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23rd International Conference on Modelling in Mechanics 2025 Analysis of Damping of a Ball Absorber from Experimentally Obtained Data Marek Kawulok a,b, *, Stanislav Pospíšil a,b a Department of Structural Mechanics, Faculty of Civil Engineering, VSB- Technical University of Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava-Poruba, Czech Republic b Institute of Theoretical and Applied Mechanics of the Czech Academy of Sciences, Prosecká 809/76, 190 00 Prague 9, Czech Abstract This paper focusses on the analysis of the damping of a ball absorber's motion based on experiments with free oscillations. The experiments were conducted with a ball absorber restricted to planar motion, where its free oscillations were recorded using a video camera. The video footage was then analysed with an algorithm that tracked the centre position of the ball over time, allowing the creation of displacement curves of the ball over time. On the basis of this position information, the angular displacement of the centre relative to its equilibrium position was subsequently calculated. The paper investigates the influence of viscous damping and friction on the motion of the ball, with a particular emphasis on the interaction between the rolling track surface and the ball. A key aspect of the analysis involves comparing two surface variants, one featuring rubber tubing applied to the surface and the other remaining smooth. The findings of this study improve the understanding of damping mechanisms and provide valuable information to select the appropriate damping type in the design of new absorbers or to choose damping models in numerical simulations. © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers 23rd International Conference on Modelling in Mechanics 2025 Analysis of Damping of a Ball Absorber from Experimentally Obtained Data Marek Kawulok a,b, *, Stanislav Pospíšil a,b a Department of Structural Mechanics, Faculty of Civil Engineering, VSB- Technical University of Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava-Poruba, Czech Republic b Institute of Theoretical and Applied Mechanics of the Czech Academy of Sciences, Prosecká 809/76, 190 00 Prague 9, Czech Abstract This paper focusses on the analysis of the damping of a ball absorber's motion based on experiments with free oscillations. The experiments were conducted with a ball absorber restricted to planar motion, where its free oscillations were recorded using a video camera. The video footage was then analysed with an algorithm that tracked the centre position of the ball over time, allowing the creation of displacement curves of the ball over time. On the basis of this position information, the angular displacement of the centre relative to its equilibrium position was subsequently calculated. The paper investigates the influence of viscous damping and friction on the motion of the ball, with a particular emphasis on the interaction between the rolling track surface and the ball. A key aspect of the analysis involves comparing two surface variants, one featuring rubber tubing applied to the surface and the other remaining smooth. The findings of this study improve the understanding of damping mechanisms and provide valuable information to select the appropriate damping type in the design of new absorbers or to choose damping models in numerical simulations. © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the event organizers Keywords: Ball absorber; Damping; Viscous damping; Friction damping; Logarithmic decrement; Amplitude Introduction

Keywords: Ball absorber; Damping; Viscous damping; Friction damping; Logarithmic decrement; Amplitude Introduction

* Corresponding author. Tel.: +420-596-991-391. E-mail address: marek.kawulok@vsb.cz * Corresponding author. Tel.: +420-596-991-391. E-mail address: marek.kawulok@vsb.cz

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers 2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the event organizers 10.1016/j.prostr.2025.10.009

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1. Introduction Mechanical vibrations are one of the important external factors to which building structures are exposed. Their effects may cause discomfort to people who use the structure, limit the operation of sensitive equipment, or, in extreme cases, damage or collapse the structure. The increasing size of structures, the greater emphasis on human comfort, and stricter standards require effective solutions to the problem of vibrations. Unfortunately, vibration cannot be completely avoided, but its effects can be significantly reduced by using appropriate damping mechanisms. Damping mechanisms can generally be categorised as passive, active, semi-active, and hybrid systems (Elias and Matsagar, 2017). Among these, passive absorbers remain the most widely used type. A representative example is the tuned mass damper, which includes the ball vibration absorber. This device consists of two main components, namely a supporting bowl that serves as the surface for the rolling motion of a ball. The absorber has been gaining popularity mainly due to its robustness, low maintenance requirements, and relatively compact design. Additional advantages include its tunability over a wide frequency range and the possibility of various configurations. However, this type of absorber also has difficulties such as motion instability, bifurcation, and auto-parametric oscillations. (Náprstek et al., 2013; Náprstek and Fischer, 2020). Numerical simulations of the absorber response under harmonic excitation further reveal the growing influence of nonlinear effects and the emergence of unstable solution regions. The results of the simulations also reveal a softening effect in the resonance curves (Kawulok et al., 2024). These findings underscore the need for further analysis of the absorber behaviour, both through numerical modelling and experimental investigations. Numerical simulations represent an effective tool for obtaining information about the behaviour of dynamic systems defined by equations of motion. However, their formulation often involves some simplifications that simplify the work during analytical derivation but may cause inaccuracies when it comes to describing the behaviour of the physical system. Therefore, it is essential to validate the simulation results using experimental measurements. These measurements provide data on the actual response of physical models and can also reveal dynamic characteristics not fully captured by the mathematical model. Based on such experimental insights, the numerical models can be further refined, improving their accuracy and predictive capabilities. One of the key parameters affecting the response of a dynamic system is damping, which primarily affects the rate and magnitude of the amplitude decay. Among the most used damping models in engineering applications are viscous damping and Coulomb (friction) damping (Chopra, 1995). To accurately describe and predict the complex behaviour of vibration absorbers, it is essential to precisely identify the damping characteristics involved. This identification is critical for developing mathematical models that reflect the true physics of the system, moving beyond oversimplified assumptions. Viscous damping models, although widely used for their mathematical simplicity, constitute an extreme simplification and may fail to accurately represent the complex energy dissipation mechanisms occurring in real absorbers. Therefore, a more precise understanding and modelling of damping can significantly improve the reliability of analytical and numerical predictions. The aim of this contribution is to identify the damping model that most accurately captures the amplitude decay observed in free oscillation experiments. The experiment was recorded on video, and the position of the ball centre was obtained through post-processing of the footage. The motion was restricted to a planar trajectory and two types of surfaces, a smooth wooden track and a modified track with attached rubber tubing, were examined. In the first stage of the analysis, the logarithmic decrement and damping ratio were calculated based on the full response from the initial to the final amplitude. The analysis was further extended by fitting the sequence of peak amplitudes using both viscous and Coulomb damping models. The quality of each fit was evaluated using the coefficient of determination and the root mean square error, allowing a comparison of the models in terms of their ability to describe the damping

behaviour of the system. 2. Experimental setup

The experimental setup designed for this study comprises three primary components. The first is a steel frame fitted with linear rails and a movable trolley, which has two wheels on one side and one wheel on the other side. This carriage can be connected to a harmonic exciter during forced vibration experiments. However, for the purposes of this contribution, which focusses on free vibration, mechanical stoppers were installed to prevent the trolley from

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moving. The steel base structure is versatile and has previously been used in other experimental measurements, including studies involving pendulums with viscous dampers (Pospíšil et al., 2014). The second component of the experimental setup is a support structure composed of two wooden plates. The shape of the plates was modified using a laser so that the edges form a rolling track for the ball with a curvature radius of 0.2 m. The edge of the plate that guides the rolling motion was modified to allow the installation of rubber coated tubes, which were added to the structure to improve friction and maintain consistent rolling behaviour. However, the installation of the radius of tubing alters the curvature of the rolling track. The plates are connected by threaded rods, with their precise spacing maintained by 3D-printed spacers. The gap between the plates can be adjusted by varying the spacer length. The entire structure is mounted on a movable trolley and fixed in place with screws. In the experiments presented in this contribution, the spacing between the tracks was set to 0.04 m. The last component of the setup was a red ball with a diameter of 0.0615 m and a mass of 0.219 kg. The experimental setup can be seen in Fig.1. Experimental measurements were recorded using a mobile phone camera at a resolution of 1920×1080 displayed pixels, with a frame rate of 120 frames per second (fps). To ensure consistency and reproducibility, the camera was securely mounted on a tripod to maintain a fixed position throughout all experiments.

Fig. 1. Experimental setup.

3. Determining the position of the ball from video footage To determine the position of the centre of the ball, an algorithm was used to obtain information about the position of the tracked circular points by post-processing the video footage. This algorithm is applied not only to the ball itself, but also to reference points, whose function will be explained in more detail later in the text. Video processing is carried out in several steps. First, the video is divided into individual frames from which distortion caused by optical distortion is removed. For this purpose, tools from the MATLAB Toolboxes for Image Processing and Computer Vision (MathWorks, 2025) were used. To remove distortion, the lens parameters must be known. These were determined before the experiment began. The camera calibration was accomplished using images of the calibration checkerboard. Subsequently, a colour filter is applied to the undistorted images to remove any distracting background. The filter works on the principle of colour contrast between the tracked object and a uniform background, where pixels corresponding to the selected colour are displayed white after filtering, while other areas are suppressed (blackened). The modified images are further analysed using the Hough Circle Transform (Duda and Hart, 1972), which detects the centre of the spherical object in each frame. However, the resulting coordinates are expressed in pixels and, therefore, must be converted into physical units. The conversion is accomplished via the three previously mentioned reference points, which are tracked together with the ball in each frame. These points are positioned at known distances along two perpendicular axes. Based on

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this information, the size of a single pixel along both axes is determined, and the resulting displacements are converted to metres. In this way, the trajectory of the centre is obtained over time. Further data processing allows the determination of the angle of displacement of the sphere relative to its equilibrium position. In the case of free vibration, this equilibrium position is defined as the position where the sphere comes to rest after the motion has been damped. To reduce measurement noise and smooth the results, a Savitzky-Golay filter (Schafer, 2011) is applied in the final step.

Fig. 2. Example of filter application on the frame, showing the ball in a steady position.

4. Analysis of the damping behaviour One of the key parameters influencing the response of a dynamic system is motion damping. It determines the rate at which the mechanical energy released during the oscillation is dissipated, and thus the rate at which the oscillations decay. In engineering practice, two basic types are most commonly considered. The first is viscous damping, which is proportional to the velocity of motion, and the second is frictional damping, which is based on Coulomb's model of shear friction and is characterised by a constant resistive force acting against the direction of motion. For numerical simulations, accurately identifying the type and magnitude of damping is essential for a realistic response of the system. Choosing the appropriate damping model and parameters is therefore crucial. Experimental measurements were made for two different rolling track surfaces. As mentioned in the description of the experimental setup, the application of rubber tubing slightly reduced the radius of curvature of the trajectory. The initial displacement was imparted manually by simply releasing the ball without giving it any initial velocity. Damping is analysed based on the angular displacement from the ball’s equilibrium position. As a first step, the value of the logarithmic decrement of damping δ for the entire recording was determined. It is given by the following relation: = 1 + , (1) where and + represent the amplitudes of the first and the last selected cycle, respectively, and n is the number of cycles between them. In both cases, the first amplitude was taken as the initial displacement at the beginning of the motion, and the last was defined as the final amplitude greater than 0.05 °. This threshold was chosen to avoid incorrect identification of additional oscillations in the low-amplitude region near the end of the motion. The logarithmic decrement can be used to calculate the damping ratio ξ . When assuming low damping, the following relation can be applied: ≈ 2 . (2) The input values for the calculations according to (1) and (2), as well as the resulting values, are presented in Table 1.

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(°) + (°) 29.10 0.13

(s -1 ) 5.68

Table 1. Input values and calculated results of logarithmic decrement and damping ratio for two different rolling surfaces.

Rolling surface

n

δ

Without rubber tubing With rubber tubing

26 13

0.21 0.40

0.03 0.06

31.46 6.09 The damping ratio values were further used to generate the viscous damping curves directly on the response graphs. Alongside this curve, a linear line connecting the amplitudes and + is also added to the graphs. This line represents the damping caused by dry friction. The envelope of viscous damping is described by the following equation: = − , (3) where is the natural angular frequency of the absorber and t is time. The natural frequency of the absorber was determined as the average period length of the first five free oscillations; the corresponding values are listed in Table 1. The response graphs are shown in Fig. 3(a) and Fig. 3(b). Figure 3(a) depicts the response of the ball moving along the track without the tubing, while Fig. 3(b) shows the response of the track with the rubber tubing applied. 0.18

Fig. 3. Free oscillation response of the centre of the ball with added envelopes representing viscous and frictional damping. Graph (a) shows the case without rubber tubing on the track, while graph (b) corresponds to the case with rubber tubing.

Experiments show that oscillations decay at slower rates when the spherical absorber moves over a surface without tubes. This conclusion is supported both by visual observation and by quantitative analysis using the logarithmic decrement. In this case, approximately 26 cycles are needed for the oscillations to fully decay. On the contrary, for motion along the surface with rubber tubing, the damping is significantly faster, with stabilisation occurring after only 13 cycles. The higher damping level is also confirmed by a higher value of the logarithmic decrease in this case. When analysing the amplitude decay graphs, it is evident that, for the surface without tubing, the response curve lies between the envelope characterising viscous damping and the straight line corresponding to friction damping. This suggests that the actual damping is a combination of both mechanisms. In contrast, for the absorber moving on the surface with tubing, the amplitudes clearly converge to the line representing the friction model. To evaluate which of the considered models provides the best representation of the real damping process, a quantitative analysis was performed in the form of a functional approximation (fitting) to the measured data. This procedure allowed a more accurate determination of the nature of the amplitude decay and which models best fit the experimental observations.

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In the first step of the fitting process, all peak amplitudes were extracted from the time response and then ordered according to their sequence (peak index). For viscous damping, equation (3) was modified to the following form (Al Hababi et al., 2020): , = − (4) where and are parameters adjusted during the fitting. The value represents the maximum amplitude at the start of the motion, and corresponds to the logarithmic decrement of damping. In the case of the linear dry friction model, the damping is described by a line: , = − (5) in which friction is the initial amplitude and is the damping coefficient indicating how much the amplitude decreases with each successive cycle. The quality of the fit was evaluated using the coefficient of determination (R 2 ) and the root mean square error (RMSE). The resulting values are presented in Table 2, while the graphical representation of the fits can be seen in Figure 5 for the ball moving on the untreated surface and in Figure 6 for the ball moving on the surface with applied rubber tubing.

Damping model (°) (°/cycle) R 2 Viscous 32.37 0.09 -

Table 2. Results of the fitting analysis.

Rolling surface

RMSE (°)

Without rubber tubing

0.97 0.98 0.96 0.99

1.37 1.18 1.92 0.98

Friction Viscous Friction

26.16 39.52 31.73

-

1.04

With rubber tubing

0.17

-

-

2.40

Fig. 4. Sorted amplitudes of free oscillations with fitted curves for viscous and frictional damping. Graph (a) shows the case without rubber tubing on the track, while graph (b) corresponds to the case with rubber tubing.

From the quantitative fitting analysis, it is evident that, for the ball moving on the surface without tubing, the damping cannot be attributed to a single model. Although both the viscous and friction models have high coefficients of determination (R 2 = 0.97 and 0.98), the difference in the RMSE values (1.37 vs. 1.18) indicates that the friction model better fits the experimental data. This is also confirmed by visual analysis of the graph, where the response amplitudes lie between the viscous and friction damping curves, suggesting that the actual damping may be a combination of both mechanisms. The initial amplitudes estimated by the models also differ, with the friction model predicting a lower initial amplitude (26.16) compared to the viscous model (32.37), possibly due to differences in the approximation of damping.

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For the surface with rubber tubing, the situation is much clearer. The friction model achieves not only a higher R² value (0.99) but also a significantly lower RMSE (0.98 compared to 1.92), confirming a better fit to the data. This is consistent with the graphical representation, where the response amplitudes closely follow the line characteristic of friction damping. The initial amplitude predicted by the friction model (31.73) also appears to be more realistic than the overestimated value predicted by the viscous model (39.52). These results confirm that, while a combined model may be appropriate for a smooth surface, friction damping alone best describes the damping on the surface with the tubing. 5. Conclusion This contribution focusses on the experimental investigation of damping in the motion of a ball absorber, with particular attention to the influence of the surface material on the damping characteristics. Two surface conditions were analysed, one without modification and the other with rubber tubing attached to the track surface. The main goal was to determine which mathematical damping model best describes the observed decay in the oscillation amplitude of the absorber. In the case of the surface without tubing, it was not possible to clearly determine whether the primary damping mechanism was viscous or frictional, as the results indicated a combination of both. However, the damping behaviour showed a slightly stronger alignment with the friction model. For the surface with rubber tubing, the correspondence with the friction model was even more evident, while the contribution of viscous damping appeared negligible. The results provide a starting point for a deeper analysis, demonstrating that surface treatment significantly influences damping behaviour. To clearly identify the dominant damping mechanism, further investigation is needed, taking into account possible experimental uncertainties, such as potential surface irregularities of the track or geometric imperfections in the shape of the ball, among other possible factors. Acknowledgements The financial support of the grant programme financed by the Ministry of Education, Youth and Sports of the Czech Republic through VSB–TUO SGS SP2025/067 and from the budget for conceptual development of science, research and innovations is highly acknowledged. References Náprstek, J., Fischer, C., Pirner, M., Fischer, O., 2013. Non-linear Model of a Ball Vibration Absorber. In: Papadrakakis, M., Fragiadakis, M., Plevris, V. (eds) Computational Methods in Earthquake Engineering. Computational Methods in Applied Sciences 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6573-3_18 Náprstek, J., Fischer, C., 2020. Stable and unstable solutions in auto-parametric resonance zone of a non-holonomic system. Nonlinear Dynamics 99, 299–312. https://doi.org/10.1007/s11071-019-04948-0 Kawulok, M., Čermák, M., Pospíšil, S., Juračka, D., 2024. Numerical Procedure for Solving the Nonlinear Behaviour of a Spheri cal Absorber. Periodica Polytechnica Civil Engineering 68 (4), 1367–1377. https://pp.bme.hu/ci/article/view/25903 Chopra, A. K., 1995. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Ed. 1. New Jersey: Prentice Hall College Div. Pospíšil, S., Fischer, C., Náprstek, J., 2014. Experimental analysis of the influence of damping on the resonance behavior of a spherical pendulum. Nonlinear Dynamics 78 (1), 371-390. https://doi.org/10.1007/s11071-014-1446-6 MathWorks, 2025. Help Center " Computer Vision Toolbox: Design and test computer vision systems", [online] URL https://www.mathworks.com/products/computer-vision.html (Accessed: 05.18.2025) Duda, R. O., Hart, P. E., 1972. Use of the Hough transformation to detect lines and curves in pictures. Communications of the ACM 15 (1), 11-15. https://doi.org/10.1145/361237.361242 Schafer, R., 2011. What Is a Savitzky-Golay Filter? [Lecture Notes]. IEEE Signal Processing Magazine 28 (4), 111-117. https://doi.org/10.1109/MSP.2011.941097 Al-Hababi, T., Cao, M., Saleh, B., Alkayem, N. F., Xu, H., 2020. A Critical Review of Nonlinear Damping Identification in Structural Dynamics: Methods, Applications, and Challenges. Sensors 20(24):7303. https://doi.org/10.3390/s20247303 Elias, S., Matsagar, V., 2017. Research developments in vibration control of structures using passive tuned mass dampers. Annual Reviews in Control 44, 129-156. https://doi.org/10.1016/j.arcontrol.2017.09.015

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23rd International Conference on Modelling in Mechanics 2025 Analysis of the clinch method for joining structural components Martin Krejsa a *, Petr Lehner a , Jakub Flodr a a Department of Structural Mechanics, Faculty of Civil Engineering, VSB-Technical University of Ostrava, Ludvika Podeste 1875/17, 708 00 Ostrava-Poruba, Czech Republic Abstract Due to the wider use of thin-walled steel structures, alternative methods of connection are currently being sought to replace bolts, nuts, or welds. One attractive option is cold joining, known as clinching. This is a mechanical connection of two or more thin walled steel plates formed by cold forming using a punch and die. The present paper provides background information on the ongoing research on the mechanical behavior of clinch joints. Tensile tests of the raw material were performed for the purpose of inverse analysis of steel. Furthermore, tensile tests of a set of specimens with one clinch joint were carried out and analyzed and compared with the outputs of a numerical model in finite element method-based software at the force-displacement diagram and limit mode levels. The verification of the models provides detailed information about the behavior of the joint, which is a suitable basis for more complex numerical models. © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers Keywords: analytical solutions; clinch technology; steel; thin-walled sections; 1. Introduction Thin-walled cold-formed (TWCF) profiles are often used in construction and clinch is a potential method of joining these components. Clinching is a process that joins materials without adhesives or fasteners; instead, specialized tools are used to mechanically join the components. Typically, this involves a punch and die. The punch compresses the materials into the die, creating a permanent joint (Lambiase and Di Ilio, 2014). Clinching is an attractive option for joining thin-walled profiles, particularly due to its cost-saving potential, ability to reduce structural weight and © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the event organizers 23rd International Conference on Modelling in Mechanics 2025 Analysis of the clinch method for joining structural components Martin Krejsa a *, Petr Lehner a , Jakub Flodr a a Department of Structural Mechanics, Faculty of Civil Engineering, VSB-Technical University of Ostrava, Ludvika Podeste 1875/17, 708 00 Ostrava-Poruba, Czech Republic Abstract Due to the wider use of thin-walled steel structures, alternative methods of connection are currently being sought to replace bolts, nuts, or welds. One attractive option is cold joining, known as clinching. This is a mechanical connection of two or more thin walled steel plates formed by cold forming using a punch and die. The present paper provides background information on the ongoing research on the mechanical behavior of clinch joints. Tensile tests of the raw material were performed for the purpose of inverse analysis of steel. Furthermore, tensile tests of a set of specimens with one clinch joint were carried out and analyzed and compared with the outputs of a numerical model in finite element method-based software at the force-displacement diagram and limit mode levels. The verification of the models provides detailed information about the behavior of the joint, which is a suitable basis for more complex numerical models. © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers Keywords: analytical solutions; clinch technology; steel; thin-walled sections; 1. Introduction Thin-walled cold-formed (TWCF) profiles are often used in construction and clinch is a potential method of joining these components. Clinching is a process that joins materials without adhesives or fasteners; instead, specialized tools are used to mechanically join the components. Typically, this involves a punch and die. The punch compresses the materials into the die, creating a permanent joint (Lambiase and Di Ilio, 2014). Clinching is an attractive option for joining thin-walled profiles, particularly due to its cost-saving potential, ability to reduce structural weight and

* Corresponding author. E-mail address: martin.krejsa@vsb.cz * Corresponding author. E-mail address: martin.krejsa@vsb.cz

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers 2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the event organizers 10.1016/j.prostr.2025.10.013

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suitability for automated production. TWCF structures are therefore a key area of research (Bernuzzi and Maxenti, 2015; Lei et al., 2019). Much of the existing research focuses on understanding how individual factors influence the final shape of the clinch joint. This is because clinching is highly repeatable in manufacturing, making it ideal for automation. Other studies are investigating different types of damage and their impact on failure modes. This particular research investigates how clinch joints behave under typical building loads, assuming a reliable pre-manufacturing process that ensures flawless joints and predictable failure patterns (Flodr et al., 2017). Verification of experiments and numerical models is also a major problem (Atia and Jain, 2018; Hamel et al., 2000; Varis and Lepistö, 2003). The paper builds on a pilot study previously published, where the basic concept of joint testing, evaluation and numerical modelling of such a joint was introduced (Flodr et al., 2020). In the original paper, a connection with a plate thickness of 2.67 mm was presented, while here the results of tensile testing of the material, clinch connection and numerical modeling of the 3.45 mm thick material are presented. 2. Experimental and numerical program 2.1. Material and test setup Physical experiments of tensile tests of clinch joints loaded by shear have been carried out for several thicknesses, in this paper the results are presented for a thickness of 3.45 mm. A clinch tool with punch designation P8184 and die SR603 was used to join the plates. The static experiment scheme corresponds to the shear loading of the joint, since it is two separate sheets clamped outside the specimen center of gravity. In Fig. 1. (a), the fixed test specimen in the electromechanical press can be seen. The test specimens were made of S390GD material. According to the data sheet, the declared minimum yield strength is 390 MPa. Prior to the experiment, tear tests were performed on the material for the purpose of calibrating the numerical model. The physical experiment was designed as a tensile test of the specimen and was terminated by failure. In the laboratory test, the dependence of the imposed force on the deformation was monitored and 5 tests were performed. Fig. 1. (b) shows photographs of one specimen before and one after the test, as well as a detail of the clinch joint.

(a)

(b)

Fig. 1. (a) Test specimen fixed in electromechanical machine; (b) specimens before and after testing; detail of clinched specimen.

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2.2. Numerical model The numerical model, created in ANSYS (ANSYS, 2020), corresponded to the full extent of the physical experiment. The model used SOLID186 volume elements with many internal nodes (see Fig. 2). On one side, the specimen was fixed at a length of 100 mm, corresponding to the fixation in the press. On the other side, the specimen was also fixed, but with an additional displacement of the nodes in the direction of movement of the pressmachine. The numerical model was materially, structurally and geometrically non-linear. The aim was to mimic the real specimen as closely as possible to make its behaviour and geometry as close to reality as possible. The material model was multilinear. The load was applied in several steps and a special load condition was created to simulate the slip of the electromechanical press head. The numerical model was conceived as a mathematical model without destruction, and cracks.

Fig. 2. Top view of the whole numerical model, cut through the clinch joint in the model and top detail.

3. Results 3.1. Experiment

The output of the tensile tests are force-displacement diagrams for individual specimens. The results from the tests on all types of specimens are shown graphically in Fig. 3. The resulting failure mode was similar in all cases (see Fig. 4). The joint fails at the neck, which is the weakest point of the clinch connection under the shear loading considered. Failure of the connection occurs by a combination of shear and partial bending stresses from eccentricity. The maximum forces ( F ult ) achieved during testing and the corresponding deformations ( u ult ) are given in Table 1 and the average force F el , which is the limiting force up to which the behaviour of the connection is evaluated as elastic, is also given. Specimen S1 was incorrectly seated during the test and the results must therefore be excluded from the file.

Table 1. Limit values read from the force-displacement diagram from the experiments. Mark F ult u ult

F el

S1 S2 S3 S4 S5 S6

9.81 kN 9.62 kN 9.61 kN 9.64 kN 9.51 kN 9.52 kN

1.09 mm 1.63 mm 1.94 mm 1.89 mm 1.74 mm 1.73 mm

8.31 kN 8.39 kN 8.81 kN 8.24 kN 8.26 kN 8.98 kN

Martin Krejsa et al. / Procedia Structural Integrity 73 (2025) 81–86 Martin Krejsa, Petr Lehner, Jakub Flodr / Structural Integrity Procedia 00 (2025) 000–000

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Fig. 3. Force-displacement diagram obtained from experiments.

Fig. 4 shows photographs of the clinch section of the specimen marked S4 after the test and cutting. From the cross section of the clinch joint, it can be seen that the failure occurred at the neck of the clinch joint. The lower part separated in the left part due to a combination of shear and bending stresses and initiated the failure. In the right part of the cross section, tensile failure occurs.

Fig. 4. Cross-sectional dimensions of the clinch connection (top); cross-sectional detail of the clinch connection (bottom left); detail of the clinch connection after the test (bottom right).

Martin Krejsa et al. / Procedia Structural Integrity 73 (2025) 81–86 Martin Krejsa, Petr Lehner, Jakub Flodr / Structural Integrity Procedia 00 (2025) 000–000

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3.2. Numerical model Fig. 5 shows a force-displacement diagram with a representative record from the physical experiment labeled S6, which showed average values against the whole set. Furthermore, two diagrams obtained from the numerical model are shown: the first one, labelled FEM_S, is without electromechanical machine slip and the second one, labelled FEM_S_corr, takes into account the slip in the jaws. After considering the slip, there is a good agreement between the results of the numerical model and the physical experiment. In the graphical output of the numerical model (see Fig. 6) the equivalent plastic strain can be seen. The locations with the highest degree of plasticity was detected in the left part of the neck. The numerical model does not consider finite element failure, the failure of the clinch joint occurs from the left part towards the right. The left part is broken by a combination of shear and bending and once this part fails, the right part of the joint is broken by tension.

Fig. 5. Comparison of experimental results and numerical models.

Fig. 6. Equivalent plastic strain from the numerical model of the clinch joint.

Martin Krejsa et al. / Procedia Structural Integrity 73 (2025) 81–86 Martin Krejsa, Petr Lehner, Jakub Flodr / Structural Integrity Procedia 00 (2025) 000–000

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4. Conclusions The paper presents the results of an investigation into the mechanical behavior of clinch joints in 3.45 mm thick S390GD steel. Through a combination of experimental tests and finite element modelling, the research has provided valuable insights into the failure mechanisms and force-displacement characteristics of these joints. Comparison of the experimental results and the numerical models, especially after accounting for electromechanical machine slip, revealed a good correlation, confirming the accuracy of the numerical approach. The analysis confirmed that the typical failure mode under shear loading occurs at the neck of the lap joint and is caused by a combination of shear and bending stresses. Acknowledgements This contribution has been developed as part of the research project of the Czech Science Foundation 25-15763S ”Structural steels behavior of thin-walled load-bearing elements during cold joining”. References ANSYS, 2020. ANSYS Meshing User’s Guide [WWW Document]. ANSYS User Guide. URL https://customercenter.ansys.com/ (accessed 10.29.20). Atia, M.K.S.S., Jain, M.K., 2018. Finite element analysis of material flow in die-less clinching process and joint strength assessment. Thin-Walled Structures 127, 500–515. https://doi.org/10.1016/j.tws.2018.03.001 Bernuzzi, C., Maxenti, F., 2015. European alternatives to design perforated thin-walled cold-formed beam-columns for steel storage systems. J Constr Steel Res. https://doi.org/10.1016/j.jcsr.2015.02.021 Flodr, J., Kałduński, P., Krejsa, M., Pařenica, P., 2017. Innovative Connection of Steel Profiles, Experimental Verification and Application. Procedia Eng 190, 215–222. https://doi.org/10.1016/j.proeng.2017.05.329 Flodr, J., Lehner, P., Krejsa, M., 2020. Experimental and numerical evaluation of clinch connections of thin-walled building structures. Sustainability (Switzerland) 12. https://doi.org/10.3390/su12145691 Hamel, V., Roelandt, J.M., Gacel, J.N., Schmit, F., 2000. Finite element modeling of clinch forming with automatic remeshing. Comput Struct 77, 185–200. https://doi.org/10.1016/S0045-7949(99)00207-2 Lambiase, F., Di Ilio, A., 2014. An experimental study on clinched joints realized with different dies. Thin-Walled Structures. https://doi.org/10.1016/j.tws.2014.08.004 Lei, L., He, X., Yu, T., Xing, B., 2019. Failure modes of mechanical clinching in metal sheet materials. Thin-Walled Structures 144, 106281. https://doi.org/10.1016/j.tws.2019.106281 Varis, J.P., Lepistö, J., 2003. A simple testing-based procedure and simulation of the clinching process using finite element analysis for establishing clinching parameters. Thin-Walled Structures 41, 691–709. https://doi.org/10.1016/S0263-8231(03)00026-0

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23rd International Conference on Modelling in Mechanics 2025 Application of computational interface damage model to a concrete FRP shear connector Roman Vodička a *, Eva Kormaníková a , Daniel Dubecký a a Institute of Structural Engineering and Transportation Structures, Faculty of Civil Engineering, Technical University of Koši ce, Vysokoškolská 4, 042 00 Košice, Slovakia Abstract The paper explores utilisation of fibre-reinforced polymer (FRP) composites in bridge construction. Key advantages of FRP as a construction material are highlighted in relation to conventionally used materials to evaluate its effectiveness in applications to different structures. The analysis of computations for varied geometrical parameter of a jigsaw puzzle type of a continuous shear connector is also provided. For the concrete-FRP shear connector interface, a cohesive bilinear interface damage model based on a variational formulation has been chosen and implemented. The implementation introduces an interface damage variable to cope with degradation of the connector. The model also ensures the load displacement relationship which locally provides a softening zone. During debonding, it may affect the smoothness and continuity of the computed structural response for analysed quantities such as stress and the interface damage variable. Assessments based on the presented results guarantee suitability of the computational variational model as a tool for predicting failure, having potential for application in material design, in design of 23rd International Conference on Modelling in Mechanics 2025 Application of computational interface damage model to a concrete FRP shear connector Roman Vodička a *, Eva Kormaníková a , Daniel Dubecký a a Institute of Structural Engineering and Transportation Structures, Faculty of Civil Engineering, Technical University of Koši ce, Vysokoškolská 4, 042 00 Košice, Slovakia Abstract The paper explores utilisation of fibre-reinforced polymer (FRP) composites in bridge construction. Key advantages of FRP as a construction material are highlighted in relation to conventionally used materials to evaluate its effectiveness in applications to different structures. The analysis of computations for varied geometrical parameter of a jigsaw puzzle type of a continuous shear connector is also provided. For the concrete-FRP shear connector interface, a cohesive bilinear interface damage model based on a variational formulation has been chosen and implemented. The implementation introduces an interface damage variable to cope with degradation of the connector. The model also ensures the load displacement relationship which locally provides a softening zone. During debonding, it may affect the smoothness and continuity of the computed structural response for analysed quantities such as stress and the interface damage variable. Assessments based on the presented results guarantee suitability of the computational variational model as a tool for predicting failure, having potential for application in material design, in design of specific construction details, and structural elements. © 202 5 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organisers Keywords: cohesive damage model, energy formulation, concrete, FRP composite, shear connector, interface 1. Introduction The fibre-reinforced polymer (FRP) composites play the significant role across various industries, particularly in construction and bridge engineering. Their growing use in structural applications, including repair and strengthening © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the event organizers specific construction details, and structural elements. © 202 5 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organisers Keywords: cohesive damage model, energy formulation, concrete, FRP composite, shear connector, interface 1. Introduction The fibre-reinforced polymer (FRP) composites play the significant role across various industries, particularly in construction and bridge engineering. Their growing use in structural applications, including repair and strengthening

* Roman Vodička . Tel.: +421-55-602-4388 E-mail address: roman.vodicka@tuke.sk * Roman Vodička . Tel.: +421-55-602-4388 E-mail address: roman.vodicka@tuke.sk

2452-3216 © 202 5 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers 2452-3216 © 202 5 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of 23rd International Conference on Modelling in Mechanics 2025 organizers

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the event organizers 10.1016/j.prostr.2025.10.025