Eigenvalue and Eigenvector Problems in Applied Mechanics

262 Pages·2019·5.49 MB·English

by  Sorin Vlase

Advanced Structured Materials
Sorin Vlase
Marin Marin
Andreas Öchsner
Eigenvalue and
Eigenvector
Problems in
Applied Mechanics
Advanced Structured Materials
Volume 96
Series editors
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University
of Applied Sciences, Esslingen, Germany
Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of
Engineering, University of Porto, Porto, Portugal
Holm Altenbach, Otto-von-Guericke University, Magdeburg, Sachsen-Anhalt,
Germany
Common engineering materials reach in many applications their limits and new
developments are required to fulfil increasing demands on engineering materials.
The performance ofmaterials can beincreasedby combiningdifferent materials to
achieve better properties than a single constituent or by shaping the material or
constituents in a specific structure. The interaction between material and structure
mayariseondifferentlengthscales,suchasmicro-,meso-ormacroscale,andoffers
possible applications in quite diverse fields.
Thisbookseriesaddressesthefundamentalrelationshipbetweenmaterialsandtheir
structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic
etc.) and applications.
The topics of Advanced Structured Materials include but are not limited to
(cid:129) classical fibre-reinforced composites (e.g. class, carbon or Aramid reinforced
plastics)
(cid:129) metal matrix composites (MMCs)
(cid:129) micro porous composites
(cid:129) micro channel materials
(cid:129) multilayered materials
(cid:129) cellular materials (e.g. metallic or polymer foams, sponges, hollow sphere
structures)
(cid:129) porous materials
(cid:129) truss structures
(cid:129) nanocomposite materials
(cid:129) biomaterials
(cid:129) nano porous metals
(cid:129) concrete
(cid:129) coated materials
(cid:129) smart materials
Advanced Structures Material is indexed in Google Scholar and Scopus.
More information about this series at http://www.springer.com/series/8611
Sorin Vlase Marin Marin
(cid:129)
Ö
Andreas chsner
Eigenvalue and Eigenvector
Problems in Applied
Mechanics
123
Sorin Vlase Andreas Öchsner
Department ofMechanical Engineering Fakultät Maschinenbau
Transilvania University of Braşov Esslingen University of Applied Sciences
Braşov,Romania Esslingen, Germany
Marin Marin
Department ofMathematics
andComputer Science
Transilvania University of Braşov
Braşov,Romania
ISSN 1869-8433 ISSN 1869-8441 (electronic)
AdvancedStructured Materials
ISBN978-3-030-00990-8 ISBN978-3-030-00991-5 (eBook)
https://doi.org/10.1007/978-3-030-00991-5
LibraryofCongressControlNumber:2018957055
©SpringerNatureSwitzerlandAG2019
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Preface
This volume presents, in a unitary way, some problems of applied mechanics,
analyzed using the matrix theory and the properties of eigenvalues and eigenvec-
tors.Problemsandsituationsofdifferentnaturearestudied.Differentproblemsand
studies inmechanical engineeringleadtopatternsthataretreatedinasimilarway.
Thesamemathematicalapparatusallowsthestudyofmathematicalstructuressuch
as the quadratic forms but also mechanical problems such as multibody rigid
mechanics, continuum mechanics, vibrations, elastic and dynamic stability or
dynamic systems. A substantial number of engineering applications illustrate this
volume.
Braşov, Romania Sorin Vlase
Braşov, Romania Marin Marin
Esslingen, Germany Andreas Öchsner
v
Contents
1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Vectors. Fundamental Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Vector Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Addition of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Dot Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.4 Scalar Triple Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.5 Vector Triple Product. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.2.6 Applications of Vector Calculus . . . . . . . . . . . . . . . . . . . 24
1.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1 Fundamental Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Basic Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1 Addition ðþMm(cid:2)n (cid:2) Mm(cid:2)n !Mm(cid:2)nÞ. . . . . . . . . . . . 45
2.2.2 Scalar Multiplication ð(cid:3)R (cid:2) Mm(cid:2)n ! Mm(cid:2)nÞ. . . . . . . . 46
2.2.3 Matrix Multiplication ð(cid:3)Mm(cid:2)p (cid:2) Mp(cid:2)n ! Mm(cid:2)nÞ . . . . 47
2.2.4 Inverse Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.5 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2.6 Transposed of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2.7 Trace of a Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.8 Matrix Representation of the Cross Product. . . . . . . . . . . 52
2.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Ortogonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Some Properties of Matrix Operations . . . . . . . . . . . . . . . . . . . . . 55
2.6 Block Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vii
viii Contents
2.7 Matrix Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.7.1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.7.2 Diagonalization of Symmetric Matrices. . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Extreme Values of a Real Function of Two Variables. . . . . . . . . . 63
3.3 Conics and Quadrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Quadratic Forms in a n-Dimensional Space . . . . . . . . . . . . . . . . . 65
3.5 Eigenvalues and Eigenvectors for Quadratic Forms. . . . . . . . . . . . 67
3.5.1 The Conditions for a Quadratic Form to Be Positive . . . . 67
3.5.2 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.3 Fundamental Theorems of Quadratic Forms. . . . . . . . . . . 69
3.5.4 Schur’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6 Orthogonal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Invariants of Quadratic Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Rigid Body Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 Finite Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.1 Defining the Position of a Rigid Body . . . . . . . . . . . . . . 87
4.1.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.3 Bryan (Cardan) Angles. . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.4 Finite Rotations and Commutativity . . . . . . . . . . . . . . . . 97
4.2 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.1 Fundamental Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.2 Moment of Inertia; Definitions . . . . . . . . . . . . . . . . . . . . 105
4.2.3 Rotation of the Coordinates System . . . . . . . . . . . . . . . . 108
4.2.4 Moment of Inertia of a Body Around an Axis. . . . . . . . . 110
4.2.5 Directions of Extremum for the Moments of Inertia. . . . . 111
4.2.6 A Property of the Principal Direction of Inertia . . . . . . . . 113
4.2.7 Inertia Ellipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2.9 Geometrical Moments of Inertia . . . . . . . . . . . . . . . . . . . 136
4.2.10 Moment of Inertia of Planar Plates . . . . . . . . . . . . . . . . . 137
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5 Strain and Stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Deformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.2 Lagrangian and Eulerian Description. . . . . . . . . . . . . . . . 142
5.1.3 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Contents ix
5.1.4 Infinitesimal Deformation. . . . . . . . . . . . . . . . . . . . . . . . 149
5.1.5 Eigenvalues and Eigenvectors. . . . . . . . . . . . . . . . . . . . . 150
5.1.6 The Physical Significance of the Components
of the Strain Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.1.7 Transformation Induced by the Strain Tensor . . . . . . . . . 153
5.1.8 Local Rigid Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.2 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2.1 Stress State in a Point . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2.2 Transformation of the Stress Tensor to Axis
Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.2.3 Normal Stress Corresponding to an Arbitrary
Direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2.4 Extremal Conditions for Normal Stress . . . . . . . . . . . . . . 158
5.2.5 Invariants of the Reduced Stress. . . . . . . . . . . . . . . . . . . 161
5.2.6 Conic of Normal Stress . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.2.7 Quadric of Normal Stress. . . . . . . . . . . . . . . . . . . . . . . . 163
5.2.8 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 164
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6 Modal Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2.1 Eigenvalues—Natural Frequencies . . . . . . . . . . . . . . . . . 168
6.2.2 Properties of the Eigenvalues . . . . . . . . . . . . . . . . . . . . . 170
6.2.3 Orthogonality Properties. . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2.4 Rayleigh’s Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.5 Generalized Orthogonality Relationships . . . . . . . . . . . . . 172
6.2.6 Definition of Relationships for the Damping Matrix. . . . . 174
6.2.7 Normalized Vibration Modes . . . . . . . . . . . . . . . . . . . . . 174
6.2.8 Decoupling the Motion Equations. . . . . . . . . . . . . . . . . . 175
6.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4 Vibration of Continuous Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.4.2 Transverse Vibration of a Bar. . . . . . . . . . . . . . . . . . . . . 221
6.4.3 Eigenvalues and Eigenmodes in Transverse
Vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.4.4 Axial Vibrations of Bars . . . . . . . . . . . . . . . . . . . . . . . . 226
6.4.5 Eigenvalues and Eigenfunction in Axial Vibration. . . . . . 227
6.4.6 Torsional Vibration of the Bar . . . . . . . . . . . . . . . . . . . . 231
6.4.7 Eigenvalues and Eigenfunctions in Torsional
Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
x Contents
7 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.2 Linear Systems with Two Degrees of Freedom . . . . . . . . . . . . . . 240
7.3 Free Vibration of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256