Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam

ptance curvature matrices of the cracked beam, crack depth = 10%
 10 
a) ωằω1 b) ωằω2 
Fig. 6. Receptance curvature matrices of the cracked beam, crack depth=20% 
a) ωằω1 b) ωằω2 
Fig. 7. Receptance curvatures of the cracked beam, crack depth=20% 
Influence of the noise 
In order to simulate the polluted measurements, white noise is added to the receptance 
curvature obtained at the first frequency of the beam having two cracks at 0.34L and 0. 65L 
with depths of 10%. The white noise vector is obtained as following formula [16]: 
 (22) 
Where σ2 is the variance of the receptance curvature, SNR is the desired signal to noise ratio 
and R is a standard normal distribution vector with zero mean value and unit standard 
deviation. The noisy receptance curvature is the sum of the simulated receptance curvature 
and the noise vector presented in Eq. (22). 
Fig. 8 presents the noisy receptance curvatures at the first frequency with the SNR of 30 and 
10. When the noise level SNR is of 40, the two peaks at the crack positions can be inspected 
clearly as presented in Fig. 8a. When the SNR equal to 20, the peaks at crack positions are 
presented but they are more difficult to be detected as shown in Fig. 8b. These results suggest 
that, the proposed method can be applied efficiently to detect small crack depth from noisy 
measurements. 
2
noise *R
*ln(10)exp
10
SNR
s
=
ổ ử
ỗ ữ
ố ứ
(a) ω ≈ ω1
 10 
a) ωằω1 b) ωằω2 
Fig. 6. Receptanc urvature matrices of the cracked beam, crack depth=20% 
a) ωằω1 b) ωằω2 
Fig. 7. Receptance curvatures of the cracked beam, crack depth=20% 
Influence of the noise 
In order to simulate the polluted measurements, white noise is added to the receptance 
curvature obtained at the first frequency of the beam having two cracks at 0.34L and 0. 65L 
with depths of 10%. The white noise vector is obtained as following formula [16]: 
 (22) 
Where σ2 is the variance of the receptance curvature, SNR is the desired signal to noise ratio 
and R is a standard normal distribution vector with zero mean value and unit standard 
deviation. The noisy receptance curvature is the sum of the simulated receptance curvature 
and the noise vector presented in Eq. (22). 
Fig. 8 presents the noisy receptance curvatures at the first frequency with the SNR of 30 and 
10. When the noise level SNR is of 40, the two peaks at the crack positions can be inspected 
clearly as presented in Fig. 8a. When the SNR equal to 20, the peaks at crack positions are 
presented but they are more difficult to be detected as shown in Fig. 8b. These results suggest 
that, the proposed method can be applied efficiently to detect small crack depth from noisy 
measurements. 
2
noise *R
*ln(10)exp
10
SNR
s
=
ổ ử
ỗ ữ
ố ứ
(b) ω ≈ ω2
Fig. 5. Receptance curvature matrices of the cracked beam, crack depth = 20%
Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam 359
 9 
peak positions, the receptance curvatures along the beam is extracted when the force acts at a 
fixed position. The positions of these sharp changes can be inspected clearly as shown in Figs. 
5 and 7 when the force acts at the position of 0.42L. As can be seen from these figures, the 
positions of the sharp changes are at 0.34L and 0. 65L which coincide the crack positions. It 
should be noted from that for each level of the crack depth, the heights of sharp peaks are 
greater when the crack position is closer to the maxima of the receptance curvatures. 
Meanwhile, the heights of sharp peaks in receptance curvature are smaller when the crack 
position is far from the maxima of the receptance curvatures. 
These results mean that, the sharp peaks in receptance curvatures can be used for crack 
detection. In order to detect the cracks, only one receptance curvature measured along the 
beam when the force acts at a fixed point is needed. The sharp peaks in this measured 
receptance curvature indicate the existence of cracks, the positions of these sharp peaks point 
out the positions of cracks, and the heights of sharp peaks correspond to the crack severities. 
In addition, the numerical simulations show that in order to have better results for crack 
detection purpose, the force and the response should be applied at maximum positions of the 
receptance curvature matrices. When the crack is located at the minimum position of the 
receptance curvature, the crack cannot be detected. Clearly, only the receptance curvature 
corresponding to the first mode shape can be used for detecting arbitrary cracks on the beam 
since the first mode shape do not have any node. 
a) ωằω1 b) ωằω2 
Fig. 4. Receptance curvature matrices of the cracked beam, crack depth=10%: 
a) ωằω1 b) ωằω2 
Fig. 5. Receptance curvatures of the cracked beam, crack depth=10% 
(a) ω ≈ ω1
 9 
peak positions, the receptance curvatures along the beam is extracted when the force acts at a 
fixed position. The positions of these sharp changes can be inspected clearly as shown in Figs. 
5 and 7 when the force acts at the position of 0.42L. As can be seen from these figures, the 
positions of the sharp changes are at 0.34L and 0. 65L which coincide the crack positions. It 
should be noted from that for each level of the crack depth, the heights of sharp peaks are 
greater when the crack position is closer to the maxima of the receptance curvatures. 
Meanwhile, the heights of sharp peaks in receptance curvature are smaller when the crack 
position is far from the maxima of the receptance curvatures. 
These results mean that, the sharp peaks in receptance curvatures can be used for crack 
detection. In order to detect the cracks, only one receptance curvature measured along the 
beam when the force acts at a fixed point is needed. The sharp peaks in this measured 
receptance curvature indicate the existence of cracks, the positions of these sharp peaks point 
out the positions of cracks, and the heights of sharp peaks correspond to the crack severities. 
In addition, the numerical simulations show that in order to have better results for crack 
detection purpose, the force and the response should be applied at maximum positions of the 
receptance curvature matrices. When the crack is located at the minimum position of the 
receptance curvature, the crack cannot be detected. Clearly, only the receptance curvature 
corresponding to the first mode shape can be used for detecting arbitrary cracks on the beam 
since the first mode shape do not have any node. 
a) ωằω1 b) ωằω2 
Fig. 4. Receptance curvature matrices of the cracked beam, crack depth=10%: 
a) ωằω1 b) ωằω2 
Fig. 5. Receptance curv tur s of the cracked beam, crack depth=10% 
(b) ω ≈ ω2
Fig. 6. Re ptance c vat res of the racked beam, crack t 10
 10 
a) ằ 1 b) ωằω2 
Fig. 6. Receptance curvature matrices of the cracked beam, crack depth=20% 
a) ωằω1 b) ωằω2 
Fig. 7. Receptance curvatures of the cracked beam, crack depth=20% 
Influence of the noise 
In order to simulate the polluted measurements, white noise is added to the receptance 
curvature obtained at the first frequency of the beam having two cracks at 0.34L and 0. 65L 
with depths of 10%. The white noise vector is obtained as following formula [16]: 
 (22) 
Where σ2 is the variance of the receptance curvature, SNR is the desired signal to noise ratio 
and R is a standard normal distribution vector with zero mean value and unit standard 
deviation. The noisy receptance curvature is the sum of the simulated receptance curvature 
and the noise vector presented in Eq. (22). 
Fig. 8 presents the noisy receptance curvatures at the first frequency with the SNR of 30 and 
10. When the noise level SNR is of 40, the two peaks at the crack positions can be inspected 
clearly as presented in Fig. 8a. When the SNR equal to 20, the peaks at crack positions are 
presented but they are more difficult to be detected as shown in Fig. 8b. These results suggest 
that, the proposed method can be applied efficiently to detect small crack depth from noisy 
measurements. 
2
noise *R
*ln(10)exp
10
SNR
s
=
ổ ử
ỗ ữ
ố ứ
(a) ω ≈ ω1
 10 
a) 1 ) 2 
Fig. 6. Receptance curv tur matrices of the cracked beam, crack depth=20% 
a) ωằω1 b) ωằω2 
Fig. 7. Receptance curv tur s of the cracked beam, crack depth=20% 
Influence of th noise
In order to simula e the polluted measurements, white noise is added to the recep ance 
curvature obtained at he first frequency of the beam aving two cracks at 0.34L and 0. 65L 
with depths of 10%. The white noise vector is obtained as following formula [16]: 
 (22) 
Where σ2 is the variance of the r ceptance urv ture, SNR is the des red signal to noise ratio 
and R is standard normal dist ibution vector with zero mean value and unit standard 
deviation. The n isy receptance curv tur is the sum of the simulat d receptance curv tur 
and the noise v ctor presented in Eq. (22). 
Fig. 8 presents the noisy receptance urv tures at the first frequency with the SNR of 30 and 
10. When the noise level SNR is of 40, the two peaks at th crack positions can be inspected 
clearly as presented in Fig. 8a. When the SNR equal to 20, the peaks at crac positions are 
presented but they are more difficult to be detected as shown in Fig. 8b. These results uggest 
that, the proposed method can be applied efficiently to d tect small crack depth from noisy 
measurements. 
2
noise *R
*ln(10)exp
10
SNR
s
=
ổ ử
ỗ ữ
ố ứ
(b) ω ≈ ω2
Fig. 7. Receptance curvatures of the crack d beam, crack dept
Influence of the noise
In order to simulate the polluted measurements, white noise is added to the recep-
tance curvature obtained at the first frequency of the beam having two cracks at 0.34L and
0.65L with depths of 10%. The white noise vector is obtained as following formula [18]
noise =
√√√√√ σ2
exp
(
SNR ∗ ln(10)
10
) ∗R, (23)
where σ2 is the variance of the r c ptance curvature, SNR is the desired signal to noise
ratio and R is stan ard normal distribution ct r with zero mean v lue and unit stan-
dard deviation. The noisy receptance curvature is the sum of the simulated receptance
curvature and the noise vector presented in Eq. (23).
360 Nguyen Viet Khoa, Cao Van Mai, Dao Thi Bich Thao
Fig. 8 presents the noisy receptance curvatures at the first frequency with the SNR of
30 and 10. When the noise level SNR is of 40, the two peaks at the crack positions can be
inspected clearly as presented in Fig. 8(a). When the SNR equal to 20, the peaks at crack
positions are presented but they are more difficult to be detected as shown in Fig. 8(b).
These results suggest that, the proposed method can be applied efficiently to detect small
crack depth from noisy measurements.
 11 
a) SNR=40 b) SNR=30 
Fig. 8. Receptance curvatures at the first natural frequency, crack depth=10%, =0.42L. 
4. Conclusions 
In this paper, exact formula of the receptance of a clamped-clamped cracked beam is 
presented. The advantage of the exact receptance function is that the frequency response at 
any point of a cracked beam can be easily predicted while the present receptance matrix 
methods can only predict the response at specific points. 
The exact receptance curvature functions of cracked beams are also derived. When there are 
cracks, the receptance curvatures of the beam are changed significantly at the crack positions. 
The sharp peaks in the receptance curvatures can be detected clearly when the crack depth is 
as small as 10% of the beam height. This is an advantage of using the receptance curvature for 
early crack detection. The sharp peaks in the receptance curvature is the indicator of the 
existence of cracks and the positions of the sharp peaks show the positions of the cracks. The 
crack can be estimated by using noisy measurements of the receptance curvatures. 
Acknowledgements 
This research is funded by Vietnam National Foundation for Science and Technology 
Development (NAFOSTED) under grant number 107.02-2017.300. 
References 
[1] R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, 
Cambridge, 1960. 
[2] B. Yang, Exact receptances of nonproportionally damped systems, Transactions of 
American Society of Mechanical Engineers Journal of Vibration and Acoustics 115 (1993) 
47–52. 
[3] J.E. Mottershead, On the zeros of structural frequency response functions and their 
sensitivities, Mechanical Systems and Signal Processing 12 (1998) 591–597. 
[4] M. Gurgoze, Receptance matrices of viscously damped systems subject to several 
constraint equations, Journal of Sound and Vibration (2000) 230(5), 1185-1190. 
[5] A. Karakas, M. Gurgoze, A novel formulation of the receptance matrix of non-
proportionally damped dynamic systems. Journal of Sound and Vibration 264 (2003) 733–
740. 
xˆ
(a) SNR = 40
 11 
a) SNR=40 b) SNR=30 
Fig. 8. Receptance urv tures at the first natural frequency, crack depth=10%, =0.42L. 
4. Conclusions 
In this paper, exact formula of the recep ance of lamped-cla e r cked be m is 
presented. Th advantage of the exact recept nce function is that the frequency respo se at 
any point of a cracked be m can be easily predicted while the present rec ptance ma rix 
methods can only predict the r spons at ecific points. 
The exact recept nce urv ture f nctions of racked beams are also derived. When there are
cracks, the receptance urv tures of the beam ar ch nged significantly at the crack positions. 
The sharp p aks in th receptance urv tures can be detected cl arly when the crack d pth is 
as small as 10% of the beam height. This is an advant ge of usin the receptance urv ture fo 
early crack detection. The sharp peaks in the recep ance urvature is the indicator of the 
existence of cracks and the positions of the sharp peaks show the positions of the cracks. The 
crack can be estimated by using noisy measurements of the receptance urv tures. 
Acknowledgements 
This research is funded by Vietnam National Found tion for Science and Technology 
Development (NAFOSTED) under grant number 107.02-2017.300. 
References 
[1] R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibrati n, Cambridge University Press, 
Cambridge, 1960. 
[2] B. Yang, Exact receptances of onpr portionally damped systems, Transactions of 
American Society of M chanical Engineers Journal of Vibration and Acoustics 115 (1993) 
47–52. 
[3] J.E. Mottershead, On the zeros of st uctural f equency response functions a d their 
sensitivities, Mechanical Systems and Signal Processing 12 (1998) 591–597. 
[4] M. Gurgoze, Receptance ma rices of viscously damped systems ubj ct to several 
constraint equatio s, Journal of Sound and Vibration (2000) 230(5), 1185-1190. 
[5] A. Karakas, M. Gurgoze, A novel formulation of the receptance ma rix of non-
proportionally damped dynamic s ste s. Journal of Sound and Vibration 264 (2003) 733–
740. 
xˆ
(b) SNR = 30
Fig. 8. Receptance curvatures at the first natural frequency, crack depth = 10%, xˆ = 0.42L
4. CONCLUSIONS
In this paper, exact formula of the receptance of a clamped-clamped cracked beam is
present d. The advantage of the exact receptance functi n is th t th frequency respons
at any point of cracked beam can be easily predi ed while the present receptance matrix
methods can only predi t the response at spe ific points.
The exact receptance curvature functions of cracked beams are also derived. When
there are cracks, the receptance curvatures of the beam are changed significantly at the
crack ositions. The sharp peaks in the receptance curvature can be det cted clearly
wh n the crack depth is as small a 10% of t e be m height. Th s is an adv ntage of us-
ing the receptance curvature for early crack d tectio . The sharp peaks in the receptance
curvature is the indicator of the existence of cracks and the positions of the sharp peaks
show the positions of the cracks. The crack can be estimated by using noisy measure-
ments of the receptance curvatures.
ACKNOWLEDGEMENTS
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.02-2017.300.
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