Superposition Principle applied to Natural Frequency Changes

of a Beam with Multiple Cracks

ZENO-IOSIF PRAISACH

“Eftimie Murgu” University of Resita

P-ta Traian Vuia 1-4, 320085 Resita

ROMANIA

zpraisach@yahoo.com

GILBERT-RAINER GILLICH

“Eftimie Murgu” University of Resita

P-ta Traian Vuia 1-4, 320085 Resita

ROMANIA

gr.gillich@uem.ro http://www.uni-resita.eu/FH-Sites/uem/index.php?id=1556

PENTRU-FLORIN MINDA

“Eftimie Murgu” University of Resita

P-ta Traian Vuia 1-4, 320085 Resita

ROMANIA

f.minda@uem.ro

NICOLETA GILLICH

“Eftimie Murgu” University of Resita

P-ta Traian Vuia 1-4, 320085 Resita

ROMANIA

n.gillich@uem.ro

Abstract: - The paper presents the phenomenon that appears in beams with multiple damages in respect to natural

frequencies. Damaged beams with different boundary conditions were analyzed by the authors, both in analytical way

and using the finite element method. The considered damages are open cracks, affecting the whole width of the beam

and have various levels of depth. In this paper the case of a cantilever beam with multiple damages is presented; the

results were compared with those obtained by experiments. The investigation lead to the conclusion that the

superposition principle can be used to find out the frequency changes of a beam with multiple cracks, when the effect

of each crack is known.

Key-Words: - vibration, natural frequency, finite element method, superposition principle, damages

1 Introduction

Damage detection using natural frequency shifts is

largely presented in literature [1], [2], [3] and [4]. The

methods based on frequency change can be classified in

two groups: methods limited to damage detection and

methods destined to detect, locate and quantify damages.

Literature reviews, [3] and [5], affirm that all methods

based on natural frequency changes belong to the second

group are model-based, typically relying on the use of

finite element models. These models can be categorized

into three main categories: local stiffness reduction;

discrete spring models; and complex models [6].

Damages influence the dynamic behavior of

structures, changing their mechanical and dynamic

characteristics such as natural frequencies, mode shapes,

damping ratio, and stiffness or flexibility. These most

common features that are used in damage detection are

identified from measured response time-histories (most

often accelerations or strains) or spectra of these time-

histories [7], [8] and [9].

It is known that all approaches fit particular cases,

given by boundary conditions or by damage location.

Our research intended to find a general method, able to

detect, locate and evaluate the severity of open cracks in

all types of beams.

Recent Advances in Signal Processing, Computational Geometry and Systems Theory

ISBN: 978-1-61804-027-5 233

While the aim of the research was to develop a

simple method to detect, localize and quantify the

severity of damages with the least equipment possible,

bending vibrations were considered. In our research we

have studied the natural frequency changes due to

damages on cantilever, simple supported and double

clamped beams, for a larger number of vibration modes.

In previous papers the analysis was performed using

analytic calculus and the finite element method (FEM),

for the undamaged beam and for the beam having one

crack placed one-by-one in a lot of locations along it

[10] and [11].

2 Numeric and analytic investigation The real analyzed beam was a steel one, having the

following geometrical characteristics: length L = 1000

mm; width B = 50 mm and height H = 5 mm.

Consequently, for the undamaged state the beam has the

cross-section A = 250·10-6 m2 and the moment of inertia

I = 520.833·10-12

m4. The material parameters of the

specimens are: mass density ρ = 7850 kg/m3; Young’s

modulus E = 2.0·1011

N/m2 and Poisson’s ratio µ = 0.3.

Three types of beams were analyzed: cantilever

beam, double clamped beam and simple supported

beam. Figure 1 presents the cantilever beam in damaged

state, while figure 2 and 3 present the double clamped

and simple supported beam respectively.

Fig. 1. Cantilever beam

Fig. 2. Double clamped beam

Fig. 3. Simple supported beam

The beams were analyzed, both in the undamaged

and damaged case, using analytic and the finite element

methods (FEM). The 3D beam was meshed by 2 mm

elements for the undamaged beam and the same mesh

but with finer elements in the vicinity of damage for the

damaged beams. The first ten natural frequencies of the

weak-axis bending modes for the undamaged beams

were determined; the values are presented in table 1, for

cantilever beam, in table 2 for double clamped beam and

in table 3 for simple supported beam.

Afterwards, a series of damages placed separately

one after the other on 190 locations along the whole

length of the beam were modeled. We selected

uncomplicated damage geometry, easy to reproduce on

the real structure by saw cuts, with the constant width of

2 mm and 9 levels of depth, reducing the cross-section

by 8, 17, 25, 33, 42, 50, 58, 67 and 75% respectively.

Table 1

Vibration

Mode

i

Natural

frequency

fi [Hz]

Vibration

Mode

i

Natural

frequency

fi [Hz]

1 4.08986 6 347.4518

2 25.6266 7 485.4578

3 71.7545 8 646.5624

4 140.6275 9 830.7827

5 232.5200 10 1038.1089

Table 2

Vibration

Mode

i

Natural

frequency

fi [Hz]

Vibration

Mode

i

Natural

frequency

fi [Hz]

1 26.099 6 486.421

2 71.927 7 647.735

3 140.991 8 832.150

4 233.071 9 1039.671

5 348.206 10 1270.255

Table 3

Vibration

Mode

i

Natural

frequency

fi [Hz]

Vibration

Mode

i

Natural

frequency

fi [Hz]

1 11.444 6 411.960

2 45.779 7 562.200

3 103.010 8 734.707

4 183.140 9 930.335

5 286.150 10 1149.068

For all the resulting damage cases, the first ten

natural frequencies were determined.

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ISBN: 978-1-61804-027-5 234

Figure 4 present analytical results, [10] and [12], in

form of network, for the first four modes of damaged

cantilever beams. The free end of the beam is on the left

side of the figures.

Fig. 4. Frequency shift for damaged cantilever beam

Fig. 5. Frequency shift for damaged double clamped

beam

Figure 5 present analytical results, in form of

network, for the first four modes of damaged double

clamped beams and figure 6 the analytical results for the

damaged simple supported beam.

Fig. 6. Frequency shift for damaged simple supported

beam

In these pictures (fig. 4, 5 and 6), the vertical axis

represents the natural frequency [Hz], the first horizontal

axis represents the length of the beam [%] and the

second horizontal axis represents the depth of the

damage [%].

3 Description of the method For all ten vibration mode, at each location of the

damage and for each level of damage depth we consider

the relative shift in frequency, given with formula (1):

100),(

),( ⋅∂−

=∂∆

U

iUi

f

xffxf [%] (1)

where,

Uf - represent the natural frequency for the

undamaged beam;

ixf ),( δ - represent the natural frequency for the

damaged beam, with damage at location x and damage

depth δ, for vibration mode No. i.

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ISBN: 978-1-61804-027-5 235

Representing the relative shift in frequency versus

vibration mode i, is obtained the relative shift in

frequency tendency, or the “genetic algorithm” which

permits to localize and assess the damage on the beam.

Fig. 7. Relative shift in frequency tendency

Considering a cantilever beam with a damage located

at 0.5L (L is the length of the beam) from the clamped

end, the damage having the depth 0.5h (h is the height of

the beam), the relative shift in frequency for the first ten

vibration modes can be represented like in the figure 7.

Fig. 8. Relative shift in frequency tendency

Then, considering a cantilever beam with a damage

located at 0.892L from the clamped end and a depth

0.5h, the relative shift in frequency for the first ten

vibration modes can be represented like in the figure 8.

4 Superposition principle We consider a cantilever beam having simultaneously

two damages like that presented in the section above. By

using the FEM there are obtained the natural frequencies

for the first ten vibration modes. Afterwards, using

relation (1) one can obtain the relative shift in frequency

for those modes.

Figure 9 represents the relative shift in frequency

tendency for cantilever beam with damage at 0.5L

(continuous black line), relative shift in frequency

tendency for cantilever beam with damage at 0.892L

(dashed gray line), relative shift in frequency tendency

for cantilever beam with the two damages (continuous

gray line) and overlapping line (dot black line) that

represents the sum of relative shift in frequency

tendency for damage located at 0.5L and 0.892L.

It is obvious that the FEM analysis for the beam with

two damages present the same the shift in frequency like

the sum of the shifts in frequency for the beam having

the two damages independently. Consequently, the

superposition principle can be used.

Fig. 9. Superposition principle applied on the cantilever

beam with two damages

In the second example it is considered a cantilever

beam with three damages, located as follows: the first

damage located at 0.77L from the clamped end, having a

depth of 0.25h, the second damage located at 0.351L

having a depth of 0.5h and third damage located at 0.5L

from the clamped having a depth of 0.5.h.

Fig. 10. Relative shift in frequency tendency

Recent Advances in Signal Processing, Computational Geometry and Systems Theory

ISBN: 978-1-61804-027-5 236

The relative shift in frequency tendency for the

cantilever beam with damage at 0.77L and 0.25h depth

is presented in figure 10.

The relative shift in frequency tendency for the

cantilever beam with damage at 0.351L and 0.5h depth

is presented in figure 11, and the relative shift in

frequency tendency for the cantilever beam with damage

at 0.5L and 0.5h depth is presented in figure 7.

Fig. 11. Relative shift in frequency tendency

Considering a cantilever beam with three damages,

located in the same position and the same damage depth

as the above mentioned, by using FEM analyses there

are obtained the natural frequencies for the first ten

vibration modes. Using relation (1) one can obtain the

relative shift in frequency for this modes.

Representing the relative shift in frequency tendency

(fig. 12) for the FEM analyzed beam with three damages

(continuous black line) and making the sum of the

relative frequency shift of each of the three damage

cases (dashed black line) it can be observed that two

lines overlaps.

Fig. 12. Superposition principle applied on the cantilever

beam with three damages

This is a new confirmation that the superposition

principle can be involved in the analysis of beams with

multiple cracks.

4 Conclusion The method presented in the paper, applicable to beams

with open cracks, is based on certain phenomena

characteristic to the dynamic behaviour of beams,

highlighted as a result of several analytical, numerical

and experimental studies developed by the authors.

Analyzing the figures 4, 5 and 6 it can be observed

that, for all types of supporting beams, at each vibration

mode, there are locations on the damaged beam that

natural frequency remains unchanged, irrespective of the

damage depth, respectively the relative shift in

frequencies are zero. These locations correspond to the

inflexion points (IP) from the mode shape, presented in

figure. 13.

Fig. 13. Fifth vibration mode, double clamp beam. Mode

shape's critical point versus the frequencies changes for

the damaged beam

This conclusion is valid also for the case of multiple

cracks, by applying the superposition principle. For

example, in the up mentioned case, with the beam with

three damages (fig. 12), at vibration mode 7, the relative

shift in frequency has zero value because in all the three

cases, at seventh vibration mode, with beam with single

damage at 0.77L from clamped end (fig. 10) and damage

at 0.351L from clamped end (fig. 11), and damage at

0.5L from clamped end (fig. 7), the relative shift in

frequency has zero value.

Analyzing figures 4, 5 and 6 it can be observed that,

for beams with all types of support, at each vibration

mode, there are locations on the damaged beam where

natural frequency exhibits local minima, amplified by

the depth of the damage, respectively the relative shift in

frequencies have local maxima.

These locations correspond to the maximum points

(MP) and minimum points (mP) from the mode shape,

like it is presented in figure 13.

Recent Advances in Signal Processing, Computational Geometry and Systems Theory

ISBN: 978-1-61804-027-5 237

However, using the superposition principle, it s

possible to describe the dynamic behavior of the beam

with multiple damages, in terms of the changes in

frequency for several vibration modes.

Acknowledgements The authors gratefully acknowledge the support of the

Managing Authority for Sectoral Operational

Programme for Human Resources Development

(MASOPHRD), within the Romanian Ministry of

Labour, Family and Equal Opportunities by co-financing

the project “Excellence in research through postdoctoral

programmes in priority domains of the knowledge-based

society (EXCEL)” ID 62557 and “Investment in

Research-innovation-development for the future

(DocInvest)” ID 76813.

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ISBN: 978-1-61804-027-5 238