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

Tóm tắt Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam: ...ly that the responses of the beam can be esti- mated by using the receptance matrices when the forcing frequency is close to natural frequencies. 7 (21) Table 1. Five cases with cracks of varying depths Case Crack depth (%) 1 2 3 4 0 10 20 30 3.1. Receptance of beam In...m with the crack depths of 10% and 20% are investigated. Figs. 4 and 6 depict the normalized receptance curvatures of the cracked beam with different levels of the crack depth when the forcing frequencies are close to the first and the second natural frequencies, respectively. As can be seen f...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...

**Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam**, để tải tài liệu về máy bạn click vào nút DOWNLOAD ở trên

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. REFERENCES [1] R. E. D. Bishop and D. C. Johnson. The mechanics of vibration. Cambridge University Press, (1960). Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam 361 [2] B. Yang. Exact receptances of nonproportionally damped dynamic systems. Journal of Vibra- tion and Acoustics, 115, (1), (1993), pp. 47–52. [3] J. E. Mottershead. On the zeros of structural frequency response functions and their sensitivities. Mechanical Systems and Signal Processing, 12, (5), (1998), pp. 591–597. https://doi.org/10.1006/mssp.1998.0167. [4] M. Gurgoze. Receptance matrices of viscously damped systems subject to several constraint equations. Journal of Sound and Vibration, 230, (5), (2000), pp. 1185–1190. [5] A. Karakas and M. Guărgoăze. A novel formulation of the receptance matrix of non- proportionally damped dynamic systems. Journal of Sound and Vibration, 264, (2003), pp. 733– 740. https://doi.org/10.1016/S0022-460X(02)01507-9. [6] P. Albertelli, M. Goletti, and M. Monno. A new receptance coupling substruc- ture analysis methodology to improve chatter free cutting conditions predic- tion. International Journal of Machine Tools and Manufacture, 72, (2013), pp. 16–24. https://doi.org/10.1016/j.ijmachtools.2013.05.003. [7] G. Muscolino and R. Santoro. Explicit frequency response function of beams with crack of uncertain depth. Procedia Engineering, 199, (2017), pp. 1128–1133. https://doi.org/10.1016/j.proeng.2017.09.239. [8] Y.-S. Lee and M.-J. Chung. A study on crack detection using eigenfrequency test data. Com- puters & Structures, 77, (3), (2000), pp. 327–342. https://doi.org/10.1016/S0045-7949(99)00194- 7. [9] D. Y. Zheng and N. J. Kessissoglou. Free vibration analysis of a cracked beam by finite element method. Journal of Sound and Vibration, 273, (3), (2004), pp. 457–475. https://doi.org/10.1016/S0022-460X(03)00504-2. [10] P. Gudmundson. The dynamic behaviour of slender structures with cross-sectional cracks. Journal of the Mechanics and Physics of Solids, 31, (4), (1983), pp. 329–345. https://doi.org/10.1016/0022-5096(83)90003-0. [11] J. Thalapil and S. K. Maiti. Detection of longitudinal cracks in long and short beams using changes in natural frequencies. International Journal of Mechanical Sciences, 83, (2014), pp. 38– 47. https://doi.org/10.1016/j.ijmecsci.2014.03.022. [12] N. V. Khoa. Monitoring a sudden crack of beam-like bridge during earthquake excitation. Vietnam Journal of Mechanics, 35, (3), (2013), pp. 189–202. https://doi.org/10.15625/0866- 7136/35/3/2561. [13] S. Caddemi and I. Calio. Exact closed-form solution for the vibration modes of the Euler– Bernoulli beam with multiple open cracks. Journal of Sound and Vibration, 327, (3-5), (2009), pp. 473–489. https://doi.org/10.1016/j.jsv.2009.07.008. [14] S. Caddemi and I. Calio`. Exact reconstruction of multiple concentrated damages on beams. Acta Mechanica, 225, (11), (2014), pp. 3137–3156. https://doi.org/10.1007/s00707-014-1105-5. [15] T. V. Lien, N. T. Duc, and N. T. Khiem. Mode shape analysis of multiple cracked functionally graded beam-like structures by using dynamic stiffness method. Vietnam Journal of Mechanics, 39, (3), (2017), pp. 215–228. https://doi.org/10.15625/0866-7136/8631. [16] K. V. Nguyen. Mode shapes analysis of a cracked beam and its application for crack detection. Journal of Sound and Vibration, 333, (3), (2014), pp. 848–872. https://doi.org/10.1016/j.jsv.2013.10.006. [17] R. F. Hoskins. Delta function. Horwood Publishing Limited, second edition, (2009). [18] R. Lyons. Understanding digital signal processing. Prentice Hall, Boston, USA, third edition, (2011).

File đính kèm:

- exact_receptance_function_and_receptance_curvature_of_a_clam.pdf