The acoustic and structural dynamic behavior of technical products—often referred to as Noise, Vibration, and Harshness (NVH)—is an important criterion in customers’ purchasing decisions. Therefore, NVH optimization is a crucial goal of the development process. NVH is influenced by dynamic excitations, caused by time-varying forces e.g. in gears or electric machines, as well as the transfer path of sound from the location of excitation to a receiver such as a driver’s ear and the sound perception. The behavior of the transfer path is determined by amplification or isolation of structure-borne and airborne sound between the location of the excitation and the location of the receiver. As a result, eigenfrequencies and damping in the transfer path form crucial structural dynamic properties.
Within jointed structures, friction between the surfaces of lap joints are the most dominant influencing factor on the transfer path’s damping. In order to increase the efficiency of NVH optimization in the development process, methods of virtual product development have become an established tool. Especially finite element method (FEM) and elastic multi-body simulation (eMBS) have allowed to numerically evaluate the structural dynamic behavior of technical products before manufacturing physical prototypes. While numerous methods for modeling jointed structures exist in FEM and eMBS environments, the choice of suitable eMBS models of joints remains challenging today. This is because the frictional behavior between the jointed partners in higher frequencies is largely influenced by the deformation of the joint surface. However, the deformation of the jointed parts is simplified in eMBS models compared to quasi-continuous FEM models. Accordingly, the choice of a suitable spatial discretization of eMBS joint models is a key factor in the model quality of eMBS models. Thus, this work proposes a modeling guideline for eMBS models of jointed structures based on the deformation of the structure using the criterion of rigidness. The method is demonstrated on an eMBS model of the academic Brake Reuß beam, which consists of two beam elements jointed with three screws in an overlap lap joint. Good agreement between eMBS simulation and measurement with an error of 1 dB is achieved using the proposed modeling method.
Hinweise
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1 Introduction
The purchasing decision of customers of technical products is heavily influenced by subjective criteria such as the acoustic and structural dynamic properties of technical products, often referred to as Noise, Vibration and Harshness (NVH) [1]. Therefore, optimizing the NVH behavior of technical products is paramount during the development process. With the advent of methods of virtual product development (VPD), simulation methods have become a well-established tool for NVH optimization of products before time-consuming and expensive manufacturing of first physical prototypes. Thus, NVH simulation models form a core aspect for resource-efficient NVH optimization during VPD. These NVH-models are usually built in elastic multibody simulation (eMBS) environments because of good computational efficiency which allows for studying the NVH behavior in the system context in time domain considering non-linear influences [2, 3].
In order to accurately model the NVH behavior in eMBS environments, the excitations mechanisms, the transfer of sound from the excitation point to the receiver as well as the sound perception of the receiver have to be modeled [4]. Within this framework, the transfer path is described by the amplification or isolation of sound and is mainly influenced by the eigenfrequencies of the sound-transmitting structure. At these eigenfrequencies, sufficient damping of vibrations is fundamental for achieving good NVH properties. Therefore, damping models of the transfer paths are crucial for accurately modeling the transfer path’s dynamic behavior. It is well known, that the damping of the transfer path of jointed structures, which are common in engineering applications, is mainly influenced by friction effects between the jointed parts compared to the material damping of the individual components [5]. Therefore, joint models need to be developed for eMBS models.
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While numerous FE models for jointed structures have been developed in the past [6‐9], the abstraction of the continuous relative movement of the jointed parts to eMBS is less studied. However, in order for damping models of jointed structures to be applicable to structural dynamic analysis of systems, suitable eMBS models are required, as FEM models are computationally too expensive to analyze system behavior. The main difference between quasi-continuous FEM models and eMBS models is that, in eMBS, the relative movement between two jointed structures is reduced to the movement between few discrete points. This abstraction can introduce errors to the eMBS model, if an insufficient, i.e. too coarse, spatial discretization is chosen. Therefore, for accurately modeling joint dynamics in eMBS, the influence of different spatial discretizations in eMBS models on the accuracy of joint models has to be analyzed. Additionally, it has to be studied, how a sufficient spatial discretization can be defined. Today, no universal guideline exists on how to define the spatial discretization of eMBS joint models.
Within this work, the influence of the spatial discretization of structural dynamic eMBS models of jointed structures is studied on the academic example of the Brake-Reuß-Beam (BRB). A criterion for the quantification of spatial distribution is chosen and its suitability is demonstrated. Furthermore, a method for parameterizing concentrated eMBS models based on a reference FE model, which is parameterized based on structural dynamic measurements, is presented. Its applicability is demonstrated by comparing the model’s output to structural dynamic measurements. The error between simulation and measurement is 1 dB with a difference in eigenfrequency of 25 Hz (corresponding to 2%).
The paper is structured into the discussion of the state of the Art in Sect. 2, the introduction of the method in Sect. 3, the presentation and discussion of results in Chap. 4 and concluding remarks in Sect. 5.
2 State of the art
In what follows, the state of the art regarding the behavior of jointed structures as well as their modelling in FEM and eMBS is presented.
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2.1 Structural dynamic behavior of jointed structures
The structural dynamic behavior of jointed structures is mainly influenced by the interaction in the joint surface [5] which can be divided into the normal and the tangential behavior of the joint, visualized in Fig. 1. The normal behavior is defined by the progressive correlation between normal pressure—induced by bolt tightening—and normal displacement. As a result, the normal pressure in the joint surfaces varies with distance from the bolt: In the region immediately next to the bolt, high pressures (denoted p in Fig. 1) occur, which decrease with increasing distance to the bolt. Far away from the bolt, little to no pressure can be observed.
Fig. 1
left: Normal pressure distribution in a bolted lap joint. right: tangential hysteresis
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The tangential behavior, see Fig. 1 right, is defined by effects of dry friction and can be visualized in form of a hysteresis. Starting from an equilibrium state without any tangential displacement, the tangential displacement is increased. The tangential force increases as the regions around the bolt with high preload transmit tangential forces according to Coulomb’s law (Eq. 1) until the slip limit FSlip is reached, which depends on the normal force FNormal and the coefficient of friction μ.
$$F_{\mathrm{Slip}}=F_{\text{Normal}}\cdot \mu$$
(1)
As the tangential displacement increases, the areas furthest away from the bolt start slipping first because of the relatively low normal pressure, which decreases the stiffness continuously as the region of slip increases in the direction towards the bolt. However, the regions close to the bolt still remain in the sticking regime. This effect of partial slip in the joint is called micro-slip. Once the tangential displacement increases so much that the regions around the bolt start slipping as well, macro-slip is reached and no further increase in tangential load with increasing tangential displacement is observed [10]. As soon as the direction of movement in tangential direction is reversed, the joint surfaces stick again and with increasing displacement, the regions furthest away from the bolt start slipping again.
During macro slip, the tangential force can be lower than the slip limiting force at the onset of macro-slip because the coefficient of friction under relative movement often is lower than the static coefficient friction [8]. This effect is neglected within the scope of this work and therefore also not visualized in Fig. 1 because, in industrial applications, jointed parts under normal operation do not experience macro-slip. Usually, the joint is designed in a way to carry the operating loads without slipping and only micro-slip induces damping to structural dynamic excitations. When micro-slip occurs, the tangential stiffness decreases with an increase in tangential displacement, while the tangential damping increases with increasing tangential displacement. This yields to a decrease in eigenfrequency and increasing damping of jointed structures under increasing tangential load and manifests itself as non-linear structural dynamic behavior [8].
The normal and tangential behavior of joints interact with each other. With an increase in joint preload, the normal stiffness and thus the tangential stiffness increases. With an increase in tangential stiffness, the displacement in tangential direction becomes smaller for any given amount of tangential force. This yields to a smaller part of the hysteresis being transversed during oscillations and thus smaller damping. Therefore, with an increase in bolt preload, the eigenfrequencies of jointed structures increase, while the damping decreases.
For complex joint surfaces, the effective stiffness and damping in a particular eigenmode depends on the mode shape [8]. Modes with a mode shape containing little relative tangential movement in the joint are less influenced by the joint damping than modes with a significant relative movement between the jointed parts. Therefore, the discretization of the movement between parts, which is often influenced by elastic deformation of the jointed parts, is crucial for achieving accurate structural dynamic models of jointed structures.
2.2 Modeling the structural dynamic behavior of jointed structures
Structural dynamic models of jointed structures have to accurately represent the non-linear stiffness and damping of the joints. As these models are usually formulated in an FEM or eMBS environment, structural dynamic models of joint behavior in FEM and eMBS are required.
First concentrated models of the structural dynamic behavior of jointed structures date back to the works of Hertz [11] for the calculation of normal contact behavior and Coulomb for linking the slipping force in tangential direction to the normal pressure. With the extension to Mindlin models [12], tangential dissipating behavior has been analytically modeled. Since then, numerous FEM models have been developed either based on phenomenological models (such as the bristle model by Dahl [13] and the LuGre-models [14]), or constitutive equations (such as Bouc-Wen [15]), linking the tangential behavior to the tangential displacement. In order to capture the locally distributed relative movement, caused by the flexible deformation of bodies, Zero-Thickness-Elements (ZTEs) have been proposed to model the normal and tangential behavior of jointed structures as well as their respective interaction [16].
With the advent of eMBS simulation, models were required to approximate the structural dynamic behavior of jointed structures as relative movement between discrete points. The simplest version of these models is the Jenkins element, which consists of a parallel connection of a linear spring and a dry friction element, see Fig. 2, left. This allows for the generation of stick-slip hysteresis with a constant stiffness in the sticking regime.
Fig. 2
left: Jenkins element, right: Iwan element
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In order to model the degressive behavior of the stiffness during micro-slip, numerous Jenkins models have to be stacked in parallel, which yields the Iwan model with discrete, degressive stiffness. The Reduced Iwan model assumes a distribution of differently parametrized Jenkins elements, thereby yielding a continuous hysteresis in micro- and macro-slip [5]. The model and its corresponding hysteresis are depicted in Fig. 2, right.
While the Reduced Iwan model is able to capture the effect of micro-slip, it is not capable of capturing the effect of different mode shapes. Therefore, in order to represent the effect of different mode shapes, multiple Reduced Iwan elements have to be applied at different points in the joint area. As a result, the influence of different spatial discretizations in eMBS models on the accuracy of structural dynamic eMBS models has to be analyzed.
3 Method
Modeling the structural dynamic behavior of jointed structures in eMBS environments requires an adequate choice of the following three modeling aspects considering both the required accuracy of the model as well as the resulting computational effort:
1st: Choosing an appropriate physical model for the joint behavior
2nd: Defining a suitable spatial discretization in eMBS, i.e. defining the amount and positioning of modeling elements required to model the behavior of a specific joint
3rd: Identifying suitable parameters for the chosen model
While numerous publications exist on physical models for structural dynamic joint behavior (see Sect. 2), the identification of suitable parameters and spatial discretization has not been the focus of recent research. However, if the goal of the eMBS model is to evaluate design changes of jointed structures, such as changing the number of bolts or their spacing in the joints, choosing the correct spatial discretization is crucial for parameterizing joint models. The main reason for this is that with a too coarse discretization, the parameterization of an eMBS model contains effects of the flexibility of the surrounding structure and thus will not appropriately represent the interaction behavior between the jointed parts. However, the discretization should not be too detailed in order to minimize the computational effort. Therefore, within this work, the approach depicted in Fig. 3 is suggested for modeling jointed structures in eMBS environments and analyzing the influence of deformation in joints on the joint’s dynamic behavior. It consists of three main steps: The choice of the Reduced Iwan model for representing the joint’s dynamics in eMBS, the definition of a suitable spatial discretization using the criterion of rigidness and a parameter identification process based on a validated FE model using quasi-static hysteresis curves. The three main aspects of the proposed model are discussed in greater detail in what follows:
Fig. 3
Suggested method for defining eMBS models of jointed structures
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Firstly, the Reduced Iwan model [5] is chosen to model the stiffness and damping behavior of the joint based on its capabilities of modeling tangential damping and amplitude-dependent stiffness in micro- and macro-slip. The choice of the Reduced Iwan model is furthermore based on the fact that it has already been successfully implemented to model the behavior of the jointed BRB using discrete modeling elements [6]. The Reduced Iwan model has been implemented in the commercial eMBS software SIMPACK. The Reduced Iwan model has, within the scope of this work, been extended to represent two-dimensional movements in the plane of the joint as well as rotational movement in the joint plane using the same formulation for the hysteresis as for the translational movement.
Secondly, the definition of suitable spatial discretization is defined using the rigidness criterion [17]. This criterion has originally been defined for frequency-based substructuring and describes how well the movement of several points can be defined by a rigid-body movement. A rigidness value of \(\rho =1\) indicates perfect rigid-body-movement and a rigidness value of 0 indicates that no rigid body movements occurs, and all movement of individual points is attributed to flexible deformation of the body, see Eq. 2.
It is proposed to use the rigidness value for the evaluation of a choice of spatial discretizations of the joint. The main idea behind choosing the rigidness as the criterion for the spatial discretization is that the main loss of information between FEM and eMBS is in the reduction of the movement of the jointed parts to few force application points, which represent a specific area of the joint surface. If the rigidness of the nodes in that specific surface is high, the reduction of the movement to one individual force application point introduces little error. With a low rigidness and thus a high discrepancy between the movements of the individual nodes and the force application point, a larger error is introduced.
In order to do define a discretization using the rigidness criterion, the following steps are executed: Firstly, a parametrized FE model of the joint is generated. A suitable method for FEM modeling and parametrization based on structural dynamic measurements has been proposed and validated in [18]. Using this FE model, an eigenfrequency analysis is carried out, which yields the mode shapes of the jointed assembly. Subsequently, a spatial discretization for the joint surface is assumed and the rigidness of the nodes of the surface to be represented by one Reduced Iwan Element is calculated. Then, iteratively, the spatial discretization is refined, until sufficiently high rigidness values are obtained. This spatial discretization is then used to build the eMBS model. This process involves the modal reduction of the jointed parts to the chosen force application points. The force application points are connected to the FE model of the part to be reduced using continuum distributing Multi-Point constraints (MPC) as this allows the flexible deformation of the FE nodes connected to one force application point, which yields to similar eigenfrequencies between the FE and the eMBS model of the reduced parts. A suitable threshold for the rigidness is identified on an application example by comparing the structural dynamic influence of different fidelity levels of spatial discretization (see Sect. 4).
Thirdly, the parameterization of the eMBS models is conducted by fitting the quasi-static hysteresis of the Reduced Iwan element to the quasi-static hysteresis of the FE model. To do so, the FE model is separated into individual joint elements, which are to be represented by one Reduced Iwan element, see the blue partial model of the joint in Fig. 4. Then, the normal load of the joint is applied to the FE model, which is derived from a quasi-static preloading analysis of the joint by bolt tightening. Subsequently, by translational and rotational movement of the FE model, the hysteresis is obtained in both translational and rotational degrees of freedom for different displacement/angular amplitudes, see Fig. 4. Lastly, the parameters of the Reduced Iwan element are optimized using a gradient-based algorithm to best match the hysteresis obtained by FE for different displacement and rotation amplitudes in micro- and macro-slip.
Fig. 4
Parametrization of eMBS model: left: FE model with corresponding hysteresis under normal load, right: eMBS modeling approach and identified hysteresis
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The suggested method allows for physically meaningful identification of eMBS modeling parameters, which only represent the relative movement of a part of the joint surface that can be assumed rigid. Therefore, the modeling element can be retained when evaluating design changes.
The presented method based on a validated FE model of the Brake Reuß beam (BRB), visualized in Fig. 5, and validated against structural dynamic measurements on the BRB.
Fig. 5
Brake-Reuß beam [8]. a monolithic beam, b jointed beam, c measurement setup
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In short, the BRB consists of two beam freely suspended elements, which are connected using a bolted lap joint of three M8 bolts. The bolts are tightened based on their axial forces which are measured using strain gauges to 1.15 kN, 5.7 kN and 11.5 kN. The comparison between the jointed assembly and the monolithic parts, as well as a reference measurement of one monolithic beam with identical dimensions to the jointed assembly, allows for identifying the structural dynamic influence of the joint. To do so, the assembly is excited on one beam element using a shaker, and the response is measured using a laser-doppler vibrometer (LDV) on the other beam element. The use of the LDV minimizes the influence of the sensors and especially the sensor cables on the structure’s damping.
4 Results
The measurement results and FE simulations on the BRB, which serve as a foundation for the eMBS modeling and the analysis of the flexible deformations in the joint surface are explained in detail in [18, 19] and will only be discussed briefly before the eMBS modeling, parametrization, validation and analysis are presented.
The structure is excited at its resonance frequencies using different force amplitudes and bolt preloads. The behavior is characterized by the frequency response function (FRF) of the mechanical admittance, which describes the ratio between the frequency-dependent velocity of the structure at the measurement point of the LDV and the excitation force. In Fig. 6, the FRFs for different excitation force amplitudes using a nominal bolt preload of 1.15 kN per bolt for the 1st and 3rd bending modes are exemplarily shown. It can be seen that with an increase in excitation amplitude the peak value of the FRF decreases, indicating an increase in damping. Furthermore, the resonance frequency shifts to lower values with an increase in excitation force amplitude. Additionally, it can be seen from the comparison between the FRFs of the two modes that with different mode shapes the effect of varying excitation force changes: While the first bending mode shows a reduction of 9 dB in peak amplitude by increasing the excitation amplitude from 0.1 to 1 N, the third bending mode only shows a reduction of 3.5 dB. The measurements results are subsequently used as a foundation for the parametrization and validation of an FE model, which has been described and validated in detail in [18].
Fig. 6
Measured FRFs of the first (left) and third (right) mode of the assembled BRB for different force excitations
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The FE model shown in Fig. 7 consists of the two beam elements which are represented using modal reduction according to [20], as well as the statically reduced Bolts, nuts and washers. In between the beam elements, the joint behavior is modeled using Zero Thickness Elements (ZTE) according to [16]. The model parameters of the normal contact behavior have been identified using optical measurements of the surfaces’ microgeometry to obtain the surface roughness and subsequently the relation between pressure and normal displacement. The tangential modeling parameters are obtained by minimizing the error between simulated and measured eigenfrequency as well as peak values using a gradient-based optimization algorithm. The parametrization process is described in detail in [18]. The FE model’s validity is proven by comparing the simulated and measured FRFs of the assembled BRB which differ by 2.5 dB in amplitude and 1.7% in eigenfrequency.
The validated FE model serves as a foundation for the transfer of the computationally slow FE simulation to the eMBS simulation with the aim of modeling the structural dynamic behavior of jointed structures in a computationally efficient environment, which allows for the analysis of design choices on the system level. The method presented in Sect. 3 and visualized in Fig. 3 is therefore applied. In order to choose an appropriate discretization of spatial discretization, the validated FE model is used to assess the mode shapes of the jointed assembly, linearized around a bolt preload of 1.15 kN. Two mode shapes (the second and third bending mode) are exemplarily visualized in Fig. 8.
Fig. 8
FE-generated mode shapes of assembled BRB, linearized at 1.15 kN bolt preload and 0 N tangential load
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The comparison shows, that for different mode shapes the amount of flexible deformation within the joint varies: While the second bending mode shows no deformation of the jointed structures in the area of the joint, the third mode exhibits significant deformation of the jointed parts. Therefore, to generate Reduced Iwan elements which are able to model both mode shapes, an appropriate spatial discretization has to be chosen that allows to approximate the areas represented by one instance of the Reduced Iwan element to be approximated at rigid. To study the influence of the spatial discretization and thus of the flexibility, the rigidness of three different fidelity levels of spatial discretization is evaluated, see Fig. 9.
Fig. 9
Different fidelity levels of spatial discretization on the BRB
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The coarsest spatial discretization has only one force element for the entire joint. The second fidelity level assumes one modeling element per bolt. The third representation accounts for the flexibility between the outer bolts and the outermost part of the jointed surface, where no bolt is situated. This discretization thus uses five instances of the Reduced Iwan element in the joint surface (one at each bolt and one additionally on each side of the joint at the outer most region). The corresponding rigidness values are 86.4%, 96.9% and 99.5%. They are obtained for each node and mode according to Eq. 2. Subsequently, the average value of all nodes in the joint surface that is represented by one Reduced Iwan element for one specific mode is used for quantifying the deformation.
The parameters for the Reduced Iwan models are identified by generating FE models of the areas to be represented by one iwan element in FE, applying the correct normal pressure distribution as obtained from a quasi-static bolt pre-tensioning analysis and generating hystereses in the translational and rotational degrees of freedom with different displacement amplitudes. The hystereses are exemplarily shown for one area of the jointed surface in Fig. 10.
Fig. 10
FE generated hysteresis of a part of the joint to be represented by one Reduced Iwan element
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By minimizing the error in effective stiffness and damping for different micro-slip and macro-slip displacements in rotational and translational direction, the parameters of the Reduced Iwan model are obtained. For the specific example studied, which consist of a BRB made of aluminum by milling with an arithmetical mean height (Sa) of 0.77 μm, the parameters for the three different variants are listed in the appendix A.
Subsequently, the identified parameters are used to parameterize the eMBS model of the jointed BRB and the eMBS model is excited using a frequency sweep in time-domain with different excitation amplitudes. The result of the finest spatial discretization is compared to the measurement in Fig. 11 for the third eigenmode. It can be seen that the absolute values of resonance frequency and peak amplitude show a difference of 23 Hz (1.9%) and 1 dB, respectively. Additionally, also the nonlinear influence of the force amplitude is matched to within 0.25 Hz and 0.4 dB. Therefore, the reference model using five Reduced Iwan elements in the joints can be assumed to be sufficiently finely discretized.
Fig. 11
Comparison of measured (left) and eMBS simulated (right) FRFs for different force amplitude of BRB using the proposed modeling method
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Firstly, the influence of the spatial discretization on the computational effort is discussed. For the chosen example of the BRB, the calculation time is mainly driven by the execution of the joint model as the calculation effort of the modally reduced bodies is negligible. However, the maximum allowable time-step is defined by the frequency at which the assembly is excited and not by the amount of considered modeling elements. Therefore, an increase of Reduced Iwan elements from 3 to 5 elements in the model increases the computation time by 21%.
The comparison of the influence of the different spatial fidelity levels are shown in Fig. 12.
Fig. 12
Influence of spatial discretization on FRF of eMBS-simulated BRB
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The comparison between the reference fidelity level with five modeling elements and the use of three modeling elements, neglecting the deformation at the outer most regions of the joint, show that the absolute values only differ by 12 Hz in eigenfrequency (corresponding to 1%) and 0.6 dB in amplitude for the lowest excitation force amplitude, but that the nonlinear dependency of the force amplitude is over-estimated by 3.5 dB. It can therefore be concluded, that for this specific example, the discretization with three modeling elements, corresponding to a rigidness of more than 95%, can be used to calculate the overall damping and stiffness of the joint, but that for a detailed study of the nonlinear influences, the finer reference model with a high rigidness value of 99.5% should be used. Additionally, it can be seen that with the coarsest representation of only one modeling element in the joint, the resonance is not visible at all, as the rotational aspects of the deformation are lost in the modeling representation. Therefore, it can be concluded, that the deformation of the jointed parts has a significant influence of the joint behavior and has to be considered during the modeling process by choosing an appropriate spatial discretization. Additionally, it can be shown that the representation of the rigidness correlates with the quality of the correlation between eMBS model and measurement.
5 Conclusion & outlook
The nonlinear damping effects in bolted lap joints of jointed structures significantly influence the structural dynamics of assembled structures and therefore need to be modeled in structural dynamic modeling environments such as elastic multi body simulation (eMBS) to predict a system’s NVH behavior. Within this work, a modeling approach for modeling the structural dynamic behavior of jointed structures in eMBS consisting of three steps is proposed: Firstly, the Reduced Iwan model is proposed as the modeling element for eMBS modeling of joint dynamics. Secondly, the rigidness is applied as a criterion to analyze the influence of local deformations in the joints and define a spatial discretization of the joint. Lastly, a parameterization process for Reduced Iwan models based on quasistatic hysteresis simulations in a validated FE model of a jointed assembly is presented. The presented method is applied to the academic example of the Brake Reuß beam. It is validated against measurements and deviations of 1 dB in peak amplitude and 2% in eigenfrequency are achieved, which is sufficiently small to analyze structural dynamic properties. Furthermore, it has been shown that the quality of the correlation between simulation and measurement correlates to the rigidness criterion. It has been suggested to define spatial discretizations based on the rigidness criterion with a threshold of 95% for absolute damping values as well as 99.5% for studying nonlinear influences. The effect of the spatial discretization on the computation time has been shown to be a 21% increase when using five instead of three Reduced Iwan elements.
In further works, the applicability of the derived thresholds for defining spatial discretization of joints on more complex structures should be proven. Additionally, the effect of the spatial discretization on computational effort in more complex systems should be studied as it is to be expected that with additional modeling elements in complex systems not related to the joint dynamics the influence of the individual joint elements on the computational effort decreases. Furthermore, the influence of different friction models considering e.g. the differences between static and dynamic coefficient of friction could be analyzed.
Acknowledgements
This research has been supported by the Federal Ministry for Economic Affairs and Energy within the scope of IGF grant No. 21111 N.
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Conflict of interest
S. Wischmann, G. Jacobs and G. Höpfner declare that they have no competing interests.
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