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Rotating Machinery Vibration 2011 Part 7

Tham khảo tài liệu 'rotating machinery vibration 2011 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 130 Chapter 4 Substituting the components of Eqs. 7 into Eqs. 5 and the results then into Eq. 6 yields the following. r f ỳ Xcos f x ysin 0V i -Xsin f x ycos ớ eíw . . 8 f X cos ex - y sin ỹy i X sin ex ycos A c r t is thus expressed in terms of two rotating vectors as follows. r t Riei ol I1 R2e-i uJ - Rị I V X cos 0V y sin ớ 2 X sin 0V y cos Ớ 2 9 R2 I V X cos 0V Y sin ớ 2 X sin Qx y cos ớv 2 X sin 3X y cos 0V Pi arctan X cos 9X y sin 0V Ị IX sin 0X y cos Ớ p2 arctan a _ V a Á X cos 9X Y sin ớ Ị Equation 9 shows the elliptical orbit decomposed into two synchronously rotating vectors one corotational of radius Rị and the other counter-rotational of radius R2 both with angular speed magnitude of co Fig. 6 . At t 0 these two vectors are positioned relative to the v-axis by their respective angles pt and p2. It is then Figure 6 Elliptical orbit as the sum of two counterprecessing circular orbits. RDA Code for Lateral Rotor Vibration Analysis 131 clear as Figure 6 illustrates at t 0 that the angle V from the A-axis to the major ellipse axis is the average of these two angles as follows. 10 When Rị R2 their vector sum produces forward whirl and conversely when R R2 their vector sum produces backward whirl. The orbit is a straight line when R R2. Furthermore the semimajor axis ỉ and semiminor axis of the orbit ellipse are given by the following expressions. b I 7 1 I I R21 fl II 7 1 I I z 2 II 11 All the results developed here for the orbit ellipse properties in terms of the A and y harmonic displacement signals are applicable for steady-state imbalance response signals as well as for instability threshold modal orbits. Once a steady-state response is computed the task of visually presenting the results depends upon how much detail the user requires. A multi-DOF version of Fig. 4 Chapter 1 with plots of amplitudes and phase angles at selected rotor stations as functions of speed is often all that may be needed. However to appreciate the potentially complex contortions .