PETER W.JASINSKI Graduate Student HOCHONGLEE Adjunct Associate Professor Also Employed a t IBM Corp., Endicott, N Y Mem ASME GEORGE N.SANDOR A L C O A Foundation Professor o f Mechanical Design Chairman, Division o f Machines and Structures Fellow ASME Rensselaer Polytechnic Institute, Troy, N Y Vibrations of Elastic Connecting Rod of a High-Speed Slider-Crank Mechanism1 The research involved in this paper jails into the area of analytical vibrations applied to planar mechanical linkages Specifically, a study of the vibrations, associated with an elastic connecting-bar for a high-speed slider-crank mechanism, is made To simplify the mathematical analysis, the vibrations of an externally viscously damped uniform elastic connecting bar is taken to be hinged at each end {i.e., the moment and displacement are assumed to vanish at each end) The equations governing the vibrations of the elastic bar are derived, a small parameter is found, and the solution is developed as an asymptotic expansion in terms of this small parameter with the aid of the KrylovBogoliubov method of averaging The elastic stability is studied and the steady-state solutions for both the longitudinal and transverse vibrations are found Introduction s, IINCE the kinematics of linkages play an important role in machine design, research on the subject is extensive Some investigators considered elasticity or elastic constraints in linkages [2, 3]2 and others investigated effects of rigid mass inertia [4-18] Although a linkage member may have both rigid mass and flexibility, elasticity and inertia (using harmonic analysis or graphical methods) have generally been treated separately Only for simple mechanisms (such as cam-follower systems) have combined effects been studied [19, 20] Since at high speed a linkage is subjected to its own inertial forces and suffers elastic deformation, the combined effects must be fully investigated Thus, with the speed of machinery constantly increasing, a detailed mathematical investigation of the vibrations of linkages is needed To begin this, one naturally turns to the slider-crank mechanism which is the simplest linkage (Fig 1) For the first step of the mathematical investigation, a model must be chosen which represents the important characteristics of an actual slidercrank mechanism but which lends itself readily to solvability To accomplish this, the elastic connecting bar in Fig is assumed to be hinged at each end (i.e., the moment and displace- ments vanish at each end) These boundary conditions are satisfied exactly by the elastic bar mounted on a rigid slider-crank mechanism in Fig These boundary conditions for the connecting bar (displacement and moment being zero at each end) permit investigation more readily Thus the model consisting of a distributed-mass, externally viscously damped elastic bar with the foregoing boundary conditions is taken as a first approximation for the study of an elastic connecting bar But even this simplified model results in a fairly complicated mathematical representation The equations governing this system, in which both longitudinal and transverse vibrations are considered, are two simultaneous nonlinear periodically time- Based on part of a dissertation by P W Jasinski toward fulfillment of the requirements for the Degree of Doctor of Engineering, Division of Machines and Structures, School of Engineering, Rensselaer Polytechnic Institute, Troy, N Y Numbers in brackets designate References at end of paper Contributed by the Design Engineering Division for publication (without presentation) in the JOUBNAL or ENGINEERING FOB IN- DTJSTBY Manuscript received at ASME Headquarters, June 18, 1970 Paper No 70-DE-C Fig ? M o d e l of a slider-crank linkage with a hinged elastic connecting bar and a rigid crank 836 / MAY 971 Copyright © 1971 by ASME Transactions of the ASME Downloaded From: http://manufacturingscience.asmedigitalcollection.asme.org/ on 09/07/2013 Terms of Use: http://asme.org/terms a la3 - sm cot — sm cot L 6L3 — — sin cot Letting X cot, il = L u (3) L the equations (1) and (2) are written Fig Model of a rigid slider-crank linkage w i t h a mounted elastic bar ITT + ( - COS T vT ~ ( j sm r I v - - — (cos 2r + AE variant partial differential equations with periodic forcing functions These equations are neither readily solvable with the aid of classical methods nor readily reducible to the well-known Hill's (or Mathieu) equation Thus, one is led almost out of necessity to approximate methods among which the KrylovBogoliubov (K-B for short) asymptotic method of averaging is foremost T h e K-B method along with the Galerkin variational method enables the solution of the above stated problem to be written in terms of an asymptotic series once a small parameter is found and the equations are written in standard form [1] The nonlinear term is assumed small and disregarded Also, a small amount of external viscous damping is assumed Introduction of Dimensionless Quantities pAV-co"1 la2 L2 & p^-co la2 a + -COST + - — (COS2T - city 'dl — U —• l + pA + EI ,- v pAZ/co2 mv f a a + —r- vr = -V T sm r + - sm r p^co L L la2 + - - sm 2r ) • \ 7~ I + Reduction to Ordinary Differential Equations a c G0S ° (ui ~ ) (1) vhere d (dy diu'-\'di) v + " v + d* - EI d¥u + Mv— T h e displacement and moment are assumed to vanish at each end of the elastic bar in Fig The functions sin nirri, where n is an integer, satisfy these boundary conditions and thus the Galerkin variational method can be used to reduce the partial differential equations to ordinary differential equations Substituting (T, + pAA vt + (x + u) — + aco cos (cot — ) df ( ! ) ' neglected when equations (4) and (5) are written pA d x (5) are small compared with ( - I and f — J and thus can be Ux d(f> tit — v — — aco sili (cot — r/>) = (4) where - has been taken to be