tailieunhanh - Harris' Shock and Vibration Handbook Part 3

Tham khảo tài liệu 'harris' shock and vibration handbook part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | NONLINEAR VIBRATION ARGUMENT 0 RAD FIGURE Generalized phase-plane solution of Bessel s equation. Introducing the second of these into Eq. and employing the condition that x0 is also a solution s K 2 1 3 X2X02 S 0 Now an expression for x0 must be obtained assuming a one-term approximation of the form X0 A cos at Eq. becomes d 2s . . - . X Y cos ọ s 0 where K 2 1 WA2 4a2X and 3 K 4a2Y 2at Ọ Equation is known as Mathieu s equation. Mathieu s equation has appeared in this analysis as a variational equation characterizing small deviations from the given periodic motion whose stability is to be investigated thus the stability of the solutions of Mathieu s equation must be given periodic motion is stable if all solutions of the variational equation associated with it tend toward zero for all positive time and unstable if there is at least CHAPTER FOUR FIGURE Stability chart for Mathieu s equation Eq. . one solution which does not tend toward zero. The stability characteristics of Eq. often are represented in a chart as shown in Fig. . From the response diagram of Duffing s equation the out-of-phase motion having the larger amplitude appears to be unstable. This portion of the response diagram Fig. corresponds to unstable motion in the Mathieu stability chart Fig. and the locus of vertical tangents of the response curves considering undamped vibration for simplicity corresponds exactly to the boundaries between stable and unstable regions in the stability chart. Thus the region of interest in the response diagram is described by the free vibration á2 K 2 1 and the locus of vertical tangents 3 k 2ụA2 lA 0 The corresponding curves in the stability chart are taken as those for small positive values of Y and X which have the approximate equations 7 11 - 2X Y - 1 2X Now if Eq. is introduced into Eqs. the resulting equations expanded by the binomial .