National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

Numerical Analysis of High Speed Annular Disks with Parabolic Thickness Variation S Bhowmick, D Misra and K N Saha3 Department of Mechanical Engineering Jadavpur University, Kolkata – 700032, India [email protected], [email protected], [email protected] Abstract: The yield limit rotational speed for given geometry parameters and loading conditions of high-speed annular disks is obtained using a numerical scheme. The annular disk has parabolic thickness variation and contains radially distributed attached masses. Variational method is used to formulate the stress analysis problem and the solution of the governing partial differential equation is obtained by assuming a series solution and using Galerkin’s principle. Corresponding to various rotational speeds, the stress distribution in the disk is estimated and the limit angular speed is determined using Von Mises yield criterion. The effect of geometry parameters along with attached mass ratio and its radial location is studied with the objective of maximizing the limit angular speed. Keywords: Limit angular speed, Von Mises criterion, Annular Disk. NOMENCLATURE a, b {c} h

h0

hc m n, k rm , ξ m u

α

β

∆

µ, E ξ ρ σ0 φi ω

Ω

Inner and outer radii of the disk The vector of unknown coefficients General thickness of the disk at any radius Thickness at the root of variable thickness annular disk Thickness of equivalent uniform annular disk, βb Mass of the rotating disk Parameters controlling the thickness variation of disk Dimensional and non-dimensional radii of the attached mass Displacement field of the disk Ratio of attached mass to disk mass Thickness ratio, hc /b Parameter ( b - a ) Poisson’s ratio and elasticity modulus Normalized coordinate in radial direction, (r − a) / ∆ Density of the disk material Yield stress of the disk material The set of orthogonal polynomials used as coordinate functions Angular velocity of the disk Dimensionless angular velocity, ω b ρ / σ 0

1. INTRODUCTION AND LITERATURE REVIEW Rotating disks find widespread application as one of the significant structural components in mechanical engineering; some common examples are impellers, gears and flywheels. The stress distribution in a rotating disk depends on the rotational speed. The stress analysis of rotating disks at high rotational speeds, thus, is of utmost importance to the designer. Continuous increase in rotational speed increases the stresses developed in the disk. At a particular speed

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

these stresses exceed the yield stress of the material. This particular speed is termed as limit angular speed of the rotating disk. As stresses near the innermost fibers of the disk are high, the hub is made thicker compared to the outer periphery thus providing a variable geometry to the disk. Review work indicates that a lot of numerical and analytical work has been carried out during the past few decades to investigate the stress distribution and estimate the limit angular speed of rotating disks. Application of variational principle [1, 2] involving Galerkin’s principle and assuming a series solution has provided an efficient numerical scheme for the study of rotating disk behavior with and without attached masses. Analysis of admissible angular speed for rotation of annular and solid disks of hyperbolically varying thickness, made of incompressible material, has been carried out using large elastic deformation theory [3] and the results were compared with constant thickness disk under plane strain assumption. Analytical method have been developed based on laws of equilibrium to study the elastic deformation and establish limit angular speeds of disks of parabolically varying thickness [4] and disks having thickness variation in power function form [5]. Results indicate increase in limit angular speed with reduction in thickness at the edge of the disk and reduction in disk mass due to shape of the profile. Closed form solutions for stresses and radial displacement are presented in [6] by considering radial density gradient in rotating disks with variable thickness. The analysis was based on Tresca’s yield condition, it’s associated flow rule and linear strain hardening and the results indicate significant influence of density gradient on radial stresses, critical angular speeds and elastic-plastic interface radius. In [7], the technique of dynamic relaxation is employed for analyzing disks with variable thickness and made of anisotropic material and the results were found to closely agree with those obtained using exact analysis. A unified numerical method is proposed in [8] in which deformation and elastic-plastic stress analysis in rotating disks with arbitrary cross section and density is carried out using Von Mises yield criterion and reported results are comparable to those obtained using Finite Element Analysis. The application of variational principle provides an advantage over other methods in terms of simplicity and ease with which various complicating effects such as addition of attached masses at various radial locations, rigid inclusions, effect of thermal stresses etc. are incorporated. The present work employs variational principle to study the stress distribution and estimate the limit angular speed of rotating annular disks with parabolically varying thickness. The variables related to the geometry of the disk are identified as geometry parameters and those controlling the external loading of the disk are termed as loading parameters. The effect of geometry and loading parameters is studied individually with an objective of maximizing the limit angular speed. The results are reported in dimensionless form and presented graphically. The results provide a substantial insight in understanding the behavior of rotating disks with and without attached masses.

2. MATHEMATICAL FORMULATION The mathematical model is framed based on the assumptions that the material of disk is homogeneous, isotropic and linear-elastic. The thickness of the disk (fig.1) is given by h ( ξ ) = h [ 1 − n ( ξ ) k ] where the parameters n and k control the disk geometry. In each 0

case, disk mass is kept constant by adjusting the value of h0, which is the thickness

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

b= 0.10 b= 0.15

hr

b= 0.20

x=0.0

x=1.0

Dimensionless radius

Fig. 1. A disk with parabolic thickness variation Fig. 2. Effect of β on disk profile (n=0.4, k=0.55) at r = a. Increase in rotational speed increases the centrifugal load which in turn increases the stresses in the disk, the magnitude of which is governed by the boundary conditions. The solution for displacement field, is obtained from the minimum potential energy principle δ(π) = δ(U+V) = 0, where U is the strain energy stored in the disk and V is the potential energy arising out of centrifugal force. Substituting the expressions of U and V in the energy principle δ(π) = 0 the governing equilibrium equation is obtained in terms of the unknown displacement field [1]. 2 ⎡ π E b ⎧⎪⎛ u 2 ⎞ b du ⎤ ⎛ du ⎞ ⎫⎪ 2 2 (1) ⎟ ⎜ ⎢ + + − δ 2 µ hdr 2 πρω r u ⎟ ⎜ ∫ ∫ r uhdr ⎥ = 0 ⎨ ⎬ 2 ⎟ ⎜ dr ⎢⎣ 1 − µ a ⎪⎩⎝ r ⎠ ⎝ dr ⎠ ⎪⎭ a ⎦ The necessary boundary conditions are given by σ r = 0 and σ r = 0 . Equation (1) is (a)

(b )

normalized through transformation ξ = ( r − a )/(b − a ) to facilitate numerical computation and the displacement field is approximated ( u (ξ ) ≅ ∑ ci φ i ) by a linear combination of sets of orthogonal coordinate functions ( φ i (ξ ) ) satisfying the boundary conditions of the problem to yield the governing differential equation in normalized coordinate. ⎡⎛ ∆ E δ ⎢ ⎜⎜ ⎢⎝ 1 − µ 2 ⎣

⎞ 1 ⎧⎪ (∑ ciφ i )2 ⎤ (∆ξ + a ) ⎡ d ⎤ µ⎡ d ⎟∫ ⎨ (∑ ciφ i )⎥ ⎢ 2 ⎟ 0 ( ∆ ξ + a ) + 2 ∆ ⎢ (∑ ciφ i ) d ξ (∑ ciφ i )⎥ + ∆ ⎣ ⎦ ⎣ dξ ⎦ ⎠ ⎪⎩ 1

{

}

⎪ ⎬ hd ξ ⎪⎭

(2)

⎤ ⎥ = 0 . ⎦

− 2 ρ ∆ ω 2 ∫ ( ∆ ξ + a ) 2 ∑ ciφ i hd ξ 0

2⎫

According to Galerkin’s error minimization principle, the variational operator ‘δ’ is replaced by partial derivatives ∂ ∂ c j , j = 1, 2,…, n and the above equation is converted into a set of linear

equations.

E 1− µ 2

1⎧

n n

∑ ∑ c i ∫0 ⎨

φ iφ j

⎩ ( ∆ξ + a )

i =1 j =1

=

ρω

2 n 1 ∑ ∫0 i =1

+

µ ∆

(φ i′φ j + φ i φ ′j ) +

{(∆ξ + a) φ }hdξ 2

( ∆ξ + a ) ∆2

⎫

φ i′φ ′j ⎬hdξ ⎭

(3)

i

The above equation can be expressed as [K]{c}={R} and the required solution is obtained using IMSL routines. A brief description of the solution algorithm is given in [1]. The effect of

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

attached mass is taken into consideration through a factor α . It is assumed that the mass is concentrated at a point and ξm describes the normalized location of attached mass. After incorporating the effect of attached mass, equation (3) becomes ⎫ E n n 1 ⎧ φiφ j µ ( ∆ξ + a ) ′φ ′j ⎬ hd ξ ′φ j + φiφ ′j ) + + c φ φ ( ∑ ∑ ∫ ⎨ i i i 0 ∆2 1 − µ 2 i =1 j =1 ⎩ ( ∆ξ + a ) ∆ ⎭ . (4) n m α ⎛ ⎞ 2 = ρω 2 ∑ ∫01 ( ∆ξ + a ) 2 φ i hd ξ + ⎜ ⎟ω ξ m φi ξ m i =1 ⎝ 2π∆ ⎠ 2 2 In the present work, the yield condition considered is σ r − σ rσ θ + σ θ ≥ σ 0 2 as given by Von Mises yield criterion.

{

}

k = 0.05 k = 0.55 k = 1.00

hr

n = 0.08 n = 0.70 n = 1.00

x=0.0

x=1.0

Dimensionless radius

Fig. 3. Effect of n on disk profile β=0.10)

x=0.0

x=1.0

Dimensionless radius

(k=0.55,

Fig. 4. Effect of k on disk profile (n=0.40, β=0.10)

3. RESULTS AND DISCUSSION The numerical values of the different parameters used in computational work are, E = 207 GPa, µ = 0.3, ρ = 7850 Kg/m3 and σ0 = 350 MPa. The dimensionless angular speed is defined as Ω = ω b ρ / σ 0 , and the value corresponding to the onset of yielding is denoted by Ω1. The disk mass is calculated for a uniform thickness annular disk by setting n = 0. For the same mass (fig.1) the effect of geometry parameters n and k and loading parameters α and ξm on limit angular speed is studied in detail and extensive numerical work is carried out to identify critical geometry parameters that maximize the limit angular speed for a given set of constraints. Thickness ratio β, is the ratio of root thickness to the outer radius of a uniform disk, and it directly governs the mass of the disk. As analysis is carried out for disks having same mass as that of uniform disk, for any given values of n and k, h0 depends directly on β. Hence the effect of each geometry parameter h0 (or β), n and k on the profile of rotating disk is studied separately in figures 2-4. The effect of β on disk thickness is presented in fig.2. Since increase in β increases disk mass, it is clearly observed that for any given value of n and k, increase in β increases the thickness of the disk at any radial location uniformly to accommodate the increased mass without changing the profile. Fig.3 depicts the effect of n on disk thickness. It is observed that for given values of k and β, increase in n increases root thickness of the disk without much affecting the radial mass distribution. Fig.4 depicts the variation of disk thickness with k where it is observed that for given values of n and β increase in k marginally reduces the root thickness of the disk but controls the mass distribution and hence the disk profile along the radius.

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

3.1. Effect Of Geometry Parameters (Without Attached Mass) Effect of geometry parameters on limit angular speed without attached mass is investigated by setting α = 0 for values of n ranging from 0.0 to 0.99 and k ranging from 0.0 to 2.0. The results have been validated with some existing studies [4, 5] and are found to be in good agreement. The contour and 3-D plot in fig.5 gives a better insight into the variation of limit angular speed with n and k. It should be noted that as there is no mass attached to the disk, parameter β has no effect on limit angular speed Ω1. Effect of parameters n and k are studied separately by taking sections of the 3-D plot and the maximum value of Ω1 is reported in each case (Ω1m). 3.1.1 Effect of Parameter n Increase in parameter n causes the root thickness (fig.3) to increase thereby providing more strength at the root. It is also quite obvious that for disks without attached masses, failure always occurs at the root, irrespective of the profile. As a result, increase in n increases Ω1, as shown in fig.6. 2

k

1.5

1

0.5

0

0

0.33

0.66

0.99

n Fig. 5. Contour and 3-D plot showing effect of n and k on Ω1 (α=0.0, β =0.10)

3.1.2 Effect of Parameter k In fig.7, the effect of parameter k is shown on limit angular speed. It is observed that for each value of n there exists a critical value of k at which maximum limit angular speed is obtained. For a given value of n, as k increases, root thickness decreases within a specified range that is dictated by selection of parameter n, but it is observed (fig.7) that Ω1 initially increases, reaches a maximum value and then decreases. This is attributed to the fact that variation of k causes the profile to vary. For values of k less than 1.0 the profile is convex, while for k greater than 1.0 the profile is concave with a linearly varying profile at k = 1.0. At very small values of k, the cross sectional area of the disk changes abruptly near the root rendering it to failure at lower angular speed. With increase in k, this abruptness in change in cross sectional area of the disk reduces thereby increasing Ω1. Then again beyond the critical value of k, Ω1 is observed to decrease due to reduction in root thickness thus weakening the disk at the root. 3.2. Effect Of Geometry Parameters (With Attached Mass) Effect of geometry parameters on limit angular speed with attached masses at specified locations is investigated for the same range of values of n (0.0 to 0.99) but a reduced range of k (0.0 to 1.0). The effect of attached mass and its location is included by assigning different values to α and ξm. It is observed that inclusion of attached mass at various locations causes the location of

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

yielding to shift radially from region of larger thickness to the lower ones. Consequently lower limit angular speed is obtained as compared to that of disk without attached mass for any given profile. 1.8

1.8

k=0.05, W 1m=1.502

n=0.99, W 1m=1.595

k =0.55,.W 1m=1.569

1.6

n=0.70, W 1m=1.286

1.6

W 1

k =1.00, W 1m=1.514 1.4

n=0.40, W 1m=1.180

1.4

n=0.08, W 1m=1.110 1.2

1.0

1.2

0

0.33

0.66

0.99

1.0

0

0.5

1

1.5

2

Parameter k

Parameter n

Fig. 6. Variation of Ω1 with n (α=0.0, β =0.10)

Fig. 7. Variation of Ω1 with k =0.10)

(α=0.0, β

3.2.1 Effect of Parameter n 1.8

a =0.50,xm=0.25, W 1m=1.183

1.8

a =0.50,xm=1.00, W 1m=0.804 1.2

W 1

a =1.00,xm=0.25, W 1m=0.987 1.2 a =0.50,xm=0.25, W 1m=1.435

0.6

a =0.50,xm=1.00, W 1m=1.365

0.6

a =1.00,xm=0.25, W 1m=1.356 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 8. Variation of Ω1 with n (k=0.55, β =0.10)

0.0

0.2

0.4

0.6

0.8

1.0

Parameter k

Parameter n

Fig. 9. Variation of Ω1 with k =0.10)

(n=0.99, β

For a disk with attached mass at different locations, variation of Ω1 with parameter n for different k values is plotted in fig.8. It is clearly evident that for location of attached mass near the root of the disk, increase in n increases Ω1, but for locations near the periphery there exists an optimum value of n that maximizes Ω1 for the given parameters. Increase in α at a given location, however, only reduces Ω1 due to increased stresses.

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

3.2.2 Effect of Parameter k

Normalized stress

The effect of variation of k on Ω1 is observed only at larger values of n. One such observation is shown in fig.9. It is quite evident that the trend of variation plotted in fig.9 is similar to that of disk without attached mass except for the fact that the magnitude of Ω1 has reduced. 1.0

1.0

0.8

0.8

k=0.01,W 1=1.233 k=0.05,W 1m=1.365 k=0.40,W 1=0.390

0.6

0.6 0.4

n=0.40,W 1=0.511

0.2

n=0.99,W 1=0.390

n=0.87,W 1m=0.804

0.2 0.0

0.0

x=0.0

0.4

Dimensionaless radius

x=1.0 x=0.0

x=1.0

Dimensionaless radius

Fig. 10. Radial distribution of Von Mises stress for different n values (k=0.55, β =0.10, α=0.50, ξm=1.00)

Fig. 11. Radial distribution of Von Mises stress for different k values (n=0.99, β =0.10, α=0.50, ξm=1.00)

The shifting of location of yielding, as shown in fig.10-11, is observed to play a significant role in defining the optimum value of n for a given value of k and vice-versa. In these two figures, Von Mises stress is normalized by yield stress ( σ 0 ). It is quite obvious that the effect of yield location shifting controls the optimum values of n and k. 3.3. Effect Of Loading Parameters It is observed that for given geometry parameters, increase in attached mass ratio α at any specific location reduces limit angular speed. Inclusion of attached mass enhances the centrifugal effect, which in fact lowers the value of Ω1.However the effect of attached mass location on Ω1 is more significant than that of attached mass ratio. It is observed that Ω1 falls drastically for values of ξm near the periphery of the disk. For a given geometry parameter, Fig.12 represent the contour and 3-D plot of the variation of Ω1 with α and ξm. For any combination of α and ξm the corresponding value of Ω1 can be obtained from Fig.12.

4. CONCLUSION The present study proposes a numerical scheme using variational method to analyze and model high speed rotating disks with radially distributed attached masses. Limit angular speed is obtained for various disk geometries with and without attached masses. Optimum geometry parameters are obtained that maximizes limit angular speed. Effect of attached mass and its location on limit angular speed and optimum geometry parameters is also studied and the results are reported in dimensionless forms and presented graphically. Although the paper presents static design analysis but the formulation procedure gives a kernel for subsequent dynamic

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National Conference on Advances in Engineering Design, 29-30 April 2005, Tamil Nadu, India

analysis and study many other complicating effects such as, shrink fitted disks, elastic-plastic behavior, effect of thermal stresses in rotating disks, etc.

Attached Mass Location

1

0.75

0.5

0.25 0.25

0.5

0.75

1

Attached Mass Ratio

Fig.12. Contour and 3-D plot showing effect of α and ξm on Ω1 k=0.55, β =0.10)

(n=0.99,

5. REFERENCES 1. Bhowmick S., Pohit G., Misra D. and Saha K.N. “Design of high speed impellers”, Proc. of Int. Conf. on Hyd. Engg. Res. Prac., IIT, Roorkee, India, 2004, 229-241. 2. Bhowmick S., Misra D. and Saha K.N. “On optimum disk profile for high speed st impellers having parabolic thickness variation”, Proc. of 31 Natl. Conf. on Fluid Mech. and Fluid Power, Jadavpur University, Kolkata, India, 2004, 417-424. 3. Chaudhary H.R. and Gupta U.S. “Rotation of hyperelastic annular and solid disks of variable thickness”, International Journal of Non-Linear Mechanics, 1992, 27(3), 341346. 4. Eraslan A.N. and Apatay T. “Elastic deformation of rotating parabolic disks: analytical solutions”, J. Fac. Eng. Arch. Gazi University, 2003, 18(2), 115-135. 5. Eraslan A. N. and Argeso H. “Limit angular velocities of variable thickness rotating disks”, International Journal of Solids and Structures, 2002, 39(12), 3109-3130. 6. Güven U. “Elastic-plastic stresses in rotating annular disk of variable thickness and variable density”, International Journal of Mechanical Science, 1992, 34(2), 133-138. 7. Sherbourne A.N. and Murthy D.N.S. “Stresses in disks with variable profile”, International Journal of Mechanical Science, 1974, 16, 449-459. 8. You L.H., Tang Y.Y., Zhang J.J. and Zheng C.Y. “Numerical analysis of elastic– plastic rotating disks with arbitrary variable thickness and density”, International Journal of Solids and Structures, 2000, 37 (52), 7809-7820.

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