obr2The article describes a diagnostic method, on the basis of which it is possible to assess whether there is a loose connection in the clamping system (detection) and to determine which of them is loose (localization) and what extent of release occurred (quantification). The diagnostic method is based on comparing the drops in the natural frequencies of a measured system with respect to a reference system, which simulates the desired state. When analysing the drop in natural frequencies in order to locate the loosening, it is necessary to relate this drop to the position of the clamping joints within the considered mode shape. To determine the extent of loosening, it is necessary to know the relationship between the changes in natural frequencies and the modeled loosening of clamping joints.

In technical practice, devices are often formed by a combination of components that are exposed to vibrations, which can lead to their defects. When these failures occur, the entire system becomes more susceptible to damage of a larger scale, which can lead to malfunction or even destruction of the device. To ensure the trouble-free operation of machinery containing clamping joints, it is important to assess the quality of these joints in terms of safety and functionality [1]. All components connected in this way include inaccuracies and non-linearities, which are mainly caused by contact and friction between the parts of the system. The following chapters describe the possibilities of detecting, locating and quantifying defects implied by changing the modal parameters of the given system (modes, natural frequencies (NF), and proportional damping) [2] - [4], [25] with respect to the position of the clamping joint in relation to specific modes of the system. This assumption is justified because changes in physical parameters of the system, such as weight, damping and stiffness, have a direct effect on the modal parameters of the system, and these changes will be more significant if they are localized in antinodes, or in the nodes of individual modes. Furthermore, a comparison of the data of the clamped beam obtained from the simulation using the Finite Element Method (FEM) [5] - [8] in Ansys and the data from experiments at different pre-tensions of the clamping joints were analyzed in Matlab. By this comparison, it was possible to assess both models obtained by simulation and testing for the needs of detection, localization, and quantification of the damage of the loose clamping joint.

Tested device and measurement description
The experiment was carried out on a measured device, specifically a guide rod, cut into 3 shorter beams of the same length. In the places of the cut, the individual parts were re-connected with each other by clamping joints in the form of two collets (K1, K2), which were fastened with collet clamps. These clamps were modified on one side so they could connect the 2 parts of the beam with each other, while the modification consisted in cutting off the excess part of the clamps and milling the holes for the double-sided connection of the beams.
The device consisted of the above-mentioned components and was equipped with seven acceleration sensors for measurement purposes. Clamping of the measured device was carried out by the jaws of the lathe [9].
The Fig. 1 shows the positions of the sensors and other important dimensions of the measured device in relation to the place of clamping.

Fig 1 Measured device with significant dimensions
Fig. 1: Measured device with significant dimensions

Such clamped beam was excited using an impact hammer by hitting the area of the clamping head of the lathe jaws, and the signal from the sensors was recorded with a measuring device.
During the experiment, 7 different states of the tested device were measured (Tab. 1). The zero state represents the "whole" beam with collets and is used for comparison with the remaining states. These are represented by 3 combinations of torque magnitude by which are clamped collets (K1, K2) tightened, connecting the 3 parts into which the entire beam was divided (Fig. 1). Tightening the collet with a torque of 50Nm means the required tightening of the joint, on the other hand, tightening with a torque of 4Nm simulated the loosening of the joint.

Tab 1 Measured states
Tab. 1: Measured states

The response of the system was measured by acceleration sensors at 7 places of the beam (S0 ÷ S6) and the excitation was carried out by a hit near the jaws of the lathe (rigid attachment to the foundation) using an impact hammer (B) (Fig. 2).

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Fig. 2: Depiction of the measured device

In Fig. 3 the frequency transfer functions (H0B ÷ H6B) of the system plus their approximations obtained based on experimental modal analysis are shown. The transfer functions HjB(iω) = Sjj(iω) / SjB(iω) were obtained based on the measured spectral power densities Sjj(iω) and SjB(iω); j = 0,1, ..., 6 in the locations of responses j and excitation B. These analyzes were realized in MATLAB environment.

Fig 3 Transfer functions
Fig. 3: Transfer functions

Based on measured ambiguous values due to signal noise on sensor S0, only signals from the remaining 6 sensors (S1-S6) were considered in the further analysis. The modal parameters of the system were afterwards obtained from these transfer functions. These modal parameters are natural frequencies, mode shapes and proportional damping [10] - [18].

Outputs of measurement and FEM simulation
On the diagrams in Fig. 4a measured modes are depicted for the measured condition No. 1 – "whole" beam.
The model created using the FEM in ANSYS was composed using BEAM3 elements, which are planar elements, while MASS21 elements were inserted into some nodes of the model so that the calculation considers the weight of the collets and sensors. The Young's modulus used was E = 210GPa and Poisson's number was set to ν = 0.3. An illustrative visualization of modes 3-8 in the ANSYS environment can be seen in Fig. 4b.
The outputs from the FEM calculation were natural frequencies and mode shapes, which were used in the interpretation of the measured data. Without creating modes using FEM analysis, it would not be easy and in some cases not even possible to assign which data belongs to which bending modes [19], [20].

Fig 4a Measured mode shapes of the measured device Fig 4b Computed mode shapes of the measured device
Fig. 4: Measured (a) and computed (b) mode shapes of the measured device

Comparison of modal parameters on a basis of measurements and FEM simulations
In this chapter, data obtained from FEM calculations are compared with experimentally obtained data, while experimentally obtained data were also compared with each other. In the mutual comparison of experimentally obtained data, the data of the measured state No. 2 are considered as the reference.
Tab. 2 contains information about the natural frequencies corresponding to the specific states of the prestressing of the clamping joints and the specific modes of vibration. It also contains information on the percentual differences between natural frequencies and modes, while the measured state when both clamping joints were tightened with a torque of 50 Nm - 50 Nm (state no. 2) was considered as the reference because it simulates the desired tightening without loosening [21], [22].

Tab 2 Measured modal parameters and their comparison in respect to the reference values
Tab. 2: Measured modal parameters and their comparison in respect to the reference values

From the comparison of the modal parameters for the individual measured states, it was found that no obvious trend of change was observable for the proportional damping or mode shapes, on the other hand the natural frequencies varied for the different measured states. It is possible to observe a trend of a decrease in natural frequencies depending on the integrity of the measured device, but also on the loosening of the clamp connection and the position of the clamp connection loosening with respect to the antinode, or a node of a specific mode. For the measured condition no. 1, the bending natural frequencies had the highest values of individual frequencies, the second highest values were reached by state 2, which was considered as a reference because it simulated required tightening, resp. no loosening of clamping joints. States 3 and 4 recorded decreases in bending natural frequencies in a way, which correlated with the assumption of a higher decrease when the clamping joint is loose in regions of antinode of specific natural shapes.
The natural frequency drops give options to:
a) The detection of a loose of a clamping joint is a qualitative assessment of a system in which it is possible to claim that at least one clamping joint is loose in a system containing 1 or more clamping joints.
b) Locating a loose of a clamping joint is also a qualitative assessment of a system, in which it is possible to identify a loose clamping joint in a system containing 2 or more clamping joints. For a system containing 1 clamping joint, the detection of a loosening of a clamping joint is also localization.
c) The quantification of a loosening extent of a clamping joint is a quantitative assessment of a system, in which it is possible to determine an extent of loosening of a clamping joint based on an extent of modal frequencies drops with regard to other parameters of a system.

The influence of the position of the loose collet within a specific mode shape on the change of modal parameters
In the graphs in Fig. 5 a correlation of the modes obtained based on the FEM analysis (curves) and the modes obtained experimentally (signs in the area of the curves) is depicted.
From the comparison, it was clear that the difference in the modes was not significant, so loosening of clamping joints will not cause a significant change in the shape of the modes. Vertical solid lines show the position of the collets and sensors, and vertical dashed lines show the position of the sensors alone.
The relationship of the drop in natural frequency with the position of the loosen clamping joint within the natural shape was demonstrated by comparing the measured states 3 and 4 of significant modes (Tab. 3). Such modes included, for example, shapes 5 and 8, which have one clamping joint in the antinode region (K1) and one in the node region (K2). Analyzing the modes 5 and 8, in which the loosened clamping joint was simulated for state 3 in the region of the antinode and in the case of state 4 in the region of the node, it can be observed that there were significant decreases when the loose was present in the region of the antinode, but not in the region of the node (Fig. 5). This difference made it possible to locate the loosening in the clamping system [23].

Tab 3 Measured states 3 and 4
Tab. 3: Measured states 3 and 4

Generally, important point in the analysis of the clamping system is that the modal parameters of the desired tightened system, which is considered as a reference, must be known. Against the reference system, another system is compared, which can be called the system under assessment [24].

Fig 5 Combination of measured and computed mode shapes
Fig. 5: Combination of measured and computed mode shapes (MS)

If the modal parameters of the assessed system do not differ significantly compared to the reference system, then the loosening is not detected, and its further localization and quantification are irrelevant. On the contrary, if significant drops in natural frequencies are present, then loosening is detected. Detection is followed by localization and quantification of the extent of loosening of the clamping joint.
In a system of two clamping joints, such as the system described above, 3 types of modes can be defined. This distribution is based on the positions of the clamping joints with respect to the antinodes and the nodes of vibration:
A) Node – Node,
B) Anti-node – Anti-node,
C) Anti-node – Node / Node – Anti-node.

Tab 4 Categorization of modal shapes
Tab. 4: Categorization of modal shapes

Case A is not informative in terms of detecting or locating a loosening in a system containing clamping joints. In the case of possible loosening in one or both clamping joints, there may not be a significant decrease in the bending natural frequencies, because the decrease in natural frequencies is very little sensitive to the loosening in the nodes.
Based on the mode shape of type B, the detection of a loose clamping joint is possible, but not its localization. Such a mode shape is sensitive to the loosening of the clamping joint, but it is not possible to determine in which clamping joint the loosening occurred according to this case.
If a drop in the arbitrary bending natural frequency of the measured state indicates the detection of a loosening (Tab. 4), while considering the C-type mode, the subsequent localization of the loosening is also possible. This mode also enables the detection of loosening, but only if the loose clamping joint is located in the region of antinode of the corresponding mode. If the clamping joint was located in the area of the node, there would be no significant drop in the natural frequency, which is a necessary condition for the detection of loosening. If this loosening is detected in the system and the natural frequencies corresponding to this type of mode have dropped significantly, then it is possible to conclude that there has been a loosening in the region of the antinode of this mode. If the loosening in the system is detected and the natural frequencies corresponding to this type of natural shape have not decreased significantly, then it is possible to conclude that there has been a loosening in the region of the node of this mode shape. To increase the reliability of the localization, it is recommended to use more than just one custom shape of type C.
Quantification of the loosening extent depends on several parameters, e.g. from the scale of the decrease in the bending natural frequency and the curvature of the mode shape in the place of the loose collet. However, this publication does not contain mathematical relationships expressing quantitative measures of loose clamping.
From the characteristic modes of type C, it can be known that the antinodes with present clamps can be in phase or in anti-phase, but from the point of view of diagnostics of clamping joints, this fact is not reflected in drops of natural frequencies.

Loosening detection and localization using modal frequency drops in respect to a reference system with regard to vibration modes
According to the postulate above, it was possible to demonstrate on an example how the diagnostics of a system with a loose clamping joint is carried out. For the examples bellow the limit for significant natural frequency (NF) drop was defined to ΔNFL = -0.8%.

Loosening of the K1 collet
The loosening of the clamping joint in the K1 collet was demonstrated in the measured state no. 4. In this case, the detection of the loosening of the clamping joint is implied by the decreases in the bending natural frequencies 3, 5, 6 and 8 (Tab. 5). Localization of the loose clamping joint was made possible by the modes 5 and 8, because the collet K1 was located in the region of antinode and the collet K2 in the node region of these modes (Fig. 6). An observable decrease in the natural frequencies corresponding to these mode shapes allowed conclusion that the collet K1 was loose.

Tab 5 Measurement state 4  
Tab. 5: Measurement state 4  
Fig 6 Mode shapes used for detection red frame and localization green frame  
Fig. 6: Mode shapes used for detection (red frame) and localization (green frame)  

Loosening of the K2 collet
The loosening of the clamping joint in the K2 collet was demonstrated in the measured condition no. 3. In this case, the detection of the loosening of the clamping joint was implied by the decreases in the bending natural frequencies of 3, 6 and 7. The loosened clamping joint was made possible by the modes 5, 7 and 8, because the collet K1 was located in the region of antinode of the modes 5 and 8, and the collet K2 in the region of the node of these modes, but for mode 7 it was exactly the opposite (Fig. 7). From Tab. 5, it is clear there was no significant drop in natural frequencies for modes 5 and 8, but such a drop occurred in the case of mode 7. This finding implies that the loosening was not in the area of the K1 collet, but in the K2 collet.

Tab 6 Measurement state 3  
Tab. 6: Measurement state 3  
Fig 7 Mode shapes used for detection red frame and localization green frame  
Fig. 7: Mode shapes used for detection (red frame) and localization (green frame)  

The quantification of a loosening extent of a clamping joint
To solve the problem of quantification, a more sophisticated model of collet release with a shaft and a sensor was used. The real stiffness of the collet with the shaft was modeled by a beam with the corresponding active length l, with its equivalent stiffness in bending E J / l3 and with its equivalent mass  S l + ms (E - Young's modulus; J – second mass moment of inertia; S – cross sectional area; l – length;  – mass density; ms – mass of sensor) [10].
For the purposes of quantifying the release of the clamping joint, was chosen as a comparative measure the so-called Euclidean norm of the vector:
norm = NFN = [j (∆NFj2)]1/2, where ∆NFj is the jth element from the ∆NF vectors in the above tables.
Hence it, it is possible to assess the relationship between the defined release of the clamps and the subsequent total decrease of the measured natural angular frequencies of the system. The change in the coefficient koej (j=1 - K1, j=2 - K2) of which defines the gradual loosening of the bending stiffness of the clamps in relation to the changes corresponding to the measured natural frequencies of the system. Fig. 8 shows the course of changes in the natural frequencies of the system (NF) represented by their norms (norm) in relation to their partial changes in the opening of the clamps koe1 and koe2. The graph shows the relationships between the calculated and measured states. These dependencies make it possible to assess the release of clamping joints during the vibration of such mechanical systems. Coefficients that represent the gradual release of clamps at the following levels are:
koe1 = 1; koe2 = 1 – state 50-50 and koe1 = 0,4; koe2 = 0,5 – state 4-4.

Fig 8 Changes in the natural frequencies of the system represented by their norms norm in relation to their partial changes in the opening of the clamps koe1 K1 and koe2 K2
Fig. 8 Changes in the natural frequencies of the system represented by their norms (norm) in relation to their partial changes in the opening of the clamps (koe1 – K1 and koe2 – K2)

From a practical point of view, the mechanical system is assessed based on spectral and modal properties on two levels. At the limit lower - still "safe" level of clamping of the collets and at the upper level - maximum clamping of the collets. The second state is taken as a reference state of gradual loosening of the clamping joints. In the framework of the mentioned application, it is necessary to realize that the limit state is not only the state 4-4, but also the state 20-4 and the state 4-20, because, for example, compared to them, the state 5-5 is still on the "safe" side. From the graph (Fig. 8), the range of "safe" opening of the individual collets can be read in the range approximately given by the red lines. Their approximation results from the non-linear dependence between physical parameters (E J) and natural frequencies (NF) and from the use of the comparative Euclidean norm (norm).

Conclusion
It results from the above analyzes that it is possible to obtain information about the defectiveness of the clamping joint system using frequency and modal analysis. From the percentual decrease of the natural frequencies, it is possible to record the loosening of the clamping joints (detection) and considering the computationally and experimentally determined modes, it is possible to determine the position of the loose joint (localization) and also assess the level of their loosening (quantification). It is necessary to expand the mentioned methodology of localization of clamping joint loosening by above mentioned research fields for enabling a potential use in industry.

Acknowledgements:
The research in this document was supported by grant agency VEGA 1/0430/20 of the Ministry of Education, Science, Research and Sports of the Slovak Republic.

Text/photo: MUSIL Miloš, ÚRADNÍČEK Juraj, PÁLENÍK Marek, CHLEBO Ondrej, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava,Institute of Applied Mechanics and Mechatronics, Bratislava, Slovakia

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