Steady state stress analysis of centrifugal fan impellers

1

0045.7949187 13.00 + 0.00 1987 Pcrpmon Joumab Ltd.

STEADY STATE STRESS ANALYSIS OF CENTRIFUGAL FAN IMPELLERS V. RAMAMURTI and P. BALASUBRAMANIAN Department of Applied Mechanics, Indian Institute of Technology, Madras 600 036, India (Received 13 May 1986) Abstract-Cyclic symmetric structures such as centrifugal fan impellers are considered in this paper. One of the identical sectors is chosen for the finite element analysis. A triangular plate element is employed, The stiffness and load matrices are obtained in cylindrical coordinates. The displacements are obtained using a modified submatrices elimination scheme. The efficiency of this scheme is discussed. Impeller stresses are calculated and checked with the results obtained using a strain gauge technique.

have rotational periodicity or cyclic symmetry. Zienkiewicz and Scott [4] applied this concept to pump impeller stress analysis. Zak ef of. [S] have extended the perturbation analysis technique to the stress analysis of non-axisymmetric configurations. Recently many commercial programs, such as NASTRAN 161,have been developed which can adequately analyse cyclic symmetric structures. Modified Potters’ method, which is an out-of-core solution technique, has been discussed for the analysis of cychc symmetric structures [7], but the symmetric nature of the stiffness matrix was not utilized.

NOTATION stiffness submatrices at ith station total number of identical sectors outer radius of the annular plate inner radius of the annular plate load vector at ith station total number of stations radius of a point in consideration displacement vector at ith station Poisson’s ratio density of the material rotational speed in rad/sec radial stress 1. INTRODUCI’ION

2. PRESENT ANALYSIS

impeller is the severely stressed part in the centrifugal fan assembly. Although rotation, temperature and aerodynamics produce stresses in the impeller, the main contribution is from rotation. For many years the conventional method of estimating impeller stresses suggested by Haerle [l] has been used. Patton [2] has used this method to compare the experimental results obtained from brittle coating and strain gauge techniques. However, the conventional method requires a large factor of safety whereas experimental methods are too costly. Hence these methods are unsatisfactory for the economic design of high pressure centrifugal fans. Therefore, a reliable analytical technique is very important to enable rapid study of the various design models at marginal extra cost. Bell and Benham [3] have applied FEM to the centrifugal impeller; only one quarter of the impeller has been considered by assuming the impeller geometry to be symmetric about x and y axes. Results were compared with those of strain gauge and brittle coating techniques. in practice the actual boundary conditions could not be reahsed. In the early 1980s a new trend was initiated in the stress analysis by incorporating the structural periodicity. If a structure has identical substructures coupled together in an identical manner in the circumferential direction, then it is considered to

In this paper, an out-of-core solution technique for the stress analysis of cyclic symmetric structures is presented. This is an extension of the submatrices elimination scheme [8]. The cycIic symmetric structure contains identical sectors positioned symmetrically about the axis of rotation. Each repeated sector is assumed to be made up of a series of identical elements. Since the forces are also identical, the deflected shape and stress dist~bution of each sector is the same as in the other sectors. Hence only one sector is considered for the analysis. Coupling stiffness matrices from the adjacent sectors are also included. Using this technique a rotating disc and fan impeller are analysed and the results verified with analytical and experimental results available elsewhere. The finite element approach has been chosen for the purpose of modelling. The triangular plate element [9] is made use of. The stiffness and force matrices are obtained in cylindrical coordinates. The inertia forces due to rotation are consistently lumped at the nodes.

The

129

2.1 Solution technique The method of solution can be expIained by considering a circumferentially periodic structure as shown in Fig. 1. Each sector is having n number of

130

V. Wn

and P. BALASUEHLAMANIAN Similarly, Li, Pi and Qi can be calculated from: LiLf=[Bi-

Pf_,.P,_,]

L,P, = Ai

Liqi=[g,-Pi_,q,_,],

i=2,3

,...

n-l

(4)

and

(5)

CL,=V’.-,+Q.-,I.

@,@ ,,

2.

Cholesky factorisation of the relevant square matrices is possible since all of them are real and symmetric. The last row in eqn (2) is given by

0 kcror or Substructure

@ n

Station

or

Parl~t~on

Numbers Numbers

A,z,+B”z,+A:,_,z,_,=g”.

Fig. I. A rotationally periodic structure.

Pre-multiplying

station points (partitions). The static equilibrium is represented by the following matrix equation [7]:

(6)

the first row in eqn (2) by Q’, gives:

QlLIz, + QlP,z, + QiQ,z. = Q;q, QiLi =(L,,Q,)’ which from eqn (3)=(A:)‘=

(7) A,.(8) 1

-B,

A,

0

0

A:,

ZI

gl

A;

B,

A,

0

0

Z2

gz

0

A;

B,

A,

0

Z3

An-2

0

Z” - 2

g,-2

B,-I

An-,

G-I

g”-I

A:-.,

B.

A:_,

B,,_z AA-2

A, In the first cycle of operation,

g3

=

(1)

g”

5”

the above equation is simplified to the following form:

. LfP,

Q, Q*

Lip,

91

z2

42 6

z3

Pn-2

Qn-2

Z.-Z

4n-2

LA-1

a-1

G-1

4.-l

A:-,

A

Zl

Q3

Lip, L,:-2

-

where Ll, PI and Q, are given by

4

=

L

Z”

~

(2)

g,

Substituting eqn (8) into eqn (7) and subtracting from eqn (6) gives:

B, = LIL;

Pm-

L, PI = A, L,Q, L,q,

g1.

- QfP,z, +A:,-,T,-I=~-Q;~I).

=A:, r

QiQ,>z,

(3)

(9)

Operations on the second row similar to the ones in

Steady state stress analysis of centrifugal fan impellers eqns (7), (8) and (9)

131

Here, L, and qn are estimated from

give

L,L:, = 8,

QiU+ QSP,r,+ QiQsn=Qiqz (10) QZi=&Qd'= -VlQ,Y=-QIp, (11)

L,q, = &*

(20)

Thus for each partition values of L’, Q, P and q are calculated. A simple backward substitution in the last +A:_,z,-,=kQiq, - Q;d (12)row of eqn (19) gives z,. Then a series of similar backward substitutions in eqn (19) gives the values of This can be continued for the first (n - 2) rows and .&_I, r,_rr G-3, *. . z,. The displacements which are results in an equation similar to eqn (12), given by in cylindrical coordinates are again transformed into local coordinates to get the element stresses at the centroid of the triangle. (B.-Q;Q,-Q;Q,-...-Q:-zQ,-2,~.

M,- QlQ, - QiQd~n-QiP2~3

-QL,P,_,z,,_,

2.2 Number of multiplications involved

+A:_,z,_,

In a general case the size of each partition may be different. For the sake of simplicity, let us assume the (13) =(g.-Q1q,-Q;q2...-Q~-24.-2). partitions to be of equal size ‘s’. Then the major Premultiplying both sides of the (n - 1)th row of eqn multiplications involved can be estimated as follows: (2) by Q:_ , , we have: Reduction stage First partition s3 (7/6) + s2 (l/2) Second to (n - I)th a:,-,~:,-,z”-,+&:,-,~~-,z”=Q:,-,g,-,. (14) partition s3 (16/6) (n - 2) + s2(3/2)(n - 2) Here, nth partition s3 [(3n - 2)/6] + .S2[(2n - 1)/2]. Back substitution stage

nth partition s2(1/2) ~“-,~:,-,=(~“-,a”-,)‘. (15) (n - 1)th partition s2(3/2) (n - 2)th partition to first partition Using eqns (4) and (5), eqn (15) can be simplified and written in the following form: s2(5/2)(n - 2). Hence the total number of multiplications will be equal to: (Qn-,)'L:-,=[L,-,.(P,-,+Q,-,)I s’[(19n - 27)/6] + s2(5n - 6). =[A,,_, -P:,-r*Q,-,] If s = 80 and n = 8 then the total number of =A,_, - Q:_2*Pn-2. (16) multiplications will be around 10.9 x 106. According to the algorithm presented in [A, the number of Combining eqns (13), (14) and (16) we have: multiplications involved was s3 (8n + 2) + s2 (5n), which gave rise to the number of multiplications of g.r, = E. f (17) 34.05 x 106. It is seen that the present analysis is roughly thrice as efficient as the algorithm presented where in [7].

8,=[B,-QlQ,-Q~Q2...-Q:-2Q._22.3

Core needed

For each partition B, A and g are read from files. Then L, P, Q and q are calculated and stored in files. -8:-,&,lii=k-Qh Hence the storage area needed is only 4b2. The area -Qiqt ...-Q:-2q,-2-Q:-14.-,1. (18)for load and displacement vectors is neglected since it is comparatively small. Storage transfers are Hence eqn (2) can be written as follows: simple.

- L’IP, GP2

Q,

ZI

91

Q2

z2

42

Q3

Lip3 G-2

z3

=

q3

Pn-2

en-2

G-2

s-2

LA-,

Q.-I

G-l

h-1

G

Z”

4.

(19)

132

V. RA!.MSURTI and P.

~A~UB~A~IA~

3. APPLICATIONTO AXISS>lMETRIC

STRUCTURES

As a check, the present method is applied to a rotating annular disc. It is fixed at the inner periphery and free at the outer periphery. One-eighth of the plate is taken into account. It is shown in Fig. 2, It is divided into 70 elements with 48 nodes, but only four partitions have been chosen for the analysis. The ratio of inner to outer radius is 0.2. The displacements and stresses obtained are compared with exact solutions reported in [IO]. They are shown in Figs 3 and 4. The agreement between the results of radial displacements are good. The radial stress curve shows a slight discrepancy. Fig. 2. One sector of annular disc. 4. CENTRIFUGAL

FAN IMPELLER

STRESS ANALYSIS

This method is extended to the fat impeller stress analysis. The impeller is described in Fig. 5 and the

-Q--Q-

0

oo.L---4 2

0

Fig. 3. Comparison for radial stress.

I

I

4

6

i-

Present

method

I 8

10

Fig, 4. Comparison of radial dispiacements.

138

.-.-.

Modutus Poisson’s

of

tlosticity ratio

= 2 x

10’ N/mm2

= 0.3

Fig. 5. One sector of the fan impeller.

Mat. All

dimns.

M.5 in mm

12

Steady state stress analysis of centrifugal fan impellers

133

Fig. 6. Experimental set~up.

Fig. 7. Finite element cliscretization.

++-@-

Theoretical

-

Experimental IGauge

0

I

800

1200

I

I

2000

1600

Speed

in

27)

I

2400

J

zaoo

RPM

Fig. 8. Comparison of strain in cover plate outer surface.

experimental set-up is shown in Fig. 6. It has eight backward curved blades and hence one-eighth of the impeilet has been considered. Back plate, cover plate, blade and the hub are the four components of the impeller. The back plate is assumed to be fixed at the hub. The number of elements and nodes taken for one sector are 149 and 88 respectively and it is shown in Fig. 7. Only the centrifugal forces are included as a force vector. The stresses are calculated at different speeds. To verify the above results experiments are conducted using strain gauge technique. Strain gauges along with slip rings and a six-channel carrier frequency amplifier are employed for the experiment. The strains are measured at diRerent discrete points for different speeds. The maximum variation in strain reading is likely to be of the order of 20.~ due to variation in contact resistance in the slip ring. Strains obtained from the analysis are compared with the

200

:

!

I

I

-

I$

/’

/

Theoret,coi

-

Experlmentai IGauge

t . -+-+-heoreticai -Experimen:cI (Gauge

ii:*; 0

800

1200

I

1

1600

2023 in RW

Speed

231

2LOO

2 300 b

Fig. 9. Comparison of strain in back plate inner surface.

-360’

I

i

a00

200

,

In

! 2LOO

2800

RPM

experimental values. They are plotted in Figs 8 to 12. All the figures except Fig. I I show good agreement.

;

The discrepancy in Fig. 11 is probably due to the very coarse finite element discretization used in the shell portion. In Fig. 12 the strain plot along the blade has been done at a speed of 1800 rpm. Comparing this with the e,xperimental results, the present analysis shows a fair agreement even with the coarse discretization. The maximum radial deflection computed is 0.0155 cm around midspan of the blade. In comparison with the sheet metal thickness (0.20 cm), this deflection is small and hence non-linear effects are neglected [9].

I

800

2000

Fig. 11.Comparison of axial strain on concave surface of blade.

I

0

I

IGOO Speed

160 c

120

1200

I

I

/

I

I

1200

1600

2000

2400

Speed

2600

RPM

in

Fig. IO. Comparison of axial strain on convex surface of the blade.

A

m ---., 8 kX

2001 3

I I /

CI -

Experimental Present

1600

rpm

analysis

I /

/

~

1

-LOO1 0

2

/

I /

n

/

~

I

50

X

in

mm

100

125

Fig. 12. Strain plot on the concave side of the BCB impeller blade inlet.

Steady state stress analysis of centrifugal fan impellers When the deflection is high as compared to the material thickness geometric stiffness effect is to be included. It has been reported earlier [1 l] that the modification of the stiffness by introducing the rotational effect made only a slight change in the stress even for a fairly large deflection of the blade running at 3600 t-pm. Since the speed encounted here is low, it is felt that the rotational stiffness can be neglected.

R. G. Patton. Stresses in centrifugal fan impellers. Proc. Inst. Mech. Engrs 187, 309-315 (1973). R. Bell and P. P. Benham. Theoretical and experimental stress analysis of centrifugal fan impeller. J. Sfrclin Anal. 13, 141-147 (1978). 0. C. Zienkiewicz and F. C. Scott. On the principles of

5.

5. CONCLUSIONS

A submatrices elimination scheme has been modified to calculate the displacements of the cyclic symmetric structures. This scheme permits the solution of fairly large systems even with a small computer. Although the examples deal with small deflections and low speeds, geometric and rotational stiffnesses can be incorporated when warranted.

6.

7.

8. 9.

Acknowledgement-The

authors would like to thank Mr S.

Swamamani and Mrs C. Sujatha for the courtesy extended to them in carrying out this investigation.

10.

REFERENCES

11.

I.

H. Haerle. The strength of rotating discs. Engineering 106, 131-134 (1918).

135

repeatability and its application in analysis of turbine and pump impellers. Inr. 1. Numer. Meth. Engng 4. 445-452 (1972). A. R. Zak, J. N. Craddock and W. H. Drysdale. Approximate finite element method of stress analysis of non-axisymmetric configurations. Compur. Slruct. 9. 201-206 (1978). R. H. Macneal, R. L. Harder and J. B. Mason. NASTRAN cyclic symmetry capability. NASTRAN users experience, NASA Tech. Memo. NASA TMX2893 (1973). V. Ramamurti and P. Balasubramanian. Static analysis of circumferentially periodic structures with Potter’s scheme. Compur. Srrucr. 22, 427-431 (1986). A. Jennings. Matrix Computation for Engineers and Scientisrs. John Wiley. London (1977). 0. C. Zienkiewicz- The Finie blemenr .Wethod. McGraw-Hill, New York (1979). V. Srinivasan and V. Ramamurti. Finite element analysis of the inplane behaviour of annular disks. Compur. Sfrucr. 13, 553-561 (1981). V. Ramamurti. Further studies on automobile fans. Proceedings of the 6th World Congress on Theory of Machines and Mechanisms, New Delhi, pp. 588-591 (1983).