By H.R. Harrison and T. Nettleton (Auth.)
, Pages xi-xii
1 - Newtonian Mechanics
, Pages 1-20
2 - Lagrange's Equations
, Pages 21-45
3 - Hamilton's Principle
, Pages 46-54
4 - inflexible physique movement in 3 Dimensions
, Pages 55-84
5 - Dynamics of Vehicles
, Pages 85-124
6 - impression and One-Dimensional Wave Propagation
, Pages 125-171
7 - Waves In a three-d Elastic Solid
, Pages 172-193
8 - robotic Arm Dynamics
, Pages 194-234
9 - Relativity
, Pages 235-260
, Pages 261-271
Appendix 1 - Vectors, Tensors and Matrices
, Pages 272-280
Appendix 2 - Analytical Dynamics
, Pages 281-287
Appendix three - Curvilinear co-ordinate systems
, Pages 288-296
, Page 297
, Pages 299-301
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Extra resources for Advanced Engineering Dynamics
X 3 N ) T and mi " " N] "m3 diagonal [m] = Now Xi "- xi(ql q2 . . t) SO Yci = E J Oxi Oxi 30 Lagrange's equations We shall, in this section, use the notation ( ) to mean a column matrix and [ ] to indicate a square matrix. X2. , oa~n) T = then we may write (i) = [A](,~) + (b) where OXl , Oql Ox, [A] = , aq, ~ , 1 , 7 Ox! 6 Oq, Ox, = f(ql q2... ) ~q; OXl . . OXn]T Ot Ot (b) = and (,~) = (,~, ,~... ~. 24) Note that use has been made of the fact that [m] is symmetrical. This fact also means that [A]T[m][A] is symmetrical.
13> These are Lagrange's equations for conservative systems. It should be noted that I, = F* - V because, with reference to Fig. 2, it is the variation of co-kinetic energy which is related to the momentum. But, as already stated, when the momentum is a linear function of velocity the co-kinetic energy T* = T, the kinetic energy. The use of co-kinetic energy 52 Hamilton's principle becomes important when particle speeds approach that of light and the non-linearity becomes apparent. 5 Illustrative example One of the areas in which Hamilton's principle is useful is that of continuous media where the number of degrees of freedom is infinite.
31) 0 From the theory of homogeneous linear equations a non-trivial result is obtained when the determinant of the square matrix is zero. This leads to a cubic in k and therefore there are three roots (k~, k2 and k3) and three corresponding vectors (o3~, o32 and %). 30). There are, therefore, three equations Moment of inertia 63 O ](0')1) -- ~! 1)2) T (COl) + ~2((01 )T (0)2) = 0 Because [Io] is symmetrical the second term is the transpose of the first and as they are scalar they cancel. 35) (~2 -- ~!
Advanced Engineering Dynamics by H.R. Harrison and T. Nettleton (Auth.)