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Unformatted text preview: After Notes Lecture 2 Energy methods 2.a. Energy Method (Single Degree of Freedom System) Another possible way to determine the dynamic of a system is by starting with the same fundamental properties m , t , r , f , defining the velocity, kinetic energy and potential energy as in (1.1), (1.7) and (1.6), respectively, together with the action L V T x x t L = ) , , ( & , ( 2 . 1 ) and replace the axiomatic law (1.3) by the Euler-Lagrange equation = x L dt d x L & , ( 2 . 2 ) which ensures that the action is stationary. 2.b. Examples (i) Mass-spring system Consider the system shown below. Let ) ( t x be the displacement of the mass from its static equilibrium. Then it follows from the free-body diagram shown and (1.3) that = + kx x m & & . ( 2 . 3 ) For this case 2 2 1 x m T & = , 2 2 1 kx V = and ( ) 2 2 2 1 kx x m L = & , and (2.2) gives = x m dt d kx & , from which we obtain (2.3)....
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- Fall '08