Models of Mechanics

by Klarbring, Anders

• ISBN13: 9781402048340
• ISBN10: 1402048343
• Format: Hardcover
• Copyright: 2006-09-15
• Publisher: Springer Verlag

Summary

This is a textbook on models and modeling in mechanics. It introduces a new unifying approach to applied mechanics: through the concept of the open scheme, a step-by-step approach to modeling evolves. The unifying approach enables a very large scope on relatively few pages: the book treats theories of mass points and rigid bodies, continuum models of solids and fluids, as well as traditional engineering mechanics of beams, cables, pipe flow and wave propagation.Models of Mechanics complements existing books that deal with continuum mechanics. In contrast to such books it gives a setting that is broad enough to encompass also the mechanics of mass points, and theories of beams and other intrinsically one-dimensional bodies. An obtained knowledge of the unifying approach can be a base for advanced studies of fluid and solid mechanics, as well as specializations in mechatronics, control and structural optimization. The book is ideal to use in conjunction with equation-based finite element programs like FEMLAB and FlexPDE. It provides tools for organizing and structuring the broad discipline of mechanics, as well as a platform for deriving new models of applied use.

Table of Contents

Part I General Background
1 Introduction	3	(8)
1.1 The Concept of a Model	3	(3)
1.2 Models, Reality and The Conceptual World	6	(2)
1.3 The Idea of an Open Scheme	8	(2)
Exercises	10	(1)
2 Open scheme	11	(10)
2.1 The Physical Space epsilon, Geometric Vectors and Tensors	11	(3)
2.2 Laws of Mechanics	14	(3)
2.2.1 Universal Laws	14	(3)
2.2.2 Particular Laws	17	(1)
Exercises	17	(4)
Part II Basic Models: Geometry and Universal Laws
3 Bodies and Their Placements in epsilon	21	(10)
3.1 Discrete Body Model	21	(2)
3.2 One-Dimensional Body Model	23	(2)
3.2.1 Pipe Flow Model	25	(1)
3.3 Two-Dimensional Body Model	25	(2)
3.4 Three-Dimensional Body Model	27	(2)
3.5 Refined Geometric Models of Bodies	29	(2)
4 Discrete Model	31	(14)
4.1 Mass and its Conservation	31	(1)
4.1.1 Center of Mass	32	(1)
4.2 Linear and Angular Momentum	32	(1)
4.2.1 Motion of the Center of Mass	33	(1)
4.3 System of Forces, and Force and Torque Resultants	33	(3)
4.3.1 Change of Base Point for the Torque Resultant	35	(1)
4.4 Explicit forms of Euler's Laws	36	(1)
4.4.1 Change of Base Point for Euler's Laws	36	(1)
4.5 The Theorem of Action and Reaction	37	(1)
4.6 Newton's Laws	38	(1)
4.7 A Complete Model	38	(3)
4.7.1 Example: An Inverse Kepler Problem	39	(2)
Exercises	41	(4)
5 One-Dimensional Model	45	(18)
5.1 Mass and its Conservation	45	(1)
5.2 Linear and Angular Momentum	46	(1)
5.3 System of Forces, and Force and Torque Resultants	46	(1)
5.4 Explicit Forms of Euler's Laws	47	(1)
5.5 Local Equations	48	(1)
5.6 Equations of Motion in Natural Parameterization and Natural Frame	49	(6)
5.6.1 Natural Parameterization	49	(4)
5.6.2 Equations of Motion in Natural Base	53	(2)
5.7 A Complete Model	55	(2)
5.7.1 Example: A Circular Beam	56	(1)
5.8 Another Complete Model	57	(3)
5.8.1 Example: Suspension Bridge	58	(2)
Exercises	60	(3)
6 Pipe Flow	63	(14)
6.1 Kinematic Constraint, Velocity and Acceleration	63	(1)
6.2 Spatial and Material Representations, and Time Derivatives	64	(2)
6.3 Forces and Couple in a Pipe Bend	66	(1)
6.4 Continuity Equation and Control Domain	67	(2)
6.5 Volume of the Pipe and Isochoric Motion	69	(2)
6.6 Equation of Motion in Terms of Pressure	71	(2)
6.7 Conservative Forces and Incompressibility	73	(2)
Exercises	75	(2)
7 Three-Dimensional Model	77	(12)
7.1 Spatial and Material Representations, and Time Derivatives	77	(2)
7.2 Volume and Isochoric Motion	79	(1)
7.3 Mass and the Continuity Equation	80	(1)
7.4 Linear and Angular Momentum	81	(1)
7.5 Stress Tensor, System of Forces, and Force and Torque Resultants	81	(3)
7.5.1 Normal and Shear Stress	82	(1)
7.5.2 The Stress Tensor and the System of Forces	82	(2)
7.6 Explicit Forms of Euler's Laws	84	(1)
7.7 Equations of Motion	84	(1)
Exercises	85	(4)
Part III Complete Models by Adding Particular Laws
8 Particular Laws	89	(10)
8.1 Three Types of Particular Laws	89	(1)
8.2 General Principles for Particular Laws	90	(3)
8.3 Examples of Particular Laws for the Discrete Model	93	(1)
8.4 Examples of Particular Laws for the One-Dimensional Model	94	(1)
8.5 Examples of Particular Laws for the Pipe Flow Model	95	(1)
8.6 Examples of Particular Laws for the Three-Dimensional Model	95	(3)
Exercises	98	(1)
9 Small Displacement Models	99	(28)
9.1 Discrete Models and General Structure	99	(10)
9.1.1 Linearizing a Simple Discrete Model	99	(4)
9.1.2 Truss Model	103	(2)
9.1.3 Work Equations	105	(2)
9.1.4 Abstraction	107	(1)
9.1.5 Use of Abstract Work Equations in Modeling	108	(1)
9.2 Beam Models	109	(8)
9.3 Linear Elasticity	117	(7)
9.3.1 Elasticity Coefficients	120	(4)
Exercises	124	(3)
10 Pipe Flow Models	127	(10)
10.1 Incompressible Pipe Flow	127	(2)
10.2 Compressible Pipe Flow	129	(5)
10.3 Incompressible Pipe Flow in a Flexible Pipe	134	(1)
Exercises	135	(2)
11 Models of Fluid Mechanics	137	(12)
11.1 Inviscid Fluid	138	(2)
11.1.1 Elastic Fluid	138	(1)
11.1.2 Incompressible Fluid	139	(1)
11.2 Viscous Fluid	140	(3)
11.2.1 Navier-Stokes' Equation for Plane Steady Laminar Flow	141	(1)
11.2.2 Dimensionless Form	142	(1)
11.3 Statics of Fluids	143	(3)
11.3.1 Archimedes' Principle	144	(2)
11.3.2 Density and Pressure of the Atmosphere	146	(1)
Exercises	146	(3)
12 Kinematic Constraints, Beams and Rigid Bodies	149	(26)
12.1 Some Examples and the General Idea	149	(4)
12.1.1 Discrete Problem	150	(1)
12.1.2 Incompressible Linear Elasticity	151	(1)
12.1.3 Incompressible Viscous Fluid	152	(1)
12.2 Beam Theory	153	(10)
12.2.1 Geometric Shape and Prerequisites	153	(2)
12.2.2 Equilibrium Equations	155	(2)
12.2.3 Displacement Fields and Constitutive Assumptions	157	(6)
12.3 Rigid Body Theory	163	(10)
12.3.1 Geometry of Rigid Body Motion	163	(5)
12.3.2 Euler's Second Law for a Rigid Body	168	(5)
Exercises	173	(2)
A Sets and Functions	175	(4)
A.1 The Notion of a Set	175	(1)
A.2 The Notion of a Function	176	(3)
B Euclidean Point and Vector Spaces	179	(8)
B.1 Euclidean Vector Space	180	(5)
B.2 Euclidean Point Space	185	(2)
C Tensors and Some Mathematical Background	187	(10)
C.1 Algebra	187	(3)
C.2 Analysis	190	(5)
C.2.1 Divergence Theorems	193	(1)
C.2.2 Localization Theorems	194	(1)
C.2.3 Fundamental Theorems	194	(1)
Exercises	195	(2)
D Note on Physical Dimensions	197	(2)
E Notation	199	(2)
References	201	(2)
Index	203