Study on the Critical Rotational Speed of Hydro-Generator Shaft Influenced by Sliding Bearing Stiffness Variation

Study on the Critical Rotational Speed of Hydro-Generator Shaft Influenced by Sliding Bearing Stiffness Variation

Yong Jae Ri, Sung Hyon Ryang, Chol Song Han, Sung Il Kong

Institute of Mechanics, State Academy of Sciences, Pyongyang, DPRK

Song Hun Kim

Institute of Electric Power Information, Ministry of Electric Power Industry, Pyongyang, DPRK

Abstract – The dynamic modeling of the main shaft system of a hydro turbine generator is carried out by considering the combined effects of unbalanced mass force, magnetic pull force, hydrodynamic force and oil film force of the sliding bearing, its critical rotational speed is obtained, the effect of the sliding bearing stiffness at each journal on the critical rotational speed of the whole shaft system is evaluated, and the method is proposed to improve the stability of the hydro turbine generator by changing it, and the validity is demonstrated by the field test results.

Keywords Hydro-turbine generator · Dynamic modeling · Critical rotational speed · Sliding bearing stiffness

1. INTRODUCTION

   The determination of the critical rotational speed of a hydro-turbine generator under the combined action of mechanical, magnetic and hydraulic forces and the correct evaluation of the influence of these factors are important issues in hydro-turbine generator stability studies.

   The nonlinear coupled vibration characteristics of a four-DOF system are investigated by numerical methods under the influence of mechanical forces, such as bearing looseness, misalignment, static unbalance, and oil-bearing blockage, in a vertical rotor-shaft-support system that models the rotor system of a hydro-turbine generator [1, 2]. The influence of various factors on the vibration response of the system in a four-DOF hydro turbine generator shaft system is analyzed by considering the unbalanced magnetic pull force and the unbalance force in the hydro-dynamic system composed of the generator rotor, hydro-turbine, shaft and support bearings [3-7].

   But there are still no papers that studied effect of sliding bearing stiffness on the critical rotational speed of the hydro turbine generator shaft system.

   Technical Editor:

   speed by considering the combined action of mass unbalance force, magnetic pull force, hydrodynamic force and oil film force of the sliding bearing, and shows the method to achieve the vibration stability of the system by evaluating the effect of sliding bearing stiffness on the critical rotational speed of the system.

2. DYNAMIC MODELING OF THE HYDRO-TURBINE GENERATOR SPINDLE SYSTEM

   The hydro-turbine generator spindle system considered in this paper consists of a generator rotor, a turbine, and three sliding bearings supporting the shaft and the shaft connecting them (Fig. 1).

   Take the coordinate system as shown in Fig. 1 and neglect the torsional or axial vibration of the hydro- turbine generator shaft system and consider only the bending vibration.

   oi (i 1,2,3, 4,5) denote the center of the upper bearing 1, generator rotor 2, lower bearing 3, turbine bearing 4 , and turbine 5, respectively, an – d

   Ri, Kim, Ryang, Han and Kong have contributed equally to this

   fx1 , f y1 , fx3 , f y3 , fx4 , f y4 are the oil film components

   work.

   of the bearings, px , py is the component of the

   * Yong Jae Ri, Song Hun Kim 2 2

   1 Institute of Mechanics, State Academy of Sciences, Pyongyang,

   Democratic Peoples Republic of Korea

   magnetic tensile force,

   px5

   , py5 is the component of

   2 Institute of Electric Power Information, Ministry of Electric Power

   Industry, Pyongyang, Democratic Peoples Republic of Korea

   Therefore, in this paper we investigate the dynamic

   the hydrodynamic excitation force,

   mass of each slide bearing, m2 , m5

   the rotor and the turbine.

   m1, m3 , m4 is the is the mass of

   modeling of the hydraulic turbine generator spindle system and the method of finding the critical rotational

   The free vibration equation of the system is expressed as [1, 2] [Mr ]{Xr }[rGr Cr ]{Xr }[Kr ]{Xr } 0.

   (1)

   12EI1

   6EI1

   12EI1

   6EI1 0

   l3 l2 l3 l2

   In Eq. (1),

   Xr is the degree of freedom vector, is the

   1 1 1 1

   6EI1

   4EI1

   6EI1

   2EI1 0

   velocity vector, and is the velocity vector.

   l2 l1 l2 l1

    

   1 1

   {Xr } [X T YT ]T ,

   Kx

   12EI1

   l3

   6EI1

   l2

   12EI1

   l3

   12EI2

   l3

   6EI1

   l 2

   6EI2

   l 2

   12EI2 ,

   l3

   T 11

   1 1 1 2 1 2 2

   {X} [x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 ] ,

   6EI1

   2EI1

   6EI1

   6EI2

   4EI1

   4EI2

   6EI2

   T l2 l1

   l2 l 2

   l1 l2

   l 2

   {Y} [ y

   y

   y

   y

   y ] .

   1 1 2 2

   1 x1 2

   x2 3

   x3 4

   x4 5 x5

   0 0

   12EI2

   6EI2

   12EI2 12EI3

   l3

   l2 l3

   l3

   The displacement of vibrations in the x and y

   axes and the rotational angles around t-he x and y

   0 0

   12EI2

   2 2 2 3

   6EI2 12EI2 12EI3

   axes and of the system elements are

   xi , yi , xi , yi .

   l3

   l 2 l3

   l3

   2 2 2 3

   0 0 6EI2 2EI2 6EI2 6EI3

   l 2 l

   l2 l 2

   (a) (b)

   Kx

   2 2 2 3

   12EI3 ,

   21 0 0 0 0

   0 0 0 0

   l

    

   3

    

   3

    

   6EI3

   l

    

   2

   3

   0 0 0 0 0

   0		0			0		0
   0		0			0		0
   6EI2 l2		0			0		0
   2EI2 l2		0			0		0
   6EI2 l 2		6EI3 l 2		12EI3 l3		12EI3 l3	0

    

   0

   0

   0

    

    

   x

    

   =

    

   ,

    

   2

    

   K 12

   0

    

   Fig. 1 Mechanical model of the hydro-turbine generator

   shaft system.

   (a) Fundamental Structure, (b) Dynamic Model

   0

   2 3 3 3

    

   3

    

    

   4EI2 4EI3

   6EI3

   2EI3

   0 0

   Mr is the mass matrix of the system, r is t-he

   l2 l3 2 l3

   l

    

   angular velocity of the shaft system, G is t-he gyro

   6EI3 12EI3 12EI4 6EI3 6EI4 12EI4 6EI4

   r l2

   l3 l3

   l2 l2

   l3 l 2

   scopic matrix,

   Cr is the damping matrix, Kr

   is the

   3 3 4 3 4 4 4

   2EI3

   6EI3 6EI4

   4EI3 4EI4 6EI4 2EI4

   x

    

   stiffness matrix, and the detailed form is as follows:

   K =

   22 l

   2 2

   l l 2 l ,

   3 l3 l4 3 4

   l4 4

   4

    

   M 0

   0 G

   12EI4

   6EI4

   12EI4

   6EI

   x x 0

   Mr 0

   M , Gr G

   0 ,

   l3

   l 2 l3

   l 2

   y y

   4 4 4 4

   0 6EI4

   2EI4

   6EI4

   4EI4

   l2

   l4 l2

   l4

    

   Cx

   Cr 0

   0

   C

    

   , Kr

   y

   Kx 0

    

   K

    

   0

    

   ,

   y

   Kx

   K =

   4 4

   Kx

   .

   (2)

   11 12

   x Kx Kx

   M M , K K , G G , C C ,

   21 22

   x y x y x y x y

   Mx diag[mi Jdi ], Gx diag[0 J pi ],

   Where E is the Youngs modulus of the shaft

   material, Ii is the axial moment of inertia of the cross

   Cx diag[cei cdi ], i 1, 2,3, 4,5.

   section of shaft i , and li is the length of shaft link Ei .

   Where,

   Jdi , J pi

   are the axial and polar moments of

   Under assumption of elastic support, the sliding bearing support condition can be expressed through the

   inertia of mass i , respectively, and Kx

   matrices:

   is the following

   generalized force.

   Qh

   cxx cxy x j

   kxx kxy x j

   Linearizing the unbalanced magnetic traction force re

   1d .

   Qh cyx cyy y j kyx kyy y j

   (3)

   presented by Eq. (4) with the oil film force of the sliding b earing, we can add to the stiffness matrix

   2d

   Transferring this generalized force to the left-hand side of Eq. (1) and synthesizing it into the corresponding elements of the stiffness matrix and damping matrix, it is superimposed on the rows and columns corresponding to the bearing positions in the original stiffness matrix or damping matrix.

   The unbalanced magnetic pull force caused by various causes, such as non-concentric rotor-stator coupling, rotor or stator pole imperfections, and shaft initial bending, is expressed by the following equation, considering that the pole pair number p of the hydro-

   turbine generator is generally greater than 3 [ 3-5 ].

   Kr , and the unbalanced force of water in the turbine repr esented by Eq. (6) is also added to the stiffness matrix.

3. SIMULATION CALCULATION OF THE CRITICAL ROTATIONAL SPEED AND THE EFFECT OF STIFFNESS
   1. Calculation Method of Critical Rotational Speed

      From Eq. (1), the free vibration equation of the syste

      r j j

       

      Fx _ ump Rr L k 2 I 2

      2

      cos

      m is [8]

       

      . [Mr ]{xr } [rGr Cr ]{xr } [Kr ]{xr } 0.

      (7)

      Fy_ ump

      40

      0 1 1 2 2 3

      sin

      (4)

      Since the mass matrix, damping matrix and stiffness matrix in Eq. (7) are not all diagonal, the general method

      Where, 0 is the magnetic induction coefficient in the air, I j is the field current of the generator rotor, K j is the air gap fundamental magnetic intensity coefficient, Rr is the radius of the generator rotor, Lr is the length of the rotor, and the expansion coefficient n is

      of computing the eigenvalues by solving the characteristic equation in the form of harmonic functions and finding the roots is not applicable.

      We use the state variable method to solve this problem.

      0 1

      0 1 2

       

      2

      n n

      n 0

      . (5)

      Introducing the state variables

      V [xr , xr ]T , Eq. (7) can be written as the following eq uation [8, 9]

      1 1 1 2

      0 2

      n 0

      x

      xr

      0 1 2 2

      V r .

      (8)

      2

      xr

      M 1[( G C ) x K x ]

      r r r r r r r

      0 is the average air-gap of the generator rotor when the generator rotation axis is not eccentric, is the rotation angle of the generator rotor, i.e.

      In matrix form

      V AV .

      Here

      (9)

      cos x2 , sin y2 ,

      e2

      is the rotor relative

      x 0 I

      e2 e2

      2 0

      V r , A .

      x M 1 K M 1 ( G C )

      eccentricity, x2 , y2

      are the displacements of the

      r r r r r r r

      generator rotor, and e2

      the generator.

      is the initial static eccentricity of

      (10)

      The critical rotational speed of the system can be found by solving the singular value problem of matrix A

      It can be seen from Eq. (5) that the unbalanced

      magnetic pull force has a strong nonlinearity with respect to the rotor displacement.

      On the other hand, if the turbine is not symmetric or a

      in Eq. (9).

      Setting the solution vector to

      V = Aet .

      (11)

      periodic change occurs in the seal clearance of the machines main shaft, the water pressure fluctuation within the seal will occur, which will result in an unbalanced force of water [4-7].

      This force is

      We have dV / dt V et , and Eq. (11) is expressed in the form AV=V.

      For example, in Matlab, an eigenvalue solution of the form of Eq. (2.5) can be easily obtained for the singular

      value of the matrix A using the V, D = eig A

      px5 kwx, py5 kw y.

      The water balance coefficient is denoted by where is the water imbalance coefficient [8, 9].

      (6) .

      kw ,

      command. The computed eigenvectors are contained in the main diagonal matrix D and the eigenvectors in the square matrix A .

      Denoting the i – th eigenvalue of the system by

      i ui i , the desired eigenvalue is a complex

      conjugate, and its imaginary part i denotes the

      rotational angular frequency of the rotor at the specified rotational speed.

      Using the imaginary part of the singular value, the critical rotational speed of the rotor can be found, and when the rotational speed of the rotor is equal to each

      other, i is the rotational speed of the system i .

      The real part ui of the singular value is used to

      judge whether the system is stable or not, if all ui is

      negative, the system is stable, and if the real part of at least one singular value is zero or positive, the system is unstable.

      The procedure to calculate the critical rotational speed of the lateral vibration of the system is as follows.

      First, the initial speed of the rotor and the calculation time step are set, and the geometric and physical parameters of each element of the system are input.

      Then, the element mass matrix and stiffness matrix are obtained and the element stiffness matrix of the sliding bearing is input.

      Then, the mass and stiffness matrices of the whole system are obtained by superposition of the above element stiffness and mass matrices, and the singular values of A are obtained by using the MATLAB eigenvalue solver according to Eq. (9).

      Plot the variation of the imaginary part of the singular value with the variation of the rotational speed and find the critical rotational speed of the lateral vibration of the system.

      When the initial parameters do not meet the accuracy requirements in design and calculation, the initial parameters and calculation step time are reset to obtain the satisfied results.

      The flowchart for calculating the critical rotational speed of the lateral vibration of the hydraulic turbine generator shaft system is shown in Fig. 2.

      Fig. 2 Flow chart for calculating the critical rotational speed.

   2. Effect of Sliding Bearing Stiffness on Critical Rotational Speed

      According to the method described above, the critical rotational speed of the hydraulic turbine generator shaft system is obtained and the effect of the sliding bearing stiffness on the critical rotational speed of the system is analyzed.

      The system parameters are as follows:

      Axis 1 in Fig. 1 is considered as a hollow shaft with a constant cross-section, i.e., inner diameter i s Id 0.8 m , outer diameter i s Od 1.26 m , and

      lengh is L 1.6 m .

      Shafts 2, 3 and 4 are one axis, with inner diameter is

      Id 1.45 m and outer diameter is Od 1.9 m .

      Where

      L1

      4.5 m,

      L2

      6.2 m

      and

      L3 2.2 m .

      The elastic modulus of the shaft is

      E 206 GPa ,

      the density of the spindle is 7850 kg/m3, the rotational

      speed of the spindle is

      n 75 rpm , the maximum

      rotational speed is n 145 rpm , the mass of the

      generator rotor is m1 1.2 106 kg , the mass inertia

      is 4.5 107 kg m2 , the aberration mass is

      m2 2.8 105 kg , and the mass inertia is

      2.4 106 kg m2 [ 1 ].

      The critical rotational speed variation of the system is analyzed by varying the stiffness of the upper slide bearing, the lower slide bearing and the water wheel slide bearing.

      For the simultaneous variation of the three sliding bearing stiffness, the coefficient of linearized unbalanced

      magnetic traction is fixed at 1.36 108 N / m , and the sliding bearing is equivalent to an elastic system with stiffness and damping.

      In Fig. 3, the first and third E igen modes of the system are shown.

      From Table 1, it can be seen that when the bearing stiffness is increased from 1 108 N / m to

      11.2 108 N / m , the first, second and third critical rotational speeds are increased by 29%, 49.9% and 60%, respectively, with the increase of bearing stiffness, the three critical rotational speeds of the system increase gradually and the increments are all relatively large.

      Degree stiffness(N/m)	1th	2th	3th
      1×108	296.17	630.57	821.65
      1.2×108	305.73	659.23	832.21
      3.2×108	353.50	878.98	926.75
      5.2×108	372.61	917.19	1 022.29
      7.2×108	382.16	936.30	1 089.17
      9.2×108	383.47	939.73	1 136.94
      11.2×108	383.85	945.85	1 165.60

       

      Overall, the effect of the sliding bearing stiffness on the primary rotational speed of the system is relatively small and the effect on the tertiary critical rotational speed is relatively large.

      Table 1 Effect of three sliding bearing stiffnesses on the critical rotational speed (units: r/min)

      As can be seen from Table 2, increasing the upper slide bearing stiffness from k 1.2 108 N / m to k 11.2 108 N / m does not change the critical rotational speed of the system.

      This indicates that the stiffness value of the upper slide bearing has little effect on the critical rotational speed of the system in the range considered above.

      -When only the stiffness of the lower slide bearing changes.

      Table 3 shows the influence of the lower slide bearing stiffness variation on the critical rotational speed when the upper and the aberration slide bearing stiffness is fixed at k 7.2 108 N / m .

      When the lower slide bearing stiffness is increased

      It should be noted that the first and second critic al rotational speeds nearly do not increase after the s tiffness of 3.2 108 N / m .

      -When only the stiffness of the upper slide bearing changes.

      To analyze the effect of one slide bearing stiffness on the critical rotational speed of the system, the stiffness of the upper slide bearing is varied and the two-slide bearing stiffness are fixed at k 7.2 108 N / m .

      Table 2 shows the effect of upper slide bearing stiffness on the critical rotational speed.

      from k 1.2 108 N / m to k 11.2 108 N / m , the critical rotational speeds increase considerably, with the second critical rotational speed increasing by about 36% and the third critical rotational speed increasing by about 20%.

      When the lower slide bearing stiffness is greater than k 7.2 108 N / m , the first and second critical rotational speeds remain constant with the increase of bearing stiffn- ess.

      The first and second critical rotational speeds are alm ost constant for stiffness values above 5.2 108 N / m .

      Degree stiffness(N/m)	1th	2th	3th
      1.2×108	382.16	936.30	1 089.17
      3.2×108	382.85	936.84	1 089.53
      5.2×108	383.21	937.28	1 090.17
      7.2×108	383.95	938.14	1 090.83
      9.2×108	384.54	938.91	1 091.37
      11.2×108	385.29	939.47	1 092.12

      Fig. 3 Natural vibration modes of a hydro-turbine generator

      Tab. 2 Effect of top slide bearing stiffness on critical rotational speed (in units: r/min)

      critical rotational speed of the system, and should be chosen reasonably according to the system characteristics, which depend on the system characteristics and can be determined based on the above analysis.

      Degree stiffness(N/m)	1th	2th	3th
      1.2×108	343.94	687.89	955.41
      3.2×108	371.34	878.98	984.07
      5.2×108	372.61	926.75	1,041.40
      7.2×108	376.51	936.30	1,089.17
      9.2×108	380.31	938.48	1,127.38
      11.2×108	382.16	940.21	1 146.49

       

      Table 3 Influence of lower slide bearing stiffness on critical rotational speed (unit: r/min)

      Degree Stiffness( N / m )	1th	2th	3th
      1.2×108	334.39	831.21	1,003.18
      3.2×108	363.05	907.64	1,041.4
      5.2×108	372.61	926.75	1,070.06
      7.2×108	382.16	936.30	1,089.17
      9.2×108	384.38	938.71	1,098.72
      11.2×108	386.31	941.61	1,108.28

       

      Table 4 Critical rotational speed (in

      units: r / min ) as a function of the rotor

      sliding bearing stiffness.

      .- When only the turbine sliding bearing stiffness changes.

      When the upper and lower slide bearings stiffness is

4. FIELD TESTS AND RESULTS ANALYSIS

   fixed at

   k 7.2 108 N / m , Tab. 4 shows the effect of

   The results of the field test analysis are illustrated to

   the aberration slide bearing stiffness on the critical rotational speed: the first critical rotational speed increases by about 14.3%, the second by about 12.6%, and the third critical rotational speed increases by about 10.4%.

   The critical rotational speeds of the first, second and third orders are constant after the stiffness is

   3.2 108 N / m .

   A comprehensive analysis of the effect of a single slide bearing on the critical rotational speed of the system shows that, in general, the ritical rotational speed of the system increases with the increase of the sliding bearing stiff-ness; however, there exists a limit at which the first and second critical rotational speeds become almost constant when the increase of the sliding bearing stiffness reaches a certain extent.

   Therefore, an overall increase in the sliding bearing stiffness does not always lead to an increase in the

   demonstrate the validity of the results for the simulation study of the mathematical model.

   The hydro-turbine generator (rated power 10 MW , speed 410 r / min , maximum head 390 m ) of a hydro- electric power plant in our country was in a state of water operation where the vibration level of each sliding bearing of the generator exceeded the allowable limit (100 m ) and caused severe vibration and was difficult to operate.

   Based on the standard design dimensions of the hydro-turbine generator, the first critical rotational speed is 16.5 Hz and the second critical rotational speed is

   39.6 Hz , calculated by the aforementioned method. The vibration level of each slide bearing measured during operation is shown in Tab. 5.

   Measurement Position	Upper slide bearing	Lower slide bearing	Turbine slide bearing
   Level	150	210	138

    

   Table 5 Vibration level of vibration measurement points (in units: [ m ])

   The real waveform and spectral diagram at the measurement points are shown in Fig. 4 and the axial diagram in Fig. 5.

   1. Vibration speed signal waveform of the water wheel sli de bearing
      1. Vibration speed signal waveform of upper slide bearing
   2. Spectrum of the water wheel sliding bearing;
   1. Power spectrum of upper slide bearing vibration spee d signal(Fig.4 a)

      Fig. 4 Vibration velocity signals and spectrogram at the measurement points.

   2. Vibration speed signal waveform of lower slide bearing
      

      Fig. 5 Axial diagram of the generator-turbine shaft system.

   3. Power spectrum of lower slide bearing(Fig.4.c)

   From the vibration levels, spectral diagrams and axial diagrams of the measuring positions, it can be seen that the rotation axis has strong bending vibration and the mode of bending vibration coincides with the first natural mode of the axial system, and the first resonance occurs.

   Inspection of the assembly state of the turbo- generator showed that the assembly dimensions and radial clearances of the upper, lower and aberration sliding bearings did not reach the design level, which reduced the

   stiffness, thus reducing the system’s first critical rotational speed.

   Based on the results calculated in the previous se ction, the bearings that have a decisive influence on the critical speed of the system are the lower and th e water wheel bearings, so based on the design-level dimensions of the turbine generator assembly, the as sembly dimensions of the lower and water wheel slid ing bearings and the oil film clearance are adjusted t o the design value, and the turbine generator is reass embled to drive the turbine generator, the vibration is

   reduced to the steady level (50 m ), the shaft diagr

   am of the generator-turbine shaft system shown in Fi

   g. 5 is not shown, and the primary resonance is elim inated.

   This is because in this case the first and second critical rotational frequencies are far enough away fro m the turbine rotational frequency of 6.83 Hz , which shows the validity of the aforementioned critical rota tional frequency calculation method.

5. CONCLUSIONS

A dynamic model representing a system of nonlinear differential equations considering various coupling actions for a hydro-turbine spindle system was established, the effect of the sliding bearing stiffness on the critical rotational speed of the system was studied, and field tests demonstrated the validity of the theoretical findings.

When the stiffness of the three sliding bearings supporting the system is increased simultaneously, the critical rotational speed of the system is generally increased, but the degree depends on the order of the critical rotational speed.

Regarding the effect of the stiffness of individual bearings on the critical rotational speed, the upper slide bearing has little effect on the critical rotational speed of the system, but the increase of the stiffness of the lower bearing and turbine bearing increases the critical rotational speed relatively significantly.

However, when the sliding bearing stiffness is large to some extent, there is a limit at which the first and second critical rotational speeds no longer increase, and it is difficult to prevent resonance by increasing the critical rotational speed of the system by increasing the bearing stiffness.

Therefore, in order to achieve vibration stability by increasing the critical rotational speed of the system sufficiently, it is necessary to increase the stiffness of the support bearings and also improve other factors such as the mass of the system, the sectional properties of the shaft and damping.

This has been verified through a field application

example.

Declaration of competing interest:

The authors declare that have no conflict of interest

Funding: This research received no external funding

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