Properties of commercial MRFs [34].
\r\n\tThis project will include chapters covering the main aspects of angiographic techniques; coronary angiography, fluorescein and microangiography, peripheral angiography, miscellaneous angiography, and new concepts and advances. It will provide an insight into significant updates including hybrid imaging, new devices, contrast medium, and techniques. As the endovascular approaches have evolved over the last several years with the rapid influx of minimally invasive techniques, it is important to point out that there are many aspects which require complex medical workups and substantial preoperative decision algorithms, which have not been covered in the literature yet. The book will be a collection of chapters from world class experts contributing to this new endeavor in medical sciences.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"d6e5b06750aa89961fd7e81c3740c6bd",bookSignature:"Dr. Patricia Bozzetto Ambrosi",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9061.jpg",keywords:"Coronary Angiography, Fluorescein, Microangiography, Neurovascular Angiography, Peripheral Angiography, Leg Claudication, Renal Stenosis, Atherosclerosis, Medicolegal, New Advances, Hybrid Imaging",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 9th 2019",dateEndSecondStepPublish:"September 26th 2019",dateEndThirdStepPublish:"November 25th 2019",dateEndFourthStepPublish:"February 13th 2020",dateEndFifthStepPublish:"April 13th 2020",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"221787",title:"Dr.",name:"Patricia",middleName:null,surname:"Bozzetto Ambrosi",slug:"patricia-bozzetto-ambrosi",fullName:"Patricia Bozzetto Ambrosi",profilePictureURL:"https://mts.intechopen.com/storage/users/221787/images/system/221787.jfif",biography:"Dr. Patricia Bozzetto Ambrosi graduated in medicine from The University of Caxias do Sul, Brazil, and the University of Rome Tor\nVergata, Italy. She is a former researcher in morphophysiology at the University of Córdoba/Reina Sofia Hospital, Córdoba, Spain.\nShe graduated in Neurology/ Neurosurgery at the Hospital of Restauração, SES, in Brazil and Neuroradiology/Radiodiagnostics at Paris Marie Curie University. She holds a master’s degree in Medicine from the University of Nova Lisboa in Portugal and in Behavioral Sciences and Neuropsychiatry from the University of Pernambuco. She also has a Ph.D. in Biological Sciences from the University of Pernambuco/Paris Diderot University. She is a former Fellow in Interventional Neuroradiology in France at the Ophthalmological Foundation Adolphe de Rothschild, Beaujon Hospital, and Hospices Civils de Strasbourg. She was Praticien Associé in Interventional Neuroradiology at Neurologique Hospital Pierre Wertheimer, University of Lyon Claude Bernard in Lyon, France, and Visiting Professor of the University of Paris Diderot-Neuri Beaujon. She is actually an independent consultant/supervisor in neuroradiology, neuroendovascular, and imaging. She has been also an academic collaborator researcher in the Cardiovascular Department at the University of Leicester. She has experience in innovative research for the development of new technologies and is also an academic editor and reviewer of several scientific publications about neurological diseases.",institutionString:"Paris Diderot University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"2",institution:{name:"Paris Diderot University",institutionURL:null,country:{name:"France"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"16",title:"Medicine",slug:"medicine"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"177731",firstName:"Dajana",lastName:"Pemac",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/177731/images/4726_n.jpg",email:"dajana@intechopen.com",biography:"As a Commissioning Editor at IntechOpen, I work closely with our collaborators in the selection of book topics for the yearly publishing plan and in preparing new book catalogues for each season. This requires extensive analysis of developing trends in scientific research in order to offer our readers relevant content. Creating the book catalogue is also based on keeping track of the most read, downloaded and highly cited chapters and books and relaunching similar topics. I am also responsible for consulting with our Scientific Advisors on which book topics to add to our catalogue and sending possible book proposal topics to them for evaluation. Once the catalogue is complete, I contact leading researchers in their respective fields and ask them to become possible Academic Editors for each book project. Once an editor is appointed, I prepare all necessary information required for them to begin their work, as well as guide them through the editorship process. I also assist editors in inviting suitable authors to contribute to a specific book project and each year, I identify and invite exceptional editors to join IntechOpen as Scientific Advisors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"314",title:"Regenerative Medicine and Tissue Engineering",subtitle:"Cells and Biomaterials",isOpenForSubmission:!1,hash:"bb67e80e480c86bb8315458012d65686",slug:"regenerative-medicine-and-tissue-engineering-cells-and-biomaterials",bookSignature:"Daniel Eberli",coverURL:"https://cdn.intechopen.com/books/images_new/314.jpg",editedByType:"Edited by",editors:[{id:"6495",title:"Dr.",name:"Daniel",surname:"Eberli",slug:"daniel-eberli",fullName:"Daniel Eberli"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"67325",title:"Impact Analysis of MR-Laminated Composite Structures",doi:"10.5772/intechopen.86466",slug:"impact-analysis-of-mr-laminated-composite-structures",body:'Laminated composite materials, due to their unique characteristics such as high strength-to-weight ratio, high corrosion and impact resistance, and excellent fatigue strength, are being widely used in aerospace, automobiles, and recently civil engineering applications [1, 2, 3, 4, 5, 6, 7]. However, composite structures are highly vulnerable for failure under dynamic loading and, most importantly, impact loads. In the past few years, elements of smart and functional materials have been added to the conventional composite structures to develop a new generation of the laminated composite structures [8]. Researchers have proposed and test many types of smart materials, including piezoelectric, shape memory alloys, fiber optics, and electrorheological (ER) and magnetorheological (MR) fluids, to add the required features, such as controllability, and improve the performance for specific applications [9, 10, 11, 12, 13, 14]. These structures have the capability to adapt their response to external stimuli such as load or environmental changes. These new structures have opened new challenges in research communities. The use of MR fluids in composite structures is relatively new as embedding fluids inside a rigid, laminated structure may raise many challenges in fabrication. Figure 1 shows a typical MR-laminated beam in which some layers are partially replaced by segments of MR fluid. Yalcintas and Dai [15] investigated the dynamic vibration response of three-layered MR and ER adaptive beams both theoretically and experimentally.
MR-laminated beam.
Sapiński et al. [16, 17] explored vibration control capabilities of a three-layered cantilever beam with MR fluid and developed FEM model to describe the phenomena in MR fluid layer during transverse vibration of the beam. Sapiński et al. [18] proposed a finite element (FE) model by using ANSYS for sandwich beam incorporating MR fluid. Rajamohan et al. [19] investigated the properties of a three-layer MR beam. The governing equations of MR adaptive beam were formulated in the finite element form and also by Ritz method. Ramamoorthy et al. [20] investigated vibration responses of a partially treated laminated composite plate integrated with MR fluid segment. The governing differential equations of motion for partially treated laminated plate with MR are presented in finite element formulation. Payganeh et al. [21] theoretically investigated free vibrational behavior of a sandwich panel with composite sheets and MR layer. They studied effects of length and width of sheet and also core thickness on frequency. Aguib et al. [22] experimentally and numerically studied the vibrational response of a MR elastomer sandwich beam subjected to harmonic excitation. They studied the effect of the intensity of the current flowing through a magnet coil on several dynamic factors. Naji et al. [23] employed generalized layerwise theory to overcome this challenge and passed behind constant shear deformation assumption in MR layer that is mainly used for vibration analysis of MR beam. Based on layerwise theory, FEM formulation was developed for simulation of MR beam, and results were verified by experimental test. Naji et al. [24, 25] presented a distinctive and innovative formulation for shear modulus of MR fluid. Most recently, Momeni et al. [26, 27] developed a finite element model to investigate the vibration response of MR-laminated beams with multiple MR layers through the thickness of the laminated beam with uniform and tapered cross sections.
The present work intends to study the vibration response of MR-laminated beam with emphasis on impact loadings. The mathematical modeling of MR-laminated beam is similar to the work done by present authors in previous publications [23, 24, 26, 27]. However, here, the modeling has been used mainly to study the effects of impact loading, although for more clarification, basic results for natural vibration have also been provided. Some experimental works have been conducted to illustrate the performance of the MR-laminated beam under practical impact loadings.
In brief, laminated composite plates are composed of individual layers, which have been stacked together, usually by hand-layup techniques. Individual layers are composed of fibers, which have been derationed according to property requirement, and matrix, which serves as binder of fibers and transfers the loads to the fibers. Changing the orientation of the fibers optimizes the composite material for strength, stiffness, fatigue, heat, and moisture resistance. Modeling laminated composite structures for conventional applications mainly is conducted by considering the stacked layers as one single layer. The equivalent single-layer (ESL) theories assume continuous displacement through the thickness of the laminate. In general, the stiffness of the adjacent layers in the laminates is not equal; thus, it results in discontinuity in transverse stress through the thickness, which is contrary to the equilibrium of the interlaminar stresses as stated by ESL. In general, ESL theories provide acceptable results for relatively thin laminate. For thick laminate and laminate with material and/or geometric inhomogeneities, such as MR-laminated beams, the ESL theories lead to erroneous results for all stresses.
In smart-laminated structures, due to the material and geometric inhomogeneities through thickness, including MR-laminated beams, it is required to acquire an accurate evaluation of strain–stress at the ply level. Interlaminar stresses can lead to delamination and failure of the laminate at loads that are much lower than the failure strength predicted by the ESL theories.
The accurate modeling of interlaminar stress field in composite laminates requires the displacement field to be piecewise continuous through the thickness direction. Researchers in composite communities have proposed and developed a variety of displacement models to provide sufficient accuracy for interlaminar stresses in composite structures. Layerwise displacement theory developed by Reddy [28] developed a layerwise theory based on the piecewise displacement through the laminate thickness.
The layerwise formulation has the capability to address local through-the-thickness effect, such as the evolution of complicated stress–strain fields in MR-laminated composite structures and interfacial phenomena between the different embedded layers. In this work, the layerwise displacement theory has been used for modeling MR-laminated beam.
The displacement field for a laminated beam based on the layerwise theory is obtained by considering the axial and through-the-thickness displacements as
where u andw are displacements along x- and z-directions, respectively. N denotes the total number of nodes through the thickness. The ratio ζ is defined as ζ = z/h, in which h represents the thickness of each discrete layer. Interpolation functions
For many applications, closed-form solution for MR-laminated beams is either not available or very complex. Therefore, most of the computations are based on finite element models. Finite element formulation has been obtained by incorporating the local in-plane approximations for the state variables introduced in Eq. (1) as follows:
where Nn is the number of nodes and
Magnetorheological (MR) fluid is a class of new intelligent materials, which rheological characteristics such as the viscosity, elasticity and plasticity change rapidly (in order of milliseconds) subject to the applied magnetic field as shown in Figure 2. By applying a magnetic field, the particles create columnar structures parallel to the applied field and these chain-like structures restrict the flow of the fluid, requiring minimum shear stress for the flow to be initiated. Upon removing the magnetic field, the fluid returns to its original status, very fast.
Structure of MR fluids (a) without magnetic field, (b) with magnetic field, and (c) with magnetic field and shear force.
Overall, MRF is composed of a carrier fluid, such as silicone oil, and iron particles, which are dispersed in the fluid [29, 30]. Each of the components plays a significant role in the characteristic of MRF.
There are two types of ferromagnetic materials, used in MRFs: the soft material and the hard material [31]. The applied magnetic field intensity (H) and the magnetization of the material (B) are two parameters that show the difference between the two kinds of materials.
Figure 3 shows the H-B hysteresis loops of the soft and hard materials. It is noted that the soft ferromagnetic material has lower remanence and coercivity [32].
The comparison between the soft and hard ferromagnetic materials [33].
The shape of the magneto-soft particles is spherical with a diameter ranging between 1 and 10
The hard magnetic material can be made of chromium dioxide (CrO2), which shows high coercivity and remanence. The size of dispersed CrO2 is between 0.1 and 10
The second component of MRF is a carrier liquid providing the continuous medium for the ferromagnetic particles [32]. Although all types of fluids are suitable for this purpose, Ashour et al. [13] recommended a fluid with a viscosity ranging between 0.01 and 1 N/m2 [32]. Silicon oil and synthetic oil are samples of suitable carrier fluids [32]. Table 1 provides the samples of carrier fluids with their specifications. As noted in Table 1, water is the suitable fluid, which can carry iron particles up to 41% of volume. Hydrocarbon oil and silicone can have the suspended particles between 22 and 36% of the volume.
Commercial MRF | Percent iron by volume | Carrier fluid | Density (g/cm3) |
---|---|---|---|
MRF-122-2ES | 22 | Hydrocarbon oil | 2.38 |
MRF-132 AD | 32 | Hydrocarbon oil | 3.09 |
MRF-336AG | 36 | Silicone oil | 3.45 |
MRF-241ES | 41 | Water | 3.86 |
Properties of commercial MRFs [34].
The third element of the MRF is a stabilizer, which are “polymers (surfactants) in nonpolar media.” The main roles of the materials as stabilizer are to retain the iron alloy particles suspended in the MRF, extend the service life of the smart fluid, and increase its reliability [32, 35]. There are three types of stabilizers—the agglomerative, the sedimental, and the thermal [32]—as briefly described below:
1.An agglomerative stabilizer is used to prevent the formation of aggregates between the iron particles. Agglomeration usually occurs due to the van der Waals’ interactions between the iron particles in the MRF, causing the ferromagnetic particles to stick together. In MRF applications, surfactants should be selected in accordance with the type and concentration of particles. In fine-dispersed concentrations, when the iron particles fill up to 50% of the volume, ionic or nonionic surfactants are recommended. In lower concentrations of iron particles, which only occupy up to 10% of the volume, gel-like stabilizers are also suggested [32].
2.A sedimental stabilizer is employed to prohibit the iron particles from settling down as a result of gravity which decreases the effectiveness of MRFs [32]. The gel-forming and nonionic surfactants, as the sedimental stabilizer, are used to be added to the fluid carrier [35].
3.A thermal stabilizer is utilized to stabilize MRF over a wide temperature range particularly for the long-term applications at high temperature.
Since then, MR fluids have been utilized in various applications, including dampers, brakes and clutches, polishing devices, hydraulic valves, seals, and flexible fixtures. Recently, the application of MR fluids in vibration control has been attracted by many researchers. However, due to the nature of MR fluids, it is very difficult to integrate them with thin-laminated composite structures.
Using the displacement field given in Eq. (1) and following a finite element procedure for each element, the governing equation of motion of MR-laminated beam in the matrix form is defined as
where [me] and [ke] are the element mass and stiffness matrices, respectively, and {f e} is the element force vector. One may note that when using layerwise displacement theory, depending on the number of layers in the laminate, each node may have many degrees of freedom.
Considering axial displacement given in Eq. (2), the generalized equation of motion of the MR-laminated beam can be obtained by assembling the mass, stiffness matrices, and the force vector as
where
It should be noted that the stiffness matrix is a complex value since it is the summation of the stiffness matrix of the laminated layer [Kc] and the stiffness matrix of the MR fluid [KMR(B, f)]:
One important feature in Eq. (4) is the structural damping which is included in the stiffness matrix. For the sake of simplicity, the complex mathematical modeling is neglected here; however, complete details are available in Ref. [23].
In order to conduct experimental test for MR beams, it is essential to provide a uniform magnetic field all over the beam. For the current work, an electromagnet device shown in Figure 4, which was previously fabricated by the current authors at Sharif University of Technology, is used. The length of poles (gap) is 240 mm. The space between the poles (gap) is 40 mm. To ensure that enough magnetic strength is provided, each arm of the electromagnet has 1000 wound turns of copper wire 1.2 mm in diameter. A Hall effect Gauss meter, (Kanetec-TM701) with a suitable probe (TM-701PRB), was used to measure the magnetic flux generated by the electromagnet. The MR fluid selected for this study was MRF-132DG manufactured by Lord Corporation. For more details of the devices, one may consult Ref. [23].
Experimental setup.
In order to perform modal tests, first a laminated composite plate (800 × 800 mm) made of glass fiber has been fabricated. Glass fiber composite is chosen as its magnetic permeability is zero. The plate was then cut into several strips (250 × 30 × 2.0 mm) by water jet to make beams. We used two similar beam strips as top and bottom faces and make a box beam with a 2.4 mm gap between two strips for MR fluid. Therefore, the total thickness of the MR-laminated beam became 4.4 mm. In order to maintain the uniform gap and hold the fluid between two strips, 2.4 mm thick spacer was glued on the inside face of one strip. Some mechanical properties of the glass fiber layers are given as follows:
Density = 200 g/m3, E1 = E2 = 15.5 GPa, G12 = G21 = 6.5 GPa, ρ = 1650 kg/m3.
In this work, the complex shear modulus of the MR fluid as function of magnetic field and driving frequency has been considered from the results of the work done by Naji et al. [10].
To study the effects of magnetic field on modal response of the MR-laminated beams, different magnetic fields, from 0 to 2000 Gauss, have been applied to the specimens.
To conduct the impact tests, the same electromagnet as described in Section 5.1 has been used. However, for impact tests, the specimens are fabricated as box-aluminum beam with length 400 mm and width 30 mm filled by MR fluid. The thickness of each aluminum layer and MR layer was 1 mm. The density of aluminum was 2700 kg/m3 and that of the MR fluid was 3500 kg/m3.
The impact tests were performed by dropping a 5.0 g mass from different heights (0.5, 1.0, and 1.5 m) on the tip of the MR-aluminum cantilever beam. In order to investigate the effect of magnetic field on the impact response of the MR-aluminum beam, the impact tests have been repeated for three levels of magnetic fields, 0, 1000, and 2000 Gauss.
In the following two subsections, the analytical results and experimental ones followed by brief explanations are provided.
The results obtained by analytical approach described in Section 4 and the results extracted from the experimental work are given in Table 2. The first three natural frequencies which were extracted from the peak of vibration response spectrum subject to three levels of magnetic fields are compared with analytical ones.
Magnetic field (Gauss) | Mode | Experimental freq. (Hz) | Analytical results |
---|---|---|---|
freq. (Hz) | |||
0 | 1 | 11.0 | 11.70 |
2 | 66.5 | 68.10 | |
3 | 185.5 | 187.36 | |
400 | 1 | 11.5 | 12.03 |
2 | 68.5 | 69.73 | |
3 | 187.0 | 191.49 | |
800 | 1 | 12.0 | 12.39 |
2 | 70.0 | 71.96 | |
3 | 190.5 | 192.79 | |
1200 | 1 | 11.5 | 11.76 |
2 | 71.5 | 73.00 | |
3 | 192.0 | 194.50 | |
1600 | 1 | 11.0 | 11.165 |
2 | 73.0 | 75.34 | |
3 | 193.0 | 197.83 | |
2000 | 1 | 10.0 | 10.18 |
2 | 75.5 | 78.67 | |
3 | 194.5 | 198.58 |
Experimental and analytical natural frequencies of MR beam.
As it is shown, the analytical results provide sufficient agreement with experimental ones for most of the cases.
An important feature that one may conclude from these results is noting that intensifying magnetic field increases the natural frequencies of the MR beams.
Increasing the natural frequencies of MR beams by increasing the magnetic field can be explained by noting that increasing magnetic field increases the stiffness of the MR fluid, which leads to increasing the total stiffness of the MR beam and, in turn, increasing the natural frequencies. However, an exception is shifting the fundamental frequency of the MR beam after 800 Gauss, which is in contradiction with intuitive sense. To interpret this phenomenon, one may note that the loss factor at high magnetic field jumps up dramatically [10], so increasing damping is dominated by increasing stiffness of MR fluid, and higher damping dictates vibration behavior.
In order to investigate the effect of MR fluid thickness on the modal response of the MR beams, the first three modes have been computed for different thickness ratios of the MR layer to base material under different applied magnetic fields. The results are given in Table 3, where h1 is the thickness of composite laminated which is a summation of the upper and lower layers and h2 is the thickness of MR fluid in middle layer.
Mode | Magnetic field (Gauss) | Natural frequencies of five modes for three different magnetic fields | |||
---|---|---|---|---|---|
h2/h1 = 1/4 | h2/h1 = 1/2 | h2/h1 = 1 | h2/h1 = 2 | ||
Mode 1 | 0 | 9.588 | 9.432 | 9.261 | 9.071 |
1000 | 14.381 | 12.913 | 12.758 | 12.308 | |
2000 | 10.826 | 9.537 | 9.611 | 9.934 | |
Mode 2 | 0 | 40.349 | 36.217 | 33.051 | 30.624 |
1000 | 44.550 | 40.574 | 37.539 | 34.566 | |
2000 | 46.026 | 43.417 | 41.308 | 40.204 | |
Mode 3 | 0 | 89.108 | 77.951 | 68.818 | 61.086 |
1000 | 99.501 | 86.666 | 76.320 | 68.195 | |
2000 | 101.465 | 89.037 | 88.784 | 81.006 |
Effect of layer ratio on the natural frequencies of MR-laminated beam.
The results generally demonstrate that increase in thickness of the MR layer decreases the natural frequencies of all the three modes. This is because in general, the stiffness of MR fluid is lower than the stiffness of the base composite material.
The MR-aluminum beams are subjected to dropping mass as described in Section 5.2. The results for dropping mass from 0.5 m on the tip of the MR beam subject to different magnetic fields are shown in Figure 5. As it is observed, increasing the magnetic field makes the MR beam stiffer and thus reduces sharply the settling time.
Impact test for dropping mass from 0.5 m.
The impact test results for dropping mass from 1.0 to 1.5 m on the tip of the MR beam for different magnetic fields is shown in Figures 6 and 7, where once again it is realized that increasing magnetic field reduces the settling time of the beam response.
Impact test for dropping mass from 1.0 m.
Impact test for dropping mass from 1.5 m.
To study the effect of magnetic field on the response of the MR beam subject to different impact loads, 1000 Gauss of magnetic field is applied to the MR beam, and its response of different dropping heights is measured and shown in Figure 8. As it is observed, increasing the level of impact force increases the amplitude of oscillation.
Impact test for different dropping heights at 1000 Gauss.
The effect of magnetic field on the vibration and impact responses of the MR beams has been investigated. For modeling purposes, the layerwise displacement theory was employed to overcome some challenges in the modeling of MR beams. The MR-laminated composite beams in which the top and bottom layers are made of glass-fiber laminated composites and the middle layer, is filled with MR fluids. Experimental tests have been conducted to validate the analytical results and show the performance of the MR-laminated beam for different magnetic fields. It was observed that increasing the magnetic field up to 800 Gauss increases the natural frequencies of the MR-laminated beam. However, beyond the 800 Gauss, the fundamental frequency of the MR-laminated beam begins to drop. The influence of MR layer thickness on the vibration behavior of MR-laminated beam was examined. Adding thickness of the MR layer affected decreases the natural frequencies of the first three modes.
In another study, a three-layered aluminum beam composed of aluminum layers at the top and bottom and the middle layer filled with MR fluid has been investigated for impact loadings. It was realized that increasing the magnetic field reduces the settling time of vibration for MR-aluminum beam. Also, for a constant magnetic field, increasing the level of impact load leads to increasing the amplitude of vibration as it was expected.
The authors wish to thank the partial support provided by the Sharif University of Technology, International Campus, on Kish Island.
We present a general form of scalar conservation laws with further properties including some basic models and provide examples of computational methods for them. The equations described by
in one dimension are known as scalar conservation laws where
We start by investigating the relation of the equations in gas dynamics with conservation laws. We take into account the equation of conservation of mass in one dimension. The density and the velocity are assumed to be constant in the tube where
The last equation is called integral form of conservation law. Integrating this expression in time from
Using the fundamental theorem of calculus after reduction of Eq. (3), it follows that
As a result, we get
Here the end points of the integrations are arbitrary; that is, for any
which is said to be the differential form of the conservation law.
A general solution to a quasilinear partial differential equation of the form
where
By applying a parametrization of
In addition to these equations, if an initial condition
Observe that the scalar conservation law (1) is a particular example of Eq. (7) if we assign
This means, the quantity
We consider the initial value problem
where the initial data is assumed to be continuously differentiable, that is,
where we define characteristic curves of Eq. (12) to be the solution of
with
Along this characteristic curve,
is satisfied, that is,
Hence we can define smooth solutions by
The basic example of the scalar conservation law is the linear advection equation. It can be obtained by setting
is a linear advection equation. Similar to Eqs. (11) and (12), an initial value problem for linear advection equation is described by
Applying the method of characteristics, it follows that
where
Here
Burgers’ equation is the simplest nonlinear partial differential equation and is the one of the most common models used in the scalar conservation laws and fluid dynamics. The classical Burgers’ equation is described by
where
where
Plugging these terms in Eq. (21), we get
Taking integration with respect to
Rewriting Eq. (25) by
it follows that
As a result the explicit form of traveling wave solution of Eq. (21) becomes
where
Remark. If the initial data is smooth and very small, then the
Whenever
Observe that
Recall that the characteristic speed
and differentiating equation (30) with respect to
Thus, substituting Eqs. (31) and (32) in (29), we can recover the inviscid Burgers’ equation. Consequently, the relations (31) and (32) imply that the solutions of Eq. (1) and particularly of Eq. (29) depend on the initial value
Let the constants uL and uR are given with a linear function,
is a simple example of discontinuous solution of the conservation law (11). If
where
For the initial value uL>uR, characteristics, and shock wave.
For the initial value uR>uL, characteristics and rarefaction waves.
A rarefaction wave is a strong solution which is a union of characteristic lines. A rarefaction fan is a collection of rarefaction waves. These waves are constant on the characteristic line
If, for instance, f is convex, then the rarefaction waves are increasing. If we consider again the inviscid Burgers’ equation with the initial values, then the region without characteristics in Figure 2 will be covered by rarefaction solution which is described by
An illustration of rarefaction waves and rarefaction fan in Eq. (36) is given in Figure 3.
Rarefaction fan.
Remark. Whenever characteristics intersect, we may have multiple valued solution or no solution; but we have no more classical (strong) solution. To get rid of this situation, we introduce a more wide-ranging notion of solution, the weak solution, in the next part. By this arrangement, we may have non-differentiable and even discontinuous solutions.
Weak solutions occur whenever there is no smooth (classical) solution. These solutions may not be differentiable or even not continuous. Considering
Putting the initial condition
Observe that there are no more derivatives of
The Riemann problem is a Cauchy problem with a particular initial value which consists a conservation law together with piecewise constant data having a single discontinuity. We consider the Riemann problem for a convex flux described by
The solution is a set of shock and rarefaction waves depending on the relation between
Case 1:
is a shock wave satisfying the shock speed
Case 2: (
A jump discontinuity along the characteristic line is controlled by the Rankine-Hugoniot jump condition. Integrating the scalar conservation law (1) in
Suppose that there is a discontinuity at the point
By the fundamental theorem of calculus, the relations (41) and (42) yield
Taking the limit whenever
The relation (44) is said to be the Rankine-Hugoniot jump condition. Geometrical meaning of the Rankine-Hugoniot jump condition is that the shock speed is the slope of the secant line through the points
Entropy and entropy flux are defined for attaining physically meaningful solutions. If
is satisfied for continuously differentiable functions
which looks like to the scalar conservation law (1). Indeed, if we multiply Eq. (1) by
It follows that Eqs. (46) and (47) are equivalent with
holds for all convex entropy pairs
Weak solutions to conservation laws may contain discontinuities as a result of a discontinuity in the initial data or of characteristics that cross each other or because of the jump conditions which are satisfied across the discontinuities. Although the Rankine-Hugoniot jump condition is satisfied, the uniqueness of the solution may always not be guaranteed. In order to eliminate the nonphysical solutions among the weak solutions, we need an additional condition, so-called entropy condition. It is described by the following: A discontinuity propagating with the characteristic speed
Example 1.1. The weak solutions to conservation laws need not be unique. If we write the inviscid Burgers’ equation in quasilinear form and multiply by
The inviscid Burgers’ equation and Eq. (49) have exactly the same smooth solutions. But their weak solutions are different. A shock traveling speed for the inviscid Burgers’ equation is
Example 1.2. We first consider the initial value problem for
Applying the method of characteristics for
Next if we integrate Eq. (51) with respect to
where
which satisfies both the jump condition and the entropy condition as
For initial value uL>uR, the characteristic solutions.
Example 1.3. We now interchange the roles of
By the method of characteristics, we obtain a solution
which is a classical (strong) solution on both sides of the characteristic line
which satisfies both jump and entropy conditions. Here we can observe the rarefaction fan arising on the interval
For initial value uL<uR, characteristic solutions u1tx and u2xt with rarefaction fan.
The equation of fluid dynamics can be represented in Eulerian and Lagrangian forms. Eulerian coordinates are related to the coordinates of a fixed observer. On the other hand, Lagrangian coordinates are in usual related to the local flow velocity. That is, due to the velocity taking different values in different parts of the fluid, the change of coordinates is different from one point to another one.
The equations of gas dynamics in Eulerian coordinates can be written in the following conservative forms:
where we ignored the heat conduction. If we denote
then Eq. (57) can be written by
where
If we do not neglect the heat conduction, then the
where
where
which are real, and the eigenvectors are linearly independent implying that the Euler equations for perfect gases are hyperbolic.
Using the results in the previous part, the Rankine-Hugoniot jump conditions for the Euler system will be of the form
where the indices
The Riemann problem for the one-dimensional Euler equation (57) is represented by
The reader is addressed to the references [18, 24] for further details.
We aim to transform the equations of gas dynamics (57) given in the Eulerian coordinates into the Lagrangian coordinates for one-dimensional case. We start denoting by
We set the following change of coordinates from Euler coordinates to Lagrange coordinates for space and time as
It follows that
which gives
It follows by some algebraic manipulations that the gas dynamic equations become
In order to derive a more convenient form of the system (69), we derive firstly the equation of conservation of mass:
where
which yields
Hence the second and third equations of Eq. (69) become
Moreover, we define a mass variable
Finally, using Eqs. (69) and (73), the Euler system (57) can be written in Lagrangian coordinates with the mass variable in the form
where
which is strictly hyperbolic. This can be verified by checking the Jacobian matrix of the flux calculated with respect to the variables
with
In fact there are different versions of the gas dynamics in Lagrangian coordinates. In this part we followed the approaches stated in [9, 10, 12]. For further details we cite these works with references therein.
Similarly as in the Euler system, the Rankine-Hugoniot jump conditions for the Lagrangian system (79) are of the form
where
Remark. The Eulerian and Lagrangian Rankine-Hugoniot relations are equivalent. Moreover, Eulerian entropy relations are equivalent to all Lagrangian entropy relations (see [9] for further detail).
Example 2.1. For simplicity of notation, we take
is a one-dimensional isentropic gas dynamics in Lagrangian coordinates which is also known as
The system (79) is equivalent to
where
has two real distinct eigenvalues
The Godunov scheme deals with solving the Riemann problem forward in time for each grid cell and then taking the mean value over these cells. The Riemann problem is solved per mesh point at each time step iteratively. If there are no strong shock discontinuities, this process may cost much and will not be effective. To get rid of such a situation, we establish approximate Riemann solvers that are easier to implement and also low cost to use. Eulerian and Lagrangian Godunov schemes are current Godunov scheme in literature. Both have advantages and disadvantages depending on the structure of the problem. A brief comparison of the method for these two approaches is presented in the last part of the chapter. In this work we will not go further in numerical examples and details of these methods; instead, we aim to present a general form of Godunov schemes for gas dynamics in Eulerian and Lagrangian coordinate. Before introducing these, we present a first-order Godunov scheme for scalar conservation laws.
Consider the scalar conservation law (1). Godunov scheme is a numerical scheme which takes advantage of analytical solutions of the Riemann problem for the conservation law (1). The numerical flux functions are evaluated at the spatial steps
respectively. These two solutions to the Riemann problem will be the numerical solution
Proceeding this process, we define the solution
Dividing both parts by
Thus, Godunov method is a conservative numerical scheme. It can be restated in an alternative form. Assigning the constant value of
Therefore, a first-order Godunov method takes the form
Here the constant value of
The Godunov method is consistent with the exact solution of the Riemann problem for the conservation law (1). If we suppose that
for each
where
For numerical illustration of Godunov schemes, we cite the articles [14, 20, 27].
We consider Eq. (59) with (60). The eigenvalues of
Consider the initial condition for a quantity
The eigenvalues satisfy
Then Godunov scheme for the Lagrangian coordinates takes the form
where
If we now consider the moving coordinates, Godunov scheme can also be derived equivalently by the following. Setting
Next we deduce
by a simple induction process. Hence the Lagrangian Godunov schemes become
with
Notice that the Lagrangian Godunov schemes can be reformulated as a finite volume method. Equation (100) can be written in conservative form:
If we integrate these equations on
Here we omit the dependency of
Moreover, if
provided
In the literature there are two types of Godunov schemes: the Eulerian and Lagrangian. To compare one with the other, both have advantages and disadvantages. These are briefly listed in the following:
It is more nature; that is the properties of a flow field are described as functions of the coordinates which are in the natural physical space and time. The flow is determined by examining the behavior of the functions. Eulerian coordinates correspond to the coordinates of a fixed observer. This approach is ease of implementation and computation. The computational grids derived from the geometry constraints are generated in advance. The computational cells are fixed in space, and the fluid particles move across the cell interfaces. Since the Eulerian schemes consider the implementation at the nodes of a fixed grid, this may lead to spurious oscillations for the problems like diffusion-dominated transport equations. By adding artificial diffusion, one can get rid of these oscillations; however the nature of the problem may differ from the original one. Besides, refining the grids may also lead to remove numerical oscillations, but this process may augment the computation cost. Besides, while refining the grids, it may cause restriction of the size of time step which is limited by CFL condition. This restriction does not occur in Lagrangian case.
It is based on the notion of mass coordinate denoted by
Apart from the two main approaches, there is another method which is a combination of both, so-called Eulerian-Lagrangian methods. It combines the advantages and eliminates disadvantages of both approaches to get a more efficient method. For further details we address the reader to the reference in the next part.
We have tried to present only the theoretical aspects of scalar conservation laws with some basic models and provide some examples of computational methods for the scalar models. There are plenty of contributors to the subject; however, we just cite some important of these and the references therein. Scalar conservation laws are thoroughly studied in particular in [12]; for a more general introduction including systems, see [13, 15, 18, 19, 22] and the references therein. There are some important works related to the concept of entropy provided by [7, 15, 16]. A more precise study of the shock and rarefaction waves can be found in [23]. A simple analysis for inviscid Burgers’ equation is done by [21]. The readers who are deeply interested in systems of conservation laws and the Riemann problem should see [8, 13, 15, 22, 24]. A well-ordered work of the propagation and the interaction of nonlinear waves are provided by [26]. We refer the reader to the papers [1, 17] for the theory of hyperbolic conservation laws on spacetime geometries and finite volume analysis with different aspects. A widely introductory material for finite difference and finite volume schemes to scalar conservation laws can be found in [18]. In this chapter we have studied the one-dimensional gas dynamics on the Eulerian and Lagrangian coordinates. For the detail on the Lagrangian conservation laws, we refer [10]; moreover for both Eulerian and Lagrangian conservation laws, we cite [11]. The proof of the equivalency of the Euler and Lagrangian equations for weak solutions is given in [25]. There are several numerical works for Lagrangian approach; some of the basic works on Lagrangian schemes are given in [2, 3, 4, 5, 6]. We refer the reader to the book [7] for a detailed analysis of the mathematical standpoint of compressible flows. Moreover Godunov-type schemes are precisely analyzed in [14, 27]; whereas, Lagrangian Godunov schemes can be found in [2, 12, 20]. As a last word, we must cite [9] as a recent and more general book consisting of scalar and system approaches of both Eulerian and Lagrangian conservation laws with theoretical and numerical parts which can be a basic source for the curious readers.
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