Determination of the Natural Frequency of the Model Spindle System with Active Regulation of the Initial Tension of the Bearings

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Article Determination of the Natural Frequency of the Model Spindle System with Active Regulation of the Initial Tension of the Bearings

Paweł Turek

Department of Machine Tools and Mechanical Technologies, Wroclaw University of Science and Technology, Lukasiewicza 5 Street, Building B-4, 50-370 Wroclaw, Poland; [email protected]; Tel.: +48-71-320-2061

Abstract: This article presents a test stand with a model high-speed spindle equipped with a system of active control of the preload of the bearings. This preload was changed by means of three piezo actuators. The work presents the results of tests during which the commercial Abacus measuring equipment from Data Physics was used. Its application has shown that the spindle system with angular contact ball bearings is responsive to changes in the preload value of these bearings. The change preload resulted in a change in the value of the resonant frequency of the system and its amplitude. This article presents the dependence between the variable value of the preload of the bearings and the corresponding values of the resonance frequency and amplitude of the spindle system. The use of the Abacus measuring equipment for testing allowed for the preparation of a model showing the dynamic behavior of the spindle. The system was forced by a signal with known parameters, and the response to this excitation was recorded at eleven points located on the surface of the entire spindle.

Keywords: machine tool; angular contact bearings; preload; natural frequency

Citation: Turek, P. Determination of the Natural Frequency of the Model Spindle System with Active 1. Introduction Regulation of the Initial Tension of Modern main drive machine tools usually use electro-spindles. For their bearing, the Bearings. Lubricants 2021, 9, 68. designers are very eager to use angular contact ball bearings, which have the ability to bear https://doi.org/10.3390/ both radial and axial loads [1]. In order to function properly, angular contact bearings must lubricants9070068 be preloaded, i.e., have so-called negative clearance. Its value is a very important matter responsible for the proper functioning of the entire spindle system. The correct value of Received: 8 June 2021 this preload [2] has the main inﬂuence on the dynamic properties of the spindle. It ensures Accepted: 6 July 2021 Published: 13 July 2021 its stable operation and is related to the natural frequencies and modes of the spindle system vibrations. Due to a number of physical phenomena that occur in the spindle

Publisher’s Note: MDPI stays neutral system, in modern machine tool spindles, various types of mechanical or mechatronic with regard to jurisdictional claims in solutions are used to control and correct the value of the initial tension of the bearings published maps and institutional affil- during spindle operation. In the publicly available literature, one can find many examples iations. where mechatronic solutions [3–5] are used to change the value of the initial stress of bearing nodes. By using such additional systems, it is possible to significantly extend the operating range of a given spindle and actively influence its operating parameters [6] and dynamic properties. The analysis of the spindle behavior is often based on a model on the basis of which it is possible to determine, for example, frequencies and modes of natural Copyright: © 2021 by the author. vibrations. When developing a spindle model, it can be reduced to a structure consisting Licensee MDPI, Basel, Switzerland. This article is an open access article of a shaft supported on two supports. This type of solution is easy to model [7]. In the distributed under the terms and case of mechanical structures, such as bars, beams, or shafts, one should also be aware of conditions of the Creative Commons resonance vibrations, which can lead to a reduction in durability or damage to a given type Attribution (CC BY) license (https:// of structure. Self-excited vibrations are also particularly destructive for machine tools [8]. creativecommons.org/licenses/by/ Despite the wide possibilities of modern programs using FEM (finite element method) 4.0/). to carry out the modal analysis of spindle systems [9,10], measurements on a real object are

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still quite popular [11,12]. They allow for the most objective presentation of the behavior of, e.g., a machine tool spindle at a specific external input. Experimental studies, however, do not allow for examining a given object for all possible cases, and the very interpretation of modal analysis is quite difficult [13]. In [14], the authors present a system for the automatic implementation and interpretation of the modal analysis of machine tool spindles. This article presents a spindle model based on an actual test stand on which the spindle behavior under the influence of external force was tested. Based on the measurements, a modal analysis was carried out, and it was shown that the system of active preload in the machine tool spindle allows its frequency and the value of the vibration amplitude to be changed. The model developed in this way presented a visualization of the work of the entire spindle subjected to external forces. It allows the amplitudes of vibrations in any place on the object to be read. However, due to the nature of the work of the spindles, the main emphasis was placed on the behavior of its front end. The series of tests carried out as part of this article can be used to develop an active control system for an active support in the spindle under test. Knowing the characteristics of the spindle system and its response to extortion by external force, it is possible to develop an algorithm that controls the preload system operating in real time. Such a solution would allow the scope of the work to be increased and the functionality of the spindle system to be extended.

2. Basics of Modal Analysis In the case of grinding spindles, their dynamic properties are important parameters [15]. They determine the quality and accuracy of machining and affect the durability of the tools. In the course of modal analysis, the so-called system transition function (FRF—frequency response function) [16] and, above all, the minimum values of the real part of this function, which determine the dynamic susceptibility of the machine tool [17], are determined. The measurement method in modal analysis depends on the number of places and types of excitations and the system response [18]. For the forcing, it is possible to use a broadband signal of the sine sweep type (signals generated by an inductor) or pulse excitation (using a modal hammer). Regardless of the measurement method, two signals are recorded simultaneously—the input and the response of the system. Determin- ing modal parameters is complicated and requires determining the form of vibrations by recognizing the characteristic waveforms for the real and imaginary part of the transition function [19]. According to the theory, in order for an object to be subject to modal analysis, it must meet several important features: • must be linear and can be described using ordinary or partial differential equations, and coefﬁcients of the equations describing the dynamics of the system must be constant during the measurements; • the system must be observable and allow for the measurement of all the characteristics that are necessary to create the model; • the tested system must meet the Maxwell principle of reciprocity, which assumes symmetry of the mass, stiffness, and damping matrices; • system damping must be low and proportional to mass or elasticity. • Modal analysis can be performed theoretically and experimentally [15]. In these areas, three types of research can be distinguished: • theoretical analysis—based on the numerical model of the structure; • experimental analysis—(EMA—experimental modal analysis), based on the measurement of excitation and system response; • operational analysis—based on the measurement of the system response. It is successfully used for testing spindle systems of machine tools, in particular for its rotating elements, such as the spindle shaft [20]. It is mainly focused on the accuracy of the spindle’s work, but not only on the accuracy of machining the work pieces. Therefore, it is very important to know the operating characteristics of the machine tool spindle under Lubricants 2021, 9, x FOR PEER REVIEW 3 of 9

• experimental analysis—(EMA—experimental modal analysis), based on the measurement of excitation and system response; • operational analysis—based on the measurement of the system response. It is successfully used for testing spindle systems of machine tools, in particular for Lubricants 2021, 9, 68 its rotating elements, such as the spindle shaft [20]. It is mainly focused on the accuracy3 of of 9 the spindle’s work, but not only on the accuracy of machining the work pieces. Therefore, it is very important to know the operating characteristics of the machine tool spindle under the influence of various types of excitations that may arise in real machining condi- tionsthe inﬂuenceas a result of of, various for example, types of contact excitations of the that cutting may tool arise with in real the machining processed conditionsmaterial. Foras the a result purposes of, for of example, the presented contact work, of the a cutting wide range tool with of applications the processed of modal material. analysis For the haspurposes been limited of the only presented to determining work, a wide the rangevalue of applicationsthe fundamental of modal resonance analysis frequency has been oflimited the tested only spindle to determining and its amplitude. the value of An the important fundamental goal resonance was to check frequency whether of thethere tested is a relationshipspindle and its(and amplitude. if so, what) An between important the goal change was in to the check initial whether stress thereof the is bearings a relationship and the(and value if so, of what)this frequency. between the These change tests in were the initialalso to stress determine of the whether bearings and how the value the value of this offrequency. the vibration These amplitude tests were changes. also to determine whether and how the value of the vibration amplitude changes. The commercial Abacus measurement system with the D780 measurement card from The commercial Abacus measurement system with the D780 measurement card from Data Physics was used to conduct the modal analysis. The Abacus is a high-performance Data Physics was used to conduct the modal analysis. The Abacus is a high-performance scaled real-time analyzer with 4 to 32 inputs. It is a measuring set that can be used to scaled real-time analyzer with 4 to 32 inputs. It is a measuring set that can be used to measure vibrations. It has high-quality analog components, and 24-bit analog-to-digital measure vibrations. It has high-quality analog components, and 24-bit analog-to-digital and digital-to-analog converters, ensuring a dynamic range of up to 150 dB. This system and digital-to-analog converters, ensuring a dynamic range of up to 150 dB. This system offers various programs that are used to carry out the next stages of the research process. offers various programs that are used to carry out the next stages of the research process. It It enables the modeling of the geometry of the tested device, on which a mesh similar to enables the modeling of the geometry of the tested device, on which a mesh similar to that that observed in the numerical analysis of the FEM type is applied. Node points are cre- observed in the numerical analysis of the FEM type is applied. Node points are created ated at the intersection of the grid lines. The appropriate number of nodal points is se- at the intersection of the grid lines. The appropriate number of nodal points is selected to lecteddeﬁne to thedefine measurement the measurement points. Thepoints. greater The theirgreater number, their number, the more the accurate more accurate the mapping the mappingof the behavior of the behavior of the testedof the tested object object in the in modal the modal model model (Mod (Mod Shape) Shape) [21], [21], on the on basisthe basisof which of which the selectionthe selection of damping of damping elements elements of the of the system system and and the the broadly broadly understood under- stoodmachine machine diagnostics diagnostics can becan performed. be performed.

3.3. Description Description of of the the Test Test Stand Stand AsAs part part of of the the work, work, a ameasuring measuring stand stand was was set set up, up, the the main main element element of of which which was was thethe model model grinding grinding spindles spindles (Figure (Figure 1).1). The The view view of of the the complete complete test test stand stand is is shown shown in in FigureFigure 2.2 It. Itconsists consists of of the testedtested spindlespindle (5) (5) with with an an active active support, support, which which was was disconnected discon- nectedfrom thefrom external the external drive drive during during the tests. the test A replacements. A replacement mass mass (8) is (8) attached is attached to its to front its frontend, end, to which to which the GW-V20the GW-V20 vibration vibration exciter exciter by Data by Data Physics Physics (4) is (4) connected. is connected. The power The powerampliﬁer amplifier type PA type 100E PA by 100E Data by Physics Data (1)Physics was used (1) was to power used theto power exciter. the Two exciter. piezoelectric Two piezoelectricsensors were sensors used forwere the used measurements. for the measurements.

FigureFigure 1. 1.ArrangementArrangement diagram diagram of of accele accelerometersrometers on on the the spindle shaft,shaft, 1—substitute1—substitute weight,weight, 2—front 2— frontsupport, support, 3—piezo 3—piezo actuators, actuators, 4—rear 4—rear support. support.

The ﬁrst, uniaxial type 62551 by Br˝uel& Kjaer (6), was used to measure the exciting force signal, and the second, triaxial type T356A15 by PCB Piezotronics (7), measured the vibrations of the structure. Both sensors required a DC power supply and were used to

convert a high impedance charge signal into a low impedance voltage signal. They were equipped with a function called TEDS (Transducer Electronic Data Sheet) by the manu- facturer, which allowed technical information about the sensor to be saved directly in its memory. Such a sensor, after connecting to the measuring system, automatically provided its calibration parameters. These sensors were connected to the Abacus measurement Lubricants 2021, 9, x FOR PEER REVIEW 4 of 9

The first, uniaxial type 62551 by Brűel & Kjaer (6), was used to measure the exciting force signal, and the second, triaxial type T356A15 by PCB Piezotronics (7), measured the vibrations of the structure. Both sensors required a DC power supply and were used to convert a high impedance charge signal into a low impedance voltage signal. They were Lubricants 2021, 9, 68 equipped with a function called TEDS (Transducer Electronic Data Sheet) by the manu-4 of 9 facturer, which allowed technical information about the sensor to be saved directly in its memory. Such a sensor, after connecting to the measuring system, automatically provided its calibration parameters. These sensors were connected to the Abacus measurement sys- system using the Data Physics D780 card (3). Additionally, the system was equipped with tem using the Data Physics D780 card (3). Additionally, the system was equipped with an an external monitor (2) enabling the visualization of the entire tested object. external monitor (2) enabling the visualization of the entire tested object.

Figure 2. View of the test stand used in modal analysis. Figure 2. View of the test stand used in modal analysis.

FigureFigure3 3 additionally additionally shows shows a a diagram diagram of of the the measurement measurement path path for for modal modal analysis. analysis. TheThe PCPC (11)(11) waswas usedused toto generategenerate aa signalsignal forfor aa specificspecific valuevalue ofof thethe preloadpreload forceforce ofof thethe bearings.bearings. ThisThis signal,signal, usingusing thethe measurementmeasurement cardcard typetype 92649264 fromfromNational National Instruments Instruments (10),(10), waswas converted into into an an electrical electrical value value an andd fed fed to tothe the input input of the of thepiezo piezo actuator actuator am- amplifierplifier type type SVR SVR 150/3 150/3 by byPiezomechanik Piezomechanik GmbH GmbH (9). (9).All Allpiezo piezo actuators actuators (8) (8)of the of thePSt PSt150/40/10 150/40/10 type typeby Piezomechanik by Piezomechanik GmbH GmbH were wereconnected connected to the to amplifier. the amplifier. They had They a hadbuilt-in a built-in strain straingauge gaugesystem, system, which made which it made possible it possible to read the to readvalues the of values their displace- of their displacement.ment. After setting After the setting required the required preload value, preload the value, spindle the system spindle (5) system was made (5) was to vibrate made toby vibrate means byof meansvibration of vibrationexciter type exciter GW typeV20 by GW Data V20 Physics by Data (4) Physics powered (4) poweredby an amplifier by an amplifiertype PA 100E type PAby Data 100E byPhysics Data (1). Physics The (1).Abacus The Abacusmeasurement measurement system with system D780 with card D780 by cardData by Physics Data Physics (2) was (2)responsible was responsible for recording for recording the signals; the signals; the forcing the forcing (6), with (6), the with use the of usea one-axis of a one-axis piezoelectric piezoelectric accelera accelerationtion sensor, sensor, type 62551 type by 62551 Brű byel & Br˝uel Kjaer; & and Kjaer; the and system the systemresponse, response, using a usingthree-axis a three-axis piezoelectric piezoelectric acceleration acceleration sensor, sensor,type T356A15 type T356A15 from PCB from Pi- PCBezotronics Piezotronics (7) and (7) for and determining for determining the para themeters parameters for the for modal the modal analysis analysis performed. performed.

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Figure 3. Diagram of the measuring track for the test stand for modal analysis of the spindle sys- FigureFigure 3. 3. DiagramDiagram of of the the measuring measuring track track for for the the test test stand stand for for modal modal analysis analysis of of the the spindle spindle sys- system tem with active regulation of the initial tension of the bearings. temwith with active active regulation regulation ofthe of th initiale initial tension tension of the of the bearings. bearings. 4.4. DevelopmentDevelopment ofof aa MeasurementMeasurement ModelModel andand ResearchResearch ResultsResults 4. Development of a Measurement Model and Research Results AA comprehensivecomprehensive modal modal spindle spindle analysis analysis was was carried carried out out in in two two stages. stages. ForFor itsits properproper A comprehensive modal spindle analysis was carried out in two stages. For its proper performance,performance, it it was was necessary necessary to to build build a simpliﬁed a simplified geometric geometric model model of the of spindle. the spindle. It came It performance, it was necessary to build a simplified geometric model of the spindle. It downcame down to modeling to modeling only two only bearing two bearing supports supports and the and spindle the spindle shaft shaft with with the equivalent the equiv- came down to modeling only two bearing supports and the spindle shaft with the equiv- mass.alent mass. They wereThey givenwere given dimensions dimensions corresponding corresponding to the to actual the actual structure. structure. A mesh A mesh was alent mass. They were given dimensions corresponding to the actual structure. A mesh appliedwas applied to the to spindle the spindle elements. elements. Eleven measurementEleven measurement points were points set atwere its nodes set at ( Figureits nodes4 ). was applied to the spindle elements. Eleven measurement points were set at its nodes Three(Figure of 4). them Three were of onthem the were spindle on shaftthe spindle (start, center,shaft (start, end) andcenter, the end) remaining and the eight remaining on the (Figure 4). Three of them were on the spindle shaft (start, center, end) and the remaining topeight surface on the of top the surface bearing of supports, the bearing four supports, on each four support. on each At thesesupport. points, At these a measuring points, a eight on the top surface of the bearing supports, four on each support. At these points, a sensormeasuring was placedsensor successively,was placed successively, which registered which the registered behavior ofthe the behavior spindle systemof the spindle under measuring sensor was placed successively, which registered the behavior of the spindle thesystem inﬂuence under ofthe harmonic influence excitations. of harmonic These excita resultstions. These were results recorded were using recorded the SignalStar using the system under the influence of harmonic excitations. These results were recorded using the VectorSignalStar program. Vector program. SignalStar Vector program.

Figure 4. Geometric structure of the spindle assembly model for modal analysis with the distribu- FigureFigure 4. 4. GeometricGeometric structure structure of of the the spindle spindle assembly assembly model for modal analysisanalysis withwith thethe distributiondistribu- tion of measurement points. tionof measurementof measurement points. points. To perform the test, it was necessary to determine the parameters of the sweep, ToTo perform perform the the test, test, it it was was necessary necessary to to determine determine the the parameters parameters of of the the sweep, sweep, namely: namely:namely: • definition of minimum and maximum frequency (Min/Max Freq); • • definitiondefinition of of minimum minimum and and maximu maximumm frequency frequency (Min/Max (Min/Max Freq); Freq); • definition of the speed of the sweep (Rate); • • definitiondefinition of of the the speed speed of of the the sweep sweep (Rate); (Rate); • definition of the duration of the sweep (Time); • • definitiondefinition of of the the duration duration of of the the sweep sweep (Time); (Time); • definition of the force sensitivity (Magnitude Sensitivity); • • definitiondefinition of of the the force force sensitivity sensitivity (Magnitude (Magnitude Sensitivity); Sensitivity); • definition of the minimum gain level (Minimum Level (dB)). • • definitiondefinition of of the the minimum minimum gain gain level level (Minimum (Minimum Level Level (dB)). (dB)). From a practical point of view, during the measurements, it was important to select FromFrom a apractical practical point point of of view, view, during during the the measurements, measurements, it it was was important important to to select select the parameters of the sweeping signal in such a way that it did not exceed the predeter- thethe parameters parameters of of the the sweeping sweeping signal signal in in such such a away way that that it itdid did not not exceed exceed the the predeter- predeter- mined measurement limits. These exceedances usually occur where the system resonance minedmined measurement measurement limits. limits. These These exceedances exceedances usually usually occur occur where where the the system system resonance resonance frequencies appear. They are mainly caused by an excessive sweep speed or too high of a

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frequencies appear. They are mainly caused by an excessive sweep speed or too high of a boost gain level. During the measurements, the SignalStar Vector software provided a boost gain level. During the measurements, the SignalStar Vector software provided a visu- visualization of the adopted “tunnel”, i.e., the range in which the control signal is to be alization of the adopted “tunnel”, i.e., the range in which the control signal is to be located. located. The size of the “tunnel” was determined by the logarithm of the acceleration The size of the “tunnel” was determined by the logarithm of the acceleration (Log.Mag.g) (Log.Mag.g) (we assume how large the allowed amplitude value is on the tested object), (we assume how large the allowed amplitude value is on the tested object), and the level and the level of the control signal was based on the logarithm of the voltage value of the control signal was based on the logarithm of the voltage value (Log.Mag.V) (we (Log.Mag.V) (we assume the appropriate excitation level). In the case of this work, all assume the appropriate excitation level). In the case of this work, all values of the sweep values of the sweep parameters were selected experimentally based on the previously parameters were selected experimentally based on the previously performed measure- performed measurements of the FFT analysis. On this basis, it was possible to conclude in ments of the FFT analysis. On this basis, it was possible to conclude in what frequency rangewhat frequency the resonance range frequencies the resonance of thefrequencies spindle systemof the spindle should system be expected should and be expected at what leveland at the what speed level and the time speed of the and sweep time shouldof the sweep be determined. should be The determined. frequency The range frequency for the sweeprange for was the50–1100 sweep (Hz)was ,50–1100 and the (Hz), sweep and time the wassweep 15 time (min). was These 15 (min). measurements These measure- were performedments were for performed five preload for five states preload corresponding states corresponding to the full range to the of full supply range voltage of supply for piezoactuatorsvoltage for piezoactuators from −30–150 from (V) −30–150(the value (V) (the of the value preload of the force preload 60 (N), force 107 60 (N),(N), 200107 (N),(N), 320200 (N),(N), 430320 (N)(N),). 430 At a(N)). given At valuea given of value initial of stress, initial the stress, data the from data all from eleven all measurement eleven meas- pointsurement were points successively were successively recorded. The recorded same measurement. The same measuremen parameters weret parameters maintained were at eachmaintained point. Atat each the same point. time, At the at eachsame measuring time, at each point, measuring the response point, of the the response system toof thethe excitationsystem to wasthe excitation recorded threewas recorded times to increasethree times the to accuracy increase of the the accuracy results. of the results. TheThe obtainedobtained resultsresults werewere importedimported intointo thethe MSMS ScopeScope program,program, whichwhich allowedallowed forfor modalmodal analysis. In In this this case, case, it itwas was very very important important to correctly to correctly define define the degrees the degrees of free- of freedomdom and and to define to define the the directions directions for for the the measurements. measurements. After After reading reading the measurement data,data, itit waswas possiblepossible to to automatically automatically average average the the values values for for the the remaining remaining nodal nodal points points of theof the spindle spindle where where no measurementsno measurements were were made. made. Conducting Conducting a modal a modal analysis analysis based based on signalson signals recorded recorded on aon real a real object object seems seems to be to abe better a better solution solution than than only only simulation simulation tests tests in whichin which some some kinds kinds of simplifications of simplifications are alwaysare always made. made. The actualThe actual measurement measurement takes takes into account,into account, e.g., e.g., assembly assembly errors, errors, inaccuracies inaccuraci ines the in performancethe performance of components, of components, or contact or con- ontact the oncontact the contact surfaces, surfaces, which which is difficult is difficult to determine to determine by simulation by simulation (surface (surface roughness, rough- coefficientness, coefficient of friction, of friction contact, contact surface, surface, etc.). etc.). InIn thethe presented presented solution, solution, the the measurements measurements were were carried carried out out in the in frequencythe frequency domain. do- Thismain. allowed This allowed the values the ofvalues the resonance of the resona frequenciesnce frequencies of the system of the and system the correspondingand the corre- accelerationsponding acceleration amplitudes amplitudes to be determined. to be determined. The 3D model The 3D presented model inpresented the research in the was re- usedsearch to was visualize used to the visualize behavior the ofbehavior the spindle of the under spindle the under influence the influence of the forcing of the forcing forces. Itforces. was givenIt was dimensions given dimensions corresponding corresponding to the actual to the structure.actual structure. The visualization The visualization of the resultsof the results was based was based on eleven on eleven measuring measuring points. points. In this In way,this way, it is possibleit is possible to present to present the behaviorthe behavior of the of spindlethe spindle at any at frequencyany frequency within within the measuring the measuring range range (50–1100 (50–1100 (Hz)) (Hz)) and theand selected the selected preload preload value. value. Figure Figure5 shows 5 shows the the behavior behavior of of the the spindle spindle at at the the frequency frequency ofof 264264 (Hz)(Hz) andand thethe initialinitial voltagevoltage ofof thethe bearingsbearings correspondingcorresponding toto thethe supplysupply voltagevoltage ofof piezopiezo actuatorsactuators +60+60 (V) of the valuevalue ofof 200200 (N).(N). WeWe observeobserve thethe spindlespindle swingingswinging onon thethe supports.supports. The maximum amplitudeamplitude valuesvalues occuroccur inin thethe directiondirection ofof thethe systemsystem excitation.excitation.

FigureFigure 5. 5. VisualizationVisualization of the of behavior the behavior of the spindle of the spindlesystem for system the frequency for the of frequency 264 (Hz) at ofthe 264 supply (Hz) voltage at the supplyof 60 (V) voltage(200 (N)) of piezoactuators. 60 (V) (200 (N)) piezoactuators.

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Figure 6 shows an example of the amplitude frequency characteristic of the system Figure6 shows an example of the amplitude frequency characteristic of the system in the full range of tested frequencies (50–1100 Hz) for the preload of 200 (N). In the given in the full range of tested frequencies (50–1100 Hz) for the preload of 200 (N). In the example, the resonance frequency was 264 (Hz). This frequency changed with the change given example, the resonance frequency was 264 (Hz). This frequency changed with the of the preload of the bearings; therefore, such characteristics have been developed for dif- change of the preload of the bearings; therefore, such characteristics have been developed ferent values of the preload of the bearings. Based on these characteristics, it was possible for different values of the preload of the bearings. Based on these characteristics, it was to determine the relationship between the preload of the bearings and the resonant fre- possible to determine the relationship between the preload of the bearings and the resonant quency of the spindle system. frequency of the spindle system.

FigureFigure 6. 6. AnAn exemplary exemplary amplitude-frequency amplitude-frequency characteri characteristicstic of a spindle forfor thethe bearingbearing preload preload equal equalto 60V to (20060V (N))(200 measured(N)) measured at the at spindle the spindle tip in tip the in direction the direction of the of input. the input.

AA summary summary of of all all the the values values of of the the resonance resonance vibration vibration frequencies frequencies and and the the corre- corre- spondingsponding amplitudes amplitudes for for the the successive successive states states of of the the initial initial tension tension of of the the spindle spindle system system isis presented presented in in Table Table 1.1. Additionally, Additionally, Figures Figures 7 andand8 8 show show bar bar graphs graphs of of the the resonance resonance frequenciesfrequencies and and the the corresponding corresponding amplitudes amplitudes di dividedvided into into five five areas, areas, each each area area of of which which corresponds to a different value of the preload of the bearings. For each area, i.e., a specific corresponds to a different value of the preload of the bearings. For each area, i.e., a specific value of the preload of the bearings, the measurement was performed three times. This value of the preload of the bearings, the measurement was performed three times. This made it possible to calculate the value of the standard deviation and to determine the error made it possible to calculate the value of the standard deviation and to determine the error spread as a range between the min/max value. The value of the standard deviation and spread as a range between the min/max value. The value of the standard deviation and the spread of errors are characterized by a small value, which clearly provides information the spread of errors are characterized by a small value, which clearly provides information about the high repeatability of the obtained results. It also allows for the conclusion that about the high repeatability of the obtained results. It also allows for the conclusion that the obtained values of resonance frequencies are not the result of chance but clearly reflect the obtained values of resonance frequencies are not the result of chance but clearly reflect the behavior of the real spindle system at the given parameters. If one wishes to be more the behavior of the real spindle system at the given parameters. If one wishes to be more certain that the tests have been performed correctly, the number of measurements should certain that the tests have been performed correctly, the number of measurements should be increased; this, however, causes a significant extension of the analysis time due to the benumber increased; of measurement this, however, points. causes a significant extension of the analysis time due to the number of measurement points. Table 1. Comparison of resonant vibration frequencies and amplitudes for variable bearing preload conditions. Table 1. Comparison of resonant vibration frequencies and amplitudes for variable bearing preload conditions. −30 V/60 (N) 15 V/107 (N) 60V/200(N) 105V/320(N) 150V/430(N) −30 V/60 (N) 15 V/107 (N) 60V/200(N) 105V/320(N) 150V/430(N) Frequency FrequencyAmp. Amp.FrequencyFrequency Amp. Amp.FrequencyFrequency Amp. Amp.FrequencyFrequency Amp.Amp. FrequencyFrequency Amp.Amp.

(Hz) (Hz)(m/s2) (m/s(Hz)2) (Hz)(m/s2) (m/s2)(Hz) (Hz) (m/s2)(m/s 2) (Hz) (Hz) (m/s(m/s2) 2) (Hz)(Hz) (m/s(m/s2) 2) 247.3 247.317.5 17.5250.7 250.716.5 16.5261.7 261.714.9 14.9278.7 278.714.6 14.6283.7 283.7 12.7 12.7 s—standard s—standard7.02 0.46 4.51 0.46 2.52 0.85 3.21 0.70 3.51 0.86 deviation 7.02 0.46 4.51 0.46 2.52 0.85 3.21 0.70 3.51 0.86 deviation

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300 300 283.7 278.7 283.7 280 278.7 280 261.7 247.3 250.7 261.7 260 250.7 260 247.3 240 240

Frequency [Hz] 220 Frequency [Hz] 220 200 200 Preload force Preload force 60 [N] 107 [N] 200 [N] 320 [N] 430 [N] 60 [N] 107 [N] 200 [N] 320 [N] 430 [N]

Figure 7.7. Comparison ofof thethe averageaverage valuesvalues ofof the the resonant resonant frequency frequency of of the the spindle spindle system system measured measured at at the the spindle spindle tip tip in inFigure the direction 7. Comparison of the input of the operation average values at different of the bearing resonant preload freque conditions.ncy of the spindle system measured at the spindle tip the direction of the input operation at different bearing preload conditions. in the direction of the input operation at different bearing preload conditions.

20 17.5 20 17.5 16.5 16.5 14.9 14.6 14.9 14.6 ] 15 12.7 2

] 15 12.7 2 10 10 5 5 Amplitude [m/s Amplitude

Amplitude [m/s Amplitude 0 0 Preload force 60 [N] 107 Preload[N] 200 force [N] 320 [N] 430 [N] 60 [N] 107 [N] 200 [N] 320 [N] 430 [N]

Figure 8. Comparison of the average values of the acceleration amplitudes for the resonance frequency of the spindle Figuresystem 8.8.measured Comparison in the of direction the average of the values input ofaction the accelerationaccelerationat different bearing amplitudes preload for conditions. thethe resonance frequency of thethe spindlespindle systemsystem measuredmeasured inin thethe directiondirection ofof thethe inputinput actionaction atat differentdifferent bearingbearing preloadpreload conditions.conditions. 5. Summary and Conclusions 5. Summary and Conclusions 5. SummaryThe use andof commercial Conclusions measuring equipment by DataPhysics for research facilitates The use of commercial measuring equipment by DataPhysics for research facilitates this typeThe useof measurement. of commercial The measuring possibility equipment of developing by DataPhysics a model of for the research tested facilitatesstructure this type of measurement. The possibility of developing a model of the tested structure makesthis type it possible of measurement. to visualize The its possibility behavior in of real developing working aconditions. model of theThe tested use of structure this sys- temmakes in itthe it possible possible case of tothe to visualize visualizetested spindle its its behavior behavior system in inarealllows real working workingit to be conditions. stated, conditions. on theThe Thebasis use useof of this the of thissys-ob- tainedtemsystem in results,the in thecase casethat of the the of the testeddesigned tested spindle spindle system system allows is allows dynamically it to it be to stated, be sensitive stated, on the on to thebasis changes basis of the ofin theob-the initialtainedobtained tensionresults, results, ofthat the that the bearings. the designed designed This spindle spindlesensitivity system system is is characterized dynamically is dynamically sensitiveby sensitivean increase to changes to changesin the in fun- the in damentalinitialthe initial tension resonant tension of the of frequency bearings. the bearings. toThis an Thissensitivityincrease sensitivity in isthe characterized ispreload characterized of the by bearingsan by increase an increase in thein the system in fun- the anddamentalfundamental a decrease resonant resonant in the frequency corresponding frequency to toan an increase amplitude. increase in in the This the preload preloadmakes ofit of possiblethe the bearings bearings to use in in the the preload system stateand aof decrease the bearings in the to corresponding control changes amplitude. in the frequency This makes and itform possible of spindle to use vibrations. the preload It alsostate makesof the the bearings bearingsit possible to to control controlto develop changes changes an algorithmin in the the frequency frequency that will and and actively form form of of spindlereact spindle to vibrations. momentary vibrations. It changesalsoIt also makes makes in the it itpossiblespindle possible operating to to develop develop parameters. an an algorithm algorithm that that will will actively actively react to momentarymomentary changes in the spindle operating parameters. Funding: This research received no external funding. Funding: This research received no external funding. Data Availability Statement: The presented research work was carried out on the basis of the stat- utoryData Availability funding of the Statement: Statement: Wrocław UniversityTheThe presented presented of Technology. research research work workAll wasresults was carried carriedand analyzesout out on the on are thebasis not basis madeof the of pub-stat- the liclyutorystatutory available funding funding inof anythe of Wroc thedatabase. Wrocławław University University of Technology. of Technology. All results All results and analyzes and analyzes are not are made not made pub- publiclylicly available available in any in anydatabase. database. Acknowledgments: Not applicable. Acknowledgments: Not applicable.

Lubricants 2021, 9, 68 9 of 9

Acknowledgments: Not applicable. Conﬂicts of Interest: The author declares no conﬂict of interest.

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