure \(p\), is measured to quantify the fluid sound. Kinematic variables, such as the sound velocity \(v\), are rarely measured [Cremer et al. (2005)].
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Solids, on the other hand, absorb not only normal stresses but also shear stresses. As a result, both longitudinal waves and transversal waves occur and propagate independently of each other in solids. In contrast to fluid sound, kinematic variables such as deflection (i.e. the relative displacement or elongation \(\epsilon\), respectively), velocity (speed \(v\)) or acceleration a are mainly measured to quantify structureborne sound. Dynamic variables such as stresses \(б\) and forces \(F\) are – if required – determined indirectly from the derivatives of kinematic variables and corresponding material properties [Cremer et al. (2005)].
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Coupled electric and magnetic fields in the form of nonmatterbound electromagnetic waves are considered in the domain of electricity and magnetism by listing their characterizing variables wavelength \(\lambda\)[4] and intensity \(I\).The complete outline section of the developed effect matrix is included in Vorwerk-Handing (2021) in Appendix C1 in Figure C.1.
4.2 Effect Catalog
The overall objective is to identify effect chains that can be used metrologically in an (existing) technical system. The effect catalog contains the relevant information about the physical effects listed in the effect matrix. Of particular relevance is the collection of selection criteria for certain effects that can already be applied at this level of abstraction and the resulting opportunity to develop targeted and promising effect chains.
The one-dimensional effect catalog systematically lists information and references about the physical relationships included in the effect matrix. For this purpose, the two catalogs have an interface in the form of a consistent and unambiguous designation of the physical effects contained. This is implemented by means of order numbers or by the literal designation of the physical effect. Since the introduced effect matrix and the effect catalog form a catalog system with a common interface, the structure of the effect catalog’s outline section depends on the effect matrix [cf. also Vorwerk-Handing (2021) Appendix C1, Figure C.1 and Figure C.2]. The contents and information of the effect catalog are subdivided into main content and access section according to the structure shown schematically in Fig. 16.
4.2.1 Main Content of the Effect Catalog
In the main content of the effect catalog, the relevant information and references to the physical effects are listed. The basis for the selected information goes back to the catalog system by Koller (1998). Accordingly, as far as possible, a sketch, an application example and references to further literature are listed for each effect. Assumed simplifications are not listed in a separate property table as in the catalog system according to Koller (1998), but are included directly in the effect catalog (cf. Figure 17).
Since the equations according to Koller (1998) and Roth (2000) are either generally valid and abstract or describe a certain case or a certain characteristic of the physical effect and thus already contain certain assumptions, the column “characteristic” is added compared to the original structure from Koller (1998). This column describes the principal characteristic(s) of the relationship between cause and effect qualitatively, e.g. in bullet points or a graph (cf. Figure 17). In this way, it is possible for the user to quickly and intuitively grasp the basic relation or possible characteristics of the relation.
The essential extension compared to the main contents of existing effect catalogs lies in the differentiated consideration of the quantities that establish the relationship between cause and the effect. According to section 3.1 and 4.1, the relationship between a causal function variable and a resulting effect is established by design parameters and/or a temporal relationship. Design parameters are composed of material and/or geometric properties of the component under consideration. The listing of both superordinate design parameters as well as material and geometric properties is consistent with the effect matrix (cf. section 4.1) and is justified by the degree of abstraction selected by the user in the consideration of the context. In addition, other function variables can be (passively) involved in a causeeffect relationship. An example therefore is the Lorentz force, cf. Eq. 5 [Meschede (2015)]:
\(\overrightarrow{F}=Q \overrightarrow{v}\times \overrightarrow{B} .\) (5)
Equation 5 describes the force \(\overrightarrow{F}\) which acts on an electric charge \(Q\) moving in a magnetic field (magnetic flux density \(\overrightarrow{B}\)) at the speed \(\overrightarrow{v}\). According to the exemplarily illustrated relationship, both the speed \(\overrightarrow{v}\) and the magnetic flux density \(\overrightarrow{B}\) can be regarded as the cause of the acting Lorentz force \(\overrightarrow{F}\), depending on the point of view. Depending on the definition of the considered causeeffect relationship in the effect matrix, the further function variable involved in the physical relationship is listed in the main content of the effect catalog. The extensions shown are summarized in the column “Relationship via …” in Fig. 17.
The presented extension of the effect catalog is useful and necessary for the intended purpose for two reasons:
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Through the differentiated listing of the variables that establish the relationship between cause and effect, it becomes obvious which dependencies exist and which variables must be known in order to be able to establish a metrologically usable, unambiguous relationship between cause and effect.
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Functionally relevant design parameters are of particular interest in the field of condition monitoring, but cannot be used as input variables in the effect matrix according to section 4.1. In contrast to function variables, design parameters are not changed or converted. They establish the relationship between function variables and thus influence the relationship between cause and effect.
By determining the cause and effect of a physical effect, the (changed) relationship and thus the (temporal) change of the involved design parameters can be inferred. Consequently, all physical relationships in which the soughtafter design parameter occurs are of interest, as potentially influenced relationships between function variables can be identified in this way. Proceeding from an initial relation between the soughtafter design parameter and an effect in the form of a function variable, effect chains can be developed by means of the effect matrix introduced in section 4.1.
In order to be able to establish a relationship between a design parameter to be determined and potential measurands by means of the effect matrix, an intermediate step is therefore necessary. In this intermediate step, the presented extension of the effect catalog is used to structure the catalog. A filtering of the effect catalog according to the design parameter to be determined in the category “Relationship via …” leads to function variables or physical effects, which are potentially usable for the determination of the design parameter. These can then be used as input variables in the effect matrix. The principle procedure therefore is illustrated in Fig. 18.
4.2.2 Access Section of the Effect Catalog
The aim of the access section is to enable the user to preselect potentially usable physical effects. In order to enable a comparison between requirements and boundary conditions of the individual physical effects and the considered technical system, necessary properties of the technical system are derived from the effectspecific requirements and boundary conditions. This step is based on the statement from Pahl et al. (2007) described in section 2.2 that the necessary requirements of a technical system for the realization of a physical effect can be described via geometric, material and kinematic properties of the technical system. In order to be able to carry out the desired comparison systematically and automatically, generally valid properties are defined, which preferably have a hierarchical structure. The starting point is the differentiation between geometric, material and kinematic properties of a technical system.[5]
Geometric properties can be categorized in a generally valid way, e.g. according to Pahl et al. (2007) into the characteristics type, shape, position, size and number as well as correspondingly associated characteristics [Pahl et al. (2007)]. However, since physical effects do not usually depend on a defined geometry, it is not possible to define necessary geometric properties for physical effects using such a generally valid characterization approach. Since every material object has geometric properties, but a description of these cannot be brought into a generally valid relationship with the requirements of physical effects, the geometric properties will not be included any further in the following effect catalog. Independently of this, existing information about geometric properties of the technical system offer potentials to be used individually in addition to the approaches introduced in this work. In particular against the background of the primary function fulfilment by function-relevant design parameters as well as installation space restrictions the geometric properties of the initial system are compelling to be considered individually.
In order to enable a systematic preselection of potentially usable physical effects, material and kinematic properties are considered. On the one hand, material properties can be considered in isolation, related to the respective component of the technical system. Kinematic properties, on the other hand, always require a temporal and/or spatial reference and can therefore only be considered at the system level [Sena (1972)]. From this fact follows the differentiation of the necessary properties listed in the effect catalog into componentdependent and systemdependent properties (cf. Figure 19).
The componentdependent properties are considered in a first step as general material properties and in a second step as applicationspecific material properties of the respective component. The state of aggregation (solid, liquid, gaseous) is used for a general differentiation (cf. Figure 19). Building on this, the properties in the main content “Relationship via …” are adopted under the heading “material property”. It must be checked for each specific application whether the corresponding material property is available in a usable form.
As systemdependent properties, generally valid kinematic properties of the component in relation to the technical system are considered in the first step [Pahl et al. (2007)]. According to Pahl et al. (2007), the type and form of the movement are used for this purpose. Possible specifications of the mentioned properties are quiescent, translatory or rotatory as well as uniform, nonuniform and oscillating (cf. Figure 19). In the second step, the function variables listed in the main content in the category “Relationship via …” under the heading “Other function variables” are analyzed. In this analysis, it must be investigated for each individual application whether the corresponding function variable is available in a usable form in the considered technical system.
In general, the information listed in the effect catalog in the columns “Simplifications made” and “Characteristic” can be used to preselect potentially usable physical effects.
The aim of the access part shown in Fig. 20 is to provide the opportunity of selecting effects based on a comparison between necessary properties of the component or system under consideration from the effect point of view and existing or producible properties. Here, it is explicitly pointed out that currently existing properties of a component can potentially be changed in order to enable certain physical effects. In particular, material properties have this potential, as long as the functionrelevant properties are not negatively influenced. Basically, there is the possibility of a replacement or a local modification of the considered component. Since kinematic properties are usually directly functionally relevant, they do not usually offer this potential. An incompatibility between the properties of the technical system and necessary properties from the perspective of a potential physical effect leads to the exclusion of the respective effect.
[2] The distance r between the two charges Q and Q‘, which are assumed to be point‑shaped, corresponds to the design parameter length l.
[3] A comparable account for the consideration of magnetic effects in the context of multipole-based modeling is also described by Grabow (2018). Beyond that, Grabow (2018) does not provide a physical justification.
[4] The wavelength λ can optionally also be expressed indirectly by means of the media‑dependent phase velocity cmed via the frequency f. The relationship is established by .
[5] It should also be noted that Roth (1982) deals extensively with access and structuring features for catalogs and includes a collection of corresponding features.