In the procedure of the dynamic property test, the cantilever is equally divided into 12 parts, that is, there are 13 test points measured. In detail, the dynamic test is carried out in three batches due to the lack of acceleration sensors, in which the seventh test point at 0. A device similar to the exciting hammer is adopted to exert the pulsed excitation on the cantilevered beam. The acceleration time history response of the dynamic test point is collected by the vibration test system of the DH type produced by the Jiangsu Donghua Testing Technology Co.
The field test is shown in Figure 5. Reduction coefficients of the first two natural frequencies: It can be seen from Figure 6 that the reduction coefficients of the first two natural frequencies decrease with the increase in crack depth. The reduction coefficients obtained by both the presented method and the experiment show good agreement, which demonstrates the validity and reliability of this method.
In addition, the calculated relation curves between the reduction coefficients of the natural frequency and the relative crack depth have a sudden change. The first-order normalized mode shapes case 2: The second example shown in Figure 8 is a fixed—free beam with double I-section. Two cases of cracked double I-section beam are investigated:. A cracked fixed—free beam with double I cross section.
FEM model of a cracked fixed—free beam with double I cross section: It is worth mentioning that the boundary condition shown in Figure 3 is used in the calculation process by the proposed method in order to verify the validity of the proposed method more fully. The reduction coefficients of the first two natural frequencies under two cases are calculated by the proposed method and FEM, respectively. And the corresponding results are presented in Figure Reduction coefficients of the first two natural frequencies for two cases: The first two-order normalized mode shapes case 2: As can be seen from Figure 10 , the reduction coefficients for the first two natural frequencies of the fixed—free beam with double I-section decrease as the relative crack depth increases.
The more the number of cracks, the more significant the changes. One sees that the good agreement between the results of the presented method and FEM is achieved. It also can be seen from Figure 11 that the first two mode shapes obtained by the presented method and FEM are very close. Thus, the validity and reliability of the proposed method are verified. The geometrical parameters of the double I-section beam shown in Figure 12 are given as follows: A cracked beam of double I-section with linear spring and rotational spring supports.
The lowest two circular natural frequencies of different crack numbers are shown in Table 1. And the mode shapes of the intact beam are shown in Figure The first two-order mode shapes of the intact double I-section beam: From the calculated circular natural frequencies in Table 1 , one sees that the calculated circular natural frequencies decrease with the increase in crack number.
For the cracked beam under pinned—pinned boundary condition, the lowest two circular natural frequencies are The corresponding lowest two circular natural frequencies are calculated and are presented in Figure Natural frequencies with respect to linear springs with different stiffnesses: As can be seen from Figure 14 , for a cracked beam with double I-section which has only one crack, the first two circular natural frequencies increase nonlinearly with the increase in stiffness of linear springs. At first, the lowest two circular natural frequencies increase rapidly and then slowly and gradually approaching the natural frequencies of the cracked beam under pinned—pinned boundary condition.
The geometric parameters of the beam shown in Figure 15 are given as follows: In this article, the effect of crack parameters on the natural frequency of the beam with fixed—free and pinned—pinned boundary conditions is discussed. The first two natural frequencies of the beam are calculated by the proposed method. Corresponding results are shown in Figures 16 and Effect of the crack depth on the natural frequencies of the fixed—free beam. Effect of the crack depth on the natural frequencies of the pinned—pinned beam. It can be seen from Figures 16 and 17 that the first two natural frequencies of the fixed—free beam and the first natural frequency of the pinned—pinned beam decrease as the depth of crack increases.
In addition, the relation curves between the reduction coefficients of the natural frequency and the relative crack depth exhibit a sudden change. However, the second-order natural frequency of the pinned—pinned beam does not change with the increase in the depth of the crack, and the reason is that the bending moment generated by the second-order mode shape is zero at the midspan section. This phenomenon is demonstrated by equation The results are shown in Figures 18 and Effect of the crack position on the natural frequencies of the fixed—free beam.
Effect of the crack position on the natural frequencies of the pinned—pinned beam. It can be seen from Figures 18 and The effect of cracks on the natural frequency of the same order is different for the beam with different boundary conditions, even if the cracks are in the same position. The effect of the cracks at the same position on the natural frequency of the different order is different under the same boundary conditions.
The effect of the cracks at the different locations on the natural frequency of the same order is also different under the same boundary conditions. The reason is analyzed that the bending moment generated by the same order mode is different under the different boundary conditions, and the distribution of the bending moment generated by different order modes is different under the same boundary condition, that is, the bending moment at the crack position is different, which leads to the different effects of the crack on the natural frequency.
The presented equation 15 verifies the above viewpoints. For the fixed—free beam shown in Figure 15 in the range of 0. Thereby, in the range of 0. Similarly, the value of the moment generated by the second-order mode reaches the maximum at about 1. While the bending moment at 0. For the pinned—pinned beam in the range of 0. Therefore, the effect of the cack position on the first two-order natural frequencies of the pinned—pinned beam is determined as shown in Figure The first-order natural frequency and normalized mode shapes obtained from modal testing of the cantilever I-section beam Figure 4 are used as input data to identify the crack positions and depths.
The actual and predicted crack parameters calculated by the proposed identification algorithm are listed in Tables 2 and 3. Crack identification of the cantilever I-section beam with single open crack. Crack identification of the cantilever I-section beam with double open cracks. From the reported results in Tables 2 and 3 , the proposed method successfully identified the position and depth of the cracks in all case studies.
It is evident that the presented method is powerful in crack iditification. It can be seen that the absolute value of maximum relative error for crack position is 2. There are nine kinds of crack cases, in which there are three cases for single crack, double cracks, and triple cracks, respectively. The first natural frequency and mode shape are adopted as the input data in all cases of the crack diagnose.
Furthermore, in order to investigate the anti-noise ability of the crack identification algorithm in this article, different levels of noise is added into the first-order mode shape calculated by the present method. The expression of the first-order mode shape with noise D input is written as. The results of crack identification of the double I-section beam are presented in Tables 4 — 6. Crack identification of the multiple I-section beam with single open crack. Crack identification of the multiple I-section beam with double open cracks.
Crack identification of the multiple I-section beam with triple open cracks. According to the calculated results in Tables 4 — 6 , crack parameters of multiple I-section beam with an arbitrary number of cracks and different boundary conditions can be determined effectively and accurately by the presented method in this article. It is also observed that the reliability of the present method is influenced by the level of noise pollution. The absolute value of maximum relative error for crack position is 3.
It is an interesting topic for the crack diagnose based on the vibration test, which is the basis of free vibration analysis of crack. Multiple I-section hollow structures are the most widely used types in engineering fields, so it is significant to carry out analysis both in direct and inverse problems of a cracked multiple I-section beam. First, this article derives the local flexibility caused by the cracks of the multiple I-section beam and then presents an approach to the free vibration analysis of the beam with an arbitrary number of cracks under any general form of boundary conditions.
And the inverse problem of multiple I-section beam is also solved by a powerful method. Experimental study and numerical simulation on I-section and multiple I-section beams are carried out in this article, and some conclusions are obtained as follows:. Whether the natural frequency and mode shape calculated or the crack parameters location and depth identified by the present method, all have good agreement with the experimental results and FEM simulation results, which fully demonstrates the validity and reliability of this approach. For the beam only supported by linear springs, natural frequencies of the beam increase slowly with the increase in spring stiffness and gradually approaching the natural frequencies of the beam under pinned—pinned boundary condition.
Whether the fixed—free beam or the pinned—pinned beam, the natural frequency of the multiple I-section beam decreases with the increase in crack depth. And the relation curves between the reduction coefficients of the natural frequency and the relative crack depth exhibit a sudden change.
It reveals that shallow open crack and deeper open crack have different local flexibility coefficients of the cracked beam with multiple I-section. For some specific modes, the effects of crack position on this mode depend on the distribution of the bending moment along the beam length. That is to say, the different bending moments at crack location lead to the different effects on natural frequency.
This further illustrates the reliability and validity of the crack diagnose algorithm in this article, and it can be thought that the algorithm has strong anti-noise capability. Skip to main content. Advances in Mechanical Engineering. Direct and inverse problems on free vibration of cracked multiple I-section beam with different boundary conditions. Download Citation If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Via Email All fields are required. Send me a copy Cancel. See all articles by this author Search Google Scholar for this author.
Wensheng Wang Wensheng Wang. Article first published online: November 19, ; Issue published: November 1, Received: June 20, ; Accepted: This article is distributed under the terms of the Creative Commons Attribution 4. Keywords Multiple I-section beam , free vibration , damage identification , crack , different boundary conditions.
Local flexibility due to a crack in multiple I-section beam Section:. Open in new tab. Multiple I cross section shallow open crack. Multiple I cross section deeper open crack.
Inverse Problems in Engineering Mechanics II
Examples of direct problem Section:. Reliability of the proposed method. Comparison with experimental results. A cracked cantilever beam with I cross section. Comparison with FEM results. Numerical simulation for a cracked beam of double I-section with spring supports. The lowest two circular natural frequencies. Results and discussion for the parameters of crack. A three-I-section beam with single crack. Examples of inverse problem Section:. Crack identification of experimental cantilever beam. Crack identification of cracked beam supported by linear springs and rotational springs.
Tips on citation download. Vibration of cracked structures: Eng Fract Mech ; A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions. Comput Math Appl ; A simplified method for natural frequency analysis of a multiple cracked beam. J Sound Vib ; Zheng, T, Ji, T. An approximate method for determining the static deflection and natural frequency of a cracked beam. A general beam element for use in damage assessment of complex structures. J Vib Acoust ; Google Scholar , Crossref. Identification of crack location in vibrating simply supported beams.
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Probabilistic models of impedance matrices. Applications to seismic soil-structure interaction. Nonparametric stochastic modeling of geometrically nonlinear structural dynamic systems. Uncertainty modeling for robust design optimization in computational mechanics. Reliability analysis of large scale structures using a non-parametric approach. Numerical simulation of the axial transmission technique for bone evaluation: Journal of Biomechanics, 39 Supplement 1: Le Maitre, Christian Soize.
Probabilistic model of uncertainties in two-phase flow through porous media. TU Delft, The Netherlands, pp. Vibroacoustics of a cavity coupled with an uncertain composite panel. Soize, C; Schueller, GI. Uncertain rotating dynamical systems with cyclic geometry. Construction of a probabilistic model for the soil impedance matrix using a non-parametric method.
Non-parametric modelling of vibroacoustic coupling interface uncertainties. Thermo-mechanical model of a cardboard-plaster-cardboard composite plate submitted to fire load and experiments. Wit Transactions on Engineering Sciences. Application au dimensionnement sismique d'un ouvrage. Nonparametric modeling of the variability of vehicle vibroacoustic behavior. On measures of nonlinearities for dynamical systems with uncertainties. Identification of a random elastic medium by vibration tests. Probabilistic models for computational stochastic mechanics and applications.
Millpress, Rotterdam, Netherlands, pp. Uncertainties in structural dynamics for composite sandwich panels. Random uncertainties modelling for vibroacoustic frequency response functions of cars. Non Gaussian matrix-valued random fields for nonparametric probabilistic modeling of elliptic stochastic partial differential operators. Stochastic conditioner for accelerating convergence of Monte Carlo simulations. Lombard, Celine Dupont, E. Analytical model of a wall acoustic impedance and experimental comparisons. Transient dynamics induced by shocks in stochastic structures.
Uncertain nonlinear dynamical systems subjected to seismic loads. Random uncertainties modeling for the medium-frequency dynamics. Model uncertainty issues for predictive models. Cambier, Christophe Desceliers, Christian Soize. Prise en compte probabiliste des incertitudes dans l'estimation du comportement sismique d'un circuit primaire. Zermout, Ahmed Mebarki, Christian Soize. Specifying manufacturing tolerances for a given amplification factor: University of Southamption, England, pp. Nonlinear dynamical systems with data and model uncertainties subjected to seismic loads. Algebraic model of a wall acoustic impedance constructed using experimental data.
Non-Gaussian simulation using Hermite polynomial expansion. Spanos, PD; Deodatis, G. Random matrix theory and random uncertainties modeling. Sas, P; VanHal, B. Random uncertainties modeling in dynamical systems. Random matrix theory for modeling random uncertainties in transient elastodynamics. A nonparametric model of random uncertainties in dynamic substructuring. Balkema Publishers, Lisse, The Metherlands, pp. Sas, P; Moens, D. Monte Carlo simulation of positive random matrices and time responses of structural dynamical systems with random uncertainties.
Balkema Publishers, Lisse, The Netherlands, pp. Main difficulties in the mid-frequency range and reduced matrix models for structural-acoustic problems. Monte Carlo construction of Karhunen-Loeve expansion for non Gaussian random fields. The Johns Hopkins University, pp. Reduced models for computational structural acoustics in the medium-frequency range. Nonlinear structural dynamics equations in finite displacements for three-dimensional viscoelastic rotating structures with cyclic symmetry and for small geometrical perturbations.