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Profile optimization of variable thickness rotating disc
0045-7949/92 SS.00 + 0.00 0 1992 Fmgmotl Press plc
Computers & Slmcfures Vol. 42, No. 5, pp. W-813, 1992 Plintcd in Great Britain.
PROFILE OPTIMIZATION OF VARIABLE THICKNESS ROTATING DISC G. S.
RAY
and B. K.
SINHA
Central Mechanical Engineering Research Institute, Mahatma Gandhi Avenue, Durgapur-713 209, India (Received 12 December 1990) Abstract-A procedure of obtaining an optimum configuration of an asymmetric disc, which rotates at high speed, is presented. The objective of strength reliability is achieved through analytical consideration within given constraints. The application of the computation technique for optimization and selection of characteristic disc profile is illustrated. The program has a special feature of evaluating the centrifugal stresses which are constrained within the strength of the disc material.
NOTATION
g
L N r u .7 a a’ Y 6 Y a* 0
The objective of the present paper is to optimize the geometrical configuration of the disc profile between the inner and outer diameters in such a way that the stress distribution throughout the disc is controlled within the strength of material.
acceleration due to gravity thickness of the disc element in the Z-direction number of elemental rings radial co-ordinate radial displacement axial co-ordinate turning angle angular strain specific weight strain Poisson’s ratio non-dimentional stress angular speed
STRESS CALCULATION
Figure l(a) shows the cross-sectional half-view in elevation of a rotating disc of axially varying thickness. Figure l(b) shows the discretization of the same disc for the purpose of analysis. The numerical method of stress calculation as proposed by Schilhansl[4] has been suitably modified for the analysis. The stress parameter is made non-dimensional on normalizing by yo*r$/g, the stress parameter in outer discretized thin ring, as if rotating individually. Thus the non-dimensional stresses in radial and tangential directions in the disc are [S]
Subscripts 0 r
t
T N
referring to r = 0 radial tangential referring to outer boundary (tip) partaining to the last elemental ring
(1) (2)
INTRODUCTION
In the recent years, considerable attention has been paid to the asymmetrical disc which is often used as an impeller in radial flow machines. The distribution of centrifugal stresses constitutes a major constraining factor for a disc rotating at high speed because failure has a catastrophic effect. In the reported methods of disc profile optimization, symmetrical configuration about the centroidal plane normal to the axis of rotation has generally been assumed. Seireg and Surana [l] adopted the optimization technique for a rational design of symmetric disc with high centrifugal stresses within given constraints. Chern and Prager [2] solved the problem of minimum weight design of axially symmetric, rotating, elastic discs in terms of uniform strength under the influence of centrifugal force and uniform radial traction along the circular edge. Malkov and Salganskaya [3] developed a technique for obtaining numerical results for an optimal distribution of material in a symmetric disc in rotation. 809
A computational scheme has been developed for calculation of stresses through the numerical solution of the equations of equilibrium and compatibility. The conditions of continuity in terms of strain compatibility and the equilibrium of forces as well as moments when considered at the interface of the two consecutive elements give rise to a set of recurrence relations between stress and strain. In conjunction with boundary conditions, a set of simultaneous equations thus obtained may be solved for stresses in each element on a digital computer using matrix method. Accuracy of discretization
The discretization or replacement of the continuous disc as a finite number of ring elements in the
G. S. RAY and B. K. SINI-M
810
surface
.-. Axis of rotation
(a)
(b)
Fig. 1. (a) Disc of variable thickness. (b) Replacement of the disc by a system of rings.
present scheme may lead to inaccuracies. The choice of the number of elements for a specific problem may be intuitive in the beginning but the number of elements, in general, should be more near the region of sharp stress gradient, i.e. near the inner and outer boundaries of the disc. Use of non-uniform element size will, therefore, be economical for successive iterations. The size of the elements or girds considered for the case investigated is shown in Fig. l(b). However, a rational way for the choice of an optimum number of elements is to test the sensitivity of the solution on the number of elements considered. Figure 2 shows the effect of the number of elements considered on the maximum value of radial and tangential stresses. The result is fairly insensitive to element number beyond N = 18.
0.1t
0
2
4
6
8
10
12 14 16
18
20
8
22
Number of grids
Fig. 2. Effect of maximum radial and tangential stress coefficients on the number of stepped rings.
Hence, 18 elements are considered appropriate for the analysis. THE OPTIMIZATION
PROBLEM
A computerized procedure is helpful in analysing the stresses of a given disc. While designing a rotor for a given duty, the objective will be to arrive at an optimum decision for the dimension of the disc under the various constraints of space, cost and other physical limitations. A rational design procedure requires the formulation of the problem in mathematical terms and development of logic and decision steps to facilitate the search for optimum solutions. The first step in the process is to identify all the significant parameters influencing the system under consideration. These parameters are then divided into given input and required output. The latter may have independent as well as dependent variables. The independent output parameters are generally known as decision parameters. The next task is to define the objective function in terms of mathematical description and also to identify the constraint or limitation under which the optimization is intended. The objective function may vary according to the designer’s objective and experience. A systematic search is then initiated to seek the solution within the domain of constraints with the highest feasible rating according to the chosen criterion. In the present paper, the Rosenbrock Hill Climb procedure [6] for optimization is coupled with the stress computation procedure in order to formulate a generalized computer code for the required dimensions of an optimized profile of the disc. The algorithm for the computational scheme of optimization with given constraints is presented in the form of a flow chart in Fig. 3.
811
Profile optimization of a rotating disc
ONE FAILURE IN
Fig. 3. Flow chart for optimum solution search algorithm.
System parameters (i) Radius
of
i=l,2,3
,...,
the
elemental
ring,
ri,
N.
(ii) Axial thicknesses of the elemental i = 1,2,3, . . . , N.
ring, Li,
The given input parameters are the radii of elemental rings and the axial thickness at the boundaries (ri, i = 1,2, 3, . . . , N), (L,, 15~). The required independent output parameters (decision parameters) are the axial thickness of the ring elements other than those at the boundaries (L,,i=2,3 ,... , N - 1) which defines the optimum configuration. Constraints The maximum and minimum limits for the axial thickness of disc are L, and L, respectively which sets the relation. L,
i=2,3 ,...,
N-l.
(3)
For a rotating disc of uniform thickness, the magnitude of the maximum tangential stress or hoop stress at any point is always observed to be greater than that of the maximum radial stress. This has been ascertained from the stress analysis carried out on various impellers of radial flow machines as well as the test results of other researchers. The same physical condition may thus be used as an implicit constraint as given below for the present problem to avoid an unrealistic configuration of the disc during optimization (e,ro)ln,, < (G),,,
.
(4)
The above constraint is described as an implicit constraint since the parameters do not appear as a simple explicit relation of a single decision parameter at a time. The stresses (a$,) and (a$,), on the other hand, are involved mathematical function of the geometrical dimensions of the disc which are taken as decision parameters in the present problem.
G. S. RAY and B. K. SINHA
812
Design objective
Initial values
Several design objectives can be envisaged in the present problem. The selection of any of these, for appropriate objectives, depends on the requirements and the working experience. The following design criteria have been used here.
The variable to be optimized is the axial thickness of the disc with discretization (N = 18). The parameters held constant during optimization and expressed in mm are
1. Minimization of the difference between the maximum and minimum tangential stresses at any point in the disc. To obtain the optimum profile of the rotor,
the difference between the maximum and minimum tangential stresses is to be minimized for maximum utilization of disc material. This criterion is important when the tip thickness of the disc is specified from a consideration of functional aspect. The objective function, F,, for the optimization may therefore, be expressed as F1 = (a ?D),~Ix- (c ;fDIknin.
(3
2. Minimization of the maximum level of tangential stress in the disc. In the case of rotating disc, the maximum tangential stress invariably becomes the maximum principal stress that determines the failure criterion. The objective function F2, may, therefore, be written as Fr = (0 :D ),a,.
(6)
Search of optimum solution
The stress distributions in the disc are calculated from eqns (1) and (2). Optimization of the profile requires starting values of the variables and physical and geometrical constraints specified as an input. On evaluating the objective functions, search procedure is carried out to locate a new feasible profile design with the improved values. The search is then continued until a point is reached where from no further improvement in the objective function is obtained. The corresponding geometrical parameters provide an optimum solution.
L,=37.5,
L,=2.0,
The geometrical therefore
r,=6.0,
constraint
2mm
i=2,
r,=82.5.
for the problem
3 ,...,
N-l.
Optimum disc projile
Figure 4 shows the optimum disc profile and the corresponding stress distributions along nondimensional radius R( = ri/rr) according to the objective function, F, . It is observed that the difference between maximum and minimum tangential stresses is reduced for the optimum disc profile. It is also interesting to note that the disc volume is reduced by 17% as the original profile is changed to the optimum one. This observation shows that the selected design criterion tends to use the disc material under stressed condition in an optimum manner through redistribution along the disc radius. Figure 5 compares the optimum profile and the stress distributions along R according to the objective function, F2. Similar to the objective function, F, , the maximum tangential stress is reduced in this case. The volume of the disc is also reduced as the profile tends to be the optimum one and the reduction in volume is 17% the same as that obtained with the objective function, F, . CONCLUSIONS
1. The formulation and logic of optimization is so presented that the numerical solution produces the required configuration of the disc from the strength point of view. The computed configurations may
Volume of the disc referred to Fig 1 (a) = 176.6 cm3 Optimum volume = 145.58 cm3 Poisson’s ratio = 0.33
On rear surface
On front surface
is
On front surface
Fig. 4. Optimum disc profile and corresponding stress distribution for design criterion 1.
813
Profile optimization of a rotating disc Volume of the disc referred to Fig 1 (a) = 176.6 cm3 Optimum volume = 145.56 cm3 Poisson’s ratio = 0.33
On rear surface
-CL-
:
On front surface
-0.2 +Bore
6.21-Bore i
On front surface
Fig. 5. Optimum disc profile and corresponding stress distribution for design criterion 2. have cross-sectional irregularities at times owing to the discretized nature of the analytical scheme. The profile can lx smoothed to facilitate fabrication without seriously affecting the constraints. 2. The accuracy of the solution depends on the choice of the optimum number of grids or segments. It is, therefore, interesting to track the sensitivity of the computed solutions with increasing number of elements. To save computation time, further decretization is inhibited as soon as the results become fairly insensitive to any increase in the number of elements. 3. Two design stress criteria are considered to illustrate their influence on the optimum profile. Both the criteria suit the problem with respect to the laid down objective and reduction in volume of the disc is also identical. However, there may be some more design criteria for which careful attention must be given for their selection. 4. A complete optimization of a radial flow impeller may be achieved by superimposing the presence of laterally attached blade segments on the ring elements of the disc. The effects such as aerodynamic,
thermodynamic, vibration, etc. may be further superimposed step-by-step to improve the profile. HOWever, containment of stress within the strength of the disc material under the influence of centrifugal force developed due to high speed is of utmost importance. Acknowledgement-The
authors are grateful to the Director of the Central Mechanical Engineering Research Institute for his kind permission to publish this paper. REFERENCES
1. A. Seireg and K. S. Surana, Optimum design of rotating disks. J. Engng Ind. Trans ASME 92, l-10 (1970). 2. J. M. Chem and W. Prager, Optimal design of rotating disk for given radial displacement of edge. J. Optimiz. Theory ipplic.
6, 161-l?O (1970).
_
3. V. P. Malkov and E. A. Salaanskava. Ontimal material distribution in rotating di&s for maximal strength. Sov. Aeronaut. 19, 46-50 (1976). 4. M. J. Schilhansl, Stress analysis of a radial flow rotor. Tram ASME,
J. Engng Power 84, 124-130 (1962).
5. G. S. Ray, Stress analysis and optimum design of rotors in radial flow machines. Ph.D. thesis, University of Burdwan (1984). 6. D. Wilde, Optimum Seeking Methods, p. 107. PrenticeHall (1984).
Computers & Slmcfures Vol. 42, No. 5, pp. W-813, 1992 Plintcd in Great Britain.
PROFILE OPTIMIZATION OF VARIABLE THICKNESS ROTATING DISC G. S.
RAY
and B. K.
SINHA
Central Mechanical Engineering Research Institute, Mahatma Gandhi Avenue, Durgapur-713 209, India (Received 12 December 1990) Abstract-A procedure of obtaining an optimum configuration of an asymmetric disc, which rotates at high speed, is presented. The objective of strength reliability is achieved through analytical consideration within given constraints. The application of the computation technique for optimization and selection of characteristic disc profile is illustrated. The program has a special feature of evaluating the centrifugal stresses which are constrained within the strength of the disc material.
NOTATION
g
L N r u .7 a a’ Y 6 Y a* 0
The objective of the present paper is to optimize the geometrical configuration of the disc profile between the inner and outer diameters in such a way that the stress distribution throughout the disc is controlled within the strength of material.
acceleration due to gravity thickness of the disc element in the Z-direction number of elemental rings radial co-ordinate radial displacement axial co-ordinate turning angle angular strain specific weight strain Poisson’s ratio non-dimentional stress angular speed
STRESS CALCULATION
Figure l(a) shows the cross-sectional half-view in elevation of a rotating disc of axially varying thickness. Figure l(b) shows the discretization of the same disc for the purpose of analysis. The numerical method of stress calculation as proposed by Schilhansl[4] has been suitably modified for the analysis. The stress parameter is made non-dimensional on normalizing by yo*r$/g, the stress parameter in outer discretized thin ring, as if rotating individually. Thus the non-dimensional stresses in radial and tangential directions in the disc are [S]
Subscripts 0 r
t
T N
referring to r = 0 radial tangential referring to outer boundary (tip) partaining to the last elemental ring
(1) (2)
INTRODUCTION
In the recent years, considerable attention has been paid to the asymmetrical disc which is often used as an impeller in radial flow machines. The distribution of centrifugal stresses constitutes a major constraining factor for a disc rotating at high speed because failure has a catastrophic effect. In the reported methods of disc profile optimization, symmetrical configuration about the centroidal plane normal to the axis of rotation has generally been assumed. Seireg and Surana [l] adopted the optimization technique for a rational design of symmetric disc with high centrifugal stresses within given constraints. Chern and Prager [2] solved the problem of minimum weight design of axially symmetric, rotating, elastic discs in terms of uniform strength under the influence of centrifugal force and uniform radial traction along the circular edge. Malkov and Salganskaya [3] developed a technique for obtaining numerical results for an optimal distribution of material in a symmetric disc in rotation. 809
A computational scheme has been developed for calculation of stresses through the numerical solution of the equations of equilibrium and compatibility. The conditions of continuity in terms of strain compatibility and the equilibrium of forces as well as moments when considered at the interface of the two consecutive elements give rise to a set of recurrence relations between stress and strain. In conjunction with boundary conditions, a set of simultaneous equations thus obtained may be solved for stresses in each element on a digital computer using matrix method. Accuracy of discretization
The discretization or replacement of the continuous disc as a finite number of ring elements in the
G. S. RAY and B. K. SINI-M
810
surface
.-. Axis of rotation
(a)
(b)
Fig. 1. (a) Disc of variable thickness. (b) Replacement of the disc by a system of rings.
present scheme may lead to inaccuracies. The choice of the number of elements for a specific problem may be intuitive in the beginning but the number of elements, in general, should be more near the region of sharp stress gradient, i.e. near the inner and outer boundaries of the disc. Use of non-uniform element size will, therefore, be economical for successive iterations. The size of the elements or girds considered for the case investigated is shown in Fig. l(b). However, a rational way for the choice of an optimum number of elements is to test the sensitivity of the solution on the number of elements considered. Figure 2 shows the effect of the number of elements considered on the maximum value of radial and tangential stresses. The result is fairly insensitive to element number beyond N = 18.
0.1t
0
2
4
6
8
10
12 14 16
18
20
8
22
Number of grids
Fig. 2. Effect of maximum radial and tangential stress coefficients on the number of stepped rings.
Hence, 18 elements are considered appropriate for the analysis. THE OPTIMIZATION
PROBLEM
A computerized procedure is helpful in analysing the stresses of a given disc. While designing a rotor for a given duty, the objective will be to arrive at an optimum decision for the dimension of the disc under the various constraints of space, cost and other physical limitations. A rational design procedure requires the formulation of the problem in mathematical terms and development of logic and decision steps to facilitate the search for optimum solutions. The first step in the process is to identify all the significant parameters influencing the system under consideration. These parameters are then divided into given input and required output. The latter may have independent as well as dependent variables. The independent output parameters are generally known as decision parameters. The next task is to define the objective function in terms of mathematical description and also to identify the constraint or limitation under which the optimization is intended. The objective function may vary according to the designer’s objective and experience. A systematic search is then initiated to seek the solution within the domain of constraints with the highest feasible rating according to the chosen criterion. In the present paper, the Rosenbrock Hill Climb procedure [6] for optimization is coupled with the stress computation procedure in order to formulate a generalized computer code for the required dimensions of an optimized profile of the disc. The algorithm for the computational scheme of optimization with given constraints is presented in the form of a flow chart in Fig. 3.
811
Profile optimization of a rotating disc
ONE FAILURE IN
Fig. 3. Flow chart for optimum solution search algorithm.
System parameters (i) Radius
of
i=l,2,3
,...,
the
elemental
ring,
ri,
N.
(ii) Axial thicknesses of the elemental i = 1,2,3, . . . , N.
ring, Li,
The given input parameters are the radii of elemental rings and the axial thickness at the boundaries (ri, i = 1,2, 3, . . . , N), (L,, 15~). The required independent output parameters (decision parameters) are the axial thickness of the ring elements other than those at the boundaries (L,,i=2,3 ,... , N - 1) which defines the optimum configuration. Constraints The maximum and minimum limits for the axial thickness of disc are L, and L, respectively which sets the relation. L,
i=2,3 ,...,
N-l.
(3)
For a rotating disc of uniform thickness, the magnitude of the maximum tangential stress or hoop stress at any point is always observed to be greater than that of the maximum radial stress. This has been ascertained from the stress analysis carried out on various impellers of radial flow machines as well as the test results of other researchers. The same physical condition may thus be used as an implicit constraint as given below for the present problem to avoid an unrealistic configuration of the disc during optimization (e,ro)ln,, < (G),,,
.
(4)
The above constraint is described as an implicit constraint since the parameters do not appear as a simple explicit relation of a single decision parameter at a time. The stresses (a$,) and (a$,), on the other hand, are involved mathematical function of the geometrical dimensions of the disc which are taken as decision parameters in the present problem.
G. S. RAY and B. K. SINHA
812
Design objective
Initial values
Several design objectives can be envisaged in the present problem. The selection of any of these, for appropriate objectives, depends on the requirements and the working experience. The following design criteria have been used here.
The variable to be optimized is the axial thickness of the disc with discretization (N = 18). The parameters held constant during optimization and expressed in mm are
1. Minimization of the difference between the maximum and minimum tangential stresses at any point in the disc. To obtain the optimum profile of the rotor,
the difference between the maximum and minimum tangential stresses is to be minimized for maximum utilization of disc material. This criterion is important when the tip thickness of the disc is specified from a consideration of functional aspect. The objective function, F,, for the optimization may therefore, be expressed as F1 = (a ?D),~Ix- (c ;fDIknin.
(3
2. Minimization of the maximum level of tangential stress in the disc. In the case of rotating disc, the maximum tangential stress invariably becomes the maximum principal stress that determines the failure criterion. The objective function F2, may, therefore, be written as Fr = (0 :D ),a,.
(6)
Search of optimum solution
The stress distributions in the disc are calculated from eqns (1) and (2). Optimization of the profile requires starting values of the variables and physical and geometrical constraints specified as an input. On evaluating the objective functions, search procedure is carried out to locate a new feasible profile design with the improved values. The search is then continued until a point is reached where from no further improvement in the objective function is obtained. The corresponding geometrical parameters provide an optimum solution.
L,=37.5,
L,=2.0,
The geometrical therefore
r,=6.0,
constraint
2mm
i=2,
r,=82.5.
for the problem
3 ,...,
N-l.
Optimum disc projile
Figure 4 shows the optimum disc profile and the corresponding stress distributions along nondimensional radius R( = ri/rr) according to the objective function, F, . It is observed that the difference between maximum and minimum tangential stresses is reduced for the optimum disc profile. It is also interesting to note that the disc volume is reduced by 17% as the original profile is changed to the optimum one. This observation shows that the selected design criterion tends to use the disc material under stressed condition in an optimum manner through redistribution along the disc radius. Figure 5 compares the optimum profile and the stress distributions along R according to the objective function, F2. Similar to the objective function, F, , the maximum tangential stress is reduced in this case. The volume of the disc is also reduced as the profile tends to be the optimum one and the reduction in volume is 17% the same as that obtained with the objective function, F, . CONCLUSIONS
1. The formulation and logic of optimization is so presented that the numerical solution produces the required configuration of the disc from the strength point of view. The computed configurations may
Volume of the disc referred to Fig 1 (a) = 176.6 cm3 Optimum volume = 145.58 cm3 Poisson’s ratio = 0.33
On rear surface
On front surface
is
On front surface
Fig. 4. Optimum disc profile and corresponding stress distribution for design criterion 1.
813
Profile optimization of a rotating disc Volume of the disc referred to Fig 1 (a) = 176.6 cm3 Optimum volume = 145.56 cm3 Poisson’s ratio = 0.33
On rear surface
-CL-
:
On front surface
-0.2 +Bore
6.21-Bore i
On front surface
Fig. 5. Optimum disc profile and corresponding stress distribution for design criterion 2. have cross-sectional irregularities at times owing to the discretized nature of the analytical scheme. The profile can lx smoothed to facilitate fabrication without seriously affecting the constraints. 2. The accuracy of the solution depends on the choice of the optimum number of grids or segments. It is, therefore, interesting to track the sensitivity of the computed solutions with increasing number of elements. To save computation time, further decretization is inhibited as soon as the results become fairly insensitive to any increase in the number of elements. 3. Two design stress criteria are considered to illustrate their influence on the optimum profile. Both the criteria suit the problem with respect to the laid down objective and reduction in volume of the disc is also identical. However, there may be some more design criteria for which careful attention must be given for their selection. 4. A complete optimization of a radial flow impeller may be achieved by superimposing the presence of laterally attached blade segments on the ring elements of the disc. The effects such as aerodynamic,
thermodynamic, vibration, etc. may be further superimposed step-by-step to improve the profile. HOWever, containment of stress within the strength of the disc material under the influence of centrifugal force developed due to high speed is of utmost importance. Acknowledgement-The
authors are grateful to the Director of the Central Mechanical Engineering Research Institute for his kind permission to publish this paper. REFERENCES
1. A. Seireg and K. S. Surana, Optimum design of rotating disks. J. Engng Ind. Trans ASME 92, l-10 (1970). 2. J. M. Chem and W. Prager, Optimal design of rotating disk for given radial displacement of edge. J. Optimiz. Theory ipplic.
6, 161-l?O (1970).
_
3. V. P. Malkov and E. A. Salaanskava. Ontimal material distribution in rotating di&s for maximal strength. Sov. Aeronaut. 19, 46-50 (1976). 4. M. J. Schilhansl, Stress analysis of a radial flow rotor. Tram ASME,
J. Engng Power 84, 124-130 (1962).
5. G. S. Ray, Stress analysis and optimum design of rotors in radial flow machines. Ph.D. thesis, University of Burdwan (1984). 6. D. Wilde, Optimum Seeking Methods, p. 107. PrenticeHall (1984).