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Analysis of nonlinear behaviors of a cracked beam with nonlinear stiffness

Number of pages: 124 File Format: word File Code: 32582
Year: Not Specified University Degree: Master's degree Category: Facilities - Mechanics
Tags/Keywords: axis - Crack depth change - cracked beam - Dynamic behavior of the structure - free vibration - Frequency - natural frequency - Nonlinear behaviors - Nonlinear stiffness - Tir
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  • Summary of Analysis of nonlinear behaviors of a cracked beam with nonlinear stiffness

    Presented for obtaining a master's degree

    Department of Mechanical Engineering - Applied Design Orientation

    Abstract

    In this research, we investigated the free vibration of a cracked beam and found natural frequencies. In the first part, by using the open crack model, by modeling the cracked part as a beam-shaped element and satisfying the boundary conditions, we have obtained the natural frequencies for different boundary conditions and showed the natural frequency changes due to the change of different crack parameters. In the next part, with the same multi-crack beam modeling, we solved the problem for cracks with different geometric shapes, such as oval, triangle, and parabola, using the Galerkin method, and obtained the natural frequencies. Finally, using the open and closed crack model, we modeled the cracked beam under the assumption that the crack opens and closes during vibration, and we derived the equations of motion for two states of curved crack and v-shaped crack. To solve these non-linear equations, we used the averaging method and analyzed the natural frequency changes as well as the crack angle change during the crack vibration and the natural frequency changes per crack depth change for different boundary conditions.          Keywords: cracked beam, natural frequency, free vibration These cracks are often caused by fatigue and are one of the factors in the failure of axles. The presence of cracks causes changes in vibration characteristics and researchers have found that most of the amplitude of these vibrations of cracked structures depends on the depth, position and shape of the crack relative to the shape of the system modes. Therefore, a comprehensive and detailed research on the vibrations of cracked structures is very necessary. In recent years, many efforts have been made in order to analytically solve such problems. The occurrence of sudden failures in structures has led researchers to research on the dynamics of structures. Therefore, investigation and research on the time response, mode shape, slope shape and most importantly the measurement of the natural frequency of the cracked structure can be considered as an index in the research.

    The aim of this project is to provide a simple and practical method to find the natural frequencies of a cracked beam by considering the effect of all the parameters related to the crack, without using the relationships related to fracture mechanics. Also, in the part of open and closed crack modeling, the goal of modeling is to provide a model that takes into account the effect of all crack parameters and can be used to obtain the natural frequency of the beam during vibration. be [1 and 2]. In 1998, Chandros[1] and his colleagues comprehensively investigated the problem for one-edge and two-edge cracked beam[3]. Some have tried to solve the problem with numerical methods [4-10]. In the numerical solution of the problem, parameters such as natural frequency, stress concentration coefficient and mode shapes have been investigated. To explain the dynamic behavior of damaged structures, several analytical methods have been used. Many models have been presented to model cracks; These models can be classified into three groups. Some have tried to solve the problem by considering the reduction of local stiffness in the structure [11]. This model is based on finite element method. Heisty and Springer[2][12], Gounaris and Dimaruginas[3][13] have done good research on the investigation of the crack effect on a region of a cracked beam using the generalization of the finite element method. Ebrahim [14] presented an elasto-plastic model to investigate the deformation of the crack at its tip. Others have tried to estimate the local relative softness created [15]. In this approach, the undamaged part of the structure is modeled using standard methods such as [4] FEM or partial strain equations and the combination of finite elements and the crack is shown with a flexible spring. Expressions and equations for the amount of released strain energy or stress concentration coefficient are obtained by using compatibility matrix.Kiesemser[5][16] and his colleagues were among the first investigators of this topic. They investigated the crack effect by equivalent forces or moments at the discontinuity position of the structure. Dimaruginas [17] obtained the consistency matrix constants based on fracture mechanics relationships and used it for vibration analysis. Kristidsmed and Bar [6] [18] mainly developed different models to investigate the vertical vibrations of a symmetric bilobed cracked beam and to investigate the torsional vibrations of a cracked bar based on the Euler-Bernoulli theory. The models presented by Christides and Barr were improved and expanded in recent years by Shen and Pierre[7][19-20-21]. First, they have provided an estimate of the damaged parameter using two-dimensional finite element models, and then by applying similar ideas, they have extended and expanded the previous models for cracked beams. Chandros and Dimaruginas [22-23-24] presented a similar approach, but in their investigations, which is called crack function, energy relations and fracture mechanics concepts were used. In this approach, an attempt has been made to combine the ideas of Christides and Barr with Dimaruginas. The third approach is to ignore the shear effect in the cross-section and present a continuous model of the cracked beam, with the assumption that the crack can be replaced by a torsion spring. Both the relative softness method and the continuum model provide a description of the structure when the crack is considered to be open. Generally, in the analysis, the crack is considered as an open type in order to ignore the effects of non-linearity of the problem caused by the opening and closing of the crack. Some laboratory studies have also been conducted to check the correctness of the models. In most cases, cracks are created by making a small cut in the sample. Some experimental results are presented by Adams and Cooley [8] [25], as well as Rotolo [9] [26], for several cracks. The non-linear behavior of cracks has been done using the perturbation method [10] by many researchers such as Gudmundson [11] [27], Yasinski [28], Plakhtinko and Yasinski [12] [29] and Ballou [13] [30].

    Abstract

    A simple method is proposed to model the open cracked beam structures.  In this method, crack is modeled as a beam element.  Hence cracked beam can be assumed to be a beam with stepped cross sections, and problem of determining natural frequency and mode shape of cracked beam, can be solved as determining these characteristics for a beam with different length and cross section.  With this work, it is not necessary to model crack as lumped flexibility model in accordance to fracture mechanics and related sciences to obtain crack stiffness, and use this spring model of crack for further analysis. Keywords: Crack, lumped stiffness model, step beam, natural frequency, mode shape.

  • Contents & References of Analysis of nonlinear behaviors of a cracked beam with nonlinear stiffness

    List:

    Table of Contents

    Chapter 1 - Introduction and review of work done 16

    1-1 Introduction. 17

    1-2     History of studies and review of the work done 17

    1-3     Types of crack modeling. 20

    1-4     Statement of the open crack modeling problem 20

    1-5     Objectives and issues examined in the thesis. 21

    Chapter 2 - Linear and non-linear modeling of cracks and checking the equations of motion. 22

    2-1 Introduction. 23

    2-2     Equations of free vibration. 23

    2-2-1 Euler-Bernoulli theory. 23

    2-2-2          Tymoshenko's theory. 32

    2-2-3 The inspection of the beam includes several cracks. 41

    2-2-4          Cracks with different geometric shapes: 45

    2-3     Open and closed crack modeling 50

    2-3-1          Curved structure crack modeling. 51

    2-3-2          Checking v-shaped crack. 61

    2-3-3           Solving the problem with the averaging method. 66

    Chapter 3 - Modeling results. 71

    3-1 Introduction. 72

    3-2     Simple open crack results 72

    3-2-1          Beams with different crack depth ratios. 72

    3-2-2          Beams with different crack span length ratios. 75

    3-2-3           Checking the effect of changing the crack position. 78

    3-3 Examining the effect of the number of cracks. 81

    3-3-1           Checking the results for fixed opening depth and length and different positions. 82

    3-3-2           Examining the results for the position and length of the fixed opening and different depths. 83

    3-3-3          Checking the results for the fixed position and depth and the length of different openings. 85

    3-4 Examining beams with different geometric shapes. 87

    3-4-1           Oval crack. 87

    3-4-2           Parabolic crack. 91

    3-4-3           Triangular crack. 92

    3-5     Open and closed crack 95

    3-5-1          Curved crack with circular structure. 96

    3-6     Mode shape, slope shape, bending moment and shear force. 106

    3-7 Validation of the results of the proposed models. 111

    3-7-1          Simple open crack 111

    3-7-2          Triangular crack. 112

    3-7-3          Crack opening and closing 115

    Chapter 4 - Conclusion and suggestions. 117

    4-1 Conclusion. 118

     

     

    Source:

     

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Analysis of nonlinear behaviors of a cracked beam with nonlinear stiffness
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