Faculty of Machine Building. Technical University of Cluj Napoca

Transcription

1 Facuty of Machine Buiding Technica University of Cuj Napoca CONTRIBUTIONS TO THE CALCULATION AND ANALYSIS OF DYNAMIC ABSORBERS, WITH APPLICATIONS ON BALANCING MECHANICAL SYSTEMS PhD THESIS 11 Scientific coordinator, Prof.dr.ing. Mariana ARGHIR PhD Student Şef.drd.ing. Monica BĂLCĂU Contents Part one Up to date research trends in the fied of dynamic absorbers Chapter 1 INTRODUCTION IN THE STUDY OF DYNAMIC ABSORBERS Chapter REDUCING LENGTHS AND MASSES IN ORDER TO OBTAIN A REDUCED VIBRATORY SYSTEM (REPLACEMENT)... Chapter 3 GENERAL CONSIDERATIONS ON THE STUDY OF DYNAMIC ABSORBERS.. Part two The effect of dynamic absorbers on the torsion vibrations of crankshafts Chapter DYNAMIC ABSORBERS... Chapter 5 MECHANICAL SYSTEMS WITH A DYNAMIC ABSORBER ATTACHED TO ONE OF ITS EXTREMITIES AND WHICH A SINGLE ORDER X HARMONICA ACTS UPON 3 Chapter MECHANICAL SYSTEMS SUBJECTED TO A SINGLE X ORDER HARMONICA TO WHICH A DYNAMIC ABSORBER IS ATTACHED INSIDE THE SYSTEM... 3 Chapter 7 MEHCANICAL SYSTEMS SUBJECTED TO SEVERAL ORDER X HARMONICAS AND WITH A DYNAMIC ABSORBER ATTACHED AT THE END OF THE MECHANICAL SYSTEM... 3 Chapter MECHANICAL SYSTEMS WITH FOUR REDUCES MASSES SUBJECTED TO ONE OR MULTIPLE ORDER X HARMONICAS WITH A DYNAMIC ABSORBER ATTACHED... 3 Part three Experimenta researches. Mathematica modeation Chapter 9 EXPERIMENTAL RESEARCHES The experimenta stand. Genera considerations. Technica characteristics of the engine used Baancing. Genera considerations Dynamic baancing Experimenta measuring of vibrations The anaysis of experimenta data Comparative anaysis. Concusions... 5 Chapter MATHEMATICAL MODELATION. RESULTS OF MATHEMATICAL MODELATION.... The resuts of mathematica modeation....3 Concusions... Chapter11 CONCLUSIONS. FURTHER RESEARCH TRENDS 11. Persona contributions... Bibiographica references

2 CHAPTER 1 INTRODUCTION IN THE STUDY OF DYNAMIC ABSORBERS The dynamic absorbers represent one of the means of reducing vibrations and they have mutipe usages in various fieds of technoogy. The first dynamic absorber has been described in 199 and patented in 1911 by Hermenn Frahm. Then Den Hartog eaborated studies to optimize it. Since then, many studies focused on increasing the efficacy of the dynamic absorber, on perfecting designing methods and on optimizing various parameters. The vibrations of the machines, equipment and constructions are produced by a wide range of sources. The dynamic absorber is an osciating system which is attached to a structure in order to eiminate its vibrations. Usuay, the dynamic absorber is superior to any other insuating system because it can be easiy tuned so that it works in the same area of frequency as the source to be eiminated. Using a dynamic absorber is recommended when the inherent frequency of the primary system is in cose range to the one of the disruptive forces. CHAPTER REDUCING LENGTHS AND MASSES IN ORDER TO OBTAIN A REDUCED VIBRATORY SYSTEM (REPLACEMENT) The rea crankshaft must be turned into a straight shaft of a constant diameter d e, buit from a materia with the same transversa easticity modue G as the materia used for the construction of the crankshaft but whose mass can be negigibe and whose ength e must be cacuated in such a way that, shoud the rea crankshaft and the straight shaft were subjected to the same torsion torque with the moment M t, the tota twisting ange φ woud be the same [Rip 77b]. The rea crankshaft is divided into n fractions composed of: an ebow, cyinders, cones, rounding, coupes and each portion is reduced to a simiar diameter d arbitrariy chosen, representing the equivaent shaft [Pop ]. The reduced ength of the ebow of the shaft of an engine is determined by reations eaborated based on experimenta trias. The reduction of the masses aims to repace the masses of the crankshaft and the mobie gear with reduced, ring shaped masses with a very sma radius, eveny distributed on circumferences with the same radius, with centers paced on the rotationa axis of the crankshaft in such a way that the mechanica inertia moments of the rea masses in reation with the rotationa axis of the crankshaft to be equa to the mechanica inertia moments of the reduced, ring shaped masses in reation to the same radius [Pop ], [Rip 77b]. CHAPTER 3 GENERAL CONSIDERATIONS ON THE STUDY OF DYNAMIC ABSORBERS The crankshaft of a piston engine forms an osciating system together with the pieces attached to it and which beong to the mobie gear. The ampitude of the torsion osciations can become very dangerous when the resonance phenomenon occurs, that is when the rotation of the shaft becomes the same with one of its critica osciations. Out of the vast range of critica rotations, which is infinitey arge, ony some of them can produce an inadmissibe increase high enough of the torsion osciation ampitude. The design of the crankshaft must be done in such a way that the running rotations shoud not be the same with the critica ones, which can be dangerous for the crankshaft resistance. The means by which the torsion osciations can be fought are [Pop ]: - Changing the way in which the crankshaft is constructed; - If the range of the running rotations cannot be separated from the critica ones, then the range of the critica rotations shoud be modified by changing the inherent pusations corresponding to the osciating system. If the means above do not provide a reduction in the torsion osciations of the crankshaft, then the foowing devices can be empoyed: - Friction dampers for torsion osciations (with energy dissipation) - Dynamic dampers for torsion osciations (without energy dissipation) CHAPTER DYNAMIC ABSORBERS The chapter estabishes the dynamic equivaence between the mechanica system composed of two reduced masses with a dynamic absorber and the equivaent mechanica system with a dynamic absorber. If the dynamic absorber L is constructed in such a way that x, the ampitudes of the two reduced masses become ni, thus they wi not perform torsion vibrations. Aso, the anaysis performed shows that there is a free sector in which the resonance cannot

3 L take pace. If the dynamic absorber is repaced with a rotating tabe and this is constructed in such a way that x resuts m p. This situation shoud be avoided. If the penduum dynamic absorber with an extended dynamic penduum on a two string suspension and this is L constructed in such a way that x resut m p, and this situation shoud be avoided. CHAPTER 5 MECHANICAL SYSTEMS WITH A DYNAMIC ABSORBER ATTACHED TO ONE OF ITS EXTREMITIES AND WHICH A SINGLE ORDER X HARMONICA ACTS UPON In the case of mechanica systems with three, four and respectivey five reduced masses with a dynamic absorber attached at the end of the system which is subjected to a singe order x harmonica which acts upon the mass that contains the absorber, if the foowing reation is imposed L x (which represents the condition for the vibrations generated by the order x harmonica to be eiminated), the reduced masses which come from reducing the crankshaft and its mobie gears, do not perform torsion vibrations. The ony mass which performs torsion vibrations is the reduced mass which comes from dynamicay equating the absorber. CHAPTER MECHANICAL SYSTEMS SUBJECTED TO A SINGLE X ORDER HARMONICA TO WHICH A DYNAMIC ABSORBER IS ATTACHED INSIDE THE SYSTEM In the case of mechanica systems with three, four and respectivey five reduced masses with a dynamic absorber attached inside the system which is subjected to a singe order x harmonica which acts upon the mass that contains the absorber, if the foowing reation is imposed L x (which represents the condition for the vibrations generated by the order x harmonica to be eiminated), the reduced masses which come from reducing the crankshaft and its mobie gears, do not perform torsion vibrations. The ony mass which performs torsion vibrations is the reduced mass which comes from dynamicay equating the absorber. CHAPTER 7 MEHCANICAL SYSTEMS SUBJECTED TO SEVERAL ORDER X HARMONICAS AND WITH A DYNAMIC ABSORBER ATTACHED AT THE END OF THE MECHANICAL SYSTEM In the case of mechanica systems with three reduced masses and a dynamic absorber attached at the end of the L system, if the foowing reation is imposed x (which represents the condition for the vibrations generated by the order x harmonica to be eiminated), the reduced masses m 1 and m, which come from reducing the crankshaft and its mobie gears, perform torsion vibrations. The ony mass which does not perform any torsion vibrations is the reduced mass m 3 which has the dynamic absorber attached. This happens even if the mechanica system is subjected to the action of a singe order x harmonica or two such harmonicas. CHAPTER MECHANICAL SYSTEMS WITH FOUR REDUCES MASSES SUBJECTED TO ONE OR MULTIPLE ORDER X HARMONICAS WITH A DYNAMIC ABSORBER ATTACHED In the case of a mechanica system composed of four reduced masses and subjected to a singe, two or respectivey four order x harmonicas, with the dynamic absorber paced at the end of the mechanica system, if the foowing reation is imposed L x 3

4 (which represents the condition for the vibrations generated by the order x harmonica to be eiminated), ony the ampitude of the vibrations performed by the reduced mass which has the dynamic absorber attached is ni, therefore this mass does not perform torsion vibrations unike the rest of the reduced masses which perform such vibrations. CHAPTER 9 EXPERIMENTAL RESEARCH 9.1. The experimenta stand. Genera considerations. Technica characteristics of the engine used The dangerous torsion vibrations are produced inside the crankshaft system when the torsion moments acting upon that system present significant periodica vibrations. In order to be abe to evauate the torsion vibrations of an eastic system so compex, it is important that this system is turned into an equivaent dynamic system formed by a rectiinear eastic shaft with a negigibe mass and oaded with reduced circuar masses [Rip 5]. Vin order to verify the hypotheses and the theoretica resuts, the experiment wi be done on a tria stand with a DACIA 13 car engine (Renaut 1) (Fig.9.1.1). Fig Dacia 13 engine tria stand 9. Baancing. Genera considerations Baancing is the method by which one can verify the mass aocation of a rotor and, if necessary, this mass is corrected so that the vibrations of the spinde bearing and/or the forces in the stand are within the imits prescripbed for the running speed. According to the causes eading to imbaances, two types of baancing can be performed: - static baancing (in a singe pane) - dynamic baancing (in two panes) The dynamic baancing is done with the hep of Fiedpaq/Impaq measurement and monitoring anayzers and the baancing Baancing program and the measuring apparatus is caed BENSTONE. The device can perform dynamic baancing in situ or on the tria stand. 9. Dynamic baancing The dynamic baancing of the DACIA 13 engine is an in situ baancing on a tria stand and it is performed by mounting counterweights in two perpendicuar panes on the rotation axis: on the engine puey and on its fywhee. The foowing stages are performed: a. Choosing the baancing program. b. The baancing method used is dynamic in two panes. c. Setting the types of sensors medium frequency acceerometers type IEPE with a sensitivity of mv/g. d. Setting the baance toerance according to ISO 19 [**]. e. Baancing operation in two panes. This operation performs two steps: - e.1. Initia rotation - e.. Mounting the tria mass in pane P1 f. Tria rotation with the mass mounted in pane P1 at, in reaming 1. g. Mounting the tria mass in pane P. h. Tria rotation with the mass mounted in pane P at, in reaming 1. i. Correction of the rotor. j. Mounting corrective masses

5 k. Correction options. Check rotation m. The history of the baancing operations 9. Anaysis of experimenta resuts Figure 9. presents the image of the engine, indicating the five points (Point 1, Point, Point 3, Point, Point 5) where the measurement of the vibrations were performed in three situations: - Imbaanced engine - Staticay baanced engine - Dynamicay baanced engine For each point there was a set of measurements at an interva of s. Fig. 9.-a. Tria stand Fig. 9.-b. Measurement points 9.7 Comparative anaysis. Concusions This anaysis focuses on the Peak to Peak acceeration for a the five points. Figure presents the Peak to Peak acceeration [mm/s ] for Point 1 in the three study cases, according to the number of measurements done. This point is situated on the engine bock. Acceeraţia Peak to Peak [mm/s^] Acceeraţia Peak to Peak a Punctuui Număr de măsurători / s Neechiibrare Echiibrare statica Echiibrare dinamica Acceeraţia Peak to Peak [mm/s^] Acceeraţia Peak to Peak a Punctuui Număr de măsurători / s Neechiibrare Echiibrare statica Echiibrare dinamica Fig Peak to Peak acceeration for Point 1 Fig Peak to Peak acceeration for Point The data on the chart show that, in the case of Point 1, the one situated on the engine bock near the aternator, the Peak to Peak acceeration vaues are ower than in the case of static baancing and ower than in the case of an imbaanced crankshaft. The aternator is formed of a rotor and stator and the rotor is in a rotationa movement at up to rpm. An important aspect is the face that the measurements of the vibrations in Point 1 are infuenced by the vibrations transmitted by the aternator, since it is situated in its immediate vicinity. Aso, in the case of the dynamic baancing, the Peak to Peak acceeration presents vaues of approx.,5 mm/s - 7,5 mm/s. Figure 9.7. presents the Peak to Peak acceeration [mm/s ] in the case of Point, in the three cases studied (imbaanced shaft, staticay baanced one and dynamicay baanced shaft). This point is situated on the engine bock and the representation is done according to the number of measurements. The data presented on the chart revea that in the case of Point, situated on the engine bock and paced between the aternator and the oi fiter, the static baance eads to Peak to Peak acceeration vaues beow the ones obtained in the case of an imbaanced engine. The dynamic baance of the shaft and its gears presents vaues of the Peak to Peak acceeration of -7 mm/s. Figure presents the Peak to Peak acceeration [mm/s ] for Point 3 in the three cases studied (imbaanced shaft, staticay baanced shaft and dynamicay baanced shaft). This point is paced on the engine crankcase. In this case, the vaues of the Peak to Peak acceeration for both the static and dynamic baancing are ower than the ones in the case of an imbaanced shaft. Point 3 is situated in the pane which contains the distribution axis and this is an area in which there is a gearing between the camshaft and the distribution axis, which eads to a high eve of vibrations. The high vaues for the Peak to Peak acceeration are strongy infuenced by the distributor (situated in the vicinity of the measuring point). 5

6 Acceeraţia Peak to Peak [mm/s^] Acceeraţia Peak to Peak a Punctuui Neechiibrare Echiibrare statica Echiibrare dinamica Acceeraţia Peak to Peak [mm/s^] Acceeraţia Peak to Peak a Punctuui Neechiibrare Echiibrare statica Echiibrare dinamica Număr de măsurători / s Număr de măsurători / s Fig Peak to Peak acceeration for Point 3 Fig Peak to Peak acceeration for Point Figure 9.7. presents the Peak to Peak acceeration [mm/s ] for Point, in the three cases studied (imbaanced shaft, staticay baanced shaft and dynamicay baanced shaft). Point is situated on the engine crankcase. The chart anaysis shows that, in the case of Point, situated on the engine crankcase, the vaues for the Peak to Peak acceeration in the case of static baance and dynamic baance are much ower than the ones for the case in which the crankshaft is not baanced at a. Figure presents the Peak to Peak acceeration [mm/s ] for Point 5, in the three cases studied (imbaanced shaft, staticay baanced shaft and dynamicay baanced shaft). Point 5 is situated on the engine bock. The chart anaysis shows that, in the case of Point 5, situated on the engine bock, near the fywhee and the case of the cutch, the vaues for the Peak to Peak acceeration in the case of dynamic baance are much ower than the ones for the case in which the crankshaft is not baanced or it is staticay baanced. In the case of static baance, the vaues for the Peak to Peak acceeration go beow and respectivey above the Peak to Peak acceeration in the case of imbaance. Acceeraţia Peak to Peak [mm/s^] 1 Acceeraţia Peak to Peak a Punctuui Număr de măsurători / s Neechiibrare Echiibrare statica Echiibrare dianmica Fig Peak to Peak acceeration for Point 5 After the dynamic baancing and measurements performed in the three situations in the five points, the vaues of the Peak to Peak acceerations are ower in the case of the imbaance engine and respectivey in the case of a static baance. CHAPTER MATHEMATICAL MODELATION. THE RESULTS OF MATHEMATICAL MODELATION. The resuts of mathematica modeation A simpifying hypothesis [Gru ] is that the anguar speed of the shaft is constant. Aso, the force that acts upon the reduced masses is the pressure of the gasses which act on each piston. The movements, speeds and acceerations resuted from the differentia equations system (.1.1) are graphicay represented. The system is soved using Mathcad 1 with software caed Cacu presented in Annex. Figure..1 depicts the chart of the movement of the reduced mass corresponding to the fywhee in a 1 second interva. Figure.. presents the chart of the movement of the reduced mass1. In comparison to the reduced mass m v corresponding to the fywhee, the movements are higher, and this fact can ony be expained due to the higher mass of the fywhee. Fig...1 Movement of reduced mass v Fig... Movement of the reduced mass m 1

7 Figure..3 represents the chart for the movement of the reduced mass. In this case, the movement is bigger, its peak vaue being over mm, in comparison to the movement of the reduced mass corresponding to the fywhee. Fig...3 Movement of the reduced mass m Fig... Movement of the reduced mass m 3 Figure.. represents the chart for the movement of the reduced mass 3. In this case, the peak vaue of the movement decreases under mm in comparison to the movement of the reduced mass, but it is higher than the vaue of the movement of the reduced mass 1, that is the reduced mass corresponding to the fywhee. Figure..5 presents the chart for the movement of the reduced mass. In this case, the peak vaue reaches approximatey mm. Fig... Movement of the reduced mass m Fig...5 Movement of the reduced mass m 5 Figure..5 presents the chart for the movement of the reduced mass 5, corresponding to the dynamic absorber. The vaues are of tens of mm. Fig... Speed of the reduced mass m v Fig...7 Speed of the reduced mass m 1 Figure.. represents the speed corresponding to the reduced mass m v. This shows that the peak vaue of the speed reaches approx.,5 m/s. The graph was created for a 1 second interva. Figure..7 represents the speed corresponding to the reduced mass 1. This shows that the peak vaue is approx.,9 m/s. The graph was created for a 1 second interva. Aso, it shows that the speed is higher than the one corresponding to the one for the reduced mass of the fywhee. Figure.. presents the speed corresponding to the reduced mass. The graph shows that the peak vaue of the reduced mass reaches approx. 1m/s. Fig... Speed of reduced mass m Fig...9 Speed of the reduced mass m 3 Figure..9 represents the speed corresponding to the reduced mass 3 and it shows that the peak vaue is more reduced than the one corresponding to the reduced mass. 7

8 Fig... Speed for the reduced mass m Fig...11 Speed for the reduced mass m 5 Figure.. represents the speed corresponding to the reduced mass. The graph shows that the peak vaue is approx.,5 m/s. Figure..11 presents the speed corresponding to the reduced mass 5, which is the one for the dynamic absorber. Figure..1 represents the acceeration corresponding to the reduced mass 1. The chart shows that the Peak to Peak acceeration is approx. 7,7 mm/s. If we compare this vaue to the vaue of the Peak to Peak acceeration from figure 9..3., the dynamicay baanced shaft from Point, which is situated in front of the first cyinder, the vaues are between,7 and 7,595 mm/s Fig...1 Acceeration of the reduced mass m 1 Fig...13 Acceeration of the reduced mass m Figure..13 represents the acceeration corresponding to the reduced mass. The graph shows that the Peak to Peak acceeration is about 7 mm/s. If we compare this vaue with the vaue of the Peak to Peak acceeration from figure 9..3., the dynamicay baanced shaft in Point and respectivey Point 3 (the cyinder is situated between these points), we notice that the vaues are between,1 and,35 mm/s. Figure..1 represents the acceeration corresponding to the reduced mass 3. The graph shows that the Peak to Peak acceeration is about 7,5 mm/s. If we compare this vaue with the vaue of the Peak to Peak acceeration in figure 9..3., the dynamicay baanced shaft in Point 3 which corresponds to the third cyinder, we notice that the vaues are between 7,3 and,35 mm/s. Fig...1 Acceeration of reduced mass m 3 Fig...15 Acceeration of reduced mass m 5 Figure..15 represents the acceeration corresponding to the reduced mass. The graph shows that the Peak to Peak acceeration is about 7,5 mm/s. If we compare this vaue to the vaue of the Peak to Peak acceeration in figure 9..3., the dynamicay baanced shaft in Point, corresponding to the fourth cyinder, we notice that the vaues are between,599 and 7,19 mm/s..3 Concusions The measurements have been performed on a disabed tria stand, which underwent a capita repair, with the segmentation of the standard quotations; so that we can confirm that the engine was in optima operationa state, which is why the resuts have been comparabe to the theoretica ones. Initiay the engine was baanced then the vibrations have been measured, assimiating one of the masses mounted at one end with the dynamic absorber. After the measurements, the dynamic baancing (by regarding one the baancing masses as a dynamic absorber) eads to a reduction of the vibrations. The theoretica substantiation is thus being demonstrated. The anayses performed show that the mathematica mode confirms the experimenta measurements and the theoretica substantiation.

9 CHAPTER 11 GENERAL CONCLUSIONS. PERSONAL CONTRIBUTIONS. NEW RESEARCH TRENDS 11. Persona contributions The theoretica substantiations are based on the fairy extensive study regarding the infuence which the dynamic absorber has over the reduced mechanica systems creating after the reduction of engths and masses of crankshafts and their mobie gears. If the order of the harmonica at the power of two is equa to the ratio of the dimensions specific of the absorber, then the number of resonance events wi be reduced. In a the cases studied, no matter if the dynamic absorber is paced inside or outside the end of the mechanica system, the resuts show that the dynamic absorber does not produce a new resonance event. The dynamic baancing is done by mounting a baancing mass in two panes: the pane of the fywhee and the pane of the puey. On the pane of the fywhee, one of the baancing masses is assimiated as the dynamic absorber and contains: one screw and three adaptabe washers. The measurements are done in three situations: imbaanced shaft, staticay baanced shaft and dynamicay baanced shaft. The mathematica modeation is performed starting from the dimensions of the dynamic absorber used for the baancing process and taking into account the baancing mass. By comparing the resuts obtained after the dynamic baancing with the ones obtained after the mathematica modeation, we notice that they are viabe, any differences between theoretica and experimenta being within admissibe imits. The reduction of the mobie gear masses and the reduction of the engths of the crankshaft are done using a program created in MathCad. The software is used to create a cacuation program for the differentia equation system so that we coud graphicay represent the movements, speed and acceeration of the points studied. A vast anaysis of the resuts obtained competes the research. A the theoretica soutions obtained are in harmony and in accordance to the resuts obtained subsequent measurements. The anayses performed show that the mathematica mode confirms the experimenta measurements and the theoretica substantiations. A the objectives of the thesis have been met. Seective references [Arg 99] Arghir, M., Mechanics. Statics & materia point kinematics, Ed. U.T.Press, Cuj-Napoca, 1999, ISBN [Arg ] Arghir, M., Mechaics II, Rigid body cinematics&dynamics, U. T. Press, Cuj-Napoca, ISBN [Ba a] Băcău, M., Ripianu, A., Case study of a mechanic system composed of four reduced masses and thedynamic absorber is paced to one of the extremities of the mechanic system and subjected to two harmonics X, în Acta Technica Napocensis,, seria Appied Mathematics and Mechanics, nr. 51, vo. IV, ISSN 11-57, pag [Ba b] Băcău, M., Ripianu, A., Case study of the mechanica system composed of four reduced masses that is subjected to a singe harmonic x to which we attached one dynamic absorber, over upon the another mass then the given harmonic x, în Acta Technica Napocensis,, seria Appied Mathematics and Mechanics, nr. 51, vo. IV, ISSN 11-57, pag.99-. [Ba 9a] Băcău, M., Arghir, M., Case study of a mechanic system composed of four reduced masses and the dynamic absorber paced to one of the extremities of the mechanic system and subjected to four harmonic x, Annas of DAAAM for 9&PROCEEDINGS of the th Internationa DAAAM Symposium Inteigent Manufacturing&Automation: Focus on Theory, Practice and Education 5- th November 9, Vienna, Austria, pag.11-1, ISSN , ISBN [Ba 9b] Băcău, M., Arghir, M., Contributions to the study of dynamic absorber, în Acta Technica Napocensis, 9, seria Appied Mathematics and Mechanics, nr. 5, vo. IV, ISSN 11-57, pag.3-5. [Ba b] Băcău, M., Arghir, M., Particuar case to the study of dynamic absorber, în Acta Technica Napocensis,, seria Appied Mathematics and Mechanics, nr. 53, vo. I, ISSN 11-57, pag.1-. [Ba e] Băcău, M., Baidoc, M., Arghir, M., Eiminating torsion vibrations in the case of interna combustion engines- technica characteristic of the engine used, în Acta Technica Napocensis,, seria Appied Mathematics and Mechanics, nr. 53, vo. IV, ISSN 11-57, pag [Ba 11a] Băcău, M., Baidoc, M., Pop, A.F., Arghir, M., Torsion vibrations of the Dacia car engine, nd Annua Meeting, 1-1 Apri, 11, Geseschaft fur Angewandte Mathematik und Mechanik, GAMM of Internationa Association of Appied Mathematics and Mechanics at Graz University of Technoogy, Austria, ISBN [Ba 11b] Băcău, M., Arghir, M., The repacement of the penduar dynamic absorber with a rotating mass, în Acta Technica Napocensis, 11, seria Appied Mathematics and Mechanics, nr. 5, vo. I, ISSN 11-57, pag