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 - effect and One-Dimensional Wave Propagation
, Pages 125-171
7 - Waves In a 3-dimensional 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
12) t~ we have ~I = tl -~i 8qi + i i -~q. 8qi ) dt = 0 Note that there is no partial differentiation with respect to time since the variation applies only to the co-ordinates and their derivatives. Because the variations are arbitrary we can consider the case for all q~ to be zero except for qj. ) ~ d-; a~. 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.
3, the Lagrangian is ~. 25) z m t 0 Y Fig. 3 Y 32 Lagrange's equations This may be interpreted as consistent with the Lagrangian being independent of the position in space of the axes and this also leads to the linear momentum in the arbitrary x direction being constant or conserved. Consider now the same system but this time referred to an arbitrary set of cylindrical coordinates. This time we shall superimpose a rotational drift of'~ of the axes about the z axis, see Fig. 4. Now the Lagrangian is me[r~ (0i I!
For example, in this case we could have included ~,3dz = 0 to the virtual work expression as a result of the motion being confined to the xy plane. ) The equation of motion in the z direction is - m g cos a = ~3 It is seen here that --~L3 corresponds to the normal force between the wheel and the plane. However, non-holonomic systems are in most cases best treated by free-body diagram methods and therefore we shall not pursue this topic any further. 11 Lagrange'sequations for impulsive forces The force is said to be impulsive when the duration of the force is so short that the change in the position co-ordinates is negligible during the application of the force.