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pushrod (the next lowest spring rate) as a exible element The model then takes the shape of Fig 1125a, where the mass of the pushrod has been divided equally between the elements on either side of it, the lifter and the rocker arm (Chen, 1982; Hanachi and Freudenstein, 1986) The parameter values then become m1 = ml + 1 m pr 2 = 0270 lbm + ( 05)0114 lbm = 0327 lbm J m 2 1 2 m pr + 2 ra + mvs rra + cs rra lra , pr 2 3 0132 lbm in 2 0153 lbm 2 (17)2 = 110 lbm + 0251 lbm(17) + 3 (0875 in )2 K1 = K pr = 30 10 5 lb in K2 = Keq = 660 lb in Since this is exactly the same form as the suspension model discussed earlier, the natural frequencies of this two-mass system can be calculated and occur at 67 Hz (as predicted by the one degree of freedom model) and 3400 Hz This suggests two things First, the next natural frequency is signi cantly higher than the rst so there is ample justi cation for using a single-mass model for basic analysis Secondly, while the single mass
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(b) Single mass with damping
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(a) Two-mass model
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FIGURE 1125 Other simple models of the valve-gear system
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model is likely a reasonable model for frequency ranges around 0 200 Hz, it would not be appropriate to use at high frequencies near the resonance at 3400 Hz While the additional complexity of the two-mass system is not required to reproduce the low frequency behavior of the valve-gear system, damping should be added to the model if accurate reproduction of the resonance around 67 Hz is desired This could take the form of an analytical prediction of friction (Pisano, 1984; Hanachi and Freudenstein, 1986) or a simple equivalent damper added in parallel to the equivalent spring (Fig 1125b) and identi ed through experiment, Chen (1982)
REFERENCES
Barkan, P, Calculation of High-Speed Valve Motion with a Flexible Overhead Linkage, Society of Automotive Engineers Transactions, (61): 687 700, 1953 Cannon, RH, Dynamics of Physical Systems, McGraw-Hill, New York, 1967 Chen, FY, Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982 De Silva, CW, Vibration: Fundamentals and Practice, CRC Press, London, 2000 Fenster, SK, and Gould, HH, Mathematics, in Rothbart, HA (ed) Mechanical Design and Systems Handbook, second edition, McGraw-Hill, New York, 1985 Friedland, B, Advanced Control System Design, Prentice-Hall, Englewood Cliffs, NJ, 1996 Garrott, WR, and Monk, MW, Vehicle Inertial Parameters Measured Values and Approximations, Society of Automotive Engineers Paper 881767, 1988 Gillespie, TD, Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, Pa, 1992 Haessig, DA, and B Friedland, On the Modelling and Simulation of Friction, American Society of Mechanical Engineers Journal of Dynamic Systems, Measurement and Control, (113): 354 362, 1991 Hanachi, S, and Freudenstein, F, The Development of a Predictive Model for the Optimization of High-Speed Cam-Follower Systems with Coulomb Damping, Internal Friction and Elastic and Fluidic Elements, American Society of Mechanical Engineers Journal of Mechanisms, Transmissions and Automation in Design, (108): 506 515, 1986 Herbert, RG, and McWhannell, DC, Shape and Frequency Composition of Impulses from an Impact Pair, American Society of Mechanical Engineers Journal of Engineering for Industry, (99): 513 518, 1977 Hunt, KH, and Crossley, FRE, Coef cient of Restitution Interpreted as Damping in Vibroimpact, American Society of Mechanical Engineers Journal of Applied Mechanics, (97): 440 445, 1975 Karnopp, D, Margolis, DL, and Rosenberg, RC, System Dynamics: Modeling and Simulation of Mechatronic Systems, John Wiley & Sons, New York, 2000 Karnopp, D, Computer Simulation of Stick-Slip Friction in Mechanical Dynamic Systems, American Society of Mechanical Engineers Journal of Dynamic Systems, Measurement and Control, (107): 100 103, 1985 Layton, RA, Principles of Analytical System Dynamics, Springer-Verlag, New York, 1998 Meirovich, L, Fundamentals of Vibrations, McGraw-Hill, New York, 2001 Pisano, AP, and Freudenstein, F, An Experimental and Analytical Veri cation of a High-Speed Cam-Follower System, American Society of Mechanical Engineers Journal of Mechanisms, Transmissions and Automation in Design, (105): 692 704, 1983 Pisano, AP, Coulomb Friction in High-Speed Cam-Follower Systems, American Society of Mechanical Engineers Journal of Mechanisms, Transmissions and Automation in Design, (106): 470 474, 1984
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