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Case HistoryVibration due to Asymmetric Stiffness of Rotating Thin Disk Fan

Rotating machinery

(compressor)Nonlinear

Multi-blade fan (Fig.5)〔2〕

At a certain rotational speed, the fan experienced excessive vibrations featuring a frequencytwice the rotational speed measured by a displacement sensor (inertial coordinate system).Stress measured on the core plate of the fan (rotational coordinate system) proved that thevibrations had 1X frequency synchronized with the rotational speed (Fig.6).

As shown in Fig.1, a keyway causes a vibration problem due to anisotropy of shaft stiffness.The natural frequencies of the shaft in X and Y directions differ slightly. As a result, when the rotational speed reaches 1/2 of its natural frequency, the rotor experiences resonance vibrationat a frequency twice the rotational speed, which is famous as a secondary critical speed due togravity (Fig.2). It is believed that a resonance problem similar to this phenomenon occurs on rotatingstructures (impeller blades, disks) such as fans. This similarity is regarded as follows.

Element Anisotropy Natural frequency Acting forceRotational shaft Shaft deflection

stiffnessVibration of shaft deflection mode

Gravity

Fan (blade) Tilting stiffness of thin disk, called moment spring

Vibration of nodal diameter k=1, i.e.,tilting mode of blade and disk

Constant moment caused by wind pressure difference on the suction side

The equation of motion of an anisotropy shafting is given on the inertial coordinate system as:

Flexural vibration → Mass Gyro Shaft stiffness

Damping Anisotropic stiffness

Gravity

Tilt vibration → Tilting stiffness

Moment

Note that ( 1) indicates the spring force changes twice per one rotation with small magnitude.Let deformation be Zg that is subject to a constant gravity force, the anisotropic stiffnessgenerates a compelling force of Ke2iΩtZg, so that resonance occurs under the following

conditions, i.e., the conditions given in Fig.3 and Fig.4:(1) On inertial coordinate system (measured by displacement sensor):

forward natural frequency of the rotor system = rotational speed × 2 (2) On rotational coordinate system (measured by train gauge):

forward natural frequency of the rotor system = rotational speeds

Figure 7 shows the result of an FFT analysis of stresses measured by a strain gauge obtainedwhen hitting the fan during rotation, while Figure 8 summarizes the result. Certainly, theintersection point in Fig.8 corresponds to the resonance condition, which agrees with therotational speed for the resonance peak amplitude of the actual machine shown in Fig.6.

Object Machine

Observed

Phenomena

Cause Presumed

Analysis and Data

Processing Rotational speed

(★ 1)

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If a keyway is provided on the shaft, a dummy keyway seems to be cut so as to ensure symmetry with respect to the shaft center. In case of a rotating disk (referring to a core plate of blade), anisotropy occurs in terms of titing stiffness due to circumferential unevenness of plate thickness. Therefore, it is a reasonable decision to eliminate the anisotropy by precise machining of the disk plate thickness, but this is not preferred because of mounting manufacturing costs. Consequently, a measure was taken to enhance the natural frequency by applying a stiffening rib on the rear side of the disk core plate by welding. If we are during the design stage and before the manufacturing stage, it is relatively easy to raise the natural frequency of one nodal diameter in order to avoid this resonance condition by properly selecting the core plate thickness and diameter.

Rotating bodies should always be manufactured with precision in terms of axial symmetry. Since the natural frequency of one nodal diameter of a disk at standstill is divided into two natural frequencies due to gyro effect, i.e., forward one tending to increase and backward one tending to decrease according to the increase of the rotation, it is necessary to deliberately perform the predictive calculation of resonance conditions. A plan should be made to allow hitting during the rest and the rotation, so as to measure the natural frequency of rotors. It would be much better to carry out a hitting test by changing the rotational speed.

[1] Gash (translated by Miwa). 1978-9. Dynamics of Rotating Bodies: Morikita PublishingCo. 129p.

[2] Hagiwara et al. 1981-11. Resonance Phenomenon in Forward & Backward Whirl Modeof Impellers. Transactions of the JSME 47(423): 1,457

Fig. 11·11 Coordinate of asymmetric shaft

Xr

Yr

X

Y

yr

x r

k+ Δ

k− Δ

g

Ω t

Fig. 11·12 Vibration characteristics of asymmetric shaft

rotational speed

ω

0 0.5 1.0

xω

yω

2 Ω

p = Ωω

+ω ω2

1 2

ω 2

xωyω

n=ω ω1nω

2

unstablezone2X vib.

circular orbit

shaf

t vib

.

resonancedue to gravity

Ω

Ω

[A] [B] μ2

[A] [B]

[C]

ω 1

ω 2

1st 2nd

1st

2nd

mid

Countermeasures

and Results

Lesson learned

References

Fig.2: General behavior (qualitative) of non-true circle axis placed horizontally [1]

Fig.1: Asymmetric axis of rotation

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1 2 3−2

−1

1

2

3

ω/ nΩ

0

0p =

ωω /n

Ω

− Ω

1.5

1.0 0.5

0.25

γ2.01.00.50.250

=

Ω2

g

γ g2.

0=

Fig. 9·2 Inertial coordinate system

natu

ral f

requ

ency

γ =g γ

Fig. 9·3 Rotational coordinate system

1 2 3

−2

−1

1

2

ω/ nΩ

0

0p =

ωω /n

r

−3Ω2−

Ω

1.5

1.0

0.5

1.75

γ

2.0

1.01.5

1.75

0=c

γ c0=

2.0

Ω = 0

natu

ral f

requ

ency

γ =c γ2−

Fig. 9·6 Multi blade fan

blade thickness = 2.3 mm

blade number = 56

400

1,420

934 560 discharge

belt

suction

motor

blade

core plate

slip ring

315

subplate

core plate thickness = 3.2 mm

subplate number = 56

Fig. 9·7 Fan model

g

m

I p I d,

suction

discharge

center ofgravity

tiltingstiffness

core platekθ

amplitudeΩcos2 t

anisotropy

discharge

Fig.3: Natural frequency on the coordinate system at rest Fig.4: Natural frequency on the rotational coordinate

Experimental apparatus of multi-blade fan

Fig.5-1: Multi-blade fan Fig.5-2: Fan model

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Fig. 9·11 Waveform of strain gauge

5

4

3

2

1

0

vib. stress propagation ( near resonace ) ( 570 rpm )

forward wave

NO.

0 5

2 3

41

strain gauge number

Ω

Fig. 9·9 FFT of impulse response

100

(a) 0 rpm

(b) 200rpm200 rpm

400 rpm

Hz

Hz

Hz

(c) 400rpm

FF

Tam

plit

ude

FF

Tam

plit

ude

FF

Tam

plit

ude

15

.6

68

.6 94

.4

71

.06

9.0

94

.6

13

.21

8.0

37

.2

66

.87

3.4 95

.8

11

.2

20

.6

37

.2

frequency [Hz]

Fig. 9·8 Resonace in strain gauge signal

20

16

12

8

4

0500 700 900

1 × Rotational frequency component

speed [rpm]

vibr

atio

n st

ress

[ × 1

0MPa

]

3000

Fig.8: Comparison of experimental values of natural frequencies of one nodal diameter with calculated values

Resonance response of multi-blade fan due to gravity

Fig.6: Response of strain gage

Fig.7: Spectrum due to impulse excitation (numbers in the figure indicate frequencies in Hz)

Waveform at the time of resonance

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