Comparison of the eigenfrequencies \n
\\n\\n
IntechOpen Book Series will also publish a program of research-driven Thematic Edited Volumes that focus on specific areas and allow for a more in-depth overview of a particular subject.
\\n\\nIntechOpen Book Series will be launching regularly to offer our authors and editors exciting opportunities to publish their research Open Access. We will begin by relaunching some of our existing Book Series in this innovative book format, and will expand in 2022 into rapidly growing research fields that are driving and advancing society.
\\n\\nLaunching 2021
\\n\\nArtificial Intelligence, ISSN 2633-1403
\\n\\nVeterinary Medicine and Science, ISSN 2632-0517
\\n\\nBiochemistry, ISSN 2632-0983
\\n\\nBiomedical Engineering, ISSN 2631-5343
\\n\\nInfectious Diseases, ISSN 2631-6188
\\n\\nPhysiology (Coming Soon)
\\n\\nDentistry (Coming Soon)
\\n\\nWe invite you to explore our IntechOpen Book Series, find the right publishing program for you and reach your desired audience in record time.
\\n\\nNote: Edited in October 2021
\\n"}]',published:!0,mainMedia:{caption:"",originalUrl:"/media/original/132"}},components:[{type:"htmlEditorComponent",content:'With the desire to make book publishing more relevant for the digital age and offer innovative Open Access publishing options, we are thrilled to announce the launch of our new publishing format: IntechOpen Book Series.
\n\nDesigned to cover fast-moving research fields in rapidly expanding areas, our Book Series feature a Topic structure allowing us to present the most relevant sub-disciplines. Book Series are headed by Series Editors, and a team of Topic Editors supported by international Editorial Board members. Topics are always open for submissions, with an Annual Volume published each calendar year.
\n\nAfter a robust peer-review process, accepted works are published quickly, thanks to Online First, ensuring research is made available to the scientific community without delay.
\n\nOur innovative Book Series format brings you:
\n\nIntechOpen Book Series will also publish a program of research-driven Thematic Edited Volumes that focus on specific areas and allow for a more in-depth overview of a particular subject.
\n\nIntechOpen Book Series will be launching regularly to offer our authors and editors exciting opportunities to publish their research Open Access. We will begin by relaunching some of our existing Book Series in this innovative book format, and will expand in 2022 into rapidly growing research fields that are driving and advancing society.
\n\nLaunching 2021
\n\nArtificial Intelligence, ISSN 2633-1403
\n\nVeterinary Medicine and Science, ISSN 2632-0517
\n\nBiochemistry, ISSN 2632-0983
\n\nBiomedical Engineering, ISSN 2631-5343
\n\nInfectious Diseases, ISSN 2631-6188
\n\nPhysiology (Coming Soon)
\n\nDentistry (Coming Soon)
\n\nWe invite you to explore our IntechOpen Book Series, find the right publishing program for you and reach your desired audience in record time.
\n\nNote: Edited in October 2021
\n'}],latestNews:[{slug:"webinar-introduction-to-open-science-wednesday-18-may-1-pm-cest-20220518",title:"Webinar: Introduction to Open Science | Wednesday 18 May, 1 PM CEST"},{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"}]},book:{item:{type:"book",id:"2021",leadTitle:null,fullTitle:"Cutting Edge Research in New Technologies",title:"Cutting Edge Research in New Technologies",subtitle:null,reviewType:"peer-reviewed",abstract:'The book "Cutting Edge Research in New Technologies" presents the contributions of some researchers in modern fields of technology, serving as a valuable tool for scientists, researchers, graduate students and professionals. The focus is on several aspects of designing and manufacturing, examining complex technical products and some aspects of the development and use of industrial and service automation. The book covered some topics as it follows: manufacturing, machining, textile industry, CAD/CAM/CAE systems, electronic circuits, control and automation, electric drives, artificial intelligence, fuzzy logic, vision systems, neural networks, intelligent systems, wireless sensor networks, environmental technology, logistic services, transportation, intelligent security, multimedia, modeling, simulation, video techniques, water plant technology, globalization and technology. 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He is currently a full professor in\nthe Department of Automation and Applied Informatics at the\nsame university. Dr. Voloşencu is the author of ten books, seven\nbook chapters, and more than 160 papers published in journals\nand conference proceedings. He has also edited twelve books and\nhas twenty-seven patents to his name. He is a manager of research grants, editor in\nchief and member of international journal editorial boards, a former plenary speaker, a member of scientific committees, and chair at international conferences. His\nresearch is in the fields of control systems, control of electric drives, fuzzy control\nsystems, neural network applications, fault detection and diagnosis, sensor network\napplications, monitoring of distributed parameter systems, and power ultrasound\napplications. 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In 1960s, Green and Naghdi specialized their general theory of Cosserat surface to obtain the linear Cosserat plate [2], while independently Eringen proposed a complete theory of plates in the framework of Cosserat elasticity [3]. Numerous plate theories were formulated afterwards; for the review of the latest developments in the area of Cosserat plates we recommend to turn to [4].
\nThe first theory of Cosserat plates based on the Reissner plate theory was developed in [5] and its finite element modeling is provided in [6]. The parametric theory of Cosserat plate, presented by the authors in [7], includes some additional assumptions leading to the introduction of the splitting parameter. This provided the highest level of approximation to the original three-dimensional problem. The theory provides the equilibrium equations and constitutive relations, and the optimal value of the minimization of the elastic energy of the Cosserat plate. The paper [7] also provides the analytical solutions of the presented plate theory and the three-dimensional Cosserat elasticity for simply supported rectangular plate. The comparison of these solutions showed that the precision of the developed Cosserat plate theory is similar to the precision of the classical plate theory developed by Reissner [8, 9].
\nThe numerical modeling of bending of simply supported rectangular plates is given in [10]. We developed the Cosserat plate field equations and a rigorous formula for the optimal value of the splitting parameter. The solution of the Cosserat plate was shown to converge to the Reissner plate as the elastic asymmetric parameters tend to zero. The Cosserat plate theory demonstrates the agreement with the size effect, confirming that the plates of smaller thickness are more rigid than is expected from the Reissner model. The modeling of Cosserat plates with simply supported rectangular holes is also provided. The finite element analysis of the perforated Cosserat plates is given in [11].
\nThe extension of the static model of Cosserat elastic plates to the dynamic problems is presented in [12]. The computations predict a new kind of natural frequencies associated with the material microstructure and were shown to be compatible with the size effect principle reported in [10] for the Cosserat plate bending.
\nThis chapter represents an extension of the paper [12] for different shapes and orientations of micro-elements incorporated into the Cosserat plates. It is based on the generalized variational principle for elastodynamics and includes a non-diagonal rotatory inertia tensor. The numerical computations of the plate free vibrations showed the existence of some additional high frequencies of micro-vibrations depending on the orientation of micro-elements. The comparison with three-dimensional Cosserat elastodynamics shows a high agreement with the exact values of the eigenvalue frequencies.
\nThe Cosserat linear elasticity balance laws are
\nwhere the \n
We will also consider the constitutive equations as in [13]:
\nand the kinematic relations in the form
\nHere \n
The constitutive Eqs. (3)–(4) can be written in the reverse form [5].
\nwhere \n
We will consider the boundary conditions given in [12].
\nand initial conditions
\nwhere \n
The strain stored energy \n
where
\nis non-negative. The relations Eqs. (3)–(4) can be written in the form [12]:
\nThe stress energy is given as
\nwhere
\nand the relations Eqs. (6)–(7) can be written as [12].
\nWe consider the work done by the stresses \n
and
\nHere \n
The stored kinetic energy \n
The kinetic energy \n
where
\nand
\nThe work \n
Keeping in mind that the variation of \n
or
\nand therefore
\nWe modify the HPR principle [14] for the case of Cosserat elastodynamics in the following way: for any set \n
of the functional
\nat \n
\n
Let us consider the variation of the functional \n
Taking into account Eq. (5) we can perform the integration by parts
\nThen keeping in mind that \n
In this section we review our stress, couple stress and kinematic assumptions of the Cosserat plate [7]. We consider the thin plate \n
The set of points \n
In our case we consider the vertical load and pure twisting momentum boundary conditions at the top and bottom of the plate, which can be written in the form:
\nwhere \n
We will also consider the rotatory inertia \n
Let \n
for every \n
Here the plate stress and kinetic energy density by the formulas
\nwhere \n
and
\n\n\n
where \n
The plate characteristics provide the approximation of the components of the three-dimensional tensors \n
where
\nThe pressures \n
and \n
The three-dimensional displacements \n
and the three-dimensional strain and torsion tensors \n
where \n
Then zero variation of the functional
\nis equivalent to the plate bending system of equations (A) and constitutive formulas (B) mixed problems.
\nA. The bending equilibrium system of equations:
\nwhere \n
at the part \n
at the part \n
B. Constitutive formulas in the reverse form:1\n
\nand the optimal value \n
where
\nWe also assume that the initial condition can be presented in the form
\nThe Cosserat plate field equations are obtained by substituting the relations Eqs. (74)–(83) into the system of Eqs. (65)–(70) similar to [10]:
\nwhere
\nThe operators \n
The coefficients \n
For the validation purposes we provide the algorithm and computation results for the three-dimensional Cosserat elastodynamics. We also present the analysis of the numerical results based on the plate theory for the microelements of different shapes and orientations incorporated into the Cosserat plate.
\nIn our computations we consider the plates made of polyurethane foam—a material reported in the literature to behave Cosserat like—and the values of the technical elastic parameters presented in [15]: \n
Let us consider the plate \n
We solve the three-dimensional Cosserat equilibrium Eqs. (1)–(2) accompanied by the constitutive Eqs. (3)–(4) and strain-displacement and torsion-rotation relations Eq. (5) complemented by the following boundary conditions:
\nwhere the initial distribution of the pressure is given as \n
Using the method of separation of variables and taking into account the boundary conditions Eqs. (86)–(87), we express the kinematic variables in the form:
\nwhere the functions \n
If we substitute the expressions Eqs. (91)–(96) into Eqs. (3)–(4) and then into Eqs. (1)–(2), we will obtain the following eigenvalue problem
\nwhere
\nand the differential operators \n
and the coefficients \n
The system of differential Eq. (97) is complemented by the following boundary conditions \n
and the coefficients \n
The idea for the solution of the eigenvalue problem Eq. (97) is based on the following algorithm:
\n\n
We fix certain value of the frequency \n
\n
Mathematically, fixing certain value of \n
\n
We run \n
Large amplitude linear vibrations of the Cosserat body forced to vibrate close to its natural frequency
The comparison of the eigenfrequencies of the Cosserat plate with the eigenfrequencies of the three-dimensional Cosserat elasticity is given in the \nTable 1\n. The rotatory inertia principle moments used are \n
\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|
Plate theory | \n0.310 | \n17.881 | \n501.13 | \n205.62 | \n338.95 | \n
D Cosserat elasticity | \n0.309 | \n17.763 | \n530.82 | \n211.98 | \n317.87 | \n
Comparison of the eigenfrequencies \n
Ball-shaped micro-elements:
We consider a plate \n
and the following hard simply supported boundary conditions [7]:
\nSimilar to [12] we apply the method of separation of variables for the eigenvalue problem Eq. (85) to solve for the kinematic variables \n
where \n
We solve an eigenvalue problem by substituting these expressions into the system of Eq. (85). The obtained nine sequences of positive eigenfrequencies \n
We perform all our numerical simulations for \n
\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n | \n | \n |
---|---|---|---|---|---|---|---|---|
\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
Ball | \n0.001 | \n0.001 | \n0.001 | \n17.88 | \n0.31 | \n501.13 | \n205.62 | \n338.95 | \n
Vertical ellipsoid | \n0.001 | \n0.001 | \n0.0001 | \n17.88 | \n0.31 | \n501.13 | \n650.22 | \n338.95 | \n
Horizontal ellipsoid | \n0.0001 | \n0.001 | \n0.001 | \n17.88 | \n0.31 | \n1363.01 | \n205.62 | \n394.08 | \n
Eigenfrequencies \n
Let \n
The eigenfrequencies for different angles of microrotation of the microelements are given in the \nTable 3\n and the \nFigure 3\n. The rotatory inertia principle moments used are \n
Angle \n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|---|---|---|---|
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n265.37 | \n265.37 | \n450.61 | \n450.61 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n255.59 | \n279.40 | \n429.89 | \n469.93 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n247.75 | \n295.33 | \n406.70 | \n484.79 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n242.57 | \n313.65 | \n382.94 | \n495.14 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n239.99 | \n333.10 | \n360.57 | \n500.46 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n239.68 | \n338.95 | \n354.35 | \n501.13 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n239.99 | \n333.10 | \n360.57 | \n500.46 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n242.57 | \n313.65 | \n382.94 | \n495.14 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n247.75 | \n295.33 | \n406.70 | \n484.79 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n255.59 | \n279.40 | \n429.89 | \n469.93 | \n
\n\n | \n17.88 | \n17.88 | \n0.31 | \n0.31 | \n650.221 | \n265.37 | \n265.37 | \n450.61 | \n450.61 | \n
Eigenfrequencies \n
Micro frequencies
In this chapter, we presented a mathematical model of Cosserat plate vibrations. The dynamic model of the plates has been developed as a dynamic extension of the Reissner plate theory. The equations has been presented in both tensorial and the matrix forms. We also described the validation of the model, which is based on the comparison with the three-dimensional Cosserat elastodynamics exact solutions. Based on the presented results of the computer simulations we were able to detect and classify the additional high resonance frequencies of a plate. We have shown that the frequencies depend on the shape and orientation of microelements (ball-shaped elements, horizontally and vertically stretched ellipsoids) incorporated into the Cosserat plates. We also have been able to identify that micro frequencies associated with the micro rotatory inertia and its transverse variation of the ellipsoid elements have higher micro frequencies than the ball-shaped elements. We also showed the dependence of the eigenfrequencies on the angles of rotation of the horizontal ellipsoid micro-elements. These results can be used to identify the characteristics of the plate micro-elements.
\nWe use the following notation convention:
the values of the Latin subindex \n
the values of the Greek indices \n
the Einstein summation notation is used throughout the chapter
artesian coordinates
\nCosserat thin plate
\nplate thickness
\nLamé parameters
\nCosserat elasticity parameters
\nmaterial density
\nrotatory inertia
\nthe stress tensor
\nthe couple stress tensor
\nstrain tensor
\nbend-twist tensor
\ndisplacement vector
\nmicrorotation vector
\nlinear momentum
\nangular momentum
\nLevi-Civita tensor
\nstrain stored energy
\nstress energy
\nstored kinetic energy
\nwork of inertia forces
\nCosserat plate stress set
\nCosserat plate displacement set
\nCosserat plate strain set
\nsplitting parameter
\npressure
\nnatural frequency of plate vibration
\nangle of microelement orientation
\nbending and twisting moments
\nshear forces
\ntransverse shear forces
\nmicropolar bending moments
\nmicropolar twisting moments
\nmicropolar couple moments
\nrotations of the middle plane around \n
vertical deflections of the middle plate
\nmicrorotations in the middle plate around \n
rate of change of the microrotation
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