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 third-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 info for Advanced Engineering Dynamics
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. The variation in any body forces can be neglected but contact forces, whether elastic or not, are regarded as external. The Lagrangian will thus be represented by the kinetic energy only and by the definition of short duration OT/Oq will also be negligible. 48) 44 L a g r a n g e ' s equations EXAMPLE The two uniform equal rods shown in Fig.
This usually means that it is possible to write down the Lagrangian such that the number of generalized co-ordinates is equal to the number of degrees of freedom. There are situations where a constraint can only be written in terms of velocities or differentials. One often-quoted case is the problem of a wheel rolling without slip on an inclined plane (see Fig. 9). ~2 + 3)2) + 1 I1~2 + -21 I2~/2 -- mg(sinay + cosar) The equation of constraint may be written ds= rdo o r as dx= dssin~ = rsin~d~ dy = d s c o s ~ = r c o s v d o We now introduce the concept of the Lagrange undetermined multipliers ~..
23) where Qj is the generalized force not obtained from a position-dependent potential or a dissipative function. EXAMPLE For the system shown in Fig. ~1)2 The virtual work done by the external forces is 8 W = F, ~X ! 4" F 2 8x2 For the generalized co-ordinate x~ application of kagrange's equation leads to ml-~'! + klXl - - k2(x2 -- Xl) + r -- C2('1f2-- "~l) and for )(2 Fig. ~2- F2 Note that in the first case the kinetic energy has no term which involves products like q~qj whereas in the second case it does.
Advanced Engineering Dynamics by H.R. Harrison and T. Nettleton (Auth.)