THÔNG TIN TÀI LIỆU
64 5 Stability of a Rotating Shaft — Oil Whip Item 4 is the subject of this chapter and is a self-excited whirling vibration due to oil film action in journal bearings. Since this may happen over a wide range of speeds above a certain threshold and may become a severe vibration, precautions must be taken. Its frequency is almost constant and is equal to the natural frequency of the shaft. Item 5 is a self-excited whirling vibration due to fluid forces in the seal gap or blade tip gap of turbines and turbocompressors and its frequency is equal to the natural frequency of the shaft. Oil whip can be included in flow-induced vibrations in a wide sense. 5.1 Oil Whip In experiments on a rotating shaft supported by journal bearings, Newkirk and Taylor (1925) found, under certain conditions, a new kind of severe vibration (or whirling) that was different from the vibration at the critical speed [2]. Since the vibration disappeared when the oil supply to the bearings was stopped and it resumed when the oil was supplied again, they concluded that the vibration was caused by the oil film in the bearing, and named it oil whip. Whip means a severe vibration of a shaft. The phenomenon is described in more detail in Fig. 5.1. When the rotating speed of a shaft is gradually increased, a large resonant vibra- tion occurs in the shaft at the critical speed ω 1 . The vibration diminishes, however, when the rotating speed passes the critical speed. Next, when twice the critical speed 2ω 1 is reached, a large vibration (or whirling) will appear as shown in Fig. 5.1a under certain conditions. When the shaft speed is increased further, this vibration will not diminish but it may continue as it was or it may become even larger, unlike resonant vibrations. This is typical of oil whip. Oil whip was simply said to start at twice the critical speed in the early days. Later, however, some examples were reported in which the oil whip onset speed was somewhat or significantly higher than twice the critical speed as shown in Fig. 5.1b. This is often observed when the bearing pressure is high. The features of oil whip are summarized in the following list: 1. When the rotating speed of a shaft is raised from zero, oil whip starts in many cases at twice the critical speed, as shown in Fig. 5.1a, and continues to exist beyond that speed. However, when the journal does not float up easily (for ex- ample when the bearing pressure is high or when the viscosity of the oil is low), oil whip may start at a speed higher than twice the critical speed, as shown in Fig. 5.1b. Even in this case, however, when the rotating speed is lowered, oil whip usually continues to exist down to twice the critical speed, i.e., the oil whip amplitude – shaft speed plot exhibits a hysteresis loop as shown in the figure. This phenomenon is sometimes called the inertia effect in oil whip. 2. The whirling speed (frequency) of a shaft in oil whip is almost constant, and is almost equal to the critical speed of the shaft. 5.1 Oil Whip 65 (a) (b) Fig. 5.1a,b. Oil whip. a typical oil whip, b oil whip with hysteresis 3. The whirling direction of a shaft in oil whip is the same as the rotating direction of the shaft. 4. Oil whip occurs easily when the journal floats up easily (i.e., the bearing pressure is low, or the viscosity of the oil is high). 5. Below twice the critical speed, the shaft may sometimes whirl quietly with little shaft bending. The whirling speed (frequency) of the shaft in this case is propor- tional to the rotating speed and is always equal to one-half of the rotation speed. Such whirling is called oil whirl or half-speed whirl. Most of these features were found experimentally by Newkirk and Taylor and reported in their first paper [2], whereas the case shown in Fig. 5.1b was later reported by Newkirk and Lewis (1956)[9][10], Pinkus (1956) [11], and others. Items 1 and 5 seem to indicate that twice the critical speed and half the rotating speed have special meanings in a journal bearing and that they are related to the nature of oil whip. The explanation for this by Newkirk and Taylor is as follows [2]. As shown in Fig. 5.2, the velocity of the oil in the oil film is equal to zero on the metal surface of the bearing and equal to the circumferential velocity of the journal on the journal surface. If a linear distribution of velocity between the two surfaces is assumed for simplicity, the average velocity of the oil in the oil film is equal to one-half the surface velocity of the journal, and is hence constant, at any section of 66 5 Stability of a Rotating Shaft — Oil Whip Fig. 5.2. Whirling speed of oil whip the film. Therefore, the flow rate of oil at any arbitrary section of the oil film per unit width is proportional to the cross-sectional area. In other words, the volume of the oil that flows in through section A is much larger than the volume of the oil which flows out of section B and so, if incompressibility of the oil is assumed, the journal center must move in such a way that the excessive oil can be accommodated. As a result, the journal center circles around the bearing center. Now, since the difference of the volume of the oil which flows in through section A and that of the oil which flows out of section B per unit width must be equal to the volume swept by the journal in space, we have the following equation: R j ω 2 (c + e) − R j ω 2 (c − e) = (2R j ) e Ω where R j is the radius of the journal, c = R b −R j is the radial clearance of the bearing, e is the eccentricity of the journal center, ω is the rotational speed of the journal, and Ω is the whirling speed of the journal. The above equation gives the whirling speed of the journal center as follows: Ω = ω 2 (5.1) Thus, the journal will whirl in the same direction as that of rotation at an angular velocity of one-half the rotational speed. Therefore, if the rotational speed becomes twice the critical speed, the shaft will whirl just at the critical speed (natural fre- quency), and a large vibration due to resonance will take place. Taylor and Newkirk thought this was the mechanism of oil whip. Based on this idea, they referred to oil whip as oil resonance. As for the fact that large-amplitude resonance continues to exist even after the rotational speed is further increased, they explained that it is because the whirling speed does not increase any further because of the increase of friction due to the increase of amplitude. 5.2 Oil Whip Theory 67 Although their explanation describes the oil film action fairly well, there is a problem in explaining why a large vibration continues beyond twice the critical speed, because they regarded oil whip as a resonant phenomenon. Further, their the- ory cannot explain why in some cases oil whip starts only at a speed somewhat above twice the critical speed, and why in that case the amplitude change depicts a hystere- sis loop. To explain these phenomena reasonably, it is necessary to treat oil whip as a self-excited vibration. 5.2 Oil Whip Theory To explain oil whip theoretically as a self-excited vibration, it is necessary (1) to calculate the oil film force of the bearing, i.e., the force exerted by the oil film on the journal, then (2) to write down the equation of motion of the rotating shaft supported by the oil film force, and then (3) to examine the stability of the shaft (whether the shaft can rotate stably or becomes unstable, leading to a self-excited vibration) by applying a stability criteria to the equation of motion. Robertson (1933) examined stability in this sense for the first time [3]. The oil film force he obtained under the assumption of an infinitely long bearing and Som- merfeld’s boundary condition is as follows, being resolved into a component in the direction of eccentricity and that normal to it: P κ = −12µ R j c 2 LR j π˙κ (1 − κ) 3/2 (5.2) P θ = 12µ R j c 2 LR j πκ(ω − 2 ˙ θ) (2 + κ 2 ) √ 1 − κ 2 (5.3) where µ is the coefficient of viscosity, R j is the bearing radius, c is the radial clearance of the bearing, L is the bearing length, κ is the eccentricity ratio, θ is the attitude angle, and ω is the rotating speed of the shaft. The dots over ˙κ and ˙ θ show time diffrentiation. He examined the shaft stability graphically using these expressions, and concluded that the speed limit of stability is always zero, meaning that the rotating shaft is always unstable, but this is not actually the case. Later, Poritsky(1953) [4] used the same oil film force and examined the stability mathematically and more rigorously and obtained the same result. No papers presented at that time could explain the oil whip phenomenon satisfactorily. In calculating the oil film force, Hori (1955)[6, 7](1958)[14](1959)[15, 16] (1) used G ¨ umbel’s (or the half-Sommerfeld) boundary condition instead of Sommer- feld’s boundary condition, which had been used until then, and, in examining the stability, (2) divided the shaft vibrations into a small vibration and a large vibration and obtained their stability limits separately, and (3) combined them to explain the process of the occurrence of oil whip shown in Fig. 5.1. Later, more precise calculations under various conditions became possible with the development of computers. For example, Someya (1963)[17] (1964)[19] (1965) [23] performed many detailed numerical computations of this kind of problem in 68 5 Stability of a Rotating Shaft — Oil Whip finite length bearings. He calculated not only the stability but the loci of journals and rotors. He also carried out calculations on rotors with imbalances. Gotoda (1963) [18](1964) [20] also performed detailed numerical computations for the oil film characteristics and the stability in the case of finite length bearings. Funakawa and Tatara (1964) [22] calculated the oil film characteristics and stability using the short bearing approximation. Nakagawa and Aoki (1965) [24] obtained an approximate analytical solution for a finite length bearing under a certain assump- tion, and performed similar calculations. Harada and Aoki (1971) [31] studied the stability of a shaft supported by turbu- lent journal bearings. Ono and Tamura (1968) [28] studied rotor stability using the boundary condition Section 3.1.4(e). The major points of oil whip theories are explained here mainly following Hori’s papers. 5.2.1 Oil Film Pressure First, the dynamic Reynolds’ equation, which takes the journal motion into consid- eration, is solved for the dynamic oil film pressure. This pressure is then integrated over the journal surface to obtain the dynamic oil film force that acts on the journal. In so doing, it is assumed that the journal motion is sufficiently slow, considering shaft vibrations near the stability limit. Fig. 5.3. Oil film and oil film force If an infinitely long bearing is assumed in Fig. 5.3, the dynamic Reynolds’ equa- tion for the shaded part of the oil film can be written as follows: 5.2 Oil Whip Theory 69 1 R j 2 d dψ h 3 dp dψ = 6µU 1 R j dh dψ + 12µ ∂h ∂t (5.4) where R j is the radius of the journal and µ is the coefficient of vicosity. Further, h is the oil film thickness, U is the circumferential velocity of the journal, ∂h/∂t is time change of oil film thickness due to the journal motion, and these can be written as follows: h = c 1 + κ cos φ ,φ= ψ − θ (5.5) U = R j ω (5.6) ∂h ∂t = c ∂κ ∂t cos φ + κ ∂θ ∂t sin φ (5.7) where c is the radial clearance, κ is the eccentricity ratio, and ω is the rotational speed of the shaft. As the boundary condition for oil film pressure, G ¨ umbel’s condition is used, as stated before, assuming sufficiently slow motion of the journal, i.e., it is assumed that: p = 0atφ = 0 and φ = π (5.8) and, in the area where the calculated oil film pressure turns out to be negative, the pressure is replaced by zero. Integration of Eq. 5.4 twice under the above conditions gives the pressure distri- bution p(φ) as follows [16]: p(φ) 6µω(R j /c) 2 = J 2 (φ) − J 2 (π) J 3 (π) J 3 (φ) + J 3 s (φ) − J 3 s (π) J 3 (π) J 3 (φ) 2˙κ ω − J 3 c (φ) − J 3 c (π) J 3 (π) J 3 (φ) 2κ ˙ θ ω (5.9) where J k (φ) = φ 0 dφ (1 + κ cos φ) k , J k c (φ) = φ 0 cos φ dφ (1 + κ cos φ) k (5.10) J k s (φ) = φ 0 sin φ dφ (1 + κ cos φ) k (5.11) These integrals can be calculated by using Sommerfeld’s variable conversion as shown in Chapter 3, but here an alternative method by Okazaki shown below is used [5] [16]. The following integral is considered first. This can be found in handbooks of integral formulas: J 1α (φ) = φ 0 dφ α + κ cos φ = 2 √ α 2 − κ 2 tan −1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ α − κ α + κ sin φ 1 + cos φ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 70 5 Stability of a Rotating Shaft — Oil Whip From the above formula, the following two integrals are easily obtained by differen- tiation under an integral sign with respect to α: J 2α (φ) = φ 0 dφ (α + κ cos φ) 2 = − ∂J 1α ∂α = 1 α 2 − κ 2 αJ 1α − κ sin φ α + κ cos φ J 3α (φ) = φ 0 dφ (α + κ cos φ) 3 = − 1 2 ∂J 2α ∂α = 1 2(α 2 − κ 2 ) 2 (2α 2 + κ 2 )J 1α − 3ακ sin φ α + κ cos φ − (α 2 − κ 2 )κ sin φ (α + κ cos φ) 2 Letting α = 1 in the above integrals yields the following three integrals J k (φ): J 1 (φ) = φ 0 dφ 1 + κ cos φ = 2 √ 1 − κ 2 tan −1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 − κ 1 + κ sin φ 1 + cos φ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (5.12) J 2 (φ) = φ 0 dφ (1 + κ cos φ) 2 = 1 1 − κ 2 J 1 (φ) − κ sin φ 1 + κ cos φ (5.13) J 3 (φ) = φ 0 dφ (α + κ cos φ) 3 = 1 2(1 − κ 2 ) 2 × (2 + κ 2 )J 1 (φ) − 3κ sin φ 1 + κ cos φ − (1 − κ 2 )κ sin φ (1 + κ cos φ) 2 (5.14) From the above results, the following integrals J k c (φ) are easily obtained: J 1 c (φ) = φ 0 cos φ 1 + κ cos φ dφ = 1 κ φ − J 1 (φ) (5.15) J 2 c (φ) = φ 0 cos φ (1 + κ cos φ) 2 dφ = 1 κ J 1 (φ) − J 2 (φ) (5.16) J 3 c (φ) = φ 0 cos φ (1 + κ cos φ) 3 dφ = 1 κ J 2 (φ) − J 3 (φ) (5.17) It is easy to calculate the following integrals J k s (φ): J 1 s (φ) = φ 0 sin φ 1 + κ cos φ dφ = − 1 κ ln 1 + κ cos φ 1 + κ (5.18) J 2 s (φ) = φ 0 sin φ (1 + κ cos φ) 2 dφ = 1 1 + κ 1 − cos φ 1 + κ cos φ (5.19) J 3 s (φ) = φ 0 sin φ (1 + κ cos φ) 3 dφ = 1 2κ 1 (1 + κ cos φ) 2 − 1 (1 + κ) 2 (5.20) The following constants can be easily obtained from the above results: J 2 (π) J 3 (π) = 2(1 −κ 2 ) 2 + κ 2 , J 3 c (π) J 3 (π) = − 3κ 2 + κ 2 , J 3 s (π) J 3 (π) = 4(1 − κ 2 ) 1/2 π(2 + κ 2 ) (5.21) All the terms in Eq. 5.9 have thus been determined. Therefore, the pressure dis- tribution is now available. 5.2 Oil Whip Theory 71 5.2.2 Oil Film Force Multiplying the oil film pressure p(φ) of the preceding section by cos φ and sin φ, and integrating them in the range of φ = 0–π as shown below yield the dynamic oil film force under G ¨ umbel’s boundary condition. L is the bearing length and R j is the journal radius: P κ = LR j π 0 p(φ) cos φ dφ, P θ = LR j π 0 p(φ) sin φ dφ (5.22) After some calculations that are much more complicated than those for Sommerfeld’s boundary condition, the following results are obtained: P κ = −6µ R j c 2 LR j 2κ 2 (ω − 2 ˙ θ) (2 + κ 2 )(1 − κ 2 ) + 2˙κ (1 − κ 2 ) 3/2 π 2 − 8 π(2 + κ 2 ) (5.23) P θ = 6µ R j c 2 LR j πκ(ω − 2 ˙ θ) (2 + κ 2 )(1 − κ 2 ) 1/2 + 4κ˙κ (2 + κ 2 )(1 − κ 2 ) (5.24) where the dots over ˙ θ and ˙κ show time differentiation, and hence ˙ θ is equal to the whirling speed of the shaft (Ω). From a comparison of Eqs. 5.2 and 5.3 with Eqs. 5.23 and 5.24, the following can be seen. For ˙κ = 0, whereas Eq. 5.2 gives P κ = 0, Eq. 5.23 generally gives a P κ of finite value. The term (ω − 2 ˙ θ) is not included in Eq. 5.2, but it is included in Eq. 5.23. Further, whereas ˙κ is not included in Eq. 5.3, it is included in Eq. 5.24. These facts have important meanings in terms of bearing characteristics. If the time derivatives are set to zero in Eqs. 5.23 and 5.24, they will become expressions for the oil film force in a stationary state, and are naturally the same as the expressions for the oil film force obtained under G ¨ umbel’s condition in Chap- ter 3. Therefore, the equilibrium position of the journal center is also the same as that obtained previously. The stability of small vibrations is to be considered in the neighborhood of this equilibrium point. Further, it is important that ω and ˙ θ are in the expressions only in the combined form (ω −2 ˙ θ). This shows that no oil film force acts on the journal when it steadily whirls at a speed of half its rotating speed, and this is in agreement with the considerations of Newkirk and others in the previous section. The oil film force in the case of the short bearing approximation can also be cal- culated by a similar method, and the results are given in the following form, similar to the above expressions [22]. These are useful because short bearings have been more and more frequently used in recent years: P κ = − 1 2 µ R j c 2 L 3 R j 2κ 2 (ω − 2 ˙ θ) (1 − κ 2 ) 2 + π˙κ(1 + 2κ 2 ) (1 − κ 2 ) 5/2 (5.25) P θ = 1 2 µ R j c 2 L 3 R j πκ(ω − 2 ˙ θ) 2(1 − κ 2 ) 3/2 + 4κ˙κ (1 − κ 2 ) 2 (5.26) 72 5 Stability of a Rotating Shaft — Oil Whip The oil film force of a finite length bearing can also be written in the following form, similar to those of an infinitely long bearing and a short bearing [20]: P κ = − 6µ R j c 2 LR j (ω − 2 ˙ θ)P (1) κ + ˙κP (2) κ (5.27) P θ = 6µ R j c 2 LR j (ω − 2 ˙ θ)P (1) θ + ˙κP (2) θ (5.28) In this case, P (1) κ , P (2) κ , P (1) θ , and P (2) θ are functions of κ with the bearing dimensions as parameters. These are usually calculated numerically. 5.2.3 Linearization of the Oil Film Force To discuss the linear stability of a shaft, the oil film force is linearized beforehand in the neighborhood of the equilibrium point of the journal center O j0 (κ 0 ,θ 0 ). Fig. 5.4. Rectangular coordinates (radial direction, circumferential direction) Equations 5.23 and 5.24 are used to express the oil film force. Then, variable substitutions κ ⇒ κ 0 + κ, θ ⇒ θ 0 + θ are made in these expressions, and assumptions that new variables κ, θ, and their time derivatives ˙κ and ˙ θ are so small that their second power, third power, and products can be disregarded give the linearized oil film force P κ and P θ as follows: P κ = − 6µ R j c 2 LR j × 2κ 0 2 ω (2 + κ 0 2 )(1 − κ 0 2 ) 1 + κ 2 κ 0 − 2κ 0 2 + κ 0 2 + 2κ 0 1 − κ 0 2 5.2 Oil Whip Theory 73 − 4κ 0 2 ˙ θ (2 + κ 0 2 )(1 − κ 0 2 ) + 2˙κ (1 − κ 0 2 ) 3/2 π 2 − 8 π(2 + κ 0 2 ) (5.29) P θ =+6µ R j c 2 LR j × ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ πκ 0 ω (2 + κ 0 2 ) 1 − κ 0 2 1 + κ 1 κ 0 − 2κ 0 2 + κ 0 2 + κ 0 1 − κ 0 2 − 2πκ 0 ˙ θ (2 + κ 0 2 ) 1 − κ 0 2 + 4κ 0 ˙κ (2 + κ 0 2 )(1 − κ 0 2 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (5.30) Now, to consider the journal motion in the rectangular coordinate system (x, y) shown in Fig. 5.4, let us transform the above components of the oil film force to the rectangular components P x and P y using the following expressions: P x = P κ − P θ0 y j cκ 0 , P y = P θ + P κ0 y j cκ 0 (5.31) where P θ0 and P κ0 are the stationary values of the oil film force at the equilibrium point, i.e., the constant terms of Eqs. 5.29 and 5.30. Now, let us perform the variable transformation κ ⇒ x j /c, θ ⇒ y j /cκ 0 ,˙κ j ⇒ ˙x j /c, and ˙ θ j ⇒ ˙y j /cκ 0 , and assume that x j , y j ,˙x j , and ˙y j are sufficiently small, considering small vibrations. Then the oil film forces P x and P y can be written as follows: P x = P x0 + K xx P 0 c x j + K xy P 0 c y j + C xx P 0 ωc ˙x j + C xy P 0 ωc ˙y j (5.32) P y = P y0 + K yx P 0 c x j + K yy P 0 c y j + C yx P 0 ωc ˙x j + C yy P 0 ωc ˙y j (5.33) where P x0 and P y0 in the above equations are the stationary values of the oil film force at the equilibrium point, and are given as follows: P x0 = − 6µ R j c 2 R j L 2κ 0 2 ω (2 + κ 0 2 )(1 − κ 0 2 ) (5.34) P y0 =+6µ R j c 2 R j L πκ 0 ω (2 + κ 0 2 ) 1 − κ 0 2 (5.35) P 0 is their resultant P 0 = P 2 x0 + P 2 y0 and is given as follows: P 0 = 6µ R j c 2 R j L ω κ 0 π 2 − (π 2 − 4)κ 0 2 (2 + κ 0 2 )(1 − κ 0 2 ) (5.36) where c is the radial clearance of the bearing and ω is the angular velocity of the rotating shaft. The coefficients K xx , ···, C xx , ···are nondimensional coefficients, and, as shown below, are given as functions of κ 0 only. This is important: [...]... sufficient if the real part of all roots of the characteristic equation Eq 5. 57 are 5. 2 Oil Whip Theory 79 negative, and the Routh–Hurwitz criterion is known as a criterion for this In the case of Eq 5. 57, it can be written as follows: A0 , A1 , A2 , A3 , A4 , A1 A3 A5 A0 A2 A4 0 A1 A3 0 A0 A2 0 0 A1 A5 , A6 > 0 0 0 A6 0 A5 0 > 0 A4 A6 A3 A5 A1 A3 A5 A0 A2 A4 > 0 0 A1 A3 (5. 58) (5. 59) (5. 60) In other words,... coordinates of the journal center (x j , y j ) from the equation of motion Eqs 5. 47 and 5. 48 Thus, substitution of Eqs 5. 49 and 5. 50 into Eqs 5. 47 and 5. 48, respectively, use of the oil film force for small vibrations Eqs 5. 32 and 5. 33, and additional use of Eq 5. 51 give the following linearized form of Eqs 5. 47 and 5. 48 on the coordinates of the disk center (x, y) only: P1 P P P x + K xx 1 − 1 x + C... xx Kyy − K xy Kyx B4 = K xx + Kyy When K xx , · · · C xx , · · · are given by Eqs 5. 37 – 5. 42, the actual calculations show that Eq 5. 58 of the Routh–Hurwitz criterion always holds, and if Eq 5. 59 holds, Eq 5. 60 always holds Therefore, only Eq 5. 59 need be considered as a stability condition If the determinant of Eq 5. 59 is developed, and the coefficients A0 , A1 , · · · are substituted into it, a comparatively... xy , Cyy = 2 − K xy (5. 39) (5. 40) (5. 41) (5. 42) Fig 5. 5 Nondimensional spring constants and nondimensional damping constants of an oil film 5. 2 Oil Whip Theory 75 Equations 5. 32 and 5. 33 can be written in an easier form to grasp as follows: P x = P x0 + k xx x j + k xy y j + c xx x j + c xy y j ˙ ˙ (5. 43) Py = Py0 + kyx x j + kyy y j + cyx x j + cyy y j ˙ ˙ (5. 44) Or, in a matrix representation: Px P... + Kyy 1 y = 0 ¨ ˙ +Cyy kωc kc mωc mc C xx (5. 52) (5. 53) The stability of the rotor can be investigated by solving these equations simultaneously Stability Criterion The above simultaneous equations can be written in the following general form: A x + B x + C x + Dx + E y + F y + G˙ + Hy = 0 ¨ ˙ ¨ y ¨ ˙ a x +b x+c x+d x+e y + f y+gy+hy=0 ¨ ˙ (5. 54) (5. 55) When solutions of the form x = αe st and... with its eccentricity from 5. 2 Oil Whip Theory 77 Fig 5. 7 Coordinates of the system (radial and circumferential direction) the bearing center The situation is shown in Fig 5. 8a,ba In this case, the oil film force Eqs 5. 23 and 5. 24 can be approximated by the linear expressions of Eqs 5. 32 and 5. 33 in the neighborhood of the equilibrium point of the journal, as stated before Fig 5. 8a,b Small vibrations (a)... the case of Eqs 5. 52 and 5. 53, the coefficients of the characteristic equation Eq 5. 57, A0 , A1 , · · · will be as follows: P1 2 1 ω2 ω1 4 mc P1 2 P1 1 1 = B1 − B2 4 mc 2 mc ωω1 ωω1 P1 2 1 1 P1 2 1 P1 = 2B0 2 2 + B3 4 − B4 2 +1 ω ω1 mc ω1 mc ω1 mc P1 2 1 1 P1 = 2B1 − B2 ω mc ωω1 2 mc P1 1 P1 2 1 P1 2 = B0 2 + 2B3 2 − B4 mc ω mc ω1 mc P1 2 1 P1 2 = B1 , A6 = B3 ω mc mc A0 = B0 A1 A2 A3 A4 A5 where B0 , B1... criterion: 80 5 Stability of a Rotating Shaft — Oil Whip 1 P1 ω2 mc > K1 (κ0 ) K2 (κ0 ) + 1 P1 ω1 2 mc (5. 61) where K1 (κ0 ) and K2 (κ0 ) are given as follows: K1 (κ0 ) = B1 2 − B1 B2 B4 + B2 2 B3 , B0 B2 2 K2 (κ0 ) = B2 B1 (5. 62) Thus, finally, the shaft will be stable if Eq 5. 61 is satisfied Fig 5. 9 Stability chart for an infinitely long bearing [14] [ 15] Since K1 and K2 are functions of κ0 only, Eq 5. 61 can... x ≡ 0 and y ≡ 0: As3 + Bs2 + Cs + D, Es3 + F s2 + Gs + H =0 a s3 + b s2 + c s + d, e s3 + f s2 + g s + h (5. 56) This is called the characteristic equation, and it will become the sixth-order equation below, if the determinant is developed: A0 s6 + A1 s5 + A2 s4 + A3 s3 + A4 s2 + A5 s + A6 = 0 (5. 57) where it is assumed that A0 > 0 If A0 < 0, then the sign of the whole equation will be changed so that...74 5 Stability of a Rotating Shaft — Oil Whip K xx = − K xy = − C xx = − C xy = + Kyx = + Kyy = − 4κ0 π2 − (π2 − 4)κ0 2 κ0 κ0 1 − + κ0 2 + κ0 2 1 − κ0 2 π 1 − κ0 2 (5. 38) κ0 π2 − (π2 − 4)κ0 2 2(2 + κ0 2 ) κ0 1 − κ0 2 π2 − (π2 − 4)κ0 2 4 8 π − 2 π(2 + κ0 2 ) π2 − (π2 − 4)κ0 2 π 1 − κ0 2 π2 − 1 C xy , 2 (π2 − 4)κ0 2 (5. 37) 1 2κ0 κ0 − + κ0 2 + κ0 2 1 − κ0 2 Cyx = C xy , Cyy = 2 − K xy (5. 39) (5. 40) (5. 41) . Thus, substitution of Eqs. 5. 49 and 5. 50 into Eqs. 5. 47 and 5. 48, respectively, use of the oil film force for small vibrations Eqs. 5. 32 and 5. 33, and additional use of Eq. 5. 51 give the fol- lowing. given by Eqs. 5. 37 – 5. 42, the actual calculations show that Eq. 5. 58 of the Routh–Hurwitz criterion always holds, and if Eq. 5. 59 holds, Eq. 5. 60 always holds. Therefore, only Eq. 5. 59 need be. A 3 , A 4 , A 5 , A 6 > 0 (5. 58) A 1 A 3 A 5 00 A 0 A 2 A 4 A 6 0 0 A 1 A 3 A 5 0 0 A 0 A 2 A 4 A 6 00A 1 A 3 A 5 > 0 (5. 59) A 1 A 3 A 5 A 0 A 2 A 4 0
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