P. G. Student, 2,3Faculty members

Department of Mechanical Engineering, Jadavpur University, Kolkata – 700032, India [email protected], [email protected], [email protected]

Abstract In the present work, the system modelled is a rotating disk of exponentially varying thickness, containing attached masses distributed radially at various locations. The limit angular speed for given geometry is obtained for various loading conditions. The problem is formulated through a variational method, where the radial displacement field is taken as unknown variable. Assuming a series solution and using Galerkin’s error minimization principle, the solution of the governing partial differential equation is obtained. 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 and loading parameters are studied. The values of optimum geometry parameters for given loading conditions are reported. It is also observed that radial shifting of attached mass location changes the location of yield initiation. The relevant results are reported in dimensionless form, and hence one can readily obtain the corresponding dimensional value for any geometry through appropriate normalizing parameters. It is observed that the present methodology is quite robust, stable and realistic and the documented results may be used by the practicing engineers as design monograms. Keywords: Limit angular speed, Von Mises criterion, Annular Disk.

Nomenclature a, b

Inner and outer radii of the disk.

{c }

The vector of unknown coefficients.

hr

Thickness of the disk at radius r.

h0

Root thickness of the annular disk.

hc

Thickness of equivalent uniform annular disk, β b.

m

Mass of the rotating disk.

n, k

Geometry parameters controlling the thickness variation of disk.

rm

Radii of the attached mass.

u

Displacement field of the disk.

α

Ratio of attached mass to disk mass.

β

Thickness ratio, hc / b.

λ

Aspect ratio, a /b.

∆

Parameter ( b - a ).

µ, E

Poisson’s ratio and elasticity modulus.

ξ

Normalized coordinate in radial direction, (r - a ) / ∆ .

ρ

Density of the disk material.

σ0

Yield stress of the disk material.

σr

Radial stress.

σθ

Tangential stress

φi

The set of orthogonal polynomials used as coordinate functions.

ω

Angular velocity of the disk.

ξm

Non-dimensional radii of the attached mass location.

Ω

Dimensionless angular velocity, ω b ρ / σ 0 .

1. Introduction High speed rotating disks find widespread application in mechanical engineering as a precision component, some common examples being high speed impellers, gears and flywheels. The stress distribution in a rotating disk depends on the rotational speed. At a particular speed the developed stresses exceed the yield stress of the material. This particular speed is termed as limit angular speed of the rotating disk. It is well understood that stresses in a rotating disk are maximum at its root. It is this theory that initiated the idea of radially varying the thickness of a disk without hampering its axisymmetric nature. Depending on the need and method of mounting, the rotating disks can be made solid or annular. One specific advantage of annular disks arises from the fact that, since usually disks are manufactured using forging processes, very large forgings are likely to have defects in the material at the center. Hence, to eliminate uncertainties it is a common practice to bore a central hole along the axis of the rotor [1]. Analysis of elastic stress state in rotating disks having variable thickness up to the point of yielding with the objective to determine limit angular speed has been extensively carried out by many researchers. Analysis based on dynamic relaxation technique [2] has been applied effectively to study elastic stress behaviour for isotropic material possessing non-linear stress-strain relationship. Exponential variation of thickness was first discussed in [3] assuming Tresca’s criterion and its associated flow rule along with linear strain hardening material behaviour, although solution satisfying all boundary and continuity conditions was not obtained. However in [4], a similar problem was studied and an analytical solution was presented that satisfy all boundary and continuity conditions. The analytical method was further applied in [5] to obtain solution for elastic deformation of solid and annular disk having parabolically varying thickness subjected to different boundary conditions. In

another paper, application of variational principle [6] involving Galerkin’s principle and assuming a series solution has provided an efficient numerical scheme for the estimation of limit angular speed of disks with and without attached masses. The analysis is based on Von Mises stress criterion and has been extended for design and optimization of various types of disk geometries [7, 8]. 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, 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 exponentially 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.

2. Mathematical Formulation An annular disk of exponentially varying thickness subjected to centrifugal loading is shown in fig.1. The mathematical model is framed based on the assumptions that the material of disk is homogeneous, isotropic and linear-elastic.

Fig. 1. An annular disk having exponentially varying thickness The thickness variation of the disk in fig.1 is given by h (ξ ) = h 0 e xp [ − n (ξ ) k ] where the parameters n and k control the disk geometry. In each case of thickness variation, weight of the disk is kept fixed by adjusting the value of h0, i.e. the disk thickness at r = a. The purpose of this study is to maximize the operating speed range of a disk having mass equivalent to that of a uniform disk by varying its thickness. For this purpose, a parameter β is introduced which defines the mass of the equivalent uniform disk having same inner and outer radii. As the analysis is carried out under plane stress assumption, thickness ratio, β is assumed to be 0.10 in the present study. The variation of profile with geometry parameters k and n is shown in fig. 2 (a, b) for disks with constant root thickness. With increase in rotational speed the stresses increase due to enhanced effect of centrifugal force, the magnitude of which is governed by the boundary conditions of the disk. The

solution for the displacement field is obtained from the minimum potential energy principle δ(π) =

δ(U+V) = 0, where, U and V are strain energy stored in the disk and potential energy arising out of centrifugal force, respectively. Substituting the expressions of U and V in the energy principle δ(π) = 0, the governing equilibrium equation is obtained [1]. ∆ π E δ 1 − µ 2

2 1 u 2 2 µ u d u ( ∆ ξ + a) d u 2 2 + hd ξ 2 π ρ ω ( ξ a ) uhd ξ − ∆ ∆ + + =0 ( ∆ ξ + a) ∆ d ξ ∆ 2 d ξ 0 0 1

∫

∫

The displacement function u(ξ) in equation (1), is approximated by u (ξ ) ≅

(1)

∑ c φ , i=1, 2,…, n, i i

where φi is the set of orthogonal functions developed through Gram-Schmidt scheme. The necessary starting function to generate the higher order orthogonal functions is selected by satisfying the relevant boundary conditions;

σr

(a )

= 0 and σ r

( b)

= 0.

Fig. 2 (a). Profile variation keeping n fixed at 1.5

Fig. 2 (b). Profile variation keeping k fixed at 0.5

while k varies from 0.0-1.0 (λ = 0.20)

while n varies from 0.0-2.0 (λ = 0.20)

The effect of attached mass is taken into consideration through a factor α, which indicates the ratio of the attached mass to the mass of the total disk m. This attached mass will contribute to the centrifugal force by an additional amount, α m ω rm , where rm indicates the radius of the location of the attached 2

mass. The contribution of this term is included in the expression of equation (1) and substituting the assumed series solution of u (ξ), followed by Galerkin’s error minimization principle, the following equation is obtained.

E 1− µ

n

2

n

∑∑ i =1

= ρω

ci

j =1 2

∫

1

φ φ

∑ ∫ {( ∆ ξ n

i =1

µ

i j + ( φ i′φ 0 ξ a) ( ∆ + ∆ 1

0

+ a )2 φ i

}

j

+ φ i φ ′j ) +

(∆ξ + a ) φ i′φ ∆2

α m 2 hd ξ + ω ξ m φ i 2π ∆

ξm

′j hd ξ

(2)

The above equations are the set of system governing equations which are expressed in matrix form as [K] {c} = {R}. Finally the unknown coefficients are obtained from {c} = [K]-1 {R}.

3. Results and discussion The numerical values of the different system parameters, considered are, E = 207 GPa, µ = 0.3, ρ = 7850 Kg/m3 and σ 0 = 350 MPa. The dimensionless angular speed corresponding to the onset of yielding is denoted by Ω1. Various parameters considered in the present study are classified into two categories. Thickness control parameters k and n, and aspect ratio λ are considered as geometry parameters and loading effect of attached mass α and its location ξ

m

are considered as loading

parameters. Extensive numerical simulation is carried out to observe the effect of different system parameters on Ω1. In each case, the analysis is carried out for disks of identical mass. Von Mises yield criterion is considered here to establish the onset of yielding, in which the yield condition is given by σ r

2

2

2

− σ rσθ + σθ ≥ σ 0 .

3.1 Disks without attached masses The effect of geometry parameters on limit angular speed at two different aspect ratios is investigated and plotted in fig.3-4. It is observed that for given λ, with increase in n, Ω1 also increases; however for each value of n, there exists an optimum value of k (kopt) that maximizes Ω1. The variation of Von Mises stress for different profiles and the representation of the profiles are provided in fig. 5 (a, b). The figure indicates that for each case maximum stresses occur at the root of the disk. For values of k less than kopt, root thickness changes abruptly giving rise to stress concentration effects, while for larger values of k, root thickness further reduces thus lowering the strength at root. This probably explains the existence of kopt for each value of n.

Fig. 3. Contour and 3-D plot of variation of Ω1 with geometry parameters (λ = 0.20) A similar observation is made in fig. 6 (a, b) for a different value of λ. It is also observed that increase in λ is accompanied by reduction in Ω1 and higher values of kopt.

Fig.4. Contour and 3-D plot of variation of Ω1 with geometry parameters (λ = 0.40)

Fig.5 (a). Radial distribution of Von-Mises stress

Fig. 5 (b). Disk profile for different k values

Fig. 6 (a). Radial distribution of Von-Mises stress

Fig. 6 (b). Disk profile for different k values

3.2 Disks with attached masses In the present study, attached mass ratio and its location is varied from 0.00 to 1.00 and 0.25 to 1.00 respectively. Detailed investigation reveals reduced values of Ω1 as well as kopt for increasing attached mass ratio at a given location. In fig. 7 (a. b), the effect of parameter k on limit angular speed is studied under different loading conditions for two different aspect ratios. In these figures, n is kept fixed at 1.0 but its effect for a given value of aspect ratio and loading parameters is presented later. It is evident from the figures that increase in λ is accompanied by fall in limit angular speed as well as kopt.

Fig. 7. Effect of k on Ω1 under different loading parameters keeping n constant at 1.0

Fig.8. Radial distribution of Von-Mises stress at different aspect ratios (n = 1.0, k = 0.5) It is further observed that for a given attached mass ratio, an increase in the value of ξ

m

also reduces

Ω1 and kopt (fig. 7 a-b). It is due to the fact that attached masses add to the centrifugal effect thereby increasing the stresses leading to reduced values of Ω1. Reduced kopt value results in larger root thickness, thereby increasing strength at the root which takes up the additional centrifugal effect. The Von-Mises stress distribution at given values of n and k for different aspect ratios under various

combination of loading parameters is plotted in fig. 8 (a, b). It is observed that the maximum stresses occur at the root of the disk irrespective of the loading conditions. The effects of geometry parameters on limit angular speed at λ = 0.20 for given values of attached mass ratio and location is studied in detail and some sample plots are shown in figs. 9-10. A similar study is carried out for λ = 0.40 and the results are plotted in figs. 11-12. These figures indicate once again, that increase in λ and α cause a reduction in the value of limit angular speed and kopt. Thus a detail study on the interactions of λ and α may yield come interesting findings towards an optimized disk design. In figs. 13-14, the effect of loading parameters on limit angular speed is plotted for a given values of geometry parameters and aspect ratios. These variations are almost identical to that of the case of a solid disk, as reported in an earlier study [8].

Fig. 9. Contour and 3-D plot of variation of Ω1 with geometry parameters (α = 0.50, ξ

m=

0.75, λ = 0.20)

Fig. 10. Contour and 3-D plot of variation of Ω1 with geometry parameters (α = 1.00, ξ

m=

0.75, λ = 0.20)

4. Conclusion Modeling and analysis of exponentially varying high speed annular disks is carried out using variational formulation and assuming a series solution of unknown displacement field. Assuming Von -

Fig. 11. Contour and 3-D plot of variation of Ω1 with geometry parameters (α = 0.50, ξ

m=

0.75, λ = 0.40)

Fig. 12. Contour and 3-D plot of variation of Ω1 with geometry parameters (α = 1.00, ξ

m=

0.75, λ = 0.40)

Fig. 13. Contour and 3-D plot of variation of Ω1 with loading parameters (n = 1.50, k = 0.50, λ = 0.20)

Fig. 14. Contour and 3-D plot of variation of Ω1 with loading parameters (n = 1.50, k = 0.50, λ = 0.40) Mises failure criterion the effect of various geometry and loading parameters on limit angular speed is investigated and corresponding parameter values leading to the optimization of limit speed are reported. The results obtained are in dimensionless form and presented graphically. The results provide a substantial insight in understanding the behavior of rotating disks with and without attached masses. The method of formulation gives a kernel for subsequent dynamic analysis and study of many other complicating effects. Further the method developed has application potential in various other problems, e.g., shrink fitted disks, disks made of anisotropic material, etc.

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