RU2578500C1 - Method of mounting gas turbine engine rotor - Google Patents

Abstract

FIELD: machine building.

SUBSTANCE: invention relates to rotary turbine machines and can be used in assembly of their rotors, including high-speed gas-turbine engines, in which the critical frequencies of rotor rotation are in working frequency range. According to the method of mounting the rotor shaft by means of rolling bearings is installed with possibility of rotation in supports, whereupon for mounting the rotor shaft in one of the supports roller bearing is used with oval running track of its ring connected by structural elements with engine stator, and installation of the bearing in support is performed in a way so that small axis of ring oval running tread coincides with direction of the rotor gravity, support stiffness and parameters of bearing ring oval tread are set, where rotor resonance at critical frequencies of its rotation is eliminated.

EFFECT: technical result is higher engine efficiency obtained by increasing its reliability and service life due to reduction or complete damping of rotor oscillations in almost all range of rotation frequencies, including due to possibility of using elastic damper supports during rotor installation.

1 cl, 1 dwg

Description

The invention relates to rotary gas turbine machines and can be used in the installation of their rotors, including high-speed gas turbine engines, in which the critical rotational speeds of the rotors are in the operating frequency range.

It is known from the prior art that one of the main problems of reliable operation of gas turbine engines is the elimination of the resonance of their rotors by bending vibration at critical rotation frequencies, when the forces leading to the deflection of the rotor shaft (centrifugal forces from the unbalanced mass of the rotor and gyroscopic moments of the rotor disks), become equal to the forces of elastic resistance of the shaft, as a result of which its lateral displacements due to an increase in the bending of the shaft significantly increase.

Elimination of resonance of rotors at critical frequencies of their rotation is achieved in various ways, for example, by removing critical frequencies (detuning rotors) from the working range of rotor rotation.

So, for example, there are known methods for changing the critical rotational frequencies of rotors at which they lose stability (resonance) due to a change in design: mass, geometric characteristics, stiffnesses of shafts and bearings (see Birger I.A. Shorr B.F., Iosilevich G. .B. Strength analysis of machine parts. Moscow, "Mechanical Engineering", 1979, 431 pp., 433 pp.).

These methods of detuning from resonance in the bending form of oscillations for high-speed engines in the presence of critical rotational frequencies in the operating range are extremely difficult to implement. This is mainly due to the lack of the ability to carry out structural changes on existing engines.

There is a method of mounting a rotor of a gas turbine engine, according to which it is installed in rolling bearings, and one of its bearings uses a roller bearing with an oval treadmill of the bearing ring connected by power elements to the engine stator, and the bearing is mounted on a support in such a way that a large the oval axis of the treadmill of the ring coincides with the direction of gravity of the rotor, the rigidity of the support is selected from the condition C> 13.3 · δω _k ² / µ _min , and the oval parameter Δ is set in the range from the minimum and to the maximum value.

The minimum value is determined from the condition

,

,

- коэффициент, равный

; f_K и f_C - критическая частота вращения ротора и собственная частота колебаний ротора; ω_k - круговая критическая скорость вращения ротора; L - пролет ротора; С - жесткость опоры; Δ_min - минимальный параметр овала дорожки качения кольца подшипника, Δ_min=(D_max-D_min)/2; µ_min - минимальный параметр овала дорожки качения кольца подшипника, µ_min=(D_max-D_min); D_max и D_min - максимальный и минимальный диаметр овальной дорожки качения; m - масса ротора, приходящаяся на опору с овальной дорожкой качения; g - ускорение свободного падения; Т - время оборота ротора.where δ is the static imbalance of the rotor per support with an oval raceway; J _O - the moment of inertia of the rotor during its angular displacement relative to the support with circular bearing raceways; β is a coefficient equal to β = f _K / f _C ;

- coefficient equal to

; f _K and f _C - critical rotor speed and natural frequency of the rotor; ω _k is the critical circular rotor speed; L is the span of the rotor; C is the stiffness of the support; Δ _min - the minimum parameter of the oval raceway of the bearing ring, Δ _min = (D _max -D _min ) / 2; µ _min - the minimum parameter of the oval raceway of the bearing ring, µ _min = (D _max -D _min ); D _max and D _min - the maximum and minimum diameter of the oval raceway; m is the mass of the rotor per support with an oval raceway; g is the acceleration of gravity; T is the time of revolution of the rotor.

и оно равно µ_max=L²(12,896·δ-13,132·m·g/ω_K ²)/J_O, где µ_max - максимальный параметр овала дорожки качения кольца подшипника, µ_max=(D_mах-D_min) (см. патент РФ №2528789, кл. F02C 7/06, F01D 25/16, 2014 г.) - наиболее близкий аналог.The maximum value of the oval parameter is also determined by this dependence only with β = 1.35 and

and it is equal to µ _max = L ² (12,896 · δ-13,132 · m · g / ω _K ² ) / J _O , where µ _max is the maximum parameter of the oval of the raceway of the bearing ring, µ _max = (D _max -D _min ) ( see RF patent No. 2528789, class F02C 7/06, F01D 25/16, 2014) - the closest analogue.

As a result of the analysis of the known method, it should be noted that during its implementation, during the operation of the engine during rotation of the rotor, the circular movement of the centrifugal force vector from its unbalanced mass in the areas of the bearing ring with an oval raceway, where the radius vector of its contour changes in the direction of the force and this increment exceeds the radial movement of the rotor shaft on the support, leading to the fact that the reaction of the support will be absent.

The complete absence of the reaction of the support for a time equal to a quarter of the period of natural oscillations of the rotor will lead to the fact that the deflection of the shaft will be eliminated, and this phenomenon will prevent loss of stability of the rotor at a critical speed.

Elimination of shaft deflection during an oval racetrack of the bearing ring will occur in two quarters of the shaft rotation, where the radius vector of the ring contour increases, and decreases for the inner ring.

In the other two quarters, the reaction of the support will be restored, and during this period the rotor will be in a mode that is prone to loss of stability, but since this process will be extremely short-term, the loss of stability of the rotor will not occur.

However, when mounting the rotor shaft in a bearing with an oval raceway of the ring, in which the major axis of the track coincides with the direction of gravity of the rotor, the resonance of the rotor is eliminated at a critical speed of rotation only with sufficiently rigid supports, which does not allow the use of flexible elastic damping supports with low stiffness due to restrictions on the possibility of using this method, and the use of which is very promising, since their use allows damping vibrations of the rotors on almost any excitation frequencies, which increases the stability of the engine.

The technical result of the present invention is to increase the efficiency of the engine by increasing its reliability and service life by reducing or completely damping the vibrations of its rotor in almost the entire range of rotational speeds, including by making it possible to use elastic dampers when mounting the rotor.

и до максимального значения µ_max=L²(12.896·δ-2.882·m·g/ω_K ²)/J_O, где δ - статический дисбаланс ротора, приходящийся на опору с овальной дорожкой качения; J_O - момент инерции ротора при его угловом перемещении относительно опоры с круговыми дорожками качения подшипника; λ - параметр, равный λ=1,234·β²+0,761·β⁴+0,118β; η - параметр, равный

; β - коэффициент, равный β=f_K/f_C;

- коэффициент, равный

; f_K и f_C - критическая частота вращения ротора и собственная частота колебаний ротора; ω_k - круговая критическая скорость вращения ротора; L - пролет ротора; С - жесткость опоры; µ - параметр овала дорожки качения кольца подшипника, µ=(D_max-D_min); D_max и D_min - максимальный и минимальный диаметр овальной дорожки качения; m - масса ротора, приходящаяся на опору с овальной дорожкой качения; g - ускорение свободного падения.The specified technical result is ensured by the fact that in the method of mounting the rotor of a gas turbine engine, according to which the rotor shaft by means of rolling bearings is mounted to rotate in bearings, moreover, a roller bearing with an oval racetrack of its ring connected by power elements is used to mount the rotor shaft in one of the bearings with the motor stator, and the installation of this bearing in the support is carried out in such a way that one of the axes of the oval of the treadmill of the ring coincides with the direction of gravity p of the torus, and the rigidity of the support and the oval parameter of the treadmill of the bearing ring are set, at which the resonance of the rotor is eliminated at the critical frequencies of its rotation, it is new that the bearing is installed from its oval running road ring in the support in such a way that with the direction of gravity the minor axis of the treadmill oval coincides, the stiffness of the support is selected from the condition C> 13.3 · (δω _k ² -mg) / µ _min , and the oval parameter µ, equal to the difference between the maximum and minimum diameters, is set in the range from the minimum values

and up to the maximum value µ _max = L ² (12.896 · δ-2.882 · m · g / ω _K ² ) / J _O , where δ is the static imbalance of the rotor falling on the support with an oval raceway; J _O - moment of inertia of the rotor at its angular displacement relative to the support with a circular bearing race; λ is a parameter equal to λ = 1.234 · β ² + 0.761 · β ⁴ + 0.118β; η is a parameter equal to

; β is a coefficient equal to β = f _K / f _C ;

- coefficient equal to

; f _K and f _C - critical rotor speed and natural frequency of the rotor; ω _k is the critical circular rotor speed; L is the span of the rotor; C is the stiffness of the support; µ is the parameter of the oval raceway of the bearing ring, µ = (D _max -D _min ); D _max and D _min - maximum and minimum diameter of the oval raceway; m is the mass of the rotor per support with an oval raceway; g is the acceleration of gravity.

The essence of the claimed method is illustrated by the drawing, which shows a diagram of the power effects of the rotor when it rotates on a support with an oval raceway of the bearing ring, the quadrants are numbered 1-4.

The claimed method is as follows.

When mounting the engine rotor, its shaft is installed in supports made in the engine stator. To ensure rotation of the rotor relative to the stator, the installation of the rotor in the bearings is carried out by means of rolling bearings. According to the claimed method, for mounting the rotor in the bearings, a roller bearing with an oval treadmill of the outer ring connected by power elements to the motor stator is mounted in one of the bearings. To ensure the achievement of the above technical result, it is important to install a bearing with an oval treadmill of a given oval parameter in a strictly defined way, as well as to select the rigidity of the support, which ensures the elimination of the resonance of the rotor at critical frequencies of its rotation.

To suppress the resonance of the rotor at a critical frequency of its rotation, the installation of the bearing with an oval treadmill of its ring in the support is set in such a way that the minor axis of the treadmill coincides with the direction of gravity of the rotor.

Very important to achieve the specified technical result is the determination of the stiffness of the support (C), which must meet the following condition: C> 13.3 · (δω _k ² -mg) / µ _min , where ω _cr is the critical circular rotational speed of the rotor; µ _min - the minimum value of the oval parameter of the raceway of the bearing ring, at which the technical result is realized, µ _min = (D _max -D _min ); D _max and D _min - respectively the maximum and minimum diameter of the oval raceway; δ is the static imbalance of the rotor per support with an oval raceway; m is the mass of the rotor per support with an oval raceway; g is the acceleration of gravity.

As studies have shown, with a lower value of stiffness, the frequency range of the rotor rotations is significantly reduced, at which vibration damping is ensured.

, а максимальное - равенству µ_max=L²(12.896·δ-2.882·m·g/ω_K ²)/J_O, где δ - статический дисбаланс ротора, приходящийся на опору с овальной дорожкой качения; J_O - момент инерции ротора при его угловом перемещении относительно опоры с круговыми дорожками качения подшипника; λ - параметр, равный λ=1,234·β²+0,761·β⁴+0,118β; η - параметр, равный

; β - коэффициент, равный β=f_K/f_C;

- коэффициент, равный

; f_K и f_C - критическая частота вращения ротора и собственная частота колебаний ротора; ω_k - круговая критическая скорость вращения ротора; L - пролет ротора; µ - параметр овала дорожки качения кольца подшипника, µ=(D_max-D_min); D_max и D_min - максимальный и минимальный диаметр овальной дорожки качения; m - масса ротора, приходящаяся на опору с овальной дорожкой качения; g - ускорение свободного падения.When conducting research it was found that the parameter of the oval of the bearing track should be in a certain range from its minimum to maximum value, and these values should correspond to the following dependencies. The minimum value of the oval parameter must correspond to the equality

and the maximum - to the equality µ _max = L ² (12.896 · δ-2.882 · m · g / ω _K ² ) / J _O , where δ is the static imbalance of the rotor falling on the support with an oval raceway; J _O - the moment of inertia of the rotor during its angular displacement relative to the support with circular bearing raceways; λ is a parameter equal to λ = 1.234 · β ² + 0.761 · β ⁴ + 0.118β; η is a parameter equal to

; β is a coefficient equal to β = f _K / f _C ;

- coefficient equal to

; f _K and f _C - critical rotor speed and natural frequency of the rotor; ω _k - circular critical rotor speed; L is the span of the rotor; µ is the parameter of the oval raceway of the bearing ring, µ = (D _max -D _min ); D _max and D _min - the maximum and minimum diameter of the oval raceway; m is the mass of the rotor per support with an oval raceway; g is the acceleration of gravity.

When the stiffness value of the support is lower than the specified value, the frequency range of the rotor rotation frequency is substantially reduced, at which vibration damping is ensured, and if it is larger, the rotor shaft random movement in the support is observed, which impairs its operation at critical frequencies.

The implementation of the above set of features during installation of the rotor contributes to the fact that during the operation of the engine the circular movement of the centrifugal force vector from the unbalanced mass of the rotor in the ring zones, where the radius vector of the contour of the outer ring of the bearing increases, the support reaction will be absent and the shaft will straighten.

The absence of impact on the rotor shaft of the reaction of the support for a time equal to a quarter of the period of the natural oscillations of the rotor leads to the fact that the deflection of the shaft is eliminated, which prevents the loss of stability of the rotor at a critical speed. As studies have shown, to prevent loss of rotor stability it is enough to ensure that the deflection is eliminated when the shaft rotates in one quarter of its revolution.

When elastic supports with a certain flexibility are used for mounting the rotor (inverse stiffness) during the operation of the engine at point A under the action of centrifugal force from the unbalanced mass of the rotor, the elastic displacement of the support occurs. With a circular rotation of the rotor and its centrifugal force from point A for a period of time Δt, the shaft contact point goes to the unshifted ring contour and the support reaction is eliminated (see drawing).

The elastic displacement of the support on the minor axis of the oval is defined as the ratio of the centrifugal force of the rotor on the support δω _K ² from its imbalance δ, reduced by the value of the own weight of the rotor per support k - stiffness of the support "C"

.

.

The oval ring contour equation is described by a trigonometric function

,

,

where R _O is the radius of the bearing ring along the minor axis of the oval; Δ _min is the oval parameter; ω _k is the circular rotational speed of the rotor at the critical frequency.

Representing the deviation of the oval contour of the raceway of the outer ring from the radius on the smaller axis of the oval in the form Δ _min sin (ω _k t), expanding it into a Taylor series taking into account the small value of ω _K Δt, we obtain sin (ω _k t) = ω _k t.

To fulfill the condition of small argument when eliminating the reaction on the support and the condition when the limiting angle of the rotor shaft landing on the support is not overcome, we take ω _k Δt = 0.15 and equate the elastic indentation of the shaft on the minor axis to the deviation of the oval contour at the reaction elimination point

.

.

From the last expression we obtain the minimum value of the stiffness of the support to fulfill the condition of smallness of the argument when the rotor shaft enters the contour of the undeformable support

,

,

where µ _min is the minimum value of the oval parameter, µ _min = 2Δ _min = (D _max -D _min ).

With this arrangement of the oval of the raceway, the elastic displacement of the support from the centrifugal forces is reduced due to the weight of the rotor, which is directed in the opposite direction relative to the centrifugal force at the time of elimination of the reaction on the support, which just allows the use of supports with less rigidity.

, и хаотического движения вала ротора на опоре происходить не будет.Subject to the rigidity of this condition, the landing of the rotor shaft on the support until the shaft is completely straightened with an oval treadmill of the outer ring will be realized with an acceptable angle of removal from point B (see drawing) not exceeding the critical

, and chaotic motion of the rotor shaft on the support will not occur.

The minimum allowable stiffness of the support is related to the oval parameter of the bearing treadmill. This is due to the fact that with a larger parameter of the oval µ, the magnitude of the elastic displacement of the support at the initial moment of eliminating the reaction on the support of point A (see drawing) is allowed more, and therefore, the stiffness of the support can be lower.

Let us determine the minimum value of the oval parameter of the treadmill of the outer ring of the bearing, taking into account the suppleness of the support at which the shaft deflection and rotor resonance are eliminated. The centrifugal force at point A (see drawing) is compensated by the reaction of the support. In a circular motion of the centrifugal force, the rotor shaft, after a time interval Δt, enters the contour of the undeformable support, the reaction of the support disappears, and the shaft is under the action of only centrifugal force.

From this moment, the radial movement of the shaft on the support with an oval raceway begins to increase. Radial displacements from centrifugal force on a flexible support will be the same as for an absolutely rigid support only with an increase in the time before the rotor shaft fits onto the support by Δt.

The landing of the rotor shaft with the minimum parameter of the oval on absolutely rigid supports, which ensures the elimination of the deflection of the shaft for a quarter of the period of free oscillations of the rotor, occurs at an angle of landing equal to

,

,

where β is a parameter that determines the landing of the rotor shaft, provided that the deflection of the shaft is completely eliminated, β = fк / fc; fк - critical rotor speed at which it loses stability; fc is the natural frequency of oscillation of the rotor on two supports.

The landing of the rotor shaft on elastically flexible supports with a minimum oval parameter occurs when the parameter β is increased by β ′, determined from the condition α = 0.15 = β ′ · π / 2

.

.

The movement of the rotor shaft when the small axis of the oval coincides with the direction of gravity of the rotor from static and dynamic imbalance, as well as the movement of the rotor shaft from the speed that it received during the time Δt before landing on the support, will be the same as when the major axis of the oval coincides with the direction of gravity of the rotor (see RF patent No. 2528789, CL F02C 7/06, F01D 25/16, 2014).

(См. Ю.Б. Назаренко. Устранение резонанса на критической частоте вращения роторов при эллиптической траектории вращения оси вала на опоре // Авиационно-космическая техника и технология. - Харьков: ХАИ. - 2013. №10(107), 63 с.), и оно составитThe movement of the rotor shaft from its own weight is set for the time determined from the moment the reaction is eliminated and until the rotor shaft is mounted on the support. This movement can be determined with sufficient accuracy in the time interval from t ₁ = 0 to

(See Yu.B. Nazarenko. Eliminating resonance at the critical rotational speed of the rotors with an elliptical trajectory of rotation of the shaft axis on the support // Aerospace Engineering and Technology. - Kharkov: KhAI. - 2013. No. 10 (107), 63 p. ), and it will be

,

,

where m is the mass of the rotor per support with an oval raceway

,

,

; β - коэффициент, равный β=f_K/f_C; f_K и f_C - критическая частота вращения ротора и собственная частота колебаний ротора;

- коэффициент, равный

; ω_k - круговая критическая скорость вращения ротора; L - пролет ротора, g - ускорение свободного падения.m _i and x _i - mass and coordinate of the disk relative to the support with circular raceways; J _O - the total moment of inertia of the rotor when it oscillates relative to the support with circular raceways,

; β - factor equal to β = f _K / f _C; f _K and f _C - critical rotor speed and natural frequency of the rotor;

- coefficient equal to

; ω _k is the critical circular rotor speed; L is the span of the rotor, g is the acceleration of gravity.

The minimum value of the oval parameter at which resonance is eliminated, determined taking into account centrifugal forces, dead weight, and taking into account the displacement from the speed obtained at the time the reaction was eliminated, will be

,

,

where Δ _min is the minimum parameter of the oval, Δ _min = (D _max -D _min ) / 2; δ is the static imbalance of the rotor falling on the support with an oval raceway.

The minimum value of the oval parameter µ, equal to the difference between the maximum D _max and minimum D _min diameter

,

,

.where λ is the parameter determined from the expression λ = (1.234 · β ² + 0.761 · β ⁴ + 0.118β); η is a parameter equal to

.

If the oval parameter is less than the specified minimum value, then, as studies have shown, complete straightening of the shaft over a quarter of the period of the natural frequency of the oscillations of the rotor will not occur and the resonance will not be eliminated.

, при котором работа ротора будет стабильной, а при большем значении может наступить хаотичное его движение (см. патент РФ №2528789, кл. F02C 7/06, F01D 25/16, 2014 г.)The maximum value of the oval parameter Δ _max is determined at a larger value of the parameter β than when landing at point E equal to β = 1.35 and

in which the rotor operation will be stable, and with a larger value its chaotic movement can occur (see RF patent No. 2528789, CL F02C 7/06, F01D 25/16, 2014)

,

,

The static imbalance of the rotor on a support with an oval raceway is determined on a balancing installation with a technological bearing with circular rings with the diameter of the treadmill of the outer ring equal to the minimum diameter of the oval raceway.

With a larger value of the maximum value of the oval parameter, the shaft landing on the support occurs at a critical angle of distance from point B (see drawing), and there occurs a random movement of the rotor shaft on the support, which further impairs the rotor operation at a critical frequency.

The claimed method will be more clear from the example below.

An example of the implementation of the method is carried out on a model representing a hollow steel rod simulating a shaft on which three steel disks simulating a rotor spaced along its length are fixed. The rod by means of rolling bearings is mounted on bearings, in one of which a roller bearing with an oval raceway is installed.

After mounting the rod in the supports, two disks are located at a distance L / 4 from the supports and one in the middle of the hollow rod. The length of the rod is L = 0.5 m, its outer and inner diameters are 80 mm and 60 mm. The masses of each disk (m ₁ , m ₂ , m ₃ ) are the same and equal to 10.2 kg, the diameter of the disk is D = 300 mm and its thickness is t = 20 mm. The imbalance of the simulated rotor per support with an oval raceway is equal to δ = 6 g · cm.

Given the raw data received natural frequency oscillations of the rod with the disks on two supports constituted f _C = 291,4 Hz, while the critical shaft speed -. 315.6 Hz (See Nazarenko YB, AY Potapov Remedy. the resonance of the rotor of a turbojet engine at a critical frequency of its rotation using ovalization of the raceway of the fixed ring of the roller bearing // Engine. No. 1, 2012. - P. 17).

To eliminate resonance at a critical rotor speed of ω _k = 1983 rad / s with parameter β = 1.08, which determines the angle of the shaft on the support, the minimum oval parameter is µ _min = 18 μm. With a smaller oval parameter, the rod deflection and resonance of the simulated rotor were not completely eliminated.

The maximum value of the oval parameter on the support is µ _max = 57 μm. With a larger oval parameter, chaotic motion of the rotor shaft in the support was observed, which impaired the operation of the simulated rotor at critical frequencies.

The minimum allowable stiffness of the support was set, which was C> C _min = 3.1 · 10 ⁷ N / m.

The critical rotor speeds were: ω _K = 1876.6 rad / s (minimum allowable) and ω _k = 1983 rad / s (actual).

With the above-mentioned parameters, the oscillations of the simulated rotor during its rotation were practically not observed.

A similar pattern occurred with the oval raceway of the stationary inner ring, which was mounted on a support. When the angular displacement of the centrifugal force vector from the unbalanced mass of the rotor in the areas of the bearing ring in areas with a decrease in the radius of the raceway of the inner ring and with an increase in the displacement of the contour to the axis of rotation exceeding the radial movement of the rotor shaft on the support in the same direction, the reaction to the shaft from the support side was absent. For a time equal to a quarter of the period of natural oscillations of the rotor, the rotor shaft straightened. In relation to graphic materials, if the centrifugal force vector from the unbalanced mass of the rotor is directed to the fourth quadrant in the direction from the center (drawing), then the rotor shaft will be planted on the inner ring in the third quadrant. In this case, the gravity of the rotor reduces the radial movement of the rotor, and the oval parameters were the same as for a stationary outer ring when the rotor shaft was planted in the second quadrant.

The application of this method ensures guaranteed full straightening of the shaft in one of the quadrants over a quarter of the natural period of the rotor’s vibrations and, thereby, prevents the resonance of the rotor at the critical frequencies of its rotation, including when using elastic-damping supports.

Claims (1)

A method of mounting a rotor of a gas turbine engine, according to which the rotor shaft by means of rolling bearings is mounted to rotate in bearings, moreover, to install the rotor shaft in one of the bearings, a roller bearing with an oval racetrack of its ring connected by power elements to the engine stator is used, and the installation of this bearing in the support, it is carried out in such a way that one of the axes of the oval of the treadmill of the ring coincides with the direction of gravity of the rotor, and the rigidity of the support and the parameter of the oval of the treadmill are set bearing rings, at which the rotor resonance is eliminated at critical frequencies of its rotation, characterized in that the bearing is mounted with its oval running road ring in the support in such a way that the small axis of the treadmill coincides with the direction of gravity of the rotor, the support stiffness is selected from the condition C> 13.3 · (δω _k ² -mg) / µ _min , and the oval parameter µ is set in the range from the minimum value $${\mu }_{\min }=2L^2\cdot (\lambda \cdot \delta -\eta \cdot m\cdot g/{\omega }_K{}^2)/[\sin (\pi \cdot \overline{\beta }/2)\cdot J_O]$$

and to a maximum value μ _max = L ² (12.896 · δ-2.882 · m · g / ω K ²⁾ / J _O, where δ - static rotor imbalance attributable to bearing raceway with an oval; J _O - the moment of inertia of the rotor during its angular displacement relative to the support with circular bearing raceways; λ is a parameter equal to λ = 1.234 · β ² + 0.761 · β ⁴ + 0.118β; η is a parameter equal to $$\eta =-0.637\cdot {\overline{\beta }}^5+1.244{\overline{\beta }}^{\mathrm{four}}-1.42{\overline{\beta }}^3+2.016\cdot {\overline{\beta }}^2-0.161\cdot \overline{\beta }$$

; β - factor equal to β = f _K / f _C; $$\overline{\beta }$$

- coefficient equal to $$\overline{\beta }=\beta +\mathrm{0,095}$$

; f _K and f _C - critical rotor speed and natural frequency of the rotor; ω _k is the critical circular rotor speed; L is the span of the rotor; C is the stiffness of the support; µ is the parameter of the oval raceway of the bearing ring, µ = (D _max -D _min ); D _max and D _mjn - the maximum and minimum diameter of the oval raceway; m is the mass of the rotor per support with an oval raceway; g is the acceleration of gravity.

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