Modeling of Gas Lubricated Compliant Foil Bearings Using Pseudo ...

Tribology Online, 10, 5 (2015) 295-305. ISSN 1881-2198 DOI 10.2474/trol.10.295

Article

Modeling of Gas Lubricated Compliant Foil Bearings Using Pseudo Spectral Scheme Balaji Sankar* and Sadanand Kulkarni National Test Facility for Rolling Element Bearings, Propulsion division, CSIR-NAL Banglaore-560017, India *Corresponding author: [email protected] ( Manuscript received 4 April 2015; accepted 26 August 2015; published 31 October 2015 )

A methodology for using the pseudo spectral scheme for the modeling of compliant gas lubricated foil bearings is presented in this work. The pseudo spectral scheme is used to obtain numerical solution of the compressible Reynolds’s equation that models the lubricating gas film. This scheme is coupled to a structural model that models the compliant structure of the foil bearing. The coupled simulation model is applied to both one dimensional and two dimensional cases. The simulation model is validated by comparing its results with analytical solution for short bearing case and to published data for compressible two dimensional cases. The advantages of using this scheme compared to a finite volume scheme for the simulation of foil bearings are presented. Keywords: compliant foil bearings, pseudo spectral method, time marching, simulation

1. Introduction Research institutes and industries around the world are actively looking for technologies that will enable the replacement of conventional oil lubricated bearings with bearings that are more environment friendly, light weight and are cheaper to install and maintain. Air lubricated bearings and magnetically levitated bearings are major contenders that are substituting conventional oil lubricated bearings in suitable, if not all applications. Hydrodynamic air lubricated foil bearings are a class of air lubricated bearings and were initially advanced by the NASA-Glenn research center. Unlike hydrostatic air lubricated bearings, these bearings do not need external power/compressed air supply and are hence more attractive for light load and high rpm conditions. These bearings have already found applications in aerospace and electronics industry, the chief advantage being the elimination of conventional lubrication system and hence savings on maintenance and weight [1]. The front view of the compliant film gas foil bearing is shown in Fig. 1. It comprises of a cylindrical shell (sleeve) lined with corrugated bumps (bump foil). The corrugated bump foil forms the flexible foundation. The encircling foil (supported by the bump foil) encircles the shaft. In practice, the bearing diameter is often designed to have very small clearance or are interference fit to the encircling foil. When the shaft rotates, it drags air into Copyright © 2015 Japanese Society of Tribologists

the space between encircling foil and the shaft. This leads to pressure being developed in the converging portion. Upon reaching a certain speed of journal, called lift-off speed, pressure developed will be sufficient to form a layer of air film between the journal and bearing surface and completely separates them apart. The attainment of this lift-offspeed can be found by the sharp reduction in the torque required to drive the shaft. With an air film maintained between the rotating and stationary surfaces, an adequate pressure is generated which bears the load applied through the shaft.

Fig. 1 Front view of the foil bearing 295

Balaji Sankar and Sadanand Kulkarni

The bump foil provides support for the top foil and its compliance allows the top foil to deform under the action of hydrodynamic pressure. It is apparent that we need to consider the elastic deformation of the top foil and the compressibility of the gas to adequately model the bearing. This is necessary because as the foil deflects, it changes the clearance between the encircling foil and the shaft, and hence changes the flow-field. A huge volume of work has been published on the simulation and design methodologies of these bearings. On the numerical simulation front Tadjbakhsh [2], provided a scheme to numerically solve for pressure distribution considering the foil deflection and compressibility of gas in 1983. Prior to that in 1977, Langlois [3] improved his own previous foil bearing simulation model. He introduced the numerical scheme in which the fluid equations are solved first and the pressures thus obtained are then applied to structure model which gives the deflection of the structure. This deflection altered the shape of the fluid domain which is solved again. Peng [4] put forward a novel way to estimate the load carrying capacity of foil bearing using minimum film thickness as the determining parameter . A theoretical model to investigate the nonlinear dynamic behavior of bump-type foil bearings, i.e. the instability and unbalance response has also been presented by Feng [5]. In this model,the foil structure of bump-type foil bearings was simulated using the link-spring model, which was presented and validated in a previous study [6]. Through parametric studies that involved varying bump number and foil thickness, they noted that more bumps or lager length ratio lead to larger load capacity. NASA has developed an analysis and prediction tool for foil bearings named XLGFBTH©, reported in [7]. The critical speeds predicted by the model agreed well with the measurements from test rig. In Ref [8], the role of radial clearance on load carrying capacity has been studied. The results indicate that, bearings with clearances lower than the optimum value are prone to thermal runaway problem and those with clearances higher than optimum suffer reduction in load carrying

capacity. Experimental tests programs have been conducted at NASA to determine the performance of foil bearings at alternate pressures and temperatures [9]. Results show an increase in load capacity with increased ambient pressure and a reduction in load capacity with increased ambient temperature. Regarding the structural model, in Ref [6], the author has developed an analytical model of bump-type foil bearings considering the elasticity of bump foil,interaction between bumps, friction forces at the contact surfaces and local deflection of top foil.They simplified each bump into two rigid links and a horizontally spaced spring, the stiffness of which is determined from Castigliano’s theorem. The local deflection of the top foil was described using a finite-element shell model and added to the film thickness to predict the air pressure with Reynolds equation. Ref [10] presents a model that represents the foil bearing structure considering the interaction between bumps. Once the stiffness matrix of the whole foil bearing structure is obtained using spring network approach, the entire static system is solved taking friction into account. Load predictions with this model showed the influence of frictional forces on the load carrying capacity. Different coatings used on the encircling foil to reduce the friction have been presented in Ref [11-14]. Most of the numerical simulation tools mentioned above convert the Reynolds’s differential equation into a difference equation using various numerical approximations of derivatives of the dependent terms as mentioned in Ref [15]. In evaluating the finite difference approximations of the dependent terms, higher order terms are neglected. The interpolation function used in these finite difference approximations cover only a small neighborhood of grid points around the point of interest. Spectral methods (as explained in section 3) on the other hand, use interpolation functions that span the whole computation domain of the problem. Hence information from whole computation domain is used in the evaluation of the derivatives. The second major advantage is that the pseudo spectral error,

Table 1 Reference parameters for non dimensionalization Parameter

Reference parameter for non dimensionalization

Pressure

Ambient pressure

Film height

Initial concentric radial clearance

Tangential distance

Radius of the inner race of the bearing

Axial distance

Half length of the bearing.

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Tribology Online, Vol. 10, No. 5 (2015) / 296

Modeling of Gas Lubricated Compliant Foil Bearings Using Pseudo Spectral Scheme

decreases exponentially as the number of grid points is increased. As detailed in Ref [16], as the number of grid points (N) is increases, the error in pseudo spectral scheme is of the order O(1/N)N, while the error of finite difference scheme is of the order O(1/N)K, where K is the order of the finite difference approximation. Since pseudo spectral scheme gives lower error for the same number of grid points, it can also be stated that for a given minimum error requirement, the pseudo spectral scheme requires lesser number of grid points. The better estimation of derivatives and the lower requirement of grid points for given error are major advantages of the pseudo spectral scheme. The application of this scheme is very straight forward for regular continuous geometries as that of a fluid film bearing. These advantages are the main drivers for the use of pseudo spectral scheme in the solution of the Reynolds’s equation in this work. The purpose of this work is to present a methodology in which the pseudo spectral scheme can be employed to solve the Reynolds’s equation in the context of the compliant foil gas film bearings.

polynomials in the axial direction. The boundary conditions enforce ambient pressures at the front and rear end of the bearing. When multiple foils are used to encircle the shaft, as used in [17], the encircling foil are allowed to lift when pressures underneath the foil is greater than the pressure between shaft and foil. Hence any sub-ambient pressure occurring in the bearing is made equal to ambient pressure in such multi sectored bearing arrangements [4]. Since a single continuous foil is used in this analysis, the sub-ambient pressures are allowed to occur and the encircling foil does not lift away from the bump foil.

2. Reynolds’s equation and structural model

Here, h is the film thickness at location θ and Hconcentric is the original of film thickness when the journal was concentric with the bearing. ε is the distance between the center of the bearing and center of the journal. The conventions for θ and φ are shown in Fig. 2. In this convention, θ is from the vertical in the anti clockwise direction and φ0 is the angular location of the minimum clearance point from the reference line. The origin of the co-ordinate φ is offset from the origin of θ by 180°. As the rotating shaft moves to an eccentric location, the pressures in certain locations increase above ambient pressure, and hence the foil deflects and clearance increases. When the effect of pressure deforming the encircling foil is considered, the equation for film thickness is modified and is given in Eq. (4).

In this analysis, the fluid film is modeled using the two dimensional compressible Reynolds’s equation given by   3 p    3 p    ph ph      y  y     ph    ph    6µ u 2 (1)  t    The tangential and axial coordinates are denoted by θ and y. The pressure and film thickness distribution are given by p and h. U denotes the tangential velocity of the journal surface and µ represents the viscosity of air. The above Reynold’s Eq. (1) has been non dimensionalized into Eq. (2) using the reference quantities shown in Table 1. The non dimensional form is   P    1 3 P  PH 3   PH    2      L / D  y  y 

2.1. Structural model The structural model [4] is used to calculate the deflection of the flexible foundation under the action of pressure forces. This film thickness will be uniform when the pressure distribution around the journal is uniformly atmospheric and the journal is at the center of the bearing. When journal center is offset from the bearing center, the film thickness distribution is a function of θ and is given by Eq. (3). h  H concentric   cos   0  (3)

  PH      PH    2 (2)      The Harrison number or bearing number(Γ) in Eq. (2) is given by 2

  R   H  concentric  where Pambient is the ambient pressure, Hconcentric is the radial clearance, R and L are the radius and length of the bearing respectively. The angular velocity of the journal is given by ω . By non-dimensionalizing with respect to (L/2) and not L, the length to diameter ratio (L/D) term explicitly appears in the equation and also the equation becomes convenient for the use of Chebyshev

 

6µ Pambient


Fig. 2 Angle conventions used Tribology Online, Vol. 10, No. 5 (2015) / 297


h  H concentric   cos    0   k1  p  pambient  (4) Here, k1 denotes the structural compliance of the foil structure and is lesser for stiffer structure. Its units is (m/Pascal). This equation can be non-dimensionalized with respect to initial concentric clearance as given in Eq. (5). (5) K P h  1  e cos    0   1 ambient  P  1 H concentric In order to obtain the expression for the compliance term K1, consider the force acting per unit transverse length of the bump foil, given in Eq. (6). Force  sPambient  P  1 (6) transverse length If Kb is the stiffness per unit transverse length, then the deflection due to pressure can be expressed as sP  P  1 Deflection  ambient (7) transverse length Kb The expression for Kb is given by [18] Et 3 (8) Kb  12 1   2  l0 3 where l0 is the half length of the bump, s is the pitch of the bump, as indicated in the Fig 3. E is the Young’s modulus of the material of the bumps, γ is the poisson's ratio and t is the thickness of the bump foil. Thus the non dimensional compliance factor to be used in the Eq. (5) is given below sPambient  (9) Et 3 H concentric 12 1   2 l03





The film thickness can now be non-dimensional form as given below h  1  e cos   0     P  1

written

in (10)

Pseudo spectral scheme The non-dimensional Reynolds’s equation given in Eq. (2), along with the structural model given in Eq. (10) can be solved in a loosely coupled manner, either using the linearized form and the full non-linear form of the equations. Numerical methods employing finite volume and finite difference schemes to discretize the source and partial derivative terms are routinely used [4,18,7,19].

These schemes are predominantly local methods in the sense that they consider the only the information available at the neighboring points of the grid point while evaluating the derivative of the pressure at that point. Spectral methods are global methods, where the computed derivative at a point depends not only on information at neighboring points, but on the information from the entire domain. The Fourier and Chebyshev spectral methods basically give us expressions for the dependent variables (such as pressure and film thickness) that are global, which are valid everywhere in the computational grid. These expressions can then be differentiated to obtain the partial derivative of the required quantity at the point of concern. 2.2. Fourier representation of tangential pressure variation The pressure distribution (P) in the tangential direction can be expressed as a Fourier series in the 

form of

e  ik . Here k denotes the wave P  P k 

 denotes the Fourier coefficients of the number and P k pressure distribution P. The above expression would be an exact representation of the pressure around the shaft if infinite terms are used in the Fourier series. Approximating it by leaving out terms above N, where N is the number of nodes in the tangential direction of bearing, the non-dimensional pressure distribution can be approximated as given in Eq. (11). P

 N /2 1



e  ik P k

(11)

 N /2

Equation (11) is a truncated Fourier series containing N terms from (−N/2) to (N/2−1), including the 0th term. The (−N/2) term does not have a symmetrical positive counterpart and hence is usually made zero explicitly. The Fourier coefficients can be evaluated using the orthogonal property of the trigonometric system [20]. Since Fourier series have been used to represent the tangential pressure variation, the periodic boundary condition for pressure is implicitly enforced.

Fig. 3 Structural model of the bump

2.2.1 Evaluation of partial derivatives in tangential direction. The partial derivatives in the tangential direction can be computed by computing the Fourier coefficients, multiplying them by their respective wave numbers and transforming them back to the physical quantity. Consider the evaluation of the term (∂(PH)/∂θ). The transformation between Fourier and physical planes are done through fast Fourier transform and inverse fast Fourier transform algorithms, provided with Matlab®. The product term P × H is evaluated in the physical plane and is transformed into the Fourier plane through “fft” algorithm. It is then multiplied by its corresponding wave number. It is to be noted that either the wave numbers or the coefficients should be fft-shifted before multiplication due to the reverse-wrap around ordering




of the fft coefficients. The modified co-efficients of the PH term are transformed back to physical plane using “ifft” algorithm. This results in the partial derivative of the PH term in the tangential direction. This process is pictorially shown in Fig. 4. 2.3. Chebyshev series representation of axial pressure variation Once the tangential pressure distribution is represented in terms of Fourier series, the axial distribution can similarly be expressed in terms of Chebyshev series. The distribution is polynomial like in the axial direction hence a series with polynomial basis function like Chebyshev series is more likely to give a better approximation with fewer terms. The Chebyshev polynomials are a special case of the Jacobi polynomials and are defined between −1 and +1. For this reason, the non-dimensionalization in the axial direction was carried out with respect to half length of the bearing, and not the full length, as is usually the case. These polynomials also satisfy the recurrence relation and are easy to generate programmatically. Using these polynomials, the pressure distribution in the axial direction can be approximated as given in Eq. (12). Pˆ P  0  Pˆ1 x  Pˆ2 2 x 2  1  Pˆ3 4 x 3  3 x   (12) 2 Where Pˆ0 , Pˆ1 , are the Chebyshev coefficients of the expansion for pressure distribution P.









2.3.1 Choice of grid points In the tangential direction, the grid points are equally spaced. However, in the axial direction, the Gauss-Lobato grid points are chosen. The expression for grid points is given in Eq. (13).  i  (13) xi  cos   , i  0,1, 2, .k .  k  Choosing Gauss-Lobato points as the grid points in the axial direction gives us 2 main advantages. One is that a grid point is located at the front and rear face of

Fig. 5

Fig. 4 Evaluation of derivatives in the tangential direction the bearing, at +1 and −1 location. So the boundary condition that front and rear faces are at ambient pressure can be easily imposed. Second, use of Gauss-Lobato points makes it easier to estimate the Chebyshev coefficients using numerical approximations. These coefficients can be estimated in a similar way as Fourier coefficients. The Chebyshev polynomials are also orthogonal with respect to each other in the space [−1, 1] while using the weighing function,

w  1 / 1  x 2 . The numerical methods to calculate the coefficients are explained in Ref [21]. Derivative of the pressure in the axial direction can be found directly by multiplying the grid values of the pressure by the Chebyshev differentiation matrix, as given in Ref [22]. The “cheb. m” function provided in Ref [22] has been used to generate the differentiation matrix in this work. The grid used in the discretization of the fluid domain of the bearing has been shown in Fig. 5. 2.4. Application of pseudo-spectral scheme to the Reynolds’s equation The representation of the pressure distribution as a Fourier series in the tangential direction and a Chebyshev

Grid used in the pseudo spectral scheme




series in the axial direction has been detailed in the above section 3. The compressible Reynolds’s equation given in Eq. (2) can now be iteratively solved using pseudo-spectral scheme by the following procedure. Initially, a uniform distribution of pressure is assumed throughout the grid. The variation of the film thickness is obtained for this uniform pressure distribution and the fixed eccentricity value. 2.4.1 Evaluation of Convective term The convection term in the tangential direction (∂(PH)/∂θ) is evaluated first. The distribution of film thickness and pressure distribution is multiplied to obtain (PH) term and its derivative is computed by multiplying the (PH) term with the Fourier differentiation matrix. 2.4.2 Evaluation of axial diffusion term   1   3 P  The axial diffusion term,   PH  2 y    ( L / D ) y  is found next. The assumed pressure distribution is multiplied with the Chebyshev differentiation matrix to obtain the axial derivative of pressure (∂P/∂y), and this derivative is multiplied with (PH3) term and then differentiated again by multiplying the product of multiplication with Chebyshev differentiation matrix again and then finally obtaining the product of the derivative with function of length to diameter ratio.

The pressure distribution can then be obtained by dividing the PH term with current film thickness distribution. This pressure distribution is taken as the next pressure distribution and is then used to estimate the film thickness distribution for the next step by using the structural model given in Eq. (3). This process is repeated until the difference between the successive pressure distributions becomes less than the specified tolerance limit. This iterative method has been shown as a flow chart in Fig. 6. The evolution of the mid-plane pressure from a uniform distribution to the converged distribution is shown in Fig. 7. 3. Validation using short bearing approximation

For infinitely short bearings, as described in Ref [1], only the axial derivatives are significant and tangential derivative of pressure can be ignored. Further, for first generation foil bearings, the clearance is independent of axial location. Hence for a first generation, incompressible, steady state case, Reynolds’ equation reduces to Eq. (16) d  dp  dh / d (16)    6Uµ h3 dy  dy  The analytical solution [1] for the above equation is

2.4.3 Evaluation of tangential diffusion term    P   PH 3 The tangential diffusion term,   is        evaluated in a similar manner. The tangential derivative of pressure is first obtained by multiplication of current pressure distribution with Fourier differentiation matrix and then the derivative is multiplied by the (PH3) term that was evaluated earlier. Then this product is finally multiplied by the Fourier differentiation matrix again to give the tangential diffusion term. 2.4.4 Evaluation of pressure distribution in the next step Once both the diffusion terms and the convection term have been obtained, the compressible Reynolds’s equation can be recast in the following form by keeping only the time derivative term on the LHS and the other terms on the RHS.    P    1  PH 3     2   ( L / D ) y    PH  1     (14)    PH   2   3 P    PH   y     Once this time derivative term has been evaluated, the pressure distribution for the next iteration can be evaluated using the Euler time marching scheme.   PH  (15) t   PH n  PH n 1  


Fig. 6

Flow chart showing the application of pseudo spectral scheme Tribology Online, Vol. 10, No. 5 (2015) / 300


Table 2 Parameters used in validation of pressure distribution for short bearing

Fig. 7

Fig. 8

Evolution of tangential distribution of mid-plane pressure with iterations

Validation of the model for short bearing case

dh / d  L2  (17) y  3 h 4  Equation (16) is solved with pseudo spectral scheme and compared with the analytical solution given in Eq. (17). The parameters used in the simulation are given in Table 2. The comparison between analytical solution and the numerical solution is given in Fig. 8. p  3Uµ

4. Validation for foil bearing geometry The pseudo spectral scheme, having been validated for a simple infinitely short bearing case, is next applied to a rigid bearing case. The grid convergence studies were done for this geometry to determine optimum number of nodes in axial direction and tangential direction. Number of grid points in axial direction was fixed at 11 and in tangential direction, it was fixed at 60. The grid convergence study for foil bearing indicated Japanese Society of Tribologists (http://www.tribology.jp/)

Parameter

Value

Bearing length

20 mm

ε

0.5

φ

0

U

50 m/s

H concentric

20 microns

that the 11 × 60 grid predicts the pressure distribution satisfactorily for foil bearing also. The result of grid convergence study for foil and rigid bearing is plotted in Fig. 9. In this plot, the maximum mid plane pressure predicted by the simulation is plotted against the number of grid points used in the simulation. This plot also shows that the pseudo-spectral scheme is able to capture the pressure distribution with minor error (0.015% error) even with 30 grid points in the tangential direction. The pressure distribution is obtained using a time marching scheme, while keeping the shaft at a fixed location inside the bearing. The solution was computed with a non-dimensional time step Dt=10×21.6Γ/Nθ4 as given in Ref [23]. The bearing geometry used for the simulation is the same as that used Peng in Ref [4], given in Table 3. The solution took 3 lakh (3×105) time steps to converge within the tolerance limit of (1×10-6), when started from a uniform ambient pressure distribution. The tolerance limit was specified in terms of the percentage change in max pressure in the pressure distribution. This pressure distribution, computed for a rigid bearing, is shown in Fig. 10. For compressible flow between the shaft and the surface of the compliant film foil bearing, analytical solutions for pressure distributions are not available. The pressure distribution estimated through iterative solution of the equations that govern the deformation of the flexible structure and the equations that govern the pressure distribution in the fluid film. For the same bearing geometry and bearing number, the foil bearing solutions are computed for a compliance factor of 0.4. The film thickness of the foil bearing will be slightly higher than the film thickness of rigid bearing for the same eccentricity. This is because the foil deforms under pressure. This effect can be clearly seen in Fig. 11, where the eccentricity ratio of the shaft has been fixed at 0.79. Higher minimum film thickness reduces the wedging effect, thereby reducing the pressure build up due to the Tribology Online, Vol. 10, No. 5 (2015) / 301


Table 3 Parameters used in validation of pressure distribution for rigid bearing

Fig. 9

Max mid plane pressure of rigid and foil bearings as a function of number grid points

Parameter

Value

Bearing length

38.10 mm

Radius of shaft

19.05 mm

Bump foil thickness

0.101 mm

Bump pitch

4.57 mm

Bump length

3.55 mm

Young’s modulus

200 GPa

Poisson’s ratio

0.31

Fig. 10 Pressure distribution in a rigid bearing convective term in the Reynolds equation. This results in lower pressure generated in foil bearing when compared to rigid bearing of the same eccentricity ratio. This difference in pressure can be seen in Fig. 12. Hence to compare the pressure generating ability, the eccentricity ratio of the foil bearing is increased such that the minimum film thickness is maintained same as that of rigid bearing, as in Ref [4]. The mid plane pressure distribution for both rigid and foil bearing cases are compared with numerical results of Peng [4] which is widely used as a validation case [19,24]. This solution was obtained with a finite difference scheme and was validated with the experimentally obtained load carrying capacity. This comparison between pressure distributions of rigid and foil bearings obtained by the proposed scheme and results of Peng [4] is shown in Fig. 13. When rigid bearing case was simulated, the compliance factor was kept as 0. The film thickness distribution, when the minimum film thickness is kept same is shown in Fig. 14. In Peng’s study, derivatives of the pressure and film thickness’ are estimated using second order central difference approximations of the pressure and film thickness distributions. It is obvious that this assumption is valid only for linear and quadratic variations of these distributions. The distributions of pressure and film thickness are more complex than quadratic functions in the tangential direction, as can be seen in Fig. 13 and Fig. 14. Thus the use of these finite difference approximations leads to non-trivial errors that Japanese Society of Tribologists (http://www.tribology.jp/)

Fig. 11

Comparison of rigid and foil bearing film thickness distribution for same eccentricity ratio

Fig. 12

Comparison of rigid and foil bearing pressure distribution for same eccentricity ratio

accumulate, especially in high gradient regions such as peak maximum pressure and minimum pressure regions. Further in the estimation of foil bearing pressure distribution, the attitude of the shaft has been adjusted along with the eccentricity of the shaft in the simulations Tribology Online, Vol. 10, No. 5 (2015) / 302


a

b

Fig. 13 Comparison of pressure distributions from proposed scheme (a) with results of Peng [4] (b)

a

Fig. 14

b

Comparison of film thickness distributions from proposed scheme (a) with results of Peng [4] (b), foil bearing eccentricity is kept at 1.13 in the proposed scheme

of Peng. In the current study, only the clearance has been iterated such that minimum film thickness is maintained between the rigid bearing and foil bearing case. The authors believe that these two reasons cause the minor difference between the results of Peng and current results. When the minimum film thickness is maintained same, the foil bearing generates higher lifting force compared to a rigid bearing. This lifting force is obtained by integrating the pressure distribution over the area. For a rigid bearing, the load carried is 38 Newton, while the foil bearing carries a higher load of 43 Newton. This is because, as shown in Fig. 13, the higher pressure acts over a larger area in a foil bearing. For both foil and rigid bearing, as the load on the shaft is increased, eccentricity of the shaft in the bearing increases. This is because the shaft has to maintain lower film thickness to obtain sufficient pressure to counter the increased load. However this increase in eccentricity with increase in load is much higher for a foil bearing than a rigid bearing, as is shown in Fig. 15. As an additional study, to determine the maximum eccentricity that can be simulated using the pseudo spectral scheme, the eccentricity of the shaft in the simulation was gradually increased until the convergence Japanese Society of Tribologists (http://www.tribology.jp/)

of the pressure in the bearing failed to follow a monotonic pattern. This study was carried out for a range of (L/D) values. The pseudo spectral scheme converged until the shaft reached an eccentricity of 0.9 in this range of (L/D). The convergence trend of maximum pressure for different eccentricity ratios is shown in Fig 16. During this study, it was also seen that for (L/D) in the range of 0.5 to 1.1, for a given eccentricity ratio, the number of non-dimensional time steps taken for reaching a converged steady state increased as the (L/D) value increased. This trend is clearly seen in the graph shown in Fig. 17.

5. Conclusion This paper described a means to obtain the pressure distribution in a compliant film gas foil bearing. Pseudo spectral scheme was used to solve the Reynolds’ equation and a simplified structural model was used to obtain the film thickness distribution. This scheme was validated by comparing it to analytical solution obtained for short bearing case. It was also validated by comparing to published numerical solutions for compliant film bearings of practical configuration. The major advantage of using this method for continuous bump foil bearing is the increase in accuracy of result, for a given number of Tribology Online, Vol. 10, No. 5 (2015) / 303


considerably simpler to use compared to other discretization schemes. The running time of the code for a given eccentricity is also very short (few minutes on an Intel dual core computer). This would facilitate rapid design iterations. The code is currently being extended to simulate actual rotor-dynamic conditions under the influence of unbalance load.

References [1]

Fig. 15

Eccentricity of the shaft under different loads for foil and rigid bearing

Fig. 16

Convergence of pressure for different eccentricities of shaft

Fig. 17

Variation of iterations taken for convergence with respect to the L/D ratio of the bearing

grid points when compared to finite difference or finite volume scheme. Another major advantage is the ease of application of this scheme. As shown in the flow chart in Fig. 6 and explained in section 3.3, the scheme is Japanese Society of Tribologists (http://www.tribology.jp/)

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