930 Appendices

3. Eigen frequencies of simple structures

Table A6-5 gives eigen frequencies of some simple structures. If the structure vi-

brates in water (or any other liquid), the fluid lowers the eigen frequencies because

of the “added mass” effect which accounts for the inertia of the liquid set into mo-

tion by the vibrating structure.

If a liquid-filled pipe vibrates in air, the added mass is given by the liquid con-

tained in the pipe. The added mass of any structure submerged in a liquid can be

estimated as 1.5-times the fluid volume displaced by the structure.

Table A6-5 Eigen frequencies of simple structures, from [14.1]

Mass-less beam with indi-

vidual mass at free end (can-

tilever) m

L

31 Lm

IE

2

732.1f

π=

Uniform beam; one end

clamped, one end free (canti-

lever)

L

4n

nL

IE

2

kf

µπ=

k1 = 3.52

k2 = 22

k3 = 61.7

Uniform beam; one end

clamped, one end free; with

individual mass at free end

L m

431 L236.0Lm

IE

2

732.1f

µ+π≈

Mass-less beam with indi-

vidual mass in center; simply

supportedm

L

31Lm

IE

2

93.6f

π=

Uniform beam; simply sup-

ported

L

4n

nL

IE

2

kf

µπ=

k1 = 9.87

k2 = 39.5

k3 = 88.8

Uniform beam with individ-

ual mass in center; simply

supported

L

m431

L383.0Lm

IE

2

86.13f

µ+π≈

Flat circular plate of thick-

ness h and radius R vibrating

at two nodal diameters

2 R

h42

3

1R)1(12

hE

2

25.5f

µν−π≈

E = Young’s modulus of elasticity

I = area moment of inertia

f = eigen frequency [Hz]

= load per unit length [kg/m] or unit surface [kg/m2] including added mass effect if applicable

ν = Poisson’s ratio (mostly 0.3)

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10.7 Hydraulic excitation of vibrations 595

rotor/stator interactions at two or more blades occur simultaneously (i.e. they are

directly in phase) and reinforce each other. Such combinations should be avoided

altogether for pumps with diffusers and double or multiple volutes. The pressure

pulses excite the impeller shrouds with zero diameter nodes (like an umbrella).

This type of excitation can also lead to axial thrust fluctuations and is linked to

unsteady torque which may produce torsional vibrations.

Case m = 1: At m = 1 the blade forces of the impeller have a non-zero resultant.

Therefore lateral vibrations are excited at the blade passing frequency (and higher

orders). The impeller shroud is excited with a one-diameter node. To avoid diffi-

culties with shaft vibrations at ν2×zLa×fn, m = 1 should never be allowed in the first

and second order and should be avoided up to the third order if possible.

Case m 2: The impeller is excited to vibrations with a frequency of ν3×zLe×fn. If

a structural eigen frequency of the impeller gets into resonance with this excitation

frequency, fatigue fractures in the front or rear shroud could occur. The strain im-

posed on the blades by shroud vibrations can also lead to blade fractures. The

number m corresponds to the number of diameter nodes with which the impeller is

excited, Fig. 10.17: At m = 1 the impeller oscillates around one, at m = 2 around

two diameter nodes etc.

Fig. 10.17. Mode shapes of impeller vibrations with one and two diametrical modes

Forms of vibration with m > 2 are of little practical importance since the struc-

tural eigen frequencies are usually sufficiently high so that no resonances occur.

An exception are the impellers of very large pumps or pump turbines (with typi-

cally d2 > 2000 mm) and high heads, especially light-weight constructions, where

even modes with more than two diameter nodes (m > 2) can cause problems

[10.49]. To verify the risk of resonance between an impeller eigen frequency and

the excitation frequency ν3×zLe×fn, the natural frequencies must be calculated or

measured. When doing so, the movement of the water induced through the impel-

ler vibration – the “added fluid mass” – must not be neglected since it considera-

bly reduces the natural frequencies. From a diagram in [10.49] it is possible to de-

rive the relationship, Eq. (10.13a), for the ratio κ of the impeller eigen frequency

in the pump to that in air:

2

ax

air

water

d

s14.238.0

f

f+=≡κ (10.13a)

Equation (10.13a) applies to the vibration with two diameter nodes; with one

diameter node κ is still about 35% lower. The narrower the impeller sidewall gap

sax, the greater is the added mass of water (and the lower is the natural frequency)

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596 10 Noise and Vibrations

since greater amplitudes are imposed on the fluid in the narrow gap through the

deformation of the impeller. Strictly speaking, Eq. (10.13a) applies only to the ge-

ometries examined in [10.49], but it can be used for rough estimations in the ab-

sence of more precise information. From measurements in [10.65] the following

ratio of the impeller natural frequencies κ in the pump and in air can be derived:

• without a diffuser (this corresponds to open impeller sidewall gaps) κ = 0.72;

0.67; 0.64 for m = 1; 2; 3 respectively (one to three diametrical nodes)

• with a diffuser (this corresponds to closed impeller sidewall gaps) κ = 0.58;

0.59; 0.56 for m = 1; 2; 3 respectively (one to three diametrical nodes)

Closing the impeller sidewall gaps by the diffuser side plates increased the

added mass by restricting the liquid movement in the sidewall gaps. Physically the

effect is similar to reducing the axial sidewall clearance. The model tested in

[10.65] was a single stage pump with a diffuser and a volute (estimated nq = 30).

The impeller had 6 blades with 2B = 32°; the distance between impeller and dif-

fuser vanes was d3* = 1.01 only.

According to the tests in [10.65], an uneven pressure distribution in the volute

(likewise to be expected in an annular casing) causes uneven pressure fluctuations

as well. The pressure pulses were measured on the rotating impeller (as in

Fig. 10.16). At high flows (q* > 1.5), the highest pulsations were found when the

blade moved into the low pressure zone upstream of the volute cutwater. At part-

load, the highest pulses were recorded in the low pressure zone downstream of the

cutwater (see Figs. 9.20 and 9.21). Pressure fluctuations which vary over the cir-

cumference generate side bands with frequencies of ν×zLe×fn as measured by strain

gages mounted on the impeller shrouds. Thus the sidebands can excite natural fre-

quencies of the impeller.

Pressure measurements on the rotating impeller of a pump turbine model are

shown in Fig. 10.18, [5.52]. The pump had 20 adjustable guide vanes and 20 dif-

fuser vanes followed by a volute. Dominant peaks are seen at the rotational fre-

quency and at f = zLe×fn, but there were also peaks at ν×zLe×fn the amplitudes of

which decrease with increasing ν. The peaks at ν = 2 to 19 are attributed to the

uneven pressure distribution in the volute. According to Fig. 9.20 the non-unifor-

mity of the pressure in the volute increases as the flow is reduced – and so do the

pressure pulses measured in Fig. 10.18 which shows the amplitudes for two differ-

ent flow rates.

It may be surmised that the effect exerted on the pressure fluctuations by the

uneven discharge pressure distribution is due to the fact that the flow rate through

the impeller channels varies around the circumference. This flow rate variation is

induced by the local counter pressures experienced by the individual impeller

channels. The mechanism is sketched in Fig. 10.19; see Chap. 9 and Table D9.1.

The rotor/stator interaction increases as the flow at the impeller outlet becomes

less uniform. That is why pressure pulsations (Fig 10.5) and excitation forces

(Figs. 10.25 to 10.27) strongly increase at partload. The same is true if one or

more diffuser channels stall, which is the case in alternate stall or rotating stall.

Measurements reported in [10.67] well demonstrate the effect of stalled diffuser