Kinematics_of_CM_01_Motion

Kinematics_of_CM_01_Motion - Section 2.1 2.1 Motion 2.1.1...

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Section 2.1 Solid Mechanics Part III Kelly 201 2.1 Motion 2.1.1 The Material Body and Motion Physical materials in the real world are modeled using an abstract mathematical entity called a body . This body consists of an infinite number of material particles 1 . Shown in Fig. 2.1.1a is a body B with material particle X . One distinguishes between this body and the space in which it resides and through which it travels. Shown in Fig. 2.1.1b is a certain place x in familiar Euclidean space E . Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body By fixing the material particles of the body to places in space, one has a configuration of the body χ , Fig. 2.1.1c. A configuration can be expressed as a mapping of the particles X to the places x , ( ) X χ x = (2.1.1) A motion of the body is a family of configurations parameterised by time t, ( ) t , X χ x = (2.1.2) At any time t , Eqn. 2.1.2 gives the location in space x of the material particle X , Fig. 2.1.2. There are a number of different ways in which the motion can be described. Eqn. 2.1.2 is the material description , in which the motion is described in terms of the material particles which make up the body. 1 these particles are not the discrete mass particles of Newtonian mechanics, rather they are very small portions of continuous matter; the meaning of particle is made precise in the definitions which follow B X x (a) (b) E (c) χ
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Section 2.1 Solid Mechanics Part III Kelly 202 Figure 2.1.2: a motion of material The Reference and Current Configurations Choose now some reference configuration , Fig. 2.1.3. The motion can then be measured relative to this configuration. The reference configuration might be the configuration occupied by the material at time 0 = t , in which case it is often called the initial configuration . For a solid, it might be natural to choose a configuration for which the material is stress-free, in which case it is often called the undeformed configuration . However, the choice of reference configuration is completely arbitrary.
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

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Kinematics_of_CM_01_Motion - Section 2.1 2.1 Motion 2.1.1...

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