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# slides_1_sp10 - CHAPTER 1 Introductory Concepts Elements of...

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1/11/2010 1 1 CHAPTER 1 Introductory Concepts Elements of Vector Analysis Newton’s Laws Units The basis of Newtonian Mechanics D’Alembert’s Principle 2 Science of Mechanics: It is concerned with the motion of material bodies. Bodies have different scales: Microscropic, macroscopic and astronomic scales. In mechanics - mostly macroscopic bodies are considered. Speed of motion - serves as another important variable - small and high (approaching speed of light).

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1/11/2010 2 3 In Newtonian mechanics - study motion of bodies much bigger than particles at atomic scale, and moving at relative motions (speeds) much smaller than the speed of light. Two general approaches: Vectorial dynamics : uses Newton’s laws to write the equations of motion of a system, motion is described in physical coordinates and their derivatives; Analytical dynamics : uses energy like quantities to define the equations of motion, uses the generalized coordinates to describe motion. 4 1.1 Vector Analysis : Scalars, vectors, tensors: Scalar : It is a quantity expressible by a single real number. Examples include: mass, time, temperature, energy, etc. Vector : It is a quantity which needs both direction and magnitude for complete specification. – Actually (mathematically), it must also have certain transformation properties.
1/11/2010 3 5 These properties are: vector magnitude remains unchanged under rotation of axes . ex: force, moment of a force, velocity, acceleration, etc. geometrically , vectors are shown or depicted as directed line segments of proper magnitude and direction. 6 – if we use a coordinate system, we define a basis set ( ): we can write or, we can also use the three components and define A A i A j A k x y z = + + { } { , , } A A A A x y z T = A A A e = e (unit vector) A X Y Z ˆ ˆ ˆ , , i j k

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1/11/2010 4 7 – The three components A x , A y , A z can be used as 3-dimensional vector elements to specify the vector. – Then, laws of vector-matrix algebra apply. Tensors : scalar - an array of zero dimension vector - an array of one dimension 8 – quantities which need arrays of two or higher dimension to specify them completely - called tensors of appropriate rank. Again - to be a tensor, the object must also satisfy certain transformation properties of rotation and translation. Exs: Second-order tensors : stress at a point in deformable body - stress tensor has nine components (a 3x3 matrix in a representation when the basis is defined), inertia tensor (again, a 3x3 matrix in usual notation) expressing mass distribution in a rigid body.
1/11/2010 5 9 TYPES OF VECTORS: Consider a force acting on a body at point P. The force has a line of action AB. This force can lead to translation of the rigid body, rotation of the rigid body about some point, as well as deformation of the body. F

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slides_1_sp10 - CHAPTER 1 Introductory Concepts Elements of...

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