Improving Dynamic Characteristic of a Truss Structure Using the Structural Dynamic Modification Technique

ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 1 Improving Dynamic Characteristic of a Truss Structure Using the Structural Dynamic Modification Technique Ezedine G. Allaboudi*, Khaled M. Ahmida, Hasan M. Nagiar and Mohammed Ali Hjaji [email protected]* (Applied Mechanics Division, Mechanical and Industrial Engineering Department, University of Tripoli, Libya) صخلملا ةينولامجلا ةأشنملل ةيكيمانيدلا لكاشملا ضعب بنجت ةيناكما ةساردل ةقرولا هذه فدهت اهصاوخ ضعب نيسحت للاخ نم يه ةساردلا هذه يف هعبتملا تاءارجلإا .ةيكيمانيدلا عيزوت .ةينولامجلا ةأشنملا ىلع اهعيزوتو ةكرحلاو عضولا يتقاطل ليلحت ساسأ ىلع ءوبنتو ةحضاو ةروص يطعي ةأشنملا ءاضعأ لماك ىلع اهيلع لصحتملا ةقاطلا لامجلا ةأشنملا .اهتلازإ ةيناكمإ وإ اهل نيسحت ءارجإ نكمملا ءاضعلألةيئاهنلا ةينو يتقاط عيزوت ىلع ً ءانب ليلحتلا ةداعإ نم تايلمع ةدع للاخ نم اهيلع لصحتملا ح لضفلأا يه ربتعت ,ةأشنملل ةكرحلاو عضولاي للاخ نم يكيمانيدلا اهكولس نسحت ث ليدعتلا ساسأ ىلع ليلحتلا ةداعإ ةينقت .ةيعيبطلا اهتاددرت يف ثداحلا نسحتلا تمت ةأشنملل يكيمانيدلا مت ,لمعلا اذه لامكلإ .ةيهانتملا رصانعلا ةقيرط مادختسإب يتقاط عيزوت داجيلإ ةيهانتملا رصانعلا ةقيرط ساسأ ىلع رتويبمك جمانرب دادعا لضفأ ىلع لوصحلل ايعس .ةينولامجلا ةأشنملا ءاضعأ لماك ىلع ةكرحلاو عضولا يز وه يكيمانيدلا ليدعتلا ءارجلإ يساسلإا فدهلا نإف ,ميمصت ةدا تاددرتلا ةميق مت .كلذ نكمأ ام ةرواجتملا ةيعيبطلا تاددرتلا نيب ةميقلا دعابتو ةيساسلأا ةيعيبطلا اهيلع لصحتملا جئاتنلاو ,ييحيضوت لاثم للاخ نم ةحرتقملا ةقيرطلا ةأفك حيضوت .ةقيرطلا هذه نم فدهلا تززع ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 2 Abstract The present paper investigates a problem related to improving the dynamic characteristics of a truss structure with the goal of avoiding certain dynamic problems. The procedure is based on the analysis of the distribution of potential and kinetic energies in all elements of a truss structure. These energies would in turn give a prediction of the elements that needs to be re-analyzed and eventually removed. The resulting truss would be an optimized structure that has an improved dynamic behavior with respect to its natural frequencies. The Reanalysis technique based on structural dynamic modification (SDM) is accomplished using Finite Element Method (FEM). To this end, an algorithm has been built, based on FEM, to obtain the distribution of kinetic and potential energies in the truss elements. Seeking the optimum condition of design, the main aim of the dynamic modification is to increase the natural frequencies and increase the spacing between them. An example illustrating the efficiency of the procedure is demonstrated and the results has proved the goal of the proposed methodology. Keywords: dynamics characteristics, Truss Structure, FEM, SDM, kinetic energy, potential energy, reanalysis. Introduction Structural Dynamics Modification (SDM) is an effective technique to improve the dynamic characteristics of a structure, such as natural frequency, mode shape and frequency response functions. Although this topic has been widely studied in the previous decades, the methodology of modification (reanalysis) of the structure continues under investigation. Several studies have been addressed to the subject of modal reanalysis and structure dynamic modifications. Few surveys have been conducted [1, 2, 3, 4,] for this purpose. There are two opposite approaches for structural dynamic modifications, the direct structural modification and the inverse structural modification. Reference [5] presented a basic theory for determining the solution existence of frequency ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 3 optimization problems for truss structures. This theory suggests that the natural frequencies remain unchanged when a truss is modified uniformly and that the natural frequency constraint is usually the key constraint in determining the solution existence of a truss dynamic optimization problem. The optimization criterion for three dimensional truss structure with multiple constraints on its natural frequencies is considered in [6]. Nodal coordinates and cross sections of elements, although of different nature, have been treated simultaneously in unified design specification for the minimum weight of the structure. The approach which is used in this paper is direct structure modification. The direct structural modification problem is treated as prediction problem which is concerned with determining the dynamic response of a structure brought about by modification. Hang et al. [7-10] proposed an approach to predict the effects of distributed structural modifications with additional DOFs. G. Canbaloglu and H. N. Özgüven [11] presented an approach for predicting the dynamic response of a structure with distributed modifications from the response of the original structure itself and dynamic flexibility matrix of the modifying structure. The performance of this method was investigated by applying it to a real structure. Successful results were obtained. Accordingly, they concluded that the structural reanalysis method proposed can be successfully and efficiently used for structures with distributed modifications. Hanbing Liu et al. [12] proposed a new method to calculate the dynamic characteristics of the structures rapidly after each modification. In this method, the structure is decomposed into relatively simple substructures. Accordingly, the modal synthesis method (MSM) was applied to obtain the dynamic characteristics of the whole structure. The proposed method, used in this paper, is based on the distribution of potential and kinetic energies in all elements of the structure. Therefore, the elements that need modifications can be easily identified. Through reanalysis procedure, the dynamic ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 4 behavior of the structure is calculated in every step. This methodology has been proposed by Ki, I. K. [13], and later developed by Maneski, T [14] to investigate the dynamic behavior of a complex real structure using the procedure of reanalysis. Distribution of modal potential and kinetic energies For an undamped system, which is not subjected to any external force, the equation of motion in matrix form is defined by [15],           0)()(  tQKtQM  (1) The eigenvalues of this equation for i-th mode shape can be expressed as,           0 rrr QMQK  (2) Where  r is the r-th eigenvalue, and Qr is the r-th eigenvector of the structure. Multiplying the left hand side of equation (2) by the r-th eigenvector one can get,             r T rir T rQMQQKQ   2 1 2 1 (3) Equation (3) is the balance equation of potential and kinetic energies for a structure in main modes of oscillation. Furthermore, the potential energy of a structure at the r-th main mode of oscillation, having in mind the previous equation, can be written as,       r T rrp QKQE 2 1 , . (4) Analogously, the kinetic energy is given by, ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 5       r T rrrk QMQE  2 1 , , (5) Theoretically, the total energy conservation at mode shape r implies that, rrkrp EEE  ,, . (6) The total kinetic and potential energies of the structure calculated at the r-th mode shape is the sum of energies of all elements, represented by,         e s r e T e s r N er e N erkrk qmqeE    1 2 1,, 2 1             N ee s r e T e s r e N erprp qkqeE 11 ,, 2 1 (7) Where         e s r e T e s r e rp qkqe 2 1 , is the potential energy of e-th element at its r-th mode shape,       e s r e T e s rrrk qmqe 2 ,2 1   is the kinetic energy of e-th element at r-th mode shape of oscillation, and   e s r q is the corresponding r-th eigenvector of the e-th element with s degrees of freedom. Consequently, the potential and kinetic energies distribution throughout all elements of the structure can be analyzed. In addition, the natural frequencies and corresponding mode shapes are also calculated, and the procedure for the Structural Dynamic Modification (SDM) can be applied. The following points should be noted when applying the SDM algorithm: ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 6 i. Elements with negligible kinetic and potential energies, when compared to other elements ii. Elements with dominant kinetic energies, when compared to their potential energies iii. Elements with dominant potential energies, when compared to their kinetic energies Reanalysis algorithm The SDM reanalysis algorithm is described in the following steps, where a FEM code using Matlab® was built for its implementation: Step 1: The analyzed structure is divided into an appropriate number of finite elements for which the kinetic and potential energies are calculated for each individual element. Step 2: The values of potential energy and kinetic energy, as well as the corresponding energy differences, are compared with each other. Step 3: The elements with very small kinetic or potential energies, compared to the total energies of structure, do not have significant effect on the dynamic behavior of the structure, at least not in the current algorithm, which is the goal of this paper. In general, reducing the mass of those elements, or removing them, lightens the weight of the whole structure without greatly affecting its dynamical behavior. Step 4: For those elements where pr kr ee , the eigenvalues can be increased by increasing the stiffness of the structure. The modifications needed to increase these values are not arbitrary, but they are accomplished according to the principle of energy distributions through the elements of the structure. Step 5: For those elements where kr pr ee , the eigenvalues can be increased by decreasing the mass of structure. Also, this operation can be done based on the distribution of energy through ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 7 the elements of the structure. According to many criteria [14], decreasing of mass is generally a desired type of modification. Step 6: Most often, elements with close values of ekr and epr are taken into account in the reanalysis procedure. In this case, the reanalysis of the structure is done based on the gradient of potential and kinetic energies pr kr ee  between the modified and the original system. Step 7: When the desired value of increase is achieved, it is possible to conduct the check of modified structure by running the FEM code with modified parameters. If the difference of energy increase on the redesigned places is less than the previous, that means that the procedure converges. Convergence is the goal of every optimization procedure. Case study The structure used as a case study consists of 21 truss elements and its dimensions are as shown in Figure (1). The AISI 1040 steel trusses data are as follows: cross-sectional area A=0.0025m2, Young’s modulus E=200Gpa, mass density ρ=7860kg/m3. Figure (1). Truss structure used as a case study. The model was solved using FEM Matlab-code and the first five natural frequencies in rad/s are found as follows: 24 m x y 2 147811 12 1096 5 3 1 23 4 5 6 7 8 10 911 12 13 14 15 18 16 17 19 10 21 3 m ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 8 1=133.6, 2=265.4, 3=434.9, 4=670.1, 5=754.6. The kinetic and potential energies for each element are calculated. Figure (2) shows the percentages of kinetic and potential energies of each element of truss compared with the total kinetic and potential energies of whole structure. The total weight of the structure is calculated to be mst1=1670.25 kg. Figure (2). Percentages of kinetic energy (ke) and potential energy (pe) of truss elements. From the above figure, one may notice that the elements (3, 7, 11, 15 and 19) has almost zero potential energy and high kinetic energy. Therefore, as a first modification, elements (7 and 15) can be eliminated to improve the natural frequencies of the truss based on the proposed technique. Figure 3 shows the modified truss. The reanalysis technique has been done for the modified model, and the first five natural frequencies in rad/s are found as follows: 𝜔1=152.8, 𝜔2=281.5, 𝜔3=505.5, 𝜔4=675.8, 𝜔5=862.9. ددعلاVolume 24 ريانيJanuary 2021 International Science and Technology Journal ةينقتلاو مولعلل ةيلودلا ةلجملا ةظوفحم عبطلا قوقح ةينقتلاو مولعلل ةيلودلا ةلجملل Copyright © ISTJ 9 Figure (3). Modified truss structure after removing elements 7 and 15, and remeshing/renumbering. The percentage of kinetic and potential energies for each element are calculated and shown in Figure (4). The total weight of the structure is calculated to be 1552.35 kg. Figure (4). Percentages of Kinetic energy (ke) and potential energy (pe) of truss elements, after removing element 7 and 15, and remeshing. The obtained results show that the dynamic characteristic parameters have been improved. Based on the proposed method, this improvement can be increased by removing the elements which have higher kinetic energy and low potential energy. 24 m x y 2 1457910 86 3 1 23 4 5 6 8 79 10 11 12 14 13 15 17 16 3 m