ap-4-12.dvi Acta Polytechnica Vol. 52 No. 4/2012 Contributions to the Study of Dynamic Absorbers, a Case Study Monica Bălcău1, Angela Pleşa1, Dan Opruţa1 1 Technical University of Cluj-Napoca, Automotive Engineering Department, Cluj-Napoca, B-dul Muncii 103-105 Correspondence to: Monica.Balcau@auto.utcluj.ro Abstract Dynamic absorbers are used to reduce torsional vibrations. This paper studies the effect of a dynamic absorber attached to a mechanical system formed of three reduced masses which are acted on by one, two or three order x harmonics of a disruptive force. Keywords: dynamic absorber, torsional vibrations, reduced masses. 1 Introduction This paper studies the effect of dynamic absorbers on reducing torsional vibrations. Like the inertial effects, the forces and the torque produced by thermal processes in the cylinders of internal combustion engines produce forces and tor- sional moments that vary nonlinearly. These forces, applied to the crankshaft and to the engine block, produce translational and rotational oscillations of the engine block and torsional oscil- lations in the crankshaft. They have to be reduced, because they cause noise and vibrations in an engine. In order to calculate the torsional vibrations of a complex elastic system of this kind, the system must first be turned into a simpler equivalent dynamic sys- tem, formed of an elastic linear shaft of negligible mass loaded with circular reduced masses obtained by reducing the mobile gears. This study focuses on three mechanical systems formed of three reduced masses, which receive a pen- dulum dynamic absorber. These masses are acted on by one, two or three order × harmonics, resulted from the decomposition of the Fourier series of the dis- ruptive force with periodic variation. The dynamic absorber is attached at the end of the mechanical system. 2 A mechanical system acted on by one harmonic The first case deals with a mechanical system formed of three reduced masses with a dynamic absorber at- tached according to Figure 1. The dynamic absorber is placed at the end of the mechanical system. The reduced mass m3 is acted on by an order x harmonic of a disruptive force presenting a periodic variation marked with Px · cos(Ωxt − ε). The dynamic absorber is replaced by an equiv- alent reduced mechanical system formed of the re- duced mass m4 and the segment of the reduced crankshaft with an elastic constant c34. The elastic constants of the segments of crank- shafts between the two consecutive reduced masses and the mechanical axial moments of the reduced masses in relation to the symmetrical geometry axis of the shaft must be chosen in such a way that the kinetic energy and the potential energy of the real vibrating system (formed by the crankshaft and its mobile gears) is equal to the kinetic energy and the potential energy of the reduced vibrating system. Figure 1: Mechanical system This case starts from the differential equation sys- tem governing the torsional vibrations created by the mechanical system presented in Figure 2: m1 d2a1 dt2 + c12(a1 − a2) = 0 m2 d2a2 dt2 + c12(a2 − a1) + c23(a2 − a3) = 0 12 Acta Polytechnica Vol. 52 No. 4/2012 m3 d2a3 dt2 + c23(a3 − a2) + c34(a3 − a4) = Px cos(Ωxt − ε) (1) m4 d2a4 dt2 + c34(a4 − a3) = 0 Figure 2: Equivalent mechanical system The relative linear elongations (displacement) ai measured on the reduced circle of radius r (of the vibrations of the four reduced masses) are expressed in relation to the linear amplitudes Ai: ai = Ai cos(Ωxt − ε) (i = 1 ∼ 4) (2) In order for the mechanical system formed of the re- duced mass m4 and the shaft segment with elastic constant c34 to be dynamically equivalent to the dy- namic absorber attached to the reduced mass m3 (in other words for this mass m3 to be subjected to the same torque as the dynamic absorber), it is necessary and sufficient that: m4 = m (L + l)2 r2 c34 = m (L + l)2 r2 L l ω20 (3) The square value of the order × harmonic is ex- pressed by: Ω2x = x 2ω20 (4) where x represents the order of the harmonic and ω0 the angular speed of the shaft. Taking into consideration equations (2), (3), (4) and introducing the differential equation system (1), we obtain an algebraic system of equations. By solv- ing this system of equations we get the four determi- nants. If the dynamic absorber is built in such a way that: L l = x2 (5) the expressions of the four determinants become: Δ1 = − Px m3 c12 m1 c23 m2 ω20 ( x2 − L l ) Δ2 = Px m3 c23 m2 ω20 ( x2ω20 − c12 m1 ) ( x2 − L l ) Δ3 = − Px m3 ω20 ( x2ω20 − c12 m1 ) · ( x2ω20 − c23 m2 − c12 m2 ) ( x2 − L l ) Δ4 = Px m3 ω20 ( x2ω20 − c12 m1 ) · ( x2ω20 − c23 m2 − c12 m2 ) L l ω20 (6) An analysis of the four determinants shows that the only mass that executes torsional vibrations is mass m4, which is not part of the reduced crankshaft. The other reduced masses do not execute torsional vibra- tions. So: A1 = 0 A2 = 0 A3 = 0 A4 �= 0 (7) 3 A mechanical system acted on by two order × harmonics This case deals with a mechanical system composed of three reduced masses, which receives a dynamic absorber placed at the end of the system presented in Figure 3. The reduced masses m2, m3 are acted on by two order × harmonics of the disruptive force presenting a periodic vibration marked with Px cos(Ωxt − ε). Figure 3: Mechanical system Figure 4: Equivalent mechanical system 13 Acta Polytechnica Vol. 52 No. 4/2012 The reduced mass m4, together with the shaft seg- ment with elastic constant c34, forms a dynamic me- chanical system equivalent to the dynamic absorber which applies the same torque to mass m3 as the absorber (Figure 4). As in the previous case, the dy- namic absorber is replaced by an equivalent dynamic system formed of the reduced mass m4 and the re- duced crankshaft segment having elastic constant c34 (Figure 4). The study starts from the differential equation system that governs the torsion vibrations performed by the mechanical system represented in Figure 4: m1 d2a1 dt2 + c12(a1 − a2) = 0 m2 d2a2 dt2 + c12(a2 − a1) + c23(a2 − a3) = Px cos(Ωxt − ε) m3 d2a3 dt2 + c23(a3 − a2) + c34(a3 − a4) = Px cos(Ωxt − ε) (8) m4 d2a4 dt2 + c34(a4 − a3) = 0 Relation (2) gives the elongation expressions ai re- lated to the amplitudes recorded on radius r on the reduction circles. Taking into account expressions (2), (3), (4) and introducing the differential system of equations (8) we obtain a system of algebraic equa- tions. By solving this system, we obtain expressions for the five determinants. Δ = 1 m3 m (L + l)2 r2 ( L l ω20 )2 [( x2ω20 − c12 m1 ) · ( x2ω20 − c12 m2 − c23 m2 ) − c12 m1 c12 m2 ] Δ1 = − Px m2 ω40 c12 m1 1 m3 m (L + l)2 r2 ( L l )2 Δ2 = − Px m2 ω40 ( x2ω20 − c12 m1 ) 1 m3 m (L + l)2 r2 ( L l )2 Δ3 = 0 Δ4 = − Px m2 c23 m3 L l ω20 ( x2ω20 − c12 m1 ) + Px m3 [( x2ω20 − c12 m1 ) ( x2ω20 − c12 m2 − c23 m2 ) − c12 m1 c12 m2 ] L l ω20 (9) The analysis of the expressions (9) shows that the reduced masses m1, m2, and m4 — born from a re- duction operation — perform torsion vibrations. The only mass that does not perform any torsional vibra- tions is mass m3, which is the mass that has the dynamic absorber attached. A1 �= 0 A2 �= 0 A3 = 0 A4 �= 0 (10) 4 The mechanical system acted on by three order × harmonics This case investigates a mechanical system com- posed of three reduced masses which receive a dy- namic absorber placed at the end of the system (Fi- gure 5). The reduced masses m1, m2, m3 are acted on by means of three order x harmonics of disruptive forces presenting a periodic variation marked with Px cos(Ωxt − ε). Figure 5: Mechanical system Figure 6: Equivalent mechanical system Just as in the previous paragraph, the dynamic absorber is replaced by an equivalent dynamic sys- tem formed of the reduced mass m4 and the reduced shaft segment having the elastic constant c34 (Fi- gure 6). The differential equations governing the vi- bratory movements of the mechanical system repre- sented in Figure 6 are: m1 d2a1 dt2 + c12(a1 − a2) = Px cos(Ωxt − ε) m2 d2a2 dt2 + c12(a2 − a1) + c23(a2 − a3) = Px cos(Ωxt − ε) m3 d2a3 dt2 + c23(a3 − a2) + c34(a3 − a4) = Px cos(Ωxt − ε) (11) 14 Acta Polytechnica Vol. 52 No. 4/2012 m4 d2a4 dt2 + c34(a4 − a3) = 0 The expressions of elongations ai according to the amplitudes recorded on the radius r reduction circles are given by relation (2). Taking into account expressions (2), (3), (4) and introducing the differential system of equations (11), we obtain an algebraic system of equations. Solving this system provides the four determinants. Δ = − 1 m3 m (L + l)2 r2 ( L l ω20 )2 [( x2ω20 − c12 m1 ) · ( x2ω20 − c12 m2 − c23 m2 ) − c12 m1 c12 m2 ] Δ1 = Px m1 ( x2ω20 − c12 m2 − c23 m2 ) 1 m3 m (L + l)2 r2 · ( L l ω20 )2 − Px m2 c12 m1 1 m3 m (L + l)2 r2 ( L l ω20 )2 Δ2 = − Px m1 c12 m2 1 m3 m (L + l)2 r2 ( L l ω20 )2 + Px m2 1 m3 m (L + l)2 r2 ( L l ω20 )2 Δ3 = 0 (12) We observe that: A1 �= 0 A2 �= 0 A3 = 0 (13) This means that the only mass that does not exe- cute torsional oscillations is mass m3, which received a dynamic absorber. Since mass m4 is not a result of reducing the crankshaft, it was no longer useful when calculating the amplitude vibration for this mass. 5 Conclusion The three cases discussed here offer us the follow- ing information: – In the case of the mechanical system formed of three reduced masses acted on by a single order x harmonic of the disruptive force, if the dynamic absorber is built according to relation (5), the three reduced masses m1, m2 and m3 do not perform any torsional vibrations, irrespective of the value of the angular speed. – In the case of the mechanical systems formed of three reduced masses acted on by two and three order x harmonics of the disruptive force, respec- tively, if the dynamic absorber is built according to relation (5), the only mass which does not execute torsional vibrations is the mass which receives the dynamic absorber. References [1] Haddow, A. G., Shaw, S. W.: Centrifugal pen- dulum vibration absorbers: an experimental and theoretical investigation. Nonlinear Dynamics, 34, 293–307, 2003. 2004 Kluwer Academic Pub- lishers. Printed in the Netherlands. [2] Ripianu, A., Crăciun, I.: The dynamic and strength calculus of straight and crank shafts. Transilvania Press Publishing House Cluj, 1999. [3] Kraemer, O.: Drehschwingungsrechnung Berech- nung der Eigenschwingungszahlen, In Tech- nishe Hochschule Karlsruhe Lehrstuhl für Kolben- maschinen und Getriebelehre, 1960. [4] Pop, A. F., Gligor, R. M., Bălcău, M.: Analis- ing of Vibrations Measurements upon Hand-Arm System and Results Comparison with Theoretical Model, 3rd European Conference on Mechanism Science, Previous EUCOMES 2010 Conferences, September 14–18, 2010, Cluj-Napoca, Roma- nia Mechanisms and Machine Science, Springer Vol. 5, p. 277–284. ISBN 978-90-481-9688-3. 15