Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 2, pp. 465-478, Warsaw 2010 ANALYSIS OF DAMPING OF VIBRATIONS THROUGH A FRICTIONAL DAMPER Zbigniew Skup Warsaw University of Technology, Institute of Machine Design Fundamentals, Warsaw, Poland e-mail: zskup@ipbm.simr.pw.edu.pl This paper presents a study of damping of non-linear vibrations in a one-degree-of freedom model of a mechanical system containing a fric- tion damper. The vibrations of the system due to harmonic excitation is considered on the assumption of uniformly varying frequency and a constant amplitude of the exciting force. The simultaneous phenome- non of structural friction (passive damping) has been considered as well. The problem is considered on the assumption of a uniformunit pressure distribution between contacting surfaces of the friction conic inner and outer rings. The aim of the analysis is to asses the influence of angu- lar acceleration, amplitude of the exciting force and the reduced mass on resonance curves during the start-off. The equations of motion of the examined system were solved by means of the Krylov-Bogolubov- Mitropolski method and digital simulation. Key words: friction damper, structural friction, non-linear vibrations, resonance curves, starting-off vibrations 1. Introduction The starting-off and braking phases are important moments in operation of every machine. During these processes, transition through the dangerous re- sonance zone can occur. The fundamental criterion in the design of friction dampers (Figs.1, 2) or their combinations, such as frictional-elastomer ones, is to make them suitable for mechanical systems operating in dynamic con- ditions. Therefore, the appropriate selection of geometric parameters and lo- adings can serve as protection against going into dangerous resonances or it can considerably reduce the resonance amplitude by making use of the na- tural capacity of vibration damping in a given material. To determine such dependencies, a mathematical model has to be created to approximate the 466 Z. Skup real system. Traditional professional literature treats frictional torsion dam- pers, frictional clutches and brakes as joints of rigid bodies. Themicrosliding effect in the elastic range of the material of cooperating elements is called the structural friction. This phenomenon is well known and referred to as the structural hysteresis loop, see Gałkowski (1999), Giergiel (1990), Kaczmarek (2003), Kosior (2005),Mostowicz-Szulewski andNizioł (1992), Osiński (1998), Sanitruk et al. (1997), Sextro (2002), Wang and Chen (1993). More advanced models are developed based mainly on finite elements, see Grudziński and Kostek (2005), Ostachowicz (1989), Zboiński and Ostachowicz (2001). Fig. 1. Simplified model of a friction damper The model of the friction damper adopted for analytical consideration (Fig.1) is composed of cooperating conical surfaces of friction pairs consisting of inner andouter rings (Fig.2).This typeof frictiondampershasbeenapplied in ring buffers and friction dampers used to dissipate energy. Contact defor- mation, friction and damping of vibrations occurring in temporary fastenings and permanent joints have essential influence on dynamical properties of ma- chines and devices.Mathematical description of the phenomenon of structural friction is not easy due to the complexity of the friction process anddifficulties in describing the state of stresses and deformations occurring in the joints of elements. Therefore, the description is based on simplified assumptions and fundamental mechanical laws that apply to the patterns of stress and defor- mations arising in the process of bending, tension, compression, torsion, and shearing, seeGałkowski (1999),Giergiel (1990),GrudzińskiandKostek (2005), Kosior (2005), Osiński (1998), Skup (1998), Zboiński andOstachowicz (2001). The following assumptionsweremade in order to analyse the friction dam- per: the distribution of unit pressurebetween cooperating surfaces in the rings contact joint is uniformandthe friction coefficient of the contacting elements is constant for any value of unit pressure. Friction forces on the contact surfaces of cooperating elements are subject to the Coulomb law, and, consequently, the frictional resistance is proportional to pressure. The material properties are described by Hooke’s law; the friction (kinetic or static) that occurs is Analysis of damping of vibrations... 467 identical in the whole zone and depends on the state of load. Furthemore, flat sections were assumed (cross-sections remain flat after deformation of the elements). Apart from theoretical examination of the model shown in Fig.2, experimental tests were conducted on the real system (Fig.1). Fig. 2. Simplified model of the contact joint of the friction pair in a friction damper Thedenotations used inFig.2 are: r – average radius of the conical surface of the rings, rz, rw – external and internal radii of the conical surface of the ring, β – angle of tilt of the conical surface, u – displacement, α – non- dimensional parameter ranging from 0 to 1, P – axial load, p – unit pressure per unit length, µ – friction coefficient. Laboratory experiments on the test stand have shown that the rotational speed can be described quite precisely by the relation: n(t)= γt2 (parameter determined experimentally). Since the rotational speed is the square function of time, the amplitude of the resisting force is constant. It is assumed that the friction damper has been burdened with a load of constant value Ps as well as a variable component P(t) in form of harmonic excitation by a force having a constant amplitude P0 with uniformly varying angular velocity of the excitation ω(t). Therefore, the overall load Pc takes the following form Pc =Ps+P(t) P(t)=P0 sinθ θ= εt2 2 + ξ ω(t)= dθ dt = εt (1.1) where θ is the phase angle of excitation, ε – angular acceleration at the start- off, ξ – phase shift angle, t – time. The application of theKrylov-Bogolubov-Mitropolski asymptoticmethod, as presented, for instance, in Mitropolskij (1964), Osiński (1979, 1998) and Skup (1998) made it possible to eliminate the function sgn u̇, which is tro- ublesome in numerical simulations. 468 Z. Skup 2. Equation of motion of the examined system The model adopted for analytical considerations is a one-degree-of-freedom mechanical systemwith a nonlinear hysteresis loop of a duplex triangle shape, see the friction damper shown in Fig.1. The differential equation ofmotion of the examined system can bewritten as follows mü+P(u, sgnu̇)=Pc (2.1) where m is the reducedmass of the examined system, u– coordinate of displa- cement of the reducedmass, P(u, sgnu̇) – force in a cycle represented by the structural hysteresis loop (Fig.3) dependent on the relative displacement u, amplitude and sign of the velocity u̇. In order to apply the asymptotic Krylov-Bogolubov-Mitropolski method, it is necessary to transform the function P(u, sgnu̇). Thus, after somemani- pulations, the function P(u, sgn u̇) takes the form P(u, sgnu̇)= ku−νF ( u, du dt ) (2.2) Therefore, differential equation of motion (2.1) assumes a form mü+ku= νF ( u, du dt ) +Pc (2.3) where k = P0/u is the stiffness of compression, ν – small non-dimensional parameter. The solution to non-linear differential equation (2.3), which describesmo- tion of the examined system, is assumed in the first approximation as u=Acos(θ+ ξ) (2.4) By incorporating the asymptotic method (see Mitropolski, 1964; Osiński, 1979), we can obtain a system of differential equations describing the ampli- tude of the angular displacement A and the phase shift ξ as given below dA dt = νA1(t,A,ξ) dξ dt =ω0−ω(t)+νB1(t,A,ξ) (2.5) Therefore, thenatural frequency ω0 of the systemcandescribedby the formula ω0 = √ k m (2.6) Analysis of damping of vibrations... 469 The first and second derivative of (2.4) takes the form du dt = νA1cos(θ+ ξ)−ω0 sin(θ+ ξ)−AνB1 sin(θ+ ξ) (2.7) d2u dt2 = ν [ (ω0−ω) ∂A1 ∂ξ −2Aω0B1 ] cos(θ+ ξ)−Aω20 cos(θ+ ξ)+ −ν [ (ω0−ω)A ∂B1 ∂ξ +2A1ω0 ] sin(θ+ ξ)+ +ν2cos(θ+ ξ) [ A1 ∂A1 ∂A +B1 ∂A1 ∂ξ −AB21 ] + −ν2 sin(θ+ ξ) [ 2A1B1+AA1 ∂B1 ∂A +AB1 ∂B1 ∂ξ ] + +ν [∂A1 ∂t cos(θ+ ξ)−A ∂B1 ∂t sin(θ+ ξ) ] The displacements and deformations of the examinedmodel are small and therefore the solution to differential equation (2.3) is adopted for considera- tions as the first approximation. Taking that into account, the expressions in which the parameter ν appears in the second or a higher power can be neglec- ted. It is necessary toaddthat, dueto the state of equilibriumof the considered system, the expressions at the secular terms cos(θ+ ξ) and sin(θ+ ξ) sho- uld amount to zero (the mechanical system is stable then). In this case, the value of the parameter ν is very small, thus it is neglected in final conside- rations. Finally, equation (2.7)2 takes the following form d2u dt2 = ν [ (ω0−ω) ∂A1 ∂ξ −2Aω0B1 ] cos(θ+ ξ)−Aω20 cos(θ+ ξ)+ (2.8) −ν [ (ω0−ω)A ∂B1 ∂ξ +2A1ω0 ] sin(θ+ ξ) Substituting formulas (2.4) and (2.8) to the left-hand side of equation (2.3), we obtain { m d2u dt2 +ku } u=Acos(θ+ξ) =mν [ (ω0−ω) ∂A1 ∂ξ −2Aω0B1 ] cos(θ+ ξ)+ (2.9) −mν [ (ω0−ω)A ∂B1 ∂ξ +2A1ω0 ] sin(θ+ ξ) 470 Z. Skup After averaging during one period of time of themain harmonic angle (θ+ξ), the right-hand side of equation (2.3) can be presented as follows { νF ( u, du dt ) +P0 sinθ+Ps } u=Acos(θ+ξ) = = ν cos(θ+ ξ) π 2π ∫ 0 F(Acos(θ+ ξ),−Aω0 sin(θ+ ξ))cos(θ+ ξ) d(θ+ξ)+ (2.10) + ν sin(θ+ ξ) π 2π ∫ 0 F(Acos(θ+ ξ),−Aω0 sin(θ+ ξ))sin(θ+ξ) d(θ+ ξ)+ +P0(cosξ sin(θ+ ξ)− sinξcos(θ+ ξ)) Having compared the coefficients at identical powers ν and, relevantly, at sines and cosines of the right-hand sides of equations (2.9) and (2.10), after some transformations of the expressions for A1(t,A,ξ) and B1(t,A,ξ), we obtain a system of equations from which the functions A1 and B1 can be determined A1(t,A,ξ) =− 1 2πmω0 2π ∫ 0 F0(A,χ)sinχdχ− P0cosξ νm[ω0+ω(t)] (2.11) B1(t,A,ξ)=− 1 2πmω0A 2π ∫ 0 F0(A,χ)cosχdχ+ P0 sinξ νmA[ω0+ω(t)] where F0(A,χ)=P(u,u̇)=F(Acosχ,−Aω0 sinχ) χ= θ+ ξ (2.12) Substituting formulae (2.11) into the systemof equations (2.5) andrearranging them, we obtain dA dt =− ν 2πmω0 2π ∫ 0 F0(A,χ)sinχdχ− P0 m(ω0+ω) cosξ (2.13) dξ dt =ω0−ω(t)− ν 2πmω0A 2π ∫ 0 F0(A,χ)cosχdχ+ P0 mA(ω0+ω) sinξ As in works by Mitropolski (1964) – formula 3.53, p.91 and Osiński (1979) – formulae 2.150, 2.151, p.65, the equivalent frequency ω0eq(A) and equivalent Analysis of damping of vibrations... 471 damping coefficient of vibrations heq(A) were introduced into the system of equations (2.13). Thus, this system takes the form dA dt =−heq(A)A− P0 m(ω0+ω) cosξ (2.14) dξ dt =ω0eq(A)−ω(t)+ P0 mA(ω0+ω) sinξ where heq(A)= ν 2πmω0 2π ∫ 0 F0(A,χ)sinχdχ (2.15) ω0eq(A)=ω0− ν 2πmω0A 2π ∫ 0 F0(A,χ)cosχdχ Because of the discontinuity of the function P(u, sgnu̇) at u̇=0, we confine the analysis to one half-period of the vibrating motion (the motion between four stops). Thus, the integration interval (0,2π) will be divided into four sub-intervals. The influence of the elasticity and frictional parameters k1 and k2, which equal tanξ1 and tanξ2 respectively, on the damping properties of the investi- gated system is shown in Fig.3. Fig. 3. Hysteretic loop for the friction pair of the ring spring The tests were conducted on the MTS testing machine at the Institute of Machine Design Fundamentals at Warsaw University of Technology with the use of the Test Ware SX software. The methodology of the experiment consisted in loading the system up to the maximum value Pmax so that not 472 Z. Skup to exceed the safe limit of the material elasticity. Next, the system was unlo- aded down to the pre-assumedvalue Pmin and re-loaded again (Fig.4). Before themeasurement began, both the extensometer and the control-measurement systemwere subject to calibration. Themethod and the results of the investi- gations are described in (Skup, 2007) k1 =tanξ1 = P1 umax k2 =tanξ2 = P2 umax (2.16) Fig. 4. Experimental hysteretic loop: loading force P max =28kN, number of loading and unloading cycles 4 Based on the graph in Fig.3, P1 =P for α=1, whereas P2 =α2P, the maximum displacement umax for the final stage of the load α = 1 (see line OA1 in Fig.3) and non-dimensional parameter α2 can be determined on the basis of the formulae derived by the author (Skup, 2007 – formulae 2.7, p.365, and 2.11, p.367), therefore umax = 2Pr πEF (ctanβ−µ tanβ+µ ) α2 = (cotβ−µ)(tanβ−µ) (tanβ+µ)(cotβ+µ) (2.17) where E is Young’s modulus, F – area of cross-section of the rings. • 1st stage of motion from 0 to π/2, P(u, sgn u̇)= k2u, u̇ < 0, u> 0 • 2nd stage of motion from π/2 to π, P(u, sgnu̇)= k1u, u̇ < 0, u< 0 • 3rd stage of motion from π to 3π/2, P(u, sgn u̇)= k2u, u̇ > 0, u< 0 • 4th stage of motion from 3π/2 to 2π, P(u, sgn u̇)= k1u, u̇ > 0, u> 0 After finding the integrals in system of equations (2.13) and having used above conditions, we obtain Analysis of damping of vibrations... 473 2π ∫ 0 F0(A,χ)sinχ dχ= π/2 ∫ 0 k2usinχdχ+ π ∫ π/2 k1usinχdχ+ + 3π/2 ∫ π k2usinχdχ+ 2π ∫ 3π/2 k1usinχ dχ=A(k2−k1) (2.18) 2π ∫ 0 F0(A,χ)cosχdχ= π/2 ∫ 0 k2ucosχdχ+ π ∫ π/2 k1ucosχ dχ+ + 3π/2 ∫ π k2ucosχ dχ+ 2π ∫ 3π/2 k1ucosχdχ= πA(k1+k2) 2 Substituting (26) into (2.15)1 and (27) into (2.15)2, we obtain heq(A)= νA(k2−k1) 2πmω0 ω0eq(A)=ω0− νπA(k1+k2) 4πmω0A (2.19) Finally, the system of differential equations (2.14), if we make use of (2.19), takes the following form dA dt = νA 2πmω0 (k1−k2)− P0cosξ m(ω0+ω) (2.20) dξ dt =ω0−ω− ν(k1+k2) 4mω0 + P0 sinξ mA(ω0+ω) Finally, in two basic differential equations (2.20) we have an expression for the relative displacement amplitude A (vibration amplitude) as function of the excitation frequency ω (the resonance curves for initial vibrations). The influence of the angular acceleration, the exciting force amplitude and reduced mass on the resonance curves during the start-off of the system has been investigated. 3. Numerical investigations Differential systemof equations (2.20) has been solved bymeans of theRunge- Kuttamethodof the fourth orderwithGill’smodification (theprogram library ofPDPRT11company,SubroutineRKGS). Inorder to compare thenumerical 474 Z. Skup results, theMathematica 4.1 software has been used to work out a comparing program in Borland C++ environment. The results obtained from computer simulations concerning the non- stationary (transient) state of forced vibrations in the friction damper are shown in Figs. 5, 6, 7. Numerical calculations carried out for the above formu- lae incorporated the basic geometrical parameters and material properties of the examined friction damper given in Table 1. Table 1.Data for numerical calculations No. Parameter [unit] Value 1 Loading force P1 =P [N] 28000 2 Non-dimensional parameter α1 1 3 Non-dimensional parameter α2 for β=12 ◦ 0.0280476 4 Non-dimensional parameter α2 for β=14 ◦ 0.0993034 5 Non-dimensional parameter α2 for β=16 ◦ 0.1588123 6 Non-dimensional parameter α2 for β=18 ◦ 0.208914 7 Young’s modulusE [N/mm2] 2.1 ·105 8 Friction coefficient µ 0.20 9 Angle of tilt of cones β [◦] 12, 14, 16, 18 10 External radius of ring rz [mm] 38.5 11 Internal radius of ring rw [mm] 31.5 12 Average radius of conical surface of ring r [mm] 37.25 13 Cross-sectional area of rings F [mm2] 132.94 14 Reducedmass m [kg] 0.443 The basic values of quantities describing the initial, non-stationary state of the system are included in Table 2. Table 2.Results of numerical calculations Angle Force Force Displacement β [◦] P1 [N] P2 [N] umax [mm] 12 28000 785.33 0.260 14 28000 2780.50 0.202 16 28000 4446.74 0.161 18 28000 5849.59 0.130 In technical cases of starting-off or braking conditionsmechanical systems, the frequency of the force which excites system vibration permanently chan- ges. Transition through the resonance zonemay appear in such circumstances Analysis of damping of vibrations... 475 (Fig.5-7). During the initial stage, resonance curves in the pre-resonance (sub- resonant) and, especially, post-resonance (super-resonant) period may differ considerably. The resonance curves show that during the starting-off stage of themechanical system, the post-resonance amplitudes are lower than the pre- resonance ones and they have an oscillating character. The graphs indicate that when the value of angular acceleration ε and the amplitude of excitation force P0 increase, the resonance amplitudesmove towards greater values of the excitation frequencies. In Fig.5, we can see a relation of the vibration ampli- tude A versus excitation frequency ω for different angular accelerations. The resonant amplitudes decrease with the increase of the angular acceleration ε. We can also observe the beating phenomenon before and after the resonance. Shifting of the resonances towards higher frequencies is brought about by the retardation effect (as mentioned in the previous section). The observed shift for increasing values of ε results from the retardation effect of the region of maximum amplitudes with respect to the natural frequency (see Fig.5). It is the known effect of transitory resonance, also observed in linear systems. Fig. 5. Graphs of resonant curves for different values of the angular acceleration ε The analysis of graphs presented in Fig.6 indicates that greater values of P0 imply higher levels of vibration amplitudes and a slight shift of the resonant frequencies toward smaller frequencies. In the case of a deterministic excitation, the existence of resonance curves corresponding to decaying super- resonant vibration confirms the peculiarity of the resonant transition. The influence of the reduced mass m on the resonance vibration ampli- tude A in function of the excitation frequency ω is essential and is shown in Fig.7. The increasing value of the reducedmass leads to a smaller value of the main vibration resonance amplitude as well as dislocation of the amplitude towards smaller values of the excitation frequency (Fig.7). 476 Z. Skup Fig. 6. Graphs of resonant curves for different excitation amplitudes P0 Fig. 7. Graphs of resonant curves for different values of the reducedmass m 4. Concluding remarks Structural friction between the contacting surfaces of internal and external rings of the examined damper increases the damping of vibrations in the exa- mined system. The damping effect is the greatest for an appropriate value of the friction force because the zone of the relative slip between the internal and external rings of the friction damper is the largest. The efficiency of vibration dampingbymeans of the friction damper is greatly influencedby the following factors: exciting force, friction force (unit pressure, friction coefficient), angu- lar acceleration, stiffness of the internal and external rings, reducedmass. The effects of structural friction can be employed to enrich the design methods of dynamical systems. On the basis of the obtained results, it has been found that all resonance curves always start from a non-zero resonance amplitude and tend asympto- Analysis of damping of vibrations... 477 tically to zero in the post-resonance range. The simulations also confirmed a certain peculiarity when the system passes through the resonance. The pe- culiarity, indicated in works byMitropolskij (1964), Osiński (1998) and Skup (1998), has form of a decaying ”beating” phenomenon. It vanishes after some time, and the amplitude rests on a level that is usually smaller than that in the sub-resonant zone. References 1. Gałkowski Z., 1999,Wpływ tarcia konstrukcyjnegona drganiawału z tuleją, Zesz. Nauk. Politechniki Rzeszowskiej, 174, 283-288 2. Giergiel J., 1990,Tłumienie drgań mechanicznych, PWN,Warszawa 3. Grudziński K., Kostek R., 2005, Influence of normal micro-vibrations in contact on sliding motion of solid body, Journal of Theoretical and Applied Mechanics, 43, 37-49 4. Kaczmarek W., 2003, Analysis of a bolted joint with elastic and frictional effects occurring between its elements, Machine Dynamics Problems, 27, 1, 157-167 5. KosiorA., 2005,Wpływparametrówwybranychpołączeń z tarciemkonstruk- cyjnym na właściwości sprężysto-tłumiące układówmechanicznych,Prace Na- ukowe, Mechanika, 209, OficynaWydawnicza PolitechnikiWarszawskiej,War- szawa 6. Mitropolskij Ju.A., 1964,Problemy asimptoticheskoj teorii nestacjonarnykh kolebanii, Izdat. Nauka, Moskwa 7. Mostowicz-Szulewski J., Nizioł J., 1992, Forced steady-state and non- stationary vibrations of a beamwith bilinear hysteresis and hysteresis in fixing, Nonlinear Vibration Problems, 24, 33-62 8. Osiński Z., 1979,Teoria drgań, PWN,Warszawa 9. Osiński Z., 1998, Damping of Vibrations, A.A. Balkema, Rotter- dam/Brookfield 10. Ostachowicz W., 1989, Forced vibrations of a beams including dry friction dampers,An Int. J. Computers and Structures, 22, 851-858 11. Sanitruk K.Y., Imregun M., Ewins D.J., 1997, Harmonic balance vibra- tion analysis of turbine blades with friction dampers, Journal of Vibration and Acoustics, 119, 96-103 12. SextroW., 2002,Dynamical Contact Problems with Friction, Springer,Berlin 478 Z. Skup 13. SkupZ., 1998,Wpływ tarcia konstrukcyjnegowwielotarczowymsprzęgle cier- nym na drgania w układzie napędowym,Prace naukowe, Mechanika, 167, Ofi- cynaWydawnicza PW (Ph. Dsc. disertation) 14. Skup Z., 2007, Theoretical and experimental studies of energy dissipation in a model of a ring spring, Journal of Theoretical and Applied Mechanics, 45, 2, 363-377 15. Wang J.H., Chen W.K., 1993, Investigation of the vibration of a blade with friction damper byHBM, J. of Engineering for Gas Turbines and Power, 115, 294-299 16. Zboiński G., Ostachowicz W., 2001, Three-dimensional elastic and elasto- plastic frictional contact analysis of turbomachinery blade attachments, J. of Theoretical Applied Mechanics, 39, 769-790 Analiza tłumienia drgań poprzez tłumik cierny Streszczenie Wpracy przedstawiono badania tłumienia drgań nieliniowych układumechanicz- nego o jednym stopniu swobody zawierającego amortyzator cierny. Rozważany jest rozruch układu przy wymuszeniu harmonicznym z jednostajnie zmienną częstością o stałej amplitudzie siły wymuszającej. Uwzględniono tłumienie drgań, wykorzystu- jąc zjawisko tarcia konstrukcyjnego (tłumienie pasywne). Zagadnienie rozpatrywane jest przy założeniu równomiernego rozkładu nacisków jednostkowych występujących pomiędzywspółpracującymi stożkowymipowierzchniami ciernymi pierścieni tłumika. Zbadano wpływ przyspieszenia kątowego, amplitudy siły wymuszającej drgania oraz masy zredukowanej na krzywe rezonansowe drgań rozruchowych. Równania ruchu badanego układu mechanicznego rozwiązano wykorzystując asymptotyczną metodę Kryłowa-Bogolubowa-Mitropolskiego i metodę symulacji cyfrowej. Manuscript received July 3, 2009; accepted for print October 27, 2009