Diagnostics of Bolted Joints in Vibrating Screens Based on a Multi-Body Dynamical Model

Citation: Krot, P.; Shiri, H.; D ˛abek, P.; Zimroz, R. Diagnostics of Bolted Joints in Vibrating Screens Based on a Multi-Body Dynamical Model. Materials 2023,1, 0. https:// doi.org/ Academic Editor: Balázs Illés Received: 18 July 2023 Revised: 15 August 2023 Accepted: 21 August 2023 Published: Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). materials Article Diagnostics of Bolted Joints in Vibrating Screens Based on a Multi-Body Dynamical Model Pavlo Krot * , Hamid Shiri , Przemysław D ˛abek and Radosław Zimroz Faculty of Geoengineering, Mining and Geology, Wroclaw University of Science and Technology, Na Grobli 15, 50-421 Wroclaw, Poland; [email protected] (H.S.); [email protected] (P.D.); [email protected] (R.Z.) *Correspondence: [email protected] Abstract: The condition-based maintenance of vibrating screens requires new methods of their elements’ diagnostics due to severe disturbances in measured signals from vibrators and falling pieces of material. The bolted joints of the sieving deck, when failed, require a lot of time and workforce for repair. In this research, the authors proposed the model-based diagnostic method based on modal analysis of the 2-DOF system, which accounts for the interaction of the screen body and the upper deck under conditions of bolted joint degradation. It is shown that the second natural mode with an out-of-phase motion of the upper deck against the main screen housing may coincide with the excitation frequency or its higher harmonics, which appear when vibrators’ bearings are in bad condition. This interaction speeds up bolt loosening and joint opening by the dynamical loading of higher amplitude. The proposed approach can be used to detune the system from resonance and anti-resonance to reduce maintenance costs and energy consumption. To prevent abrupt failures, such parameters as second natural mode frequency, damping factor, and phase space plot (PSP) distortion measures are proposed as bolt health indicators, and these are verified on the laboratory vibrating screen. Also, the robustness is tested by the impulsive non-Gaussian noise addition to the measurement data. A special diagram was proposed for the bolted joints’ strength capacity assessment and maintenance actions planning (tightening, replacement), depending on clearance in the joints. Keywords: vibrating screen; bolted joints; clearances; nonlinear stiffness; dynamical model; modal analysis; resonance; diagnostics; damping; phase space plot 1. Introduction Different vibrating screens are widely used in raw material processing and aggregate industries. Vibration screens have a significant role in separating bulk materials such as the fractions of coal and ore. Sieving screens can separate from 10 to over 1000 tons per hour depending on their design, drive power, size, and the number of decks [ 1 , 2 ]. Cyclic excitation for screen deck motion can be produced by unbalanced rotating shafts, hydraulic cylinders, or electromagnetic actuators. Using decks and particle motion criteria, screens are classified into linear, elliptical, or circular types. Several decks can be utilized to increase general productivity and the final quality of the product. 1.1. Problems in Vibrating Sieving Screen Operation and Diagnostics Sieving material is fed to the top of the screen by the belt conveyor with a specific linear speed. Some systems are under development [ 3 ] to prevent damage to equipment and improve process monitoring, but the conventional way is to design the upper deck of the screen as massive grizzly bars for providing scalping of the intake stream from the oversized components. The upper deck and lower levels of sieves are subjected to blinding, which induces screen overloading and technological process interruption. Also, Materials 2023,1, 0. https://doi.org/10.3390/ma1010000 https://www.mdpi.com/journal/materials Materials 2023,1, 0 2 of 31 large falling fragments of material generate force impacts with significant amplitude; thus, those disturbances which are stochastic by nature should be considered to prevent false alarms in fault diagnostic procedures [ 4 – 8 ]. The impulsive noise cancellation method for copper ore crusher vibration signal enhancement is proposed in [ 9 ]. The selection of the informative frequency band in a bearing fault diagnosis in the presence of non-Gaussian noise with a comparison of recently developed methods is fulfilled in [ 10 ]. Battery-powered wireless sensors are designed in [ 11 ] which are able to withstand vibratory loading and are embedded in the rubber screens for condition monitoring. Nevertheless, diagnostics of vibrating machines, in particular sieving screens, still is a challenge in theory and practice. At the design stage, by using the discrete and finite element approach [ 12 – 19 ], the natural modes of the screen are examined to confirm minimal structural stresses and required trajectories of bulk material motion. Nonetheless, these approaches need significant computing resources in optimization and research. Also, statistical fraction distribution in the input flow, particle configuration, and a 3D screen model with details are required [ 20 ]. Hence, the reduced degree-of-freedom spring–mass models can be appropriate to accomplish a dynamical analysis of a vibrating screen as a set of rigid bodies connected by links with a certain stiffness and mass [ 21 – 25 ]. To account for the non-linear features of such systems, a piecewise model and estimates of damping and natural frequency are implemented in [ 26 ], including adaptive diagnosis of the bilinear mechanical systems based on the free oscillation method with the decrement as a recognition feature [27]. Supporting springs are well-known as vital components of sieving screens, and they substantially affect particles’ trajectory and overall process efficiency due to a significant influence on their dynamics [ 28 ]. Although there are a few advantages of using elastomeric or air-filled springs, they could create nonlinear behavior of the mechanical system [29,30] . In contrast, steel springs have linear stiffness within a wide range of deformation. However, specific dynamical effects can appear due to the nonlinear relation between the side bending displacement of steel springs and vertical stiffness [ 31 , 32 ]. Coupled mode parametric resonance in a vibrating screen is investigated on the two mass models in [ 33 ]. The additional disturbances in the measured signals and dynamical effects can be produced by the inertial vibrators’ bearings [ 34 ], damage and wear from which can result in a complex vibration spectra structure. The most desirable by the energy consumption are resonance vibrating screens [ 35 ] but they are not widely used in the industry due to working mode instability and complexity of control when material mass is changing. Issues of multiple vibrators’ synchronization are considered in [ 36 ] and many other works. A chaotic vibrating screen is proposed in [ 37 ] to improve its sieving performance. In the standard far beyond resonance vibrating screens, a serious dynamical problem causing excessive energy consumption constitutes the Sommerfeld effect when the machine structure passes its main resonance at startup. About 30% of the additional power of electric drives is needed for reliable transition through the resonance, which results in excessive energy consumption at the working rotation speed. When the rotation speed of vibrators goes down, the screen structure passes resonance very slowly and its elements are subjected to excessive loading cycles of high amplitude. Some solutions are proposed in this regard based on magneto-rheological dampers (MRDs) [ 38 , 39 ] and electric drive control [ 40 ], which is not yet widely implemented in the industry. The additional dynamical loading is caused by the gaps opening, which can appear due to contact wear in joints and deformation of bolts. This effect has a potential threat to appear in non-stationary vibrating machines with reverse loading. Thus, due to the significant influence on the lifetime and reliability of different parts of the vibrating screen, clearances may be thought of as the most critical but hidden for measurement operational parameters. The bolted joints are one of the most vital factors determining the correct operation of vibrating screens. The strength capability of bolted joints supposes that the designers of machines should forecast their peak loads and prevent the contacted surfaces from opening at any joint under any forces because in this case, the additional stresses Materials 2023,1, 0 3 of 31 (torsional, bending, shear) appear in the bolt shank [ 41 ]. The greater preloading (limited by yield stress) can reduce the chance of bolt failure. Although preloading of bolted joints is the crucial factor, the specific tightening torque is not usually available during the repairing process, especially for large-size industrial systems where bolts diameter can reach 100 mm. Based on the ASTM F568M, the metric bolts used for heavy-duty applications should have property classes 8.8, 10.9, or 12.9 and be made of alloy steels. However, in practice, the strength capacity of bolts can significantly deviate from the standard values due to different factors. 1.2. Diagnostics of Bolted Joints Loosening in Structures The diagnosis of bolts loosening is a complex scientific and engineering problem considering the potentially massive number of bolted joints and the dramatic outcomes of their loosening, leading to the unexpected redistribution of inner loads between other bolts in joints, not only in rotating systems but in building, bridge, and other infrastructure that use a lot of bolts. Health monitoring of bolts can be divided into two general approaches. The first is based on smart bolts and dense sensors in the local area, and the second approach uses analysis of bolts based on modal analysis of whole structures and systems. A dense sensor approach can be performed by measuring tensile stress with ultrasonic waves [ 42 , 43 ], utilizing acoustic emission on rotating systems [ 44 ], electrical conductivity [ 45 ], vibroacoustic modulation (VM), and wave energy dissipation (WED) [ 46 – 48 ], as well as using spectral sidebands and high-order harmonics [ 49 ]. For measuring, such instrumentation can be used as piezoelectric active sensing [ 50 – 53 ], smart washer manufactured by lead zirconate titanate (PZT) [ 54 – 57 ]. Also, these methods can be combined with wireless technology for critical civil infrastructure and industrial facility monitoring. In the sieving screens, strong continuous excitation from unbalanced vibrators may have an influence on the reliability of electronic components. In recent years, due to technological improvement of cameras and related algorithms, the approach based on image processing to diagnose bolts has accumulated some favor [58–60] . This method is usually combined with deep learning technology [ 61 – 63 ], support vector machines [ 64 ], and nonlinear decomposition such as empirical mode decomposition [ 65 ]. Selecting the reference points to detect changes in the nuts’ position has a substantial effect on the reliability of these methods. Nevertheless, these methods cannot detect phenomena such as bolt creep and axial deformation in some states. Another category of bolt diagnostic approaches is based on identified modal parameters of the system. Detecting changes in a natural frequency and phases can detect the weak bolt tightening caused by changing the system’s stiffness and contact friction [ 66 ]. For instance, the vibration transmissibility function is a more reliable parameter to determine the joint state, while natural frequency and modal damping in modes with low frequency seemed less reliable. In other studies [ 67 , 68 ], experimental and theoretical methods have been accomplished to assess system modal properties and frequency response functions to detect bolted joint degradation [ 69 – 71 ]. In [ 72 ], the authors demonstrated that the first-order phase difference parameter is susceptible to the looseness of the bolts on the plate joint of the wind turbine tower. In [ 73 ], a finite element simulation is used to analyze rod flange-bolt structure unit (FBSU). Authors in [ 74 ] developed a simple method to assist bolt tightness by measuring natural frequency and damping ratio during the hammer test. In [ 75 ], non-linearity and damping ratio in bolt frequency response were studied. In [ 76 ], the authors modelled the bolt as a plane beam with two linear end springs (transverse and rotational). In this research, the authors are focused on the dynamical analysis of a vibrating screen with an emphasis on the causes of frequent failures on the heavy upper deck and methods of bolted joint diagnostics. All further calculations are regarding only those bolts that fasten the upper deck to the screen structure to restrain its vertical displacement but not the other bolts in side panels, which loosening is planned for analysis in our future Materials 2023,1, 0 4 of 31 research. We consider the most difficult case when nuts are fixed and other methods, e.g., visual inspection, are not sufficient in the bolted joint maintenance on the screens. Both practical and theoretical goals motivated this research. The practical goal was to understand the reasons for frequent failures of bolted joints in the vibrating screen and to develop remedies against such failures. The theoretical goal was to develop methods of bolted joint loosening based on a dynamical model and implement the appropriate signal processing techniques. In fact, the developed method and diagnostic parameters are applicable to any horizontal screens and other vibrating machines (feeders, transportation tables) where two main parts are joined by the bolts and susceptible to frequent failures. The scope of the conducted research covers industrial research and measurements on the fully functional laboratory screen. The novelty of the developed approach is in considering the vibrating screen as a system with a changeable structure when an additional degree of freedom appears due to bolted joint degradation. In this case, the second mode natural frequency change and the first mode damping values are used as the diagnostic parameters of bolted joint loosening. In addition, anti-resonances are identified in the 2DOF dynamical system and their possible effect on screen working efficiency and diagnostics is explained. The developed method is not sensitive to the non-Gaussian noise created by the sieved bulk material. The paper is organized as follows. In Section 1, we represented all aspects of vibrating screens’ operation and their diagnostics. In Section 2, we give a description of the investigated machines. In Section 3, the dynamical model, modal analysis, and detuning from resonances are represented. In Section 4, the results of model simulations are given, including non-Gaussian noise. In Section 5, the results of measurements on a laboratory screen are represented to validate the new diagnostic method. Finally, in Section 6, the achieved results are discussed, and in Section 7, conclusions and further research are highlighted. 2. Materials and Methods In our research, we analyze two devices. The first is an industrial vibrating sieving screen, a general view of which is depicted in Figure 1. The second machine is a fully functional sieving screen in our laboratory, a photo of which is shown in section 5 . Maintenance records and some vibration data from the industrial screen are observed for a long time to undermine its weak elements (see statistics in Figure 2). In particular, the bolted joints are determined (26% of failures), which motivated our studies on their diagnostics. In addition, the spring–mass parameters of this screen (see in section 2.2 ) are used in the dynamical model simulations. The second machine was a laboratory screen, which was used to validate our method of diagnostics since we could not conduct experiments with controlled bolt loosening and measurements in the industrial plant. Figure 1. The typical design of a vibrating sieving screen with two unbalanced vibrators [77]. Materials 2023,1, 0 5 of 31 Figure 2. Upper deck of vibrating screen: severe abrasive wear of beams (a); blinding with a near-mesh-size material (b). 2.1. Design of Industrial Sieving Screen The investigated typical industrial vibrating screen (see Figure 1) includes the body with side panels linked by reinforcement beams, one or multiple sieving decks, and helical supporting springs. All screen parts are subjected to significant abrasive wear and fatigue; thus, instead of welded joints, special huck bolts are used. 2.2. Vibrating Screen Dynamics and Failures The upper deck blinding with a near-mesh-size material can increase the weight of vibrating masses while the severe abrasive wear of beams will reduce the mass of the deck (see Figure 2). The total mass of the deck obviously includes a certain mass of sieved material (up to 30% of total mass) at every moment in time. Nevertheless, vibrating screens can be considered systems with constant lumped stiffness and mass parameters at a certain period of their operation. Instead, the bolted joints—the subject of research—can change axial stiffness in the course of machine operation, but they are also assumed as constant parameters at a certain moment in time. According to the recommendations of the vibrating screens’ producers regarding the equipment maintenance actions (see Table 1), the vibrator bearings’ and sieves’ mounting joints are the most frequently inspected units, which means they have the highest susceptibility to failures and malfunctions. This is justified by the repair data analysis (see Figure 3) where bolts, drives, bearings, and sieves constitute 72% of all issues that happen during the vibrating screen operation. Hence, the diagnostics of these elements is of great importance for the smooth operation of industrial enterprises. Table 1. Maintenance periods for vibrating screen elements. Maintenance Period Nr Action 50 h Week Month Year 2 Years 1 Lubrication of vibrator bearings 2 Control of sieve mounting 3 Control of sieves and vibrators 4 Control of springs 5 Control of belt drives 6 Inspection of sieve wear 7 Inspection of drives 8 Inspection of vibrators Materials 2023,1, 0 6 of 31 Figure 3. Statistics of elements failures in vibrating screen. 2.3. Measuring Equipment Vibration measurements were obtained by the Kistler LabAmp 5165A, a 4-channel universal laboratory charge amplifier used for dynamic signal or mechanical quantity measurements with piezoelectric sensors (IEPE) [ 78 ]. The K-Shear 8702B500 accelerometers coupled with the amplifier were used. The parameters of the sensors were as follows: sensitivity—10 mV/g; maximum acceleration range—500 g. 3. Methodology of the Bolted Joints Diagnostics in Vibrating Screens 3.1. Equation of Motion The equation of the mechanical system motion with ndegrees of freedom and viscous damping can be represented as follows: [m]¨ x+ [b]˙ x+ [k]x=F(t)(1) where, x , ˙ x , and ¨ x are displacment, velocity, and acceleration, respectively. In addition, [m] , [b] , and [k] are mass, damping, and stiffness matrix, respectively, and F is force. Also, the mentioned equation can be presented in a state space form [79]: ˙ x(t) = ˙ x(t)(2) ¨ x(t) = −[m]−1[b]˙ x(t)−[m]−1[k]x(t) + [m]−1F(t)(3) by definition of 2n-dimensional state space vector y(t)as follows: y(t) = x(t) ˙ x(t)(4) Equations (2) and (3) can be rewritten in a state space form as follows: ˙ y(t) = [A]y(t) + [B]F(t)(5) where the coefficients matrices [A] and [B] of order 2 n× 2 n and 2 n×n , respectively, are equal to: [A] = 0I −[m]−1[k]−[m]−1[b](6) [B] = [0] [m]−1(7) Materials 2023,1, 0 7 of 31 3.1.1. Modal Analysis For the modal analysis, first, we assume the free vibration problem with f(t) = 0, so Equation (5) can be rewritten as follows: ˙ y(t) = [A]y(t)(8) This equation represents a set of 2 n first-order ordinary differential equations. The solution Equation (8) is considered in the following form: y(t) = Yeλt(9) where Y and λ are constant vectors and values, respectively. By substituting Equation (9) to Equation (8) and simplifying it, we obtain [A]Y=λY(10) Equation (10) is a standard algebraic eigenvalue problem. The solution of it gives the eigenvalues and corresponding eigenvectors. Following the previous studies, the natural frequencies of the two lowest modes of vibrations in heavy industrial machines are rarely observed above 20–30 Hz and are sensitive to clearances in the dynamical system. 3.1.2. Damping Ratio Besides the natural frequencies, the damping ratio is used for bolt looseness detection. The method is based on the transient vibration curve robust nonlinear fitting solved with the least-square approach. The problem of finding coefficients xcan be defined as follows: min x∥F(x,xdata)−ydata∥2 2=min x∑ i (F(x,xdatai)−ydatai)2(11) where xdata is input data, ydata is an observed output, both represented as either matrix or vector, and F(x , xdata) is a matrix-valued or vector-valued function of the same size as ydata, which is computed instead of a standard sum of squares. Using this method of curve fitting, a damping ratio is estimated by the decaying transient vibrations to detect bolt loosening during experiments. 3.1.3. Frequency Response Functions Modal testing is an experimental technique that is used to design the modal model of linear invariant vibratory systems over time. The theoretical basis of the technique is to determine the relationship between one location’s vibration response and another location’s excitation as a function of the excitation frequency. This relationship is often a complex mathematical function, called the frequency response function (FRF) [ 80 ]. FRFs are often displayed in acceleration units (g) and force units (N) , resulting in g/N units. Also, there are other formats in which FRF can be displayed. These alternative formats are created by executing mathematical operations over the FRF—integration or differentiation, then, acceleration can be replaced with velocity and displacement. In the following, we show the FRF representation by using displacement based on Equation (1), which is known as compliance: FRF(ω) = x/F(12) where x , F , and ω are displacement, force, and frequency, respectively. More information about the driving procedure and details can be found in reference [80]. Materials 2023,1, 0 8 of 31 3.2. Impulsive Non-Gaussian Noise It is known from the theory and authors’ experience of industrial measurements that stochastic non-Gaussian impacts from the large pieces of sieved material falling on the upper deck produce a wide-band spectrum of forces. It can be described by the following formulas [81]: X=Sα,β×sinαV+Bα,β (cos V)1/α× cosV−αV+Bα,β W!(1−α)/α +µ(13) Sα,β=σ×1+βtan πα 221/2α (14) Bα,β=arctanβtan πα 2 α(15) X=σ×2 ππ 2+βVtan V−βlogπ 2Wcos V π 2+βV+2 πβσ log σ+µ(16) 3.3. Dynamical Model of the Vibrating Screen For vibrating screen analysis, the 2-DOF dynamical model is assumed, the calculation scheme for which is represented in Figure 4. The separation of the second mass corresponding to the upper deck is quite justified because its size and weight (about 5 t) are of the same order as the rest of the screen body. This is a distinctive feature of the developed dynamical model, since the screen is usually considered a single mass. Figure 4. The calculation scheme of the large-scale industrial vibrating screen with the upper sieving deck as a separate mass (see parameters’ description in Table 2). The matrices of the dynamical system parameters in Equation (1) have the following form: [m] = m10 0m2,[b] = b1+b2−b2 −b2b2,[k] = k1+k2−k2 −k2k2(17) which values are given in Table 2. The governing equations for the 2-DOF non-linear system are the same as for the linear model but the stiffness of degrading bolts is described by the discontinuous nonlinear function with a dead zone (gap) and two linear pieces of constant stiffness: k2=   0, k2, k3, δ<∆1 ∆1≤δ≤∆2 δ>∆2 (18) Materials 2023,1, 0 9 of 31 where δ= (x2−x1) is the amplitude of bolts’ tensile deformation; ∆1 is the deformation corresponding to the yield stress of bolts’ material; ∆2 is the deformation corresponding to the gap opening in the bolted joints. Material hardening is not assumed for stresses above the elastic limit for high-strength steel bolts. The composed system of non-linear differential equations is solved numerically by the Runge–Kutta method of fourth order with a constant time step of integration. Table 2. Parameters of the dynamical model. Parameter Value Units Mass of screen body m115,000 kg Mass of upper deck m25450 kg Stiffness of supporting springs k10.56 ×107N/m Stiffness of bolted joints k21.46 ×108N/m Damping in supporting springs b110 s−1 Damping in bolted joints b210 s−1 Clearance in bolted joints ∆0.0–1.2 ×10−3m This model is further used for vibrating screen dynamics analysis and diagnostic feature investigation. 4. Results of Simulation 4.1. Analysis of Frequency Response Functions The dynamics analysis included the building of Frequency Response Functions (FRFs) of the vibrating screen. Taking into account that the total exciting force F 1 from both unbalanced vibrators acts directly on mass m 1 and impacts force F 2 from the falling pieces of the material acting on mass m 2 , the following four channels are considered (see Figure 5): •FRF11: from (vibrators’ force F1) to (displacement of mass m1); •FRF12: from (vibrators’ force F1) to (displacement of mass m2); •FRF21: from (material impacts force F2) to (displacement of mass m1); •FRF22: from (material impacts force F2) to (displacement of mass m2). The exact values of natural mode frequencies of vertical vibration are as follows: 1st mode—2.6 Hz; 2nd mode—30.5 Hz. Functions FRF 11 and FRF 22 have anti-resonance frequencies of about 26.2 Hz and 16.1 Hz, respectively. This means that the working frequency of vibrators should be at least ± 5 Hz from the minimum point of FRF 11 to avoid the inefficient energy consumption by the electrical motors of vibrators. While the first mode of vibration is mainly determined by the design parameters of the screen (total mass and stiffness of springs in the supporting units), the second mode of the screen’s natural vibration depends on several factors. The most influencing factor is the gradually changing stiffness between mass m 1 and mass m 2 , which greatly depends on bolted joints’ condition (tightening and axial plastic deformations). The mass of sieved material on the screen decks has less influence on the second vibration mode. It was noted by the data obtained from the permanent vibration monitoring system that the spectrum of excitation force measured on the bearings of the vibrators’ shafts contains higher harmonics (30 Hz, 45 Hz) at certain periods of screen operation. This feature corresponds to the bad condition of one or several bearings or supporting springs (see Figure 6). After maintenance actions are undertaken on the screen, those higher harmonics disappeared, which proves their origin. Measurements of vibration in the four corners of the main screen housing on supporting springs showed that it vibrates as the rigid body. Hence, even one bearing with damaged rings or having excessive clearance can generate not only 15 Hz but as high as 30 Hz and 45 Hz harmonics and excite the second natural mode with increased tension in the bolted joints. Since the maintenance actions of bearings and bolted joints have different periods, this process can occur at any time of operation and requires new methods of damage detection in the condition monitoring system. Materials 2023,1, 0 16 of 31 graph in Figure 16), the anti-resonance frequency (15.62 Hz) is very close to the vibrators’ rotation frequency (15 Hz). Figure 16. The dependence of FRF11 and FRF22 on the bolts’ stiffness k2(grades of loosening). In the context of bolted joints’ diagnostics, the coincidence of the vibrators’ rotation and anti-resonance frequency has a positive effect on the dynamics reduction of bolted joints, because amplitudes of out-of-phase vibrations are less at these frequencies. However, this is not desirable for energy saving of the screen and has to be avoided in practice. 5. Measurements on a Laboratory Vibrating Screen Since the measurements of bolt loosening and moreover their regulation is almost impossible in industrial sieving screens, the experimental part of this research was conducted on the fully functional laboratory vibrating screen (see Figure 17) to demonstrate the possibility of bolted joint loosening detection by the vibration signals with the developed methods. Locations of mounted sensors can be seen in Figure 18. Figure 17. The laboratory vibrating sieving screen with one unbalanced vibrator and bolted joints on the upper sieving deck. The sensors’ positions are shown with arrows. Materials 2023,1, 0 17 of 31 (a)(b) (c)(d) Figure 18. Locations of the mounted sensors: (a) sensor A—upper sieving deck; (b) sensor B—screen arm; (c) sensor C—bottom part of the screen; (d) sensor D—upper part of the screen. 5.1. Results of Measurements The signals registered during the described experiment can be seen in Figures 19–22 . In the case visible in Figure 22, the measurement was ended early due to the highly increased vibrations and the possibility of damaging the vibrating screen. Figure 19. Raw signals from sensors on vibrating screen in normal condition—no loosened bolts. Materials 2023,1, 0 18 of 31 Figure 20. Raw signals from sensors on the vibrating screen, with the left upper screw loosened. Figure 21. Raw signals from sensors on the vibrating screen, with the left upper screw loosened more than in the previous case. Materials 2023,1, 0 19 of 31 Figure 22. Raw signals from sensors on the vibrating screen with two bottom screws loosened. Parameters of FFT used in signal processing procedures are as follows: sampling frequency: 25 kHz; data length: 4 s (100,000 samples); frequency resolution: 0.25 Hz; windowing: no window. The spectrum and changes of the first (about 4.2 Hz) and the second (16.7 Hz) natural modes’ frequencies are shown in Figures 23 and 24, respectively. These graphs demonstrate three investigated cases: normal state of bolted joints; one upper left screw is loosened; and two bottom screws are loosened. As an additional diagnostic parameter for bolt loosening, the damping ratio was used in the same series of experiments. The change in the damping parameter in three different cases can be seen in Figure 25 and a summary of calculated values is given in Figure 26. Materials 2023,1, 0 20 of 31 Figure 23. Change of the first natural frequency in three investigated cases (measurements from Sensor A—sieve). Figure 24. Change of the second natural frequency in three investigated cases (measurements from sensor A—sieve). Materials 2023,1, 0 21 of 31 Figure 25. The calculation of the damping ratios in three investigated cases (measurement from Sensor A—sieve) based on exponential approximation of transient signals. Figure 26. The calculated values of the damping ratio in three investigated cases. As shown above, on the dynamical model of screen vibrations, phase space plots are sensitive to bolt looseness. Based on experimental data obtained on the laboratory screen, these graphs are built for three cases and are shown in Figure 27. In the normal state, PSP is characterized by a trajectory with minimal deviations. The second case with a weak looseness of only one bolt produces visible distortions in the trajectory of the upper sieving deck. The third case demonstrates the critical state (maximal looseness) of two bolted joints when the PSP trajectory is fully degraded and transformed into unpredictable oscillations of high amplitude. Materials 2023,1, 0 22 of 31 Figure 27. Phase space plots generated on the data from three investigated cases: (a) normal state; (b) one bolt loosened; (c) two bolts loosened (critical state). The numerical parameters of PSP shape are given in Table 3. Any of them can be used as the diagnostic parameters of looseness or “health indicators” of bolted joints not only in the sieving screens but in other vibrating machines too. Alarm levels can be clearly interpreted since the clearances always increase the amplitudes of vibrations. Table 3. PSP shape parameters of the signals (×10−3) from Figure 27. Parameters Case (a) Case (b) Case (c) Dmin −0.18 −0.19 −25.68 Dmax 0.18 0.19 24.44 Vmin −24.60 −25.14 −79.57 Vmax 24.10 23.97 77.99 ∆D=Dmax −Dmin 0.36 0.38 50.12 ∆V=Vmax −Vmin 48.70 49.11 157.56 ∆D×∆V17.51 18.82 7897.11 5.2. Verification of Robustness to Impulsive Noise Additionally, the authors checked if the second natural frequency would change after simulating the additive impulsive noise. Based on previous research, authors determined that in the real working conditions, the maximum range of impulsive noise impacts usually does not exceed the amplitude of the machine vibrations more than three times and comes from the material falling on the upper sieve. Three kinds of noises—Noise1, Noise2, and Noise3—have been used in the simulation. The are assumed to have alpha-stable distribution with parameters presented in Table 4. The alpha parameter is understood as the characteristic exponent, beta as the skewness, gamma as the scale parameter, and delta as the location parameter. Noises generated with these parameters can be seen in Figure 28 with an example of the original signal (coming from the Sensor A—on the sieve) convoluted with the first noise (cut to the moment of the machine being turned off) presented in Figure 29. Materials 2023,1, 0 23 of 31 Table 4. Parameters of the generated alpha-stable noises. Parameter Noise1 Noise2 Noise3 Alpha 1.7 1.9 1.5 Beta 0.01 0.03 0.005 Gamma 0 0 0 Delta 0 0 0 Figure 28. Visual representation of the generated noises. Figure 29. Visual representation of the original signals with added first of the generated noises. Results visible in Figures 30–32 show that the second natural frequency exhibits no significant changes after adding the impulsive noise, i.e., it is robust to external disturbances, which are always acting in the vibrating machines for bulk materials processing (compare values with Figure 24). The final dependence of the second mode natural frequency on bolt looseness is shown in Figure 33. Materials 2023,1, 0 24 of 31 Figure 30. Second natural frequency after adding the first noise. Figure 31. Second natural frequency after adding the second noise. Figure 32. Second natural frequency after adding the third noise. Materials 2023,1, 0 25 of 31 Figure 33. The calculated values of second natural mode frequency in three investigated cases. 6. Discussion The new approach to bolted joint diagnostics in the vibrating screens is developed based on the dynamical model and verified experimentally on the fully functional laboratory sieving screen. No other scientific works on bolted joint diagnostics in the sieving screens were found during our research. The proposed in-the-market systems designed especially for the vibrating screens, e.g., CONiQ (Shenk) [ 83 ], SmartCheck (FAG) [ 84 ], ScreenWatch (Metso) [ 85 ], have no specific options for bolted joint diagnostics. Only a “Loose screen mesh” defect is noted in the Copperhead (SKF) system [ 86 ]. Hence, the developed method is an innovative solution for the industry. The proposed approach, unlike other known studies, e.g., [ 11 ], considers the vibrating screen as a system with a changeable structure, which depends on bolted joints loosening. In other studies, the higher-order dynamical models including DEM and FEM simulations [ 19 ] were focused on the screen performance optimization along with a diagnostics of supporting springs [ 1 ] and bearings [ 34 ], while diagnostics of bolted joints was not covered. If we are to compare the developed method with “smart” bolts or washers for loosening detection, our solution is more cost-efficient and convenient because it does not require additional time during screen maintenance for electronic component installation. Vibration signals for diagnostic purposes can be recorded during the idle period of machine work to reduce the influence of impulsive noise from bulk material; however, this is usually not desirable in the continuous technological chains of large-scale enterprises. Therefore, to speed up and make the bolted joint diagnostics reliable, in distinction to the standard methods supposing detection of spectrum parameters changes at the kinematics-related frequencies, it is proposed to use a multi-body system of the natural mode frequencies, which are absolutely robust to impulsive noise, speed, and load variations. Identification of the first two natural modes frequencies can be accomplished by the calculation or a bump test conducted on the machine. Also, the signals during the system transition through the resonance are appropriate. It was discovered, by experiments and on the model, that for certain combinations of design parameters of vibrating screens, the additional excitation (not accounted for at the development) can be generated by the higher (second, third) harmonics of the main frequency of inertial vibrators in case of their bearings’ deterioration. This fact is wellknown in diagnostics but is firstly highlighted here as an initiating factor for bolted joint failures.