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Identification of the stiffness and damping coefficients of the truss support

Number of pages: 70 File Format: word File Code: 32558
Year: 2013 University Degree: Master's degree Category: Facilities - Mechanics
Tags/Keywords: Connections - damping coefficient - Difficulty coefficients - Dynamic measurement - Engineering structures - Reverse analysis - Stiffness factor - support
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  • Summary of Identification of the stiffness and damping coefficients of the truss support

    Master's Thesis

    Mechanical Engineering - Applied Design Orientation

    Abstract

    Identification of Stiffness and Damping Coefficients of Truss Beam Support

    By:

    Majid Abedi

    Supports and connections play a fundamental and important role in engineering structures. It is necessary to identify different support parameters. Stiffness and damping parameters are the most important parameters of a support. In this thesis, an inverse method based on the dynamic measurement data of strain and acceleration has been used to identify and check the stiffness and damping coefficients of the supports of trusses and two ends involved. For this purpose, using the least squares method, an optimization problem has been defined and then solved. In the design beam, the effect of various parameters such as the amount of measurement error, number of measurement data, type of data (strain or acceleration), initial guess, sensor installation location, number of sensors, time interval of force application, values ??of stiffness and damping parameters, data collection time on the inverse analysis response have been investigated. In the double-ended beam, only the effect of the measurement error, the number of measurement data, the type of data and the number of sensors on the results have been investigated. The results show that the double-ended beam problem is much more difficult than the single-ended beam problem. The use of acceleration data in the beam problem leads to better results. By carefully examining the numerical results obtained, an attempt has been made to answer the questions and problems that may occur during the practical test.

    Key words: inverse analysis, beam support, stiffness coefficient, damping coefficient, dynamic measurement data

    - Chapter 1: Introduction

    1-1- Importance Subject

    Vibrations of various bodies have been and are being researched by researchers and researchers, especially scientists of mechanics, physics and mathematics. Identification and analysis of vibrations of mechanical systems followed by the calculation of natural frequencies and modes [1] has always presented itself as an important issue in mechanical science in the direction of designing, identifying faults and controlling these systems. On the other hand, analyzing and checking the vibrations of continuous systems requires detailed knowledge of the geometry, physical and mechanical properties, loadings, initial and boundary conditions [2] governing the system. This is while often modeling these parameters in the form of a mathematical problem can be very challenging and at the same time very effective and important. Therefore, modeling these parameters as accurately and realistically as possible is considered a great help in designing, controlling and identifying defects of a system.

    One of these components is supports[3]. Basically, the place where a structure is connected to a foundation or another structure is called a support. In general, supports can be divided into fixed joint supports [4], movable joint supports [5] (roller), fixed supports [6] (solid), spring or elastic supports [7], etc. Each of the aforementioned supports has a specific number of degrees of freedom [8]. Of course, the desired degrees of freedom that are defined for the mentioned types of supports and used in the analysis, are in fact an ideal definition of the type of supports, and these supports may behave differently in reality, which can have different effects on the response of the mechanical system. For this reason, in the design and analysis of structural systems, it is inevitable to pay attention to supports and connections and their type of function. Different supports are made and used by various connections such as welds, rivets, pins, screws, rollers, etc. with their own characteristics in order to satisfy the predefined needs in different mechanical systems such as beams, sheets, frames, wings, types of shells, etc.

    Among the widely used structures in engineering, single-ended beams [9] are involved. Basically, a beam is said to have one end fixed (solid) and the other end free and able to move freely. As we know, in the ideal state, there is no degree of freedom in the support of this type of beams, in other words, there is no translational and rotational movement [10] at the place of the support, that is, both components of translational and rotational location change. It is zero.

    Tarai beams have important applications in various industries such as military, aviation, construction, etc. For example, airplane wings, nuclear power probes, construction cranes, bridges, etc. can be considered a single-ended beam. In Figure (1-1), some of the applications of the beam are depicted.

    It is clear that the supports in a mechanical system greatly affect the amount of energy loss and flexibility [1] of that system, and since the damping and flexibility of a system strongly affect its vibration response, providing models that can calculate the amount of effects resulting from constraints as accurately and realistically as possible will be necessary and inevitable. Also, all materials have a certain amount of structural damping [2], which depends on the type and structure of that material, and the amount of this damping can also be effective depending on the type of material and the desired system [1]. It is the structure of the system. On the other hand, one of the most important effects caused by a support in a system is the amount of energy loss and flexibility derived from that support in the system. Designing, analyzing and checking, the process of controlling and identifying the defects of a mechanical system without knowing these parameters leads to unrealistic conclusions.

    In the upcoming thesis, a single-ended beam and a double-ended beam whose support parameters are unknown are considered. It is clear that the stiffness and damping parameters of the supports play a major role in the vibration response of the mentioned beams. In this thesis, each fixed support is modeled with a pin[3] along with a linear torsional spring[4] and a viscous linear torsional damper[5]. The said pin is only allowed to move around the axis perpendicular to the pin, and the other directions are fixed. In the following, an attempt is made to estimate and calculate these parameters using strain measurement data [6] or acceleration [7]. Measurement data are provided with the help of simulation [8] in Ansys [9] software. In the next chapters, there are more explanations about this simulation and how to do it. As mentioned, it is very difficult to obtain these parameters in a direct way [10] and the best option for this is to use the inverse method [11]. Therefore, the use of inverse methods, which have received a lot of attention from researchers in recent years, can be very efficient and appropriate. Basically, an inverse problem [12] is a general framework for converting observed measurements into information about a physical object or a system under investigation,

    (images are available in the main file)

     

    Abstract

     

     

    Identification of Stiffness and Damping Coefficients of a Cantilever Support

     

     

     

          Supports and joints play a basic and important role in the engineering structures. It is necessary to identify the various parameters of supports. The stiffness and damping parameters are the most important parameters of a support. In this thesis, an inverse method based on dynamic strain and acceleration measurement data is used to identify and study the stiffness and damping coefficients of the supports of cantilever and doubly clamped beams. To this end, an optimization problem using the least squares method is defined, and solved subsequently. In the cantilever beam, the effect of various parameters such as the magnitude of measurement errors, number of measured data, data type (strain or acceleration), initial guess, sensor location, number of sensors, time duration of the applied load, magnitudes of the stiffness and damping parameters, time interval of data collection, on the inverse solutions are studied. In the doubly clamped beam, the effects of the magnitude of measurement errors, number of measured data, data type and number of sensors on the results are studied. The results show that the doubly clamped beam problem is more difficult than the cantilever beam problem. Better results in the cantilever beam are obtained when the acceleration data is used.

  • Contents & References of Identification of the stiffness and damping coefficients of the truss support

    List:

    1- First chapter: Introduction. 2

    1-1- Importance of the subject. 2

    1-2- The purpose of doing this thesis and the steps to do it. 5

    2- The second chapter: an overview of previous studies. 8

    2-1- Introduction. 8

    2-2-History of beam vibrations 8

    2-3-History of inverse analysis. 9

    2-3-1-Inverse identification of shock loads. 10

    2-3-2-inverse identification of material constants. 11

    2-3-3- issues of identifying cracks and defects. 11

    2-4-History of the use of springs and dampers 12

    3-Chapter 3: Theory basics. 17

    3-1- Introduction. 17

    3-2- The general process of solving an inverse problem. 18

    3-2-1- Problem definition. 20

    3-2-2- Presenting the direct model. 20

    3-2-3-Sensitivity calculation between outputs and parameters 20

    3-2-4-Test design. 21

    3-2-5-minimizing measurement error. 21

    3-2-6-Using reverse formulation. 21

    3-2-7- Reviewing the answer. 22

    3-3-basic concepts of inverse problems. 22

    3-4-inverse formulation. 28

    3-5-Choosing outputs 30

    3-6-Smoothing for physical problems 31

    3-7- Optimization methods. 33

    3-7-1- Direct search methods. 36

    3-7-2- Search methods based on gradient. 37

    3-7-3-Nonlinear method of least squares. 37

    3-7-4-methods of finding the root. 38

    3-7-5-genetic algorithms. 38

    3-7-6-points regarding optimization methods. 39

    4- Chapter 4: How to conduct research. Error! Bookmark not defined.

    4-1-Introduction. Error! Bookmark not defined.

    4-2-Description of the proposed model. Error! Bookmark not defined.

    4-3- Inverse analysis formulation. Error! Bookmark not defined.

    4-3-1- Calculation of the sensitivity matrix. Error! Bookmark not defined.

    4-3-2-simulation of measurement data. Error! Bookmark not defined.

    4-3-3-Making calculations in the software Error! Bookmark not defined.

    4-5- Examining the influence of different parameters on the temporal response of a projectile. Error! Bookmark not defined.

    4-5-1- Investigating the influence of the force application interval on the response of a beam. Error! Bookmark not defined.

    4-5-2- Investigating the effect of the stiffness coefficient on the response of the beam. Error! Bookmark not defined.

    4-5-3- Investigating the effect of the damping coefficient on the response of the beam. Error! Bookmark not defined.

    5- The fifth chapter: Numerical examples. Error! Bookmark not defined.

    5-1-Introduction. Error! Bookmark not defined.

    5-2-Inspection of a beam with one end involved (a beam) Error! Bookmark not defined.

    5-2-1- Investigating the effect of measurement error on the inverse response in a beam. Error! Bookmark not defined.

    5-2-2-Investigation of the convergence process of the inverse response in the beam. Error! Bookmark not defined.

    5-2-3- Investigating the effect of sensor location on the reverse response in a beam. Error! Bookmark not defined.

    5-2-4-Investigation of the effect of the number of measurement data on the reverse response in the beam. Error! Bookmark not defined.

    5-2-5- Investigating the effect of the number of sensors on the inverse response in a beam. Error! Bookmark not defined.

    5-2-6- Investigating the effect of the initial guess value on the reverse response in the beam. Error! Bookmark not defined.

    5-2-7- Investigating the effect of force application time on the reverse response in a beam. Error! Bookmark not defined.

    5-2-8- Examining the effect of time difference on the response of a beam. Error! Bookmark not defined.

    5-2-9-Investigation of the effect of data collection time on the reverse response in a beam without considering the time difference Error! Bookmark not defined.

    5-2-10-Investigation of the effect of data collection time on the reverse response in the design beam by considering the time difference Error! Bookmark not defined.

    5-2-11- Investigating the influence of the place of application of force on the reverse response in a beam. Error! Bookmark not defined.

    5-2-12- Investigating the influence of the damping coefficient value on the reverse response in a beam. Error! Bookmark not defined.

    5-2-13- Investigating the influence of the stiffness coefficient on the reverse response in a beam. Error! Bookmark not defined.

    5-3-Inspection of the two-headed beam involved. Error! Bookmark not defined.

    5-3-1-Investigation of the effect of measurement error on the reverse response in the double-ended beam involved. Error! Bookmark not defined.

    5-3-2- Investigating the effect of the number of measurement data on the reverse response in the double-headed beam.

    5-3-2- Investigating the effect of the number of measurement data on the inverse response in the double-headed beam involved. Error! Bookmark not defined.

    5-3-3- Investigating the effect of the number of sensors on the reverse response in the two-headed beam involved. Error! Bookmark not defined.

    6- The sixth chapter: conclusions and suggestions. Error! Bookmark not defined.

    6-1-Introduction. Error! Bookmark not defined.

    6-2-Conclusion. Error! Bookmark not defined.

    6-3-Proposals. Error! Bookmark not defined.

    References and sources. 42

     

    Source:

     

     

    [1]. Arora, S.S., Nigam, S.P., and Naveen Kwatra, Study of vibration characteristics of cantilever beams of different materials, Mechanical Engineering Department Thapar University Patiala- 147004, India. 2012.

    [2]. Rezaee, M., and Hassannejad, R., Damped free vibration analysis of a beam with a fatigue crack using energy balance methods, International Journal of the Physical Sciences, vol. 5(6), pp. 793-803, 2010. [3]. Ke, L.L., Yang, J., Kitipornchai, S., and Xiang, Y., Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials, Journal of advanced materials and structure, vol. 16, pp. 488-502, 2009. [4]. Frisswell, M.I., and Mottershead, J.E., FRF and finite element equations with rigid body constraints and their applications in model updating. IMAC XVII - 17th International Modal Analysis Conference, pp. 654-659, 1999. [5]. Li, W.L., Free vibration of beams with general boundary conditions, United technology carrier Corporation USA, 2012.

    [6]. Nie, J., and Wei, X., On the use of material-dependent damping in ANSYS for superposition of transient analysis, ASME pressure vessels and piping division, vol. 8, pp. 239-244 2011. [7]. Prasad, D.R., and Seshu, D.R., A study on dynamic characteristics of structural materials using modal analysis, Asian Journal of Civil Engineering, vol. 9(2), pp. 141-152, 2008. [8]. Hadamard, J., Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923. [9]. Achenbach, J.D., Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973.

    [10]. Liu, G.R., and Han, X., Computational Inverse Techniques in Nondestructive Evaluation, CRC Press, Boca Raton, FL, 2003.

    [11]. Goodier, J.N., Jahsman, W.E., and Riperger, E.A., An Experimental Surface Wave Method for Recording Force-Time Curves in Elastic Impacts, Journal of Applied Mechanics, vol. 26(3), pp. 3-7, 1959. [12]. Doyle, J.F., An Experimental Method for Determining the Dynamic Contact Law, Experimental Mechanics, vol. 24(1), pp. 10-16, 1984. [13]. Doyle, J. F., Further Development in Determining the Dynamic Contact Law, Experimental Mechanics, vol. 24(4), pp. 265-270, 1984. [14]. Doyle, J. F., Determining the Contact Force During the Transverse Impact of Plates, Experimental Mechanics, vol. 27(1), pp. 68-72, 1987. [15]. Hollandsworth, P.E., and Busby, H.R., Impact Force Identification Using the General Inverse Technique, International Journal of Impact Engineering, vol. 8(4), pp. 315-322, 1989. [16]. Inoue, H., Ikeda, N., Kishimoto, K., Shibuya, T., and Koizumi, T., Inverse Analysis of the Magnitude and Direction of Impact Force, JSME international journal, Ser. A, Mechanics and material engineering, vol. 38(1), pp. 84-91, 1995. [17]. Zare, Mahmoud Reza, and Hamtian, Mohammad Rahim, and Khajepour, Salar., Estimation of aerodynamic loads entering aerial structures by means of strain measurement, in the collection of articles of the first specialized conference on aerial structures and separation systems, Aerospace Industries Organization, 2016.

    [18]. Hamtian, Mohammad Rahim., Zare, Mahmoud Reza., and Khajepour, Salar., Inverse calculation of applied loads to composite sheets with non-linear behavior, in Proceedings of the 15th Annual (International) Conference on Mechanical Engineering, Iran, Tehran, Amirkabir University of Technology, 1386.

    [19]. Kazemi, M., and Hematiyan, M.R.

Identification of the stiffness and damping coefficients of the truss support
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