Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 1, pp. 15-23, Warsaw 2007 THE TRIPLE PENDULUM WITH BARRIERS AND THE PISTON – CONNECTING ROD – CRANCKSHAFT MODEL Jan Awrejcewicz Grzegorz Kudra Department of Automatics and Biomechanics, Technical University of Lodz e-mail: awrejcew@p.lodz.pl Amodel of a flat triple pendulumwith obstacles imposed on its position is used for the modelling of a piston-connecting rod-crankshaft system of a mono-cylinder four-stroke combustion engine. The introduced self- excited system can be only treated as the first step in more advanced modelling of real processes, but some examples of numerical simulations exhibit well known six stages of the piston sliding along the cylinder surface per cycle. Key words: triple pendulum, multibody system, unilateral constraints, impact, piston dynamics 1. Introduction Single degree-of-freedom models are only the first step to understand real behaviour of either natural or engineering systems, usually modelled by a few degrees of freedom.Ontheotherhand, it iswell knownthat impactand friction accompanies almost all real behaviour, leading to non-smooth dynamics. The non-smooth dynamical systems are analysed in both pure (Kunze, 2000) and applied sciences (Brogliato, 1999). The non-classical bifurcations are analysed in systems with dry friction (Leine et al., 2000) and in systems with impacts (Ivanov, 1996; Lenci and Rega, 2000). The scope of the paper contains themodelling of a flat triple physical pen- dulumwith arbitrary situated barriers imposed on the position of the system (includingmodelling of the impact and slidingmotion), numerical schemes for system simulation, methods for stability investigation of the orbit analysis in the case of the non-smooth system (Müller, 1995) and its application in the investigated system in order to studynon-smooth dynamics aswell as classical and non-classical bifurcations (Awrejcewicz et al., 2002, 2004; Kudra, 2002). 16 J. Awrejcewicz, G. Kudra This report is devoted to another goal of the research focused on applica- tions of the investigated system and an example of a piston-connecting rod- crankshaft system of a mono-cylinder combustion engine modelled as an in- verted triple pendulumwith impacts (Kudra, 2002). Themathematical model is described and some numerical simulations are presented and discussed. 2. Mathematical model The special case of a triple pendulum with barriers: the piston – connecting rod – crankshaft system of amono–cylinder combustion engine is presented in Fig.1. Thefirst link of the pendulumrepresents crankshaft (1), the second one is connecting rod (2) and the third one is piston (3).The links (withmasses mi and with moments of inertia Jzi with respect to the principal central axes perpendicular to the plane of motion including points Oi) are connected by rotational joints with viscous damping (with real coefficients ci). The cylinder barrel imposes restrictions on the position of the piston, which moves in the cylinderwith backlash. It is assumed that in the contact zone between surfaces of the piston and the cylinder, a tangent force does not appear. It is assumed that the gas pressure force F(ϕ1) is a function of the angular position ϕ1 of the crankshaft and can be reduced to a force acting along the line parallel to the axis of the system and containing the piston pin axis O3. Moreover, the crankshaft is externally driven by the moment M0 originating from an external power receiver (brake) and acting contrary to the positi- ve sense of the angle ϕ1. We also assume that the rotational speed of the crankshaft is positive. In that way we obtaine a self-excited system. Thenon-dimensional governing equations (if noneof theobstacles is active) are as follows M(φ)φ̈+N(φ)φ̇ 2 +Cφ+p(φ)=fe(φ) (2.1) where φ=      ϕ1 ϕ2 ϕ3      φ̈=      ϕ̈1 ϕ̈2 ϕ̈3      φ̇ 2 =        ϕ̇21 ϕ̇22 ϕ̇23        φ̇=      ϕ̇1 ϕ̇2 ϕ̇3      M(φ)=    1 ν12cos(ϕ1−ϕ2) ν13cos(ϕ1−ϕ3) ν12cos(ϕ1−ϕ2) β2 ν23cos(ϕ2−ϕ3) ν13cos(ϕ1−ϕ3) ν23cos(ϕ2−ϕ3) β3    The triple pendulum with barriers and the piston... 17 Fig. 1. Piston – connecting rod – crankshaft system N(φ)=    0 ν12 sin(ϕ1−ϕ2) ν13 sin(ϕ1−ϕ3) −ν12 sin(ϕ1−ϕ2) 0 ν23 sin(ϕ2−ϕ3) −ν13 sin(ϕ1−ϕ3) −ν23 sin(ϕ2−ϕ3) 0    (2.2) C=α−11 c1    1+c21 −c21 0 −c21 c21+ c31 −c31 0 −c31 c31    fe(φ)=α −2 1    F0p(ϕ1)sinϕ1−M0 λ2F0p(ϕ1)sinϕ2 0    In the above the symbols (·̇) and (̈·) denote respectively the first and second derivatives with respect to the non-dimensional time t (such that the angular velocity ϕ1 in the post-transient motion is approximately equal to 1) and p(ϕ1) is the non-dimensional function of gas pressure distribution. 18 J. Awrejcewicz, G. Kudra The restrictions on the piston position imposed by the cylinder barrel can be described using the following set of non-dimensional inequalities h1(φ)= ∆ 2 − sinϕ1−λ2 sinϕ2−η sinϕ3− δ 2 cosϕ3 ­ 0 h2(φ)= ∆ 2 − sinϕ1−λ2 sinϕ2+(σ−η)sinϕ3− δ 2 cosϕ3 ­ 0 (2.3) h3(φ)= ∆ 2 +sinϕ1+λ2 sinϕ2+η sinϕ3− δ 2 cosϕ3 ­ 0 h4(φ)= ∆ 2 +sinϕ1+λ2 sinϕ2− (σ−η)sinϕ3− δ 2 cosϕ3 ­ 0 where λ2 = l2 l1 η= h l1 σ= s l1 δ= d l1 ∆= D l1 More details concerning described the model of the piston – connecting rod – crankshaft system can be found in work by Kudra (2002). Observe that the proposed dynamical model of the piston – connecting rod – crankshaft system can be treated as a simplifiedmodel, since some very important technological details are neglected. Themost important simplifica- tions are: • tangent forces of interaction between the piston and cylinder surfaces are neglected; • interaction of the piston-cylinder introduced by the piston rings (byme- ans of friction forces in the ring grooves in the direction perpendicular to the cylinder surface) is neglected; • simplified frictionmodel in every joint of the system (i.e. linear damping) is assumed. In addition, the modelling of impact between the piston and the cylinder, where an oil layer exists, requires an approach different from the generalized restitution coefficient rule. 3. Numerical examples In Figures 2-4, exemplary solutions of the piston – connecting rod – crank- shaft system described in the previous section are presented. The gas pressure function p(ϕ1) is developed into the Fourier series with K =25 terms using The triple pendulum with barriers and the piston... 19 Fig. 2. System response for e=0 and D=0.08004m real data presented inwork by Sygniewicz (1999) for a four stroke engine. The maximal gas pressure is choosen as pmax = 3MPa for the rotational crank- shaft speed n = 2000rot/min. The rest of real parameters are as follows: m1 = 10kg, m2 = 1kg, m3 = 0.4kg, Jz1 = 1kgm 2, Jz2 = 0.0075kgm 2, Jz3 = 0.001kgm 2, l1 = 0.04m, l2 = 0.15m, ey1 = 0m, ey2 = 0.12m, ey3 = 0.01m, d = 0.08m, s = 0.08m, h = 0.04m, c1 = 0.00432Nm −1 s, c21 = c2/c1 =0.2, c31 = c3/c1 =0.2, M0 =9.807Nm. 20 J. Awrejcewicz, G. Kudra Fig. 3. System response for e=0.9 and D=0.08004m The calculations are performed for different values of the restitution coeffi- cient e and the external diameter D. The quantities x03 and y03 describe the non-dimensional position of the piston pin axis O3. It is seen inFig.2 that the pistonmoves six times from one side of the cylinder to the second side during one cycle of the engine work, and most of the time the piston adjoins either to one or second side of the cylinder surface. This result confirms the investi- gations presented by Sygniewicz (1999). However, the piston loses its contact The triple pendulum with barriers and the piston... 21 Fig. 4. System response for e=0.9 and D=0.0802m with the cylindermoving from one side to the second side of the cylinder with a small rotation angle. This phenomenon differs from the results presented by Sygniewicz (1999), where it was assumed that the piston does not loose the contact with the cylinder. The crankshaft angular positions at the beginning and end of the phases of the piston adjoining and sliding along the cylinder also differ from the results presented by Sygniewicz (1999) up to 35◦. In the latter case, the exhibited differences follow from negligence of some essential 22 J. Awrejcewicz, G. Kudra technological details mentioned in the previous section. For a larger restitu- tion coefficient (Fig.3) and larger backlash (Fig.4), we also observe tendency of the piston to slide six times along the cylinder per one cycle of the engine work, but since multiple impacts between the piston and the cylinder occur before each of the sliding state, it happens that before the piston stablizes its motion at one cylinder side, it rapidly leaves the contact and transits into the other side of the cylinder. 4. Concluding remarks Thedeveloped generalmodel of the triple physical pendulumwithbarriers can be useful in the modelling of many real processes in nature and engineering. The presented model of the piston – connecting rod – crankshaft system mo- delled as a triple physical pendulumwith impacts (in spite of somedifferences) behaves in a way similar to that described and illustrated in the monograph by Sygniewicz (1999). In particular, six piston movements from one side of the cylinder to its opposite side (during one cycle of the engine work) have been detected. The presentedmodel can be treated as the first step to descri- be the real piston-connecting rod-crankshaft system, and after taking account of some technological details, a better convergence with the real system can be expected. Moreover, the proposed model describes full dynamics of piston motion in the cylinder, and thus it can be very useful for analysis of noise generated by impacts between the piston and cylinder barrel. Acknowledgment This work has been supported by the PolishMinistry of Science andHigher Edu- cation for years 2005-2008 under grant no. 4T07A03128. References 1. Awrejcewicz J., Kudra G., Lamarque C.-H., 2002, Nonlinear dynamics of triple pendulum with impacts, Journal of Technical Physics, 43, 97-112 2. Awrejcewicz J., KudraG., LamarqueC.-H., 2004, Investigation of triple physical pendulumwith impacts using fundamental solutionmatrices, Interna- tional Journal of Bifurcation and Chaos, 14, 12, 4191-4213 3. Brogliato B., 1999,Nonsmooth Mechanics, Springer, London 4. IvanovA.P., 1996,Bifurcations in impact systems,Chaos, Solitons and Frac- tals, 7, 1615-1634 The triple pendulum with barriers and the piston... 23 5. Kudra G., 2002,Analysis of Bifurcations and Chaos in Triple Physical Pen- dulum with Impacts, Ph.D. Thesis, Technical University of Lodz 6. Kunze M., 2000,Non-Smooth Dynamical Systems, Springer-Verlag, Berlin 7. Leine R.I., vanCampen D.H., van deVrandeB.L., 2000, Bifurcations in nonlinear discontinuous systems,Nonlinear Dynamics, 23, 105-164 8. Lenci S., Rega G., 2000, Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos, Solitions and Fractals, 11, 2453-2472 9. Müller P.C., 1995, Calculation of Lyapunov exponents for dynamic systems with discontinuities,Chaos, Solitons and Fractals, 5, 1671-1691 10. Sygniewicz J., 1999, Modeling interaction of a piston with piston rings and a barrel, Scientific Bulletin of Lodz Technical University, 615/149 [in Polish Potrójne wahadło z przeszkodami i model mechanizmu korbowego Streszczenie Model płaskiego potrójnego wahadła fizycznego z przeszkodami ograniczającymi jego ruch został użyty do zamodelowaniamechanizmukorbowego silnika spalinowego. Wprowadzony samowzbudnymodel może być traktowany jedynie jako bardzo zgrab- ne przybliżenie rzeczywistych zjawisk zachodzących w cylindrze silnika spalinowego. Pomimo tego, kilka zaprezentowanychprzykładów symulacji numerycznychwykazuje bardzo dobrą zgodność z danymi doświadczalnymi prezentowanymiw literaturze. Manuscript received August 3, 2006; accepted for print August 18, 2006