Development of user-friendly programs for static and dynamic analysis of radial impellers

Development of user-friendly programs for static and dynamic analysisof radial impellers

K. Balajia, K. Sriramb, V. Ramamurtic,*

aDepartment of Mechanical Engineering, University of California, Berkeley, CA, USAbDepartment of Mechanical Engineering, MIT, Cambridge, USA

cMachine Dynamics Laboratory, Department of Applied Mechanics, IIT Madras, Chennai 600036, India

Received 13 July 1999; received in revised form 8 February 2000; accepted 24 February 2000

Abstract

Radial backward straight-bladed (BSB) impellers and backward curved bladed (BCB) impellers are analysed using the concept of cyclic

symmetry. Impellers with single cover plate and two cover plates are considered. Both static and dynamic analyses are done and user-friendly

programs for design of these impellers are developed. q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Radial impellers; User-friendly programs; Cyclic symmetry

1. Introduction

Radial impellers, which are analysed here, are widely

used in many industries like the cement industry. Those

impellers that are employed in the cement industry are air

¯ow type impellers. Similar impellers are considered for

analysis in the present work. They may have a single

cover plate or two cover plates. Since these impellers are

mostly air-¯ow impellers, only the centrifugal force is

considered for the force vector. Analysis of the impeller

becomes very cumbersome, if the whole impeller is taken

for the ®nite element model. The repeated structures are

connected to one another by complex constraints [1]. The

concept of cyclic symmetry is used to solve both static and

dynamic problems. Utilisation of this concept reduces both

the core and the time needed to design and analyse the

impeller.

Simultaneous iteration scheme [2] is used to ®nd the

natural frequencies of the impeller. Dickens and Pool [3]

have discussed periodic time domain loading of cyclic

symmetric structures. Henry and Lalanne [4] have reported

on the vibration analysis of compressor blades. Ozacka and

Hinton [5] have worked on free-vibration analysis and opti-

misation of axisymmetric plates. Zienkiewicz and Scott [6]

have done considerable work on analysis of turbines and

pump impellers.

2. Cyclic symmetry

In the cyclic symmetry approach, only one repeating

sector of a cyclically symmetric object is considered for

analysis. This approach is detailed below:

Referring to Fig. 1, the cyclically symmetric object

shown has N repeating sectors. Only sector 1 is considered

for Finite Element (FE) analysis. Forces on the object may

be expanded in Fourier series as below:

Fk � f1 1 f2 ei�k21�C 1 ¼ 1 fN21 e i�k21��N22�C

1 fN e i�k21��N21�C �1�

Here, Fk is the force on the kth sector, f1; f2;¼; fN21; fN are

the Fourier coef®cients. The angle C � 2p=N; is the angle

between any two repeated structures. The values of these

Fourier coef®cients may be obtained by solving the follow-

ing matrix equation.

Advances in Engineering Software 31 (2000) 775±791

0965-9978/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.

PII: S0965-9978(00)00052-1

www.elsevier.com/locate/advengsoft

* Corresponding author. Tel.: 1 91-44-235-1365; fax: 1 91-44-235-

0509.

E-mail address: [email protected] (V. Ramamurti).

2.1. Matrix equation and its solution

1 1 1 ¼ ¼ ¼ 1

1 e ic e i2c ¼ ¼ ¼ e i�N21�c

1 e i2c e i4c ¼ ¼ ¼ e i2�N21�c

..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

.

1 e i�k21�c e i2�k21�c ¼ ¼ ¼ e i�N21��k21�c

2666666666666664

3777777777777775

f1

f2

f3

..

.

..

.

fN

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;�

a0

a1

a2

..

.

..

.

aN21

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;�2�

Here, the `a's are the actual forces acting in the individual

sectors of the cyclically symmetric object.

Eq. (2) on simpli®cation yields a solution of the form,

f1

f2

f3

..

.

..

.

fN

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;� 1=N

1 1 1 ¼ ¼ ¼ 1

1 e 2ic e 2i2c ¼ ¼ ¼ e 2i�N21�c

1 e 2i2c e 2i4c ¼ ¼ ¼ e 2i2�N21�c

..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

.

1 e 2i�k21�c e 2i2�k21�c ¼ ¼ ¼ e 2i�N21��k21�c

2666666666666664

3777777777777775

�

a0

a1

a2

..

.

..

.

aN21

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;(3)

We may thus interpret the force on the structure to act in

N harmonics, with the complete behaviour of the structure

being a superposition of its responses in all the N harmonics.

It may be noticed that, of the N harmonics, �N 1 1�=2 (if N is

odd) or N=2 1 1 (if N is even) are independent. Thus by

static and dynamic analyses of the sector for the N=2 1 1

or �N 1 1�=2 harmonics the analysis of the cyclically

symmetric object would be complete.

For the harmonic n � 0; uniform force of f1 is considered

as acting in all the sectors. Thus the displacements and

stresses in all the sectors would be identical for the harmo-

nic n � 0: For the harmonic n � 1;, a force f2 acts in the 1st

sector, f2eic in the 2nd sector, f2e

i2c in the 3rd sector and so

on. The displacements also follow the same rule. Similarly,

in the harmonic n � 2; force f3 acts in the 1st sector, f3ei2c in

the sector, f3ei4c in the 3rd sector and so on. Thus all the

harmonics may be easily analysed for the behaviour of the

whole structure, since, the displacements and forces in one

sector determine the displacements and forces in the

remaining sectors for each harmonic.

In essence, the following relationship holds good for the

jth sector:

K. Balaji et al. / Advances in Engineering Software 31 (2000) 775±791776

Fig. 1. A cyclically symmetric object. N� number of repeating sectors;

n� number of substructures in each sector.

Nomenclature

[Bp] Strain displacement vector for in-plane

displacement

[Bf] Strain displacement vector for bending

(¯exure)

[Dp] Elasticity matrix for in-plane displacement

[Df] Elasticity matrix for bending

Fk Force on the kth repeating sector

[K] Stiffness matrix of the structure

L1,L2,L3 Area co-ordinates for the plate element

[M] Mass matrix of the structure

N number of repeated structures

OD Outer diameter of the radial impeller

{Zp} pth ®nite Fourier harmonic eigenvector

b1,b2,b3 y-component of the length of the side of plate

element

c1,c2,c3 x-component of the length of the side of plate

element

fi Fourier coef®cient

i imaginary number��21p

[kp] Stiffness matrix for in-plane displacement

[kf] Stiffness matrix for bending (¯exure)

{q2} Bending displacement vector

u1,u2,u3 x-component of displacement of plate element

v1,v2,v3 y-component of displacement of plate element

{1 p} Strain vector for in-plane displacement

{1 f} Strain vector for bending (¯exure)

1 x Strain along x-direction

1 y Strain along y-direction

{s p} Stress vector for in-plane displacement

{s f} Stress vector for bending (¯exure)

For the harmonic n � M 2 1;

{Zj11} � {Zj} e i�M21�c �4�This is in accordance with the standing wave theory

proposed by Thomas [1].

For the dynamic analysis of the structure, the global

eigenvector for the Mth harmonic is given by

{Z} � {{{Z1}{Z2}¼{Zn}¼{Z1 e i�M21�C}

� {Z2 e i�M21�C}¼{Zn e i�M21�C}¼{Z1 e i�M21��N21�C}

� {Z2 e i�M21��N21�C}¼{Zn e i�M21��N21�C}} (5)

3. Method of analysis

3.1. Finite element model used

The element that is used to analyse the impeller is the

triangular plate element. This element has six degrees of

freedom per node, thus resulting in an elemental stiffness

matrix of size �18 £ 18�: Subroutines are written to calculate

the shape functions, elasticity matrices, strain displacement

vectors and hence the stiffness matrix for this element. The

stiffness matrix consists of two parts, viz. the in-plane stiff-

ness and bending stiffness matrices.

The in-plane strain vector {1 p} for the element is given by

{1p} �1x

1y

1z

8>><>>:9>>=>>; �

2u=2x

2v=2y

2u=2y 1 2v=2x

8>><>>:9>>=>>;

�

X�2u=2Li��2Li=2x�X�2v=2Li��2Li=2y�X

�2u=2Li��2Li=2y�1 �2v=2Li��2Li=2y�

8>>><>>>:9>>>=>>>; �6�

{1p} � 1

2A

b1 0 b2 0 b3 0

0 c1 0 c2 0 c3

c1 b1 c2 b2 c3 b3

26643775

u1

v1

u2

v2

u3

v3

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;�7�

In Eq. (6), Li refers to the area co-ordinates for the plate

element. Hence

{1p} � �Bp�{q1} �8�Bp is the in-plane strain displacement vector, and q1 the

in-plane displacement vector.

sp �sx

sy

txy

8>><>>:9>>=>>; � �Dp�{1p} �9�

[Dp] in Eq. (9) is the in-plane elasticity matrix given by

�Dp� � E

�1 2 m 2�

1 m 0

m 1 0

0 0 �1 2 m�=2

26643775 �10�

Thus the stiffness matrix for the in-plane displacements is

given by

�kp� �ZZ

A�Bp�T�Dp��Bp� dA �11�

The strain vector for the bending (¯exure) of the plate

element is given by

{1f} � �S1��S2�{q2} �12�where {q2} is the displacement vector for the bending

displacement.

[S1] is the matrix given by

b1 b2 b3 2b1b2 2b2b3 2b3b1

c1 c2 c3 2c1c2 2c2c3 2c3c1

2b1c1 2b2c2 2b3c3 �2b1c2 1 2b2c1� �2b2c3 1 2b3c2� �2b1c3 1 2b3c1�

26643775

�13�[S2] is the matrix given by

22Nb1

2L21

22Nb2

2L21

¼ ¼ 22Nb9

2L21

22Nb1

2L22

22Nb2

2L22

¼ ¼ 22Nb9

2L22

22Nb1

2L23

22Nb2

2L23

¼ ¼ 22Nb9

2L23

22Nb1

2L12L2

22Nb2

2L12L2

¼ ¼ 22Nb9

2L12L2

22Nb1

2L22L :3

22Nb2

2L22L3

¼ ¼ 22Nb9

2L22L3

22Nb1

2L32L1

22Nb2

2L32L1

¼ ¼ 22Nb9

2L32L1

2666666666666666666666666666664

3777777777777777777777777777775

�14�

Thus the stiffness matrix for bending displacements is

given by

�kf� �ZZ

A �Bf�T�Df��Bf� dA �15�

In the above expression, [Df] is the elasticity matrix for

K. Balaji et al. / Advances in Engineering Software 31 (2000) 775±791 777

plate ¯exure given by

�Df� � Eh3

12�1 2 m2�

1 m 0

m 1 0

0 0 �1 2 m�=2

26643775 �16�

The strain displacement matrix for ¯exure [Bf] is given by

�Bf� � �S1��S2� �17�

It is important to note that the displacement component of

rotation u z does not appear in either q1 or q2. But, for the

purpose of assembling the stiffness matrix for the plate

element, this may be taken into account by introducing a

®ctitious couple Mz. The associated terms in the stiffness

matrix for the element will be taken as zero.

Thus the stiffness matrix for the plate element is given by

�kp� �0� �0��0� �kf� �0��0� �0� �kuz�

26643775 �18�

Here [ku z] is an arbitrary ®ctitious stiffness coef®cient.

The elemental mass matrix is given by

�ME� �Z�N�T�N� dvol �19�

where [N] is the shape function for the element.

3.2. Skyline approach for nodal numbering

In this approach for numbering the nodes of the sector

that is chosen for analysis, the stiffness matrix and mass

matrices are stored in a linear array. The nodal numbering

is done as shown in Fig. 2.

The nodes can be classi®ed as:

Master nodes and slave boundary nodes: The slave


Fig. 2. Nodal numbering for skyline approach. Nodes 1±15� interior

nodes; nodes 16±20� adjacent nodes; nodes 21±25�master nodes;

nodes 26±30� slave nodes.

Fig. 3. Finite element model for impeller.

Fig. 4. BSB single cover plate impeller.

Fig. 5. BSB double cover plate impeller.

boundary are those boundary nodes that are connected to

the master boundary nodes by a complex constraint,

namely ei(M21)c for the Mth harmonic.

Adjacent nodes: These are the nodes that are adjacent to

the slave boundary nodes.

Interior nodes: These are the nodes that are neither

master/slave or adjacent nodes.

This approach results in the partitioning of the stiffness

matrix [7,8] as shown below

Interior

nodes

Adjacent

nodes

Master/slave

nodes

Interior

nodes

[KAA]

(real)

[KAB]

(real)

[KAC]

(real)

Adjacent

nodes

[KBA]

(real)

[KBB]

(real)

[KBC]

(complex)

Master/slave

boundary nodes

[KCA]

(real)

[KCB]

(complex)

[KCC]

(real):

3.3. Storage of the stiffness and mass matrices

From the above partition of the stiffness matrix, it can be

seen that a large portion of the stiffness matrix is real and is

not affected by the complex constraints that occur between

the slave and the adjacent nodes. Hence, when the stiffness

matrix is stored in a linear array, the complex elements are

pushed to the tail end of the array. Thus one may store the

real elements of the stiffness matrix in a linear array and the


Fig. 6. BCB single cover plate impeller.

Fig. 7. BCB double cover plate impeller.

Fig. 8. Natural frequency for BSB single cover plate impeller for harmonic n � 0 and OD # 1000 mm:


Fig. 9. Natural frequency for BSB single cover plate impeller for harmonic n � 0 and OD $ 1600 mm:

Fig. 10. Natural frequency for BSB single cover plate impeller for harmonic n � 1 and OD # 1000 mm:


Fig. 11. Natural frequency for BSB single cover plate impeller for harmonic n � 1 and OD $ 1600 mm:

Fig. 12. Natural frequency for BSB double cover plate impeller for harmonic n � 0 and OD # 1000 mm:

relatively small number of complex elements in another

linear array. The complex array is the one that depends on

the harmonic that is to be analysed, while the real array is

the same for all the harmonics. Thus we may reduce the core

required for the problem by storing the real array only once

and by generating a complex array when a particular har-

monic is being analysed.

3.4. Static analysis of the impeller

For the static analysis, as mentioned above, only the

centrifugal forces are considered. Also, even with cyclic

symmetry, one needs to consider only the ®rst harmonic,

which is real. This is because all the forces are radial and act

with equal magnitude in all the sectors. This is as good as

analysing the fundamental harmonic of the cyclic symmetry

problem, in which the force matrix is real. The stiffness

matrix factorisation is done using real Cholesky Factori-

sation of the skyline matrix.

A general problem under static loading will be of the type

�K�{u} � {f } �20�where [K] is the stiffness matrix of size �n £ n�; {u} the

displacement matrix of size �n £ 1� and { f } is the force

vector of size �n £ 1}:

The static analysis of the impeller needs to be done

because, as the impeller operates under working conditions

where ¯ue gases ¯ow through the impeller, corrosion tends

to occur over a period of time (see Fig. 3). This results in

wear of the impeller in the blades, back plate and the cover

plate(s) (Figs. 4±7). The reduction in the thickness of the

various parts of the impeller causes an increase in the load-

ing due to centrifugal forces thereby raising the stresses to

values beyond safe operating limits. A parametric study of

the stresses in the impeller would help in monitoring the

stresses in the impeller at any operating condition. The

programs that are written can be easily modi®ed to suit

the operating working conditionsÐfor e.g. if thermal

stresses are to be considered, the programs may be suitably

modi®ed to accommodate them in the force vector.

3.5. Eigenvalue analysis of the impeller

For the dynamic analysis, all the harmonics have to be

taken into account. Here, Cholesky factorisation of the real

matrix is done only once and it is stored separately, since

it will be the same for each harmonic. Only the com-

plex portion of the stiffness matrix is factorised for each

harmonic. Since the complex part of the stiffness matrix is

very small in proportion to the real part, considerable saving

in core and time is achieved even though the eigenvalue


Fig. 13. Natural frequency for BSB double cover plate impeller for harmonic n � 0 and OD $ 1600 mm:


Fig. 14. Natural frequency for BSB double cover plate impeller for harmonic n � 1 and OD # 1000 mm:

Fig. 15. Natural frequency for BSB double cover plate impeller for harmonic n � 1 and OD $ 1600 mm:

analysis is to be done for each harmonic. For the dynamic

analysis, simultaneous iteration algorithm [2] is employed.

The steps involved in the extraction of the real eigen-

values from the Hermitian matrix are as follows:

1. Trial eigenvectors on the right-hand side are assumed as

{Zp0} and are orthonormalised with respect to mass

matrix [Mp].

2. Vector

{Gp} � �Mp�{Zp0} �21�

is computed.

3. The reduced simultaneous equation of the form

�Mp�{Zp1} � {Gp} �22�

is solved.

4. The new vector {Zp} is orthonormalised with respect to

[Mp].

5. The interaction matrix

�V 2� � {Zp1}H�Kp�{Zp1} �23�

is computed.

6. The orthonormalised new vector {Zp1} is used in step 2

for the next iteration.

This is repeated until convergence of the eigenvalue is

obtained.

In the case of the impeller, the forcing function has the

frequency values which are integral multiples of the oper-

ating speed. The importance of the eigenvalue analysis is

re¯ected from the fact that one should avoid the running

speed or any of its higher multiples from falling near the

range of the eigenvalues for the impeller. Even small vari-

ation in the thickness of the plates would change the natural

frequency by a few hundred rpm (revolutions per minute).

This could very well result in drastically different dynamic

response of the system. As in the case of all engineering

problems, the main concern would be to avoid the funda-

mental frequency from falling within range of any of the

forcing frequencies, because the operating speeds generally

are not too high, except probably in very special cases like

turbochargers.

4. Development of the user-friendly routine

A user-friendly program is developed for the design of the

following four kinds of impellers:

(i) BSB single cover plate impeller (Fig. 4).

(ii) BSB double cover plate impeller (Fig. 5).


Fig. 16. Natural frequency for BCB single cover plate impeller for harmonic n � 0 and OD # 1000 mm:

(iii) BCB single cover plate impeller (Fig. 6).

(iv) BCB double cover plate impeller (Fig. 7).

The programs developed need as inputs the outer

diameter of the impeller, the thickness of the impeller

plates, the number of blades, inlet and outlet blade angles.

For the curved blade (BCB) impellers, co-ordinates of a few

points on the blade are also required as additional inputs.

The other dimensions of an impeller can be got using the

following empirical relations:

D1 � �0:5 2 0:7� p D2

Dhub � �0:2 2 0:6� p D1

Z2 � 0:25 p D2

Z1 � 0:35 p D2

b1 � 208±308

b2 � 508±758

where D2 is the outer diameter of the cover/back plate, D1

the inner diameter of the cover plate, Dhub the diameter of

the hub, Z2 the blade height at the outer end of the cover

plate, Z1 the blade height at the inner end of the cover plate,

b 1 the blade inlet angle, and b 2 is the blade outlet angle.

With these minimal inputs, i.e. basic dimensions of the

impeller, thickness of the plates used in the impeller and the

number of blades in the impeller, the programs that are

written generate the ®nite element mesh for the geometry

of the blades. The programs also generate the coordinates

of the nodes. The programs also take care of the fact that,

for the accurate Finite Element (FE) analysis using the tri-

angular plate element, the ratio of any two sides must be 4:1

or less. If better accuracy is sought, the user may easily alter

the number of nodes and the convergence criterion for the

eigenvalue analysis. For static analysis, the programs calcu-

late principal stresses. Failure criteria such as the von Mises

or Tresca's criteria may also be incorporated to calculate the

stresses.

All the programs were developed in the BORLAND

C11 environment. But due to the huge size of the problem,

there were memory allocation problems with this compiler.

So, the WATCOM C compiler was used so that a large

portion of the RAM could be allocated.

5. Parametric study

The development of the user-friendly routine for the

design and analysis of the impellers also facilitates para-

metric study. Study of the variations of the lowest frequency


Fig. 17. Natural frequency for BCB single cover plate impeller for harmonic n � 0 and OD $ 1600 mm:

of the ®rst few harmonics with the various values of both the

outer diameter and thickness of the impeller plates was

carried out and the results were plotted. This parametric

study was done for all four kinds of impellers. The higher

harmonics are not very signi®cant because they consist of

frequencies that are much higher than the normal running

speeds of the impellers and their ®rst few multiples. In all

the cases it is found that the fundamental frequency of the

impeller occurs in the second harmonic.

From the graphs that were plotted, it can be observed that

the natural frequencies (and the lowest frequency in any

harmonic) decrease with increase in outer diameter of the

impeller, for a particular thickness of the impeller plates.

For a particular outer diameter of the impeller, the natural

frequency (and the lowest frequency in any harmonic)

increases with increase in thickness of the plates.

From the parametric study it is observed that the natural

frequencies of the straight blade (BSB) impeller and curved

blade (BCB) impellers do not differ very much, since the

curvature of the blade is quite small.

The variation of the principal stress was also studied, with

the impeller running at 160 rad/s. From the graphs plotted, it

is found that for a particular blade thickness, the stress value

increases with increase in the outer diameter of the impeller.

Also, it is found that for a particular outer diameter of the

impeller, the stress value decreases with the increase in

blade thickness. It is also observed that in the present

case, it is not possible to operate the impeller even with a

small in its outer diameter at or beyond a speed of 160 rad/s.

It is because the calculated principal stress value exceeds the

yield point stress at these speeds of operation.

In the course of the parametric study, impellers with

different combinations of diameter and plate thickness

were used. The number of blades was 8. The outer diameters

of these impellers generally range from 400 to 4000 mm.

For the purpose of parametric study, steps of 600 mm were

taken for the diameter values i.e. 400, 1000, 1600, 2200,

2800, 3400 and 4000 mm were the OD values considered

for the analysis. The standard values of thickness that were

used for the plate are 5, 8, 12, 16 and 20 mm. Backplate,

blades and coverplates were assumed to have the same

thickness. But varying thickness for different parts of the

impeller can also be easily incorporated into the user-

friendly routine. It is found that for the BCB impellers

(Figs. 6 and 7), the blade curvature is very low. Hence,

the natural frequencies and stress values are quite close to

the respective values of frequencies and stress values for the

BSB impellers (Figs. 4 and 5) of the same dimensions of OD

and thickness and number of blades. Hence, not all

frequency and stress variation plots for the BCB impellers

have been shown.

It is also observed during the course of the parametric


Fig. 18. Natural frequency for BCB single cover plate impeller for harmonic n � 1 and OD # 1000 mm:

study that for the impellers whose outer diameters are

greater than or equal to 1600 mm, the value of the maximum

principal stress exceeds 150 MPa for a running speed of

160 rad/s. At this speed and beyond, not only the developed

stresses cross the allowable values, but also at such speeds,

the impeller is being subjected to a continuous, larger wear

rate by the corrosive ash in the ¯ue gases. The variation of

maximum permissible speeds for a maximum principal

stress of 150 MPa, for different impeller thickness has

been shown in Fig. 24. Hence such speeds that lie on the

higher side of the safe limits should be avoided in operation.

6. Experimental veri®cation of the analysis

For the purpose of veri®cation of the methods that were

used to analyse and design the impeller, an existing straight-

blade (BSB) impeller with a single cover plate was taken.

Dimensions of the impeller:

D2� outer diameter� 400 mm;

N� number of blades� 8;

b 1� inlet blade angle� 25.58;b 2� outlet blade angle� 54.18;h� thickness of the plates:

� 3 mm for the back plate;

� 2 mm for the cover plate;

� 2 mm for the blades.

The experimentally observed natural frequency is 65 Hz

and the theoretically evaluated natural frequency is 71 Hz. It

is seen that the theoretically determined value lies within

10% variation limits.

Possible explanations for the variation between the

experimentally and theoretically obtained values are:

(a) Thickness of the structure is not uniform throughout,

i.e. even in the back plate there are variations in the

thickness in the circumferential as well as the radial direc-

tions. Similar variations exist in the blades and the cover

plates also.

(b) Inaccuracy is also due to the assumption made that the

triangular plate elements have straight edges contrary to

the curvature of the back plates and the cover plates, and

hence the boundaries of the blades are not exactly

modelled.

7. Conclusion

From the parametric study that was conducted on the

BSB (Figs. 4 and 5) and BCB impellers (Figs. 6 and 7),

the following conclusions are drawn:


Fig. 19. Natural frequency for BCB single cover plate impeller for harmonic n � 1 and OD $ 1600 mm:

(i) The natural frequencies and principal stress values for

the BSB and BCB impellers are not greatly different from

each other, since the blade curvature is very small. The

natural frequencies of any harmonic increase with

increasing plate thickness for a particular outer diameter.

This natural frequency variation is true for BSB and BCB

impellers, with single or double cover plate. The natural

frequencies of any harmonic decrease with increasing OD


Fig. 20. Natural frequency for BCB double cover plate impeller for harmonic n � 1 and OD # 1000 mm:

Fig. 21. Natural frequency for BCB double cover plate impeller for harmonic n � 1 and OD $ 1600 mm:


Fig. 23. Maximum principal stress for BSB double cover plate impeller for OD # 1000 mm:

Fig. 22. Maximum principal stress for BSB single cover plate impeller for OD # 1000 mm:

for a particular thickness of the plate. The variation is true

for BSB and BCB impellers, with single or double cover

plate. The fundamental frequency of the impellers always

occur in the harmonic n � 0 which are shown in Figs.

8±21.

(ii) Values of principal stresses for BSB and BCB

impellers (single and double cover plated) decrease

with increasing plate thickness for a particular OD and

increase with increasing OD for a particular plate thick-

ness. Stress values for OD $ 1600 mm are not plotted as

they exceed the yield point value for steel. Stress vari-

ations are shown in Figs. 22 and 23.

(iii) Were the natural frequency of the impeller to coin-

cide with the running speed or its multiples, one method

of making the natural frequency far removed away from

the running speed is to manufacture an impeller of greater

thickness, thereby increasing the natural frequency (see

Fig. 24).

(iv) Optimal thickness of the impeller can be found out

using the parametric study, i.e. impeller plate thickness

for which, the stress values lie within safe limits and the

natural frequency does not coincide with running speed.

However, it should be borne in mind that during operation

in a hostile environment with coal of high ash content, the

thickness of the impeller is bound to decrease. Hence, an

allowance has to be made for the reduction in thickness

for continuous operation.

(v) Finally, a software package can be easily developed

using this user-friendly program, by interfacing it with

graphics. Here again, it must be noted that because of

the huge amount of RAM that is required to store the

stiffness and mass matrices, graphics interface must be

done using a compiler that supports the allocation of a

large portion of the RAM for the storage of all the vari-

ables involved.

References

[1] Thomas DL. Standing waves in rotationally periodic structures. J

Sound Vibr 1974;37:288±90.

[2] Jennings A. Matrix computations for engineers and scientists. New

York: Wiley, 1977.

[3] Dickens JM, Pool KV. Modal truncation vectors and periodic time

domain analysis applied to a cyclic symmetric structure. Comput Struct

1992;45(4):685±96.

[4] Henry R, Lalanne M. Vibration analysis of rotating compressor blades.

J Engng Ind 1974;96(5):1028±35.

[5] Ozacka M, Hinton E. Free vibration analysis and optimisation of

axisymmetric plates and shellsÐI. Finite element formulation.

Comput Struct 1994;52(6):1181±97.

[6] Zienkiwicz OC, Scott FC. On the principle of repeatability and its


Fig. 24. Allowable running speed for BSB impellers for OD � 1600 mm (max stress� 150 MPa)

application to the analysis of turbine and pump impellers. Int J Num

Methods Engng 1972;4:445±52.

[7] Ramamurti V. Computer aided mechanical design and analysis. New

York: McGraw Hill, 1998.

[8] Balasubramanian P, Jagdeesh JG, Suhas HK, Ramamurti V. Free vibra-

tion analysis of symmetric structures. Comm Appl Num Methods

1991;7:131±9.


Development of user-friendly programs for static and dynamic analysis of radial impellers

Documents

Transcript of Development of user-friendly programs for static and dynamic analysis of radial impellers