Journal of Mechanical Science and Technology 22 (2008) 2163~2170 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-008-0749-2 Journal of Mechanical Science and Technology Tolerance effects on natural frequencies of multibody systems undergoing constant rotational motion † Seung Man Eom 1 , Bum Suk Kim 2 and Hong Hee Yoo 2 1 Korea Institute of Aerospace Technology, Daejeon, Korea, 305-811 2 School of Mechanical Engineering, Hanyang University, Seoul, Korea, 133-791 (Manuscript Received February 18, 2008; Revised May 9, 2008; Accepted July 31, 2008) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Abstract A general multi-body formulation to analyze the tolerance effects on the statistical property variations of natural fre- quencies of multi-body systems undergoing constant rotational motion is proposed in this paper. To obtain the toler- ance effects, Monte-Carlo simulation method is conventionally employed. However, the Monte-Carlo simulation has serious drawbacks; spending too much computation time for the simulation and achieving very slow convergence around some dynamically unstable regions. To resolve such problems, a method employing analytical sensitivity in- formation is suggested in this paper. To obtain the sensitivities of natural frequencies the eigenvalue problem should be differentiated with respect to a design variable. The sensitivities of mass and stiffness matrices should be calculated at the dynamic equilibrium. By employing the sensitivities of natural frequencies along with the tolerance of the design variable, the statistical property variations of the natural frequencies can be calculated. Keywords: Tolerance; Natural frequency; Dynamic equilibrium; Sensitivity analysis; Multi-body system; Statistical property ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ 1. Introduction In a state of dynamic equilibrium of a multi-body system, some of the generalized coordinates remain constant while the others vary with time. The dy- namic equilibrium states often occur in multi-body systems undergoing constant rotational motion. Such systems can be found in several rotating systems such as a governor mechanism and turbo-machineries- machinery. A formulation to obtain the dynamic equi- librium position of a multi-body system undergoing rotational motion was presented by Choi et al. [1]. The modal characteristics of a multi-body system in dynamic equilibrium differ from those of the same system in static equilibrium. Methods to find the mo- dal characteristics of a multi-body system in static and dynamic equilibriums were proposed by Sohoni and Whitesell [2] and Choi et al. [3], respectively. To design a multi-body system, the static and the dy- namic equilibriums and the corresponding modal characteristics at the equilibrium position need to be found effectively as well as accurately. Very often, engineers also need to find the tolerance effects of some design parameters on the statistical property variations of modal characteristics of multi-body sys- tems. Such information is often crucial for the robust design of the a mechanical system. The effects of various manufacturing tolerances and errors on the motion errors of mechanical systems have been studied by many previous engineers. Hartenberg and Denavit [4] first addressed the issue of mechanical errors in linkages. They estimated the mechanical errors based on the maximum allowable tolerances of the link lengths in four-bar linkages. Their approach employed a deterministic method and offered a “worst case” analysis of tolerance. Garrett †This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim * Corresponding author. Tel.: +82 2 2220 0446, Fax.: +82 2 2293 5070 E-mail address: hhyoo@hanyang.ac.kr © KSME & Springer 2008