Experimental and Numerical Analysis of Transversal Open Cracked Shafts considering Beam Slenderness...

adfa, p. 1, 2011.

© Springer-Verlag Berlin Heidelberg 2011 4PhD Student

Experimental and Numerical Analysis of Transversal

Open Cracked Shafts considering Beam Slenderness and

Crack Depth

Rafael García-Illescas1,2,4, Julio C. Gomez-Mancilla1, Luigi Bregant3

1Rotordynamic and Vibrations Laboratory, Instituto Politécnico Nacional, D.F. Mexico

[email protected], [email protected] 2Turbomachinery Department, Instituto de Investigaciones Eléctricas, Cuernavaca Mor. Mexico

3Dipartimento di Ingegneria e Architettura, Università degli Studi di Trieste, Italy

[email protected]

Abstract.

Stiffness variation due to cracks in rotors is a well-known problem; plenty of

studies to prevent/avoid catastrophic accidents and rotor bursts exists. FEM nu-

merical results calculating natural modal frequencies for open cracked beams at

different notch depths and slenderness ratios, are compared vs. experimental la-

boratory measurements. Shaft beams under simply supported and free-free

boundary conditions, focusing on solids and Timoshenko beam finite elements,

are studied. Timoshenko beams employ the Mayes&Davies equivalent-length

concept for crack modeling. Notched shafts provide useful upper bound fre-

quency reductions values (breathing cracks display up to, and smaller decre-

ments). Modal frequency splitting for each natural frequency are confirmed and

validated by numerical simulation and tests.

Modal frequencies splitting functions are given, obtained by simulation and

experiments which reflect strong crack depth and slenderness ratio influences.

Vibrational coupling energy of the frequency splitting reaches a maximum when

the excitation is orthogonal to the crack orientation, conclusions are given.

Keywords: notched open crack, modal frequencies splitting, crack depth, beam

slenderness, FEM.

1 Introduction

Cracked rotating shafts are thoroughly studied by many researchers since the 1970’s,

some good examples are shown in [1,2,3,4,5,6,7]. Some works of the authors involving

crack behavior, detection and modeling analysis are presented in [3], [4], [8]. Relevant

researchers and works are numerous too long to mention. In the present study, theoret-

ical and numerical values for the first vibration modes of simple beams having an open

crack are studied. Numerical analyses are performed by NISA, a general purposes finite

element program. Several modal tests were performed in order to validate the previous

calculations.

mailto:[email protected]



2 Shaft Modeling

A shaft can be modelled by means of finite elements, using solids elements as well as

beam elements. Normally, the use of solid elements give more precise results than beam

elements, however it requires a much higher computational effort, that is why beam

elements are used but with some additional features for modelling the crack. There are

also other model approaches based on differential equations or energy methods. Alt-

hough it is possible to add a crack at any shaft axial location, in the present work all

analyses were done with the crack located at shaft midspan in order to compare calcu-

lations to with experimental tests. For upper vibration modes, there are no enough gen-

eral simple analytical models, especially when the support conditions are not simple.

In those cases Ritz method is good enough to obtain higher modal shapes preferably

for simple boundary conditions. The finite element method (FEM) is normally the best

solution for complex structures with several support conditions, different beam trans-

verse sections and multiple mass locations.

2.1 Crack Modeling

The crack can be modeled in several ways, directly modeled into the mesh if solids

finite elements are used. Alternatively, adding an artificial flexibility due to the crack

presence by means of an element having an equivalent length and special characteristics

when beam elements are used, a concept introduced by Mayes and Davis [6]. At the

cracked beam section such equivalent length, leq, is obtained from Equation (1); which

to account for the stiffness loss, modifies structural beam characteristics diminishing

the geometrical properties of the shaft cross section as follows,

Δ𝐼 𝐼⁄

1−Δ𝐼 𝐼⁄=

𝑟

𝑙𝑒𝑞(1 − 𝜈2)𝐹 (1)

whereI= I0-Icrack, represents the change in second moment of area due to the crack,

(I0 corresponds to the integral cross section; Icrack the cracked cross section, I, the weak-

est direction, I at crack orthogonal direction), refer to Figure 1; r is shaft radius, v is

Poisson ratio, F is the compliance function see [5,6,7]. Therefore, a cracked beam ele-

ment will have two different equivalent lengths, along and .

Fig. 1. Representation and parameters of the transversal crack in the shaft

The crack simulation involves placing two elements at shaft mid-span, whose sum is

equal to equivalent length leq and the orthogonal different second moments of area are

using equations (2, 3), where 𝛼 = cos−1 (𝑎

𝑟− 1) defines the transversal crack depth.

𝐼𝜉 =𝑟4

4{𝛼 − sin 𝛼 cos 𝛼 + 2𝑠𝑖𝑛3𝛼 cos 𝛼 −

16 sin6 𝛼

9(𝛼−sin 𝛼 cos𝛼)} (2)

𝐼𝜂 =𝑟4

12{3𝛼 − 3 sin 𝛼 cos 𝛼 − 2𝑠𝑖𝑛3𝛼 cos 𝛼} (3)

3 Numerical Simulations

For two different boundary conditions: simply supported beam (SSB), and free-free

beam (FFB) are considered. Different shaft length/diameter slenderness ratios (L/d) and

crack depths ratios (a/d) are analyzed: d=1cm steel shafts, having L/d= 5, 8, 12, 20, 35;

plus another shaft with d=6.5cm, L/d=7.3. For FEM-solid models, second order solid

elements are used using 6400 elements for the smallest case (L/d=5), and 18,560 ele-

ments for the most slender (L/d=35); while for FEM-beam models, 44 and 60 elements

are used, respectively. To optimize the number of elements, a convergence analysis of

the FEM mesh for each type of elements is realized; though due to space reasons are

not shown. Figure 2 shows two flexural mode shapes, 3rd and 5th. Notice that cracked

shaft modes along orientation (right side of fig.) are similar to the integral case; how-

ever, at weaker orientation , these shapes are clearly different, much bent at crack

location. Since the crack is located at shaft mid-span, high bending mainly manifests in

the first and odd modes, where the even modes are slightly affected. Though FEM-

beam elements using the equivalent length concept [6,7] provide results congruent with

FEM-solids, to obtain improve mode shapes another method needs to be developed.

a) Third mode

b) Fifth mode

Fig. 2. Two SSB mode shapes for L/d=12, deep crack a/d=0.5. Results above in each set used

FEM-solids; while FEM-beams are below; along -orientation (LHS), -orientation (RHS).

Tables 1 and 2 show some results of dimensionless natural frequencies along direc-

tion, where the normalization is done with respect to the integral case for each L/d ratio.

From previous table, it can be concluded that for both boundary conditions (FFB, SSB),

stocky beams give a better opportunity for detecting cracks. Due to damage location,

modes 1 and 3 have the major frequency changes. Very slender beams do not allow

crack identification especially for mode 2 (not shown here) even for stocky cases.

Table 1. Normalized natural frequencies obtained by FEM-Solids, first_ flexural mode

Free-Free, FFB Simply Supported, SSB

a/d L/d=5 L/d=8 L/d=12 L/d=20 L/d=35 L/d=5 L/d=8 L/d=12 L/d=20 L/d=35

0 1 1 1 1 1 1 1 1 1 1

0.1 0.991 0.994 0.996 0.997 0.999 0.987 0.991 0.996 0.994 0.999

0.2 0.958 0.972 0.980 0.988 0.993 0.955 0.960 0.981 0.984 0.994

0.3 0.893 0.925 0.947 0.967 0.980 0.896 0.901 0.952 0.966 0.983

0.4 0.804 0.858 0.895 0.932 0.959 0.810 0.817 0.904 0.935 0.964

0.5 0.674 0.749 0.807 0.867 0.916 0.698 0.700 0.832 0.884 0.932

Table 2. Normalized natural frequencies obtained by FEM-solids, third_ flexural mode

Free-Free, FFB Simply Supported, SSB

a/d L/d=5 L/d=8 L/d=12 L/d=20 L/d=35 L/d=5 L/d=8 L/d=12 L/d=20 L/d=35

0 1 1 1 1 1 1 1 1 1 1

0.1 0.998 0.998 0.998 0.998 0.999 0.994 0.995 0.997 0.998 0.999

0.2 0.971 0.982 0.989 0.992 0.995 0.970 0.977 0.983 0.989 0.993

0.3 0.961 0.965 0.970 0.979 0.987 0.929 0.947 0.961 0.972 0.983

0.4 0.916 0.928 0.939 0.958 0.973 0.875 0.909 0.937 0.944 0.966

0.5 0.818 0.824 0.885 0.922 0.947 0.819 0.870 0.919 0.903 0.938

4 Experimental Tests

Several tests are accomplished using small shafts (1cm diameter) for the case of simply

supported beam, SSB; good test results are obtained only for slender beams. Therefore

due to the difficulties on testing the previous shafts, a stocky beam with d=6.5cm and

L/d=7.3 is analyzed. Several experiments on this steel shaft with different crack depths

are carried out by means of an acquisition system based on a Pimento front end and

Test.Xpress software both from LMS. The crack was rather a slot made with a saw

machine with 1mm of thickness. The material properties are E=210GPa, density=7750

kg/m3 and Poisson ratio=0.3. The shaft is tested under free-free conditions, FFB. 12

points are excited along the shaft: 6 in the vertical crack plane and 6 in the horizontal;

one triaxial accelerometer is used. In Figure 3, a plot of matching frequencies between

FEM calculations and testing for the integral shaft are illustrated in the left side. Alt-

hough Euler-Bernoulli, Rayleigh, Shear, and Timoshenko beam theories were used to

calculate the flexural mode frequencies, the first three flexural modes are shown for the

most representative cases are shown at the right side of the same figure. A very good

match was obtained for the FEM solids, theoretical Timoshenko model and tests.

Fig. 3. Integral shaft natural frequencies comparison for FEM vs. test (left) and first three flex-

ural modes (right), both for free-free integral beam, L/d=7.3

Figures 4-8 are for different test cases under free-free boundary conditions, FFB: inte-

gral shaft and considering several slot depths. Frequency Response Function (FRF)

plots for the integral shaft are shown in Figure 4 along with different test responses at

different excitation points. The first flexural mode is at 1325 Hz; second flexural mode

at 3387 Hz; first torsional mode at 3474 Hz (small peaks); first axial mode at 5558 Hz;

third flexural mode at 6078 Hz; and second torsional mode at 6973 Hz.

Figure 5 shows the slot influences on the natural frequencies in first low modes.

However, the highest changes are mainly in the first and third flexural modes since the

damage is located at the middle of the span (where second flexural mode is not highly

affected by the slot). Other special case is the first axial mode that decreases from

5558Hz (integral) up to 4386Hz (when a/d=0.5). The coupling mode are magnified

depending on the angle and location and proximity to the crack location.

Fig. 4. FRF plot for FFB, L/d=7.3, integral shaft showing the first six natural frequencies (each

color response denotes one of the six hit-excitation points)

1st flexural

2nd flexural

1st tors 1st axial

3rd flexural

2nd tors

1000

2000

3000

4000

5000

6000

7000

1000 3000 5000 7000

Nat

ura

l Fre

qu

ency

(H

z)

Natural Frequency (Hz)

FEMSOLIDS

TEST

1000

2000

3000

4000

5000

6000

1 2 3

Freq

uen

cy (

Hz)

Mode

Timoshenko

FEM Solids

TEST

0 2000 4000 6000 8000

Frequency [Hz]

-60

-40

-20

0

20

40

dB

/1 [

(m/s

2)/

N]

Frequency Traces: 6/6 Compressed

D1: Uncrck_susp_90 M108: FRF(C1,C2) [(m/s2)/N] 7/2 -X/+X






1st flexural 2nd flexural

1st torsional 1st axial

3rd flexural

2nd torsional

Fig. 5. FRF crack depth variations, FFB, L/d=7.3: black –integral; orange-a/d=0.1; green-

a/d=0.2; blue- a/d=0.3; grey- a/d=0.4; and red- a/d=0.5

Mainly focusing on the first three flexural modes for the integral and damaged shaft

(Figs. 6,7,8), a clear splitting in each natural frequency is observed, behavior confirmed

by numerical calculation. For slotted shafts two values of natural frequencies appear

since the structure has now two different orthogonal stiffness’ being the direction the

weaker orientation. This modal frequency separation is mostly influenced by crack

depth. When the excitation is orthogonal to crack orientation (along ), important cou-

pling between the two directions occurs since vibration along the cracked plane direc-

tion is also excited. This shows how crack orientation vibrating modal coupling energy

is manifested. However this coupling is highly minimized when the excitation occurs

along the crack direction () becoming inexistent should the impact direction is exact.

Fig. 6. First flexural mode frequencies for FFB, L/d=7.3, integral (black) and slotted shaft at

crack depths: a/d=0.3 orange, a/d=0.4 green

0 2000 4000 6000 8000

Frequency [Hz]

-60

-40

-20

0

20

40 d

B/1

[(m

/s2

)/N

]

Frequency Traces: 6/6 Compressed

D1: Uncrck_susp_90 M28: FRF(C1,C3) [(m/s2)/N] 2/2 -Y/+Y

D2: Crck10_susp_90 M28: FRF(C1,C3) [(m/s2)/N] 2/2 -Y/+Y





500 1000 1500 2000

Frequency [Hz]

-60

-40

-20

0

20

40

dB

/1 [

(m/s

2)/

N]

Frequency Traces: 3/3 Uncompressed


D4: Crck30_susp_90 M142: FRF(C1,C2) [(m/s2)/N] 9/2 -X/+X


1st flexural

1st axial

3rd flexural

direction

direction

integral

Fig. 7. Second flexural mode frequencies for FFB, L/d=7.3 integral (black) and slotted shafts,

a/d=0.3 orange, a/d=0.4 green (peaks above 3400Hz are decreased 1st torsional frequencies)

Fig. 8. Third flexural mode frequencies for FFB, L/d=7.3 integral (black) and slotted shafts;

crack depths: a/d=0.3 orange, a/d=0.4 green.

5 Final Discussion

Several slenderness ratios combined with different crack depths under two boundary

conditions (simply supported and free-free) are studied. Although the first ten natural

frequencies are calculated for all cases, only the first four flexural modes are clearly

visible during tests. Fig. 9 shows the 1st mode natural frequencies for these two bound-

ary conditions. From this figure, the influence of the beam slenderness on the crack

detectability can be seen. The stockier the beam, the better for detecting cracks since

the change in frequency is bigger for the same crack depth, this is verified to both

boundary conditions. For instance, in the SSB, with 4% in frequency change, an a/d=0.2

3100 3200 3300 3400 3500 3600

Frequency [Hz]

-20

0

20

40

dB

/1 [

(m/s

2)/

N]


D1: Uncrck_susp_90 M50: FRF(C1,C4) [(m/s2)/N] 3/2 -Y/+Z

D4: Crck30_susp_90 M50: FRF(C1,C4) [(m/s2)/N] 3/2 -Y/+Z

D5: Crck40_susp_90 M50: FRF(C1,C4) [(m/s2)/N] 3/2 -Y/+Z

5600 5700 5800 5900 6000 6100 6200 6300

Frequency [Hz]

-20

0

20

40

dB

/1 [

(m/s

2)/

N]





direction

direction

direction

direction

integral

integral

crack can be detected in a stocky shaft (L/d=8), while the same crack depth in a slender

shaft (L/d=35) only reveals less than 1% in frequency change. Fortunately real rotor

machines are not so slender allowing the opportunity to detect cracks. For the FFB,

stocky beams (L/d=8) require deeper cracks to obtain the same frequency change.

Fig. 9. First mode simply supported slotted beam (left) and free-free beam (right): dimension-

less natural frequencies along plotted vs beam slenderness ratios and crack depths

The concept of modal frequencies (separation) splitting functions, obtained by simula-

tion and experiments, is introduced by means of Equation (4). These frequency func-

tions reflect strong crack depth and slenderness ratio influences and represent the dif-

ference between the two peak values that appear for a cracked shaft for each mode j.

Inspection of these functions shows that the deeper the crack and the stockier the shaft,

the bigger value of this frequency function, as shown in Figure 10 for the SSB, where

only the first mode is shown.

∆𝜔𝑗 = 100 ∗ (𝜔𝜂,𝑗 − 𝜔𝜉,𝑗)/𝜔𝑜,𝑗 (4)

Fig. 10. Frequency split functions for different crack depths and beam slenderness for SSB for

fundamental mode

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0.0 0.1 0.2 0.3

w1

/w1

FEM

-S

crack depth a/d

L/d=8 FEM solids

L/d=8 Test

L/d=20 FEM solids

L/d=20 Test

L/d=35 FEM solids

L/d=35 Test

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0.00 0.10 0.20 0.30

w1

/w1

FEM

-S

crack depth a/d

L/d=7.3 FFB FEM solidsL/d=7.3 FFB TestL/d=20 FEM solidsL/d=20 TestL/d=35 FEM solidsL/d=35 Test

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5

w1=100*(w

-w)/wo

crack depth, a/d

L/d=8, FEM-S

L/d=12, FEM-S

L/d=12, Test

L/d=35, FEM-S

L/d=35, Test

Figure 11 shows two bending natural frequencies of the stocky shaft (L/d=7.3) FFB. In

both sides of the figure it is observed that the crack is more detectable if the beam is

excited in the direction rather than direction since the stiffness change is bigger. On

the other hand, since the crack is located at half midspan, only the first flexural mode

(and odd modes) give a better opportunity to detect the crack. In the second flexural

mode (and even modes) only very deep cracks can be detected for this crack location.

Fig. 11. Natural frequencies for 1st flexural mode (left) and 2nd flexural mode (right), for FFB,

L/d=7.3 along crack direction and orthogonal direction (TB=Timoshenko Beam, S=Solids)

Fig. 12 corresponds to the first torsional and axial modes found during tests and also

numerical calculation. The first torsional mode does not show an excellent opportunity

to detect the crack since produces around 2% in frequency change even for deep cracks

(a/d=0.4 or deeper). However, the axial mode is very sensitive to the crack presence.

Shear effects on torsional vibrations are here evident since Rayleigh beam does not

follow the pattern as Timoshenko beam, FEM Solid and test do. For higher flexural

modes only some natural frequencies are reached.

Fig. 12. Natural frequencies of the 1st torsional (left) and 1st axial (right) modes (FFB)

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.0 0.1 0.2 0.3 0.4 0.5

w/w

FEM

-So

lid)

crack depth, a/d

FEM-TB orthog. to crack

FEM-S orthog. to crack

Test, orthog. to crack

FEM-TB, crack dir.

FEM-S, crack dir.

Test, crack dir.0.90

0.92

0.94

0.96

0.98

1.00

1.02

0.0 0.1 0.2 0.3 0.4 0.5

w/w

FEM

-So

lid)

crack depth, a/d

FEM-TB orthog. to crack

FEM-S orthog. to crack

Test, orthog. to crack

FEM-TB, crack dir.

FEM-S, crack dir.

Test, crack dir.

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0 0.1 0.2 0.3 0.4 0.5

w/w

FEM

-So

lid

crack depth (a/d)

FEM Solid

FEM Timoshenko Beam

FEM Rayleigh Beam

TEST 0.75

0.8

0.85

0.9

0.95

1

0 0.1 0.2 0.3 0.4 0.5

w/w

FEM

-So

lid

crack depth (a/d)

FEM Solid

TEST

6 Conclusions

Although real machines during operation are much more complex, this study help

define a relation between crack depth and slenderness of shafts to feasibility of detect-

ing cracks. Beam elements use equivalent-length to model the crack [6] and reproduce

the frequency split and reduction. Splitting is verified also during test. Modal testing

analysis provides confident results up to first 3 flexural modes, first torsional and axial

(clearly shown) modes. Beam slenderness has an important effect on the crack detec-

tion; i.e., stocky cracked beams display clear differences in natural frequencies and

mode shapes compared to slender beams, where crack identification is more complex.

Notched shafts here analyzed provide useful upper bound frequency reductions val-

ues (breathing cracks display up to, and smaller decreases). Modal frequency splitting

for each natural frequency are confirmed and validated both, by numerical simulation

and test, limit values of diminished frequencies for a cracked shaft represents the ex-

treme case of a damaged rotor by an actual crack. Breathing mechanisms have nonlinear

aspects that are out of the scope of the present study. Vibrating modal coupling ener-

gy helps identify how the shaft is excited; strong energy manifests when excitation is

along the orthogonal direction-; coupling energy magnitude greatly diminished when

the excitation occurs along crack -orientation..

Acknowledgements. The authors1 acknowledge the National Council of Science and

Technology (CONACYT) and the Instituto Politécnico Nacional IPN, MÉXICO for the

scholarships, sabbatical year and grants received to accomplish this work.

7 References

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3. Machorro-Lopez J.M., Adams Douglas E, Gómez-Mancilla J.C., Gul Kamra: Identification

of damaged shafts using active sensing – simulation and experimentation, Journal of Sound

and vibration, Elsevier, Vol. 327, number 3-5, pp. 368–390, ISSN 0022-460x (2009)

4. J.C. Gómez-Mancilla, J.J. Sinou, V R Nosov, F. Thouverez, A Zambrano: The influence of

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5. N. Bachschmid, P. Pennacchi, E. Tanzi: Cracked Rotors, A survey on static and Dynamic

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