SolSec1_8 - Problems and Solutions Section 1.8 (1.90...

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Problems and Solutions Section 1.8 (1.90 through 1.93) 1.90 Consider the system of Figure 1.90 and (a) write the equations of motion in terms of the angle, θ , the bar makes with the vertical. Assume linear deflections of the springs and linearize the equations of motion. Then (b) discuss the stability of the linear system’s solutions in terms of the physical constants, m , k , and ! . Assume the mass of the rod acts at the center as indicated in the figure. Figure P1.90 Solution: Note that from the geometry, the springs deflect a distance kx = k ( ! sin ! ) and the cg moves a distance ! 2 cos . Thus the total potential energy is U = 2 ! 1 2 k ( ! sin " ) 2 # mg ! 2 cos and the total kinetic energy is T = 1 2 J O ! 2 = 1 2 m " 2 3 ! 2 The Lagrange equation (1.64) becomes d dt ! T ! ! # $ % ( + ! U ! = d dt m " 2 3 ! # $ % ( + 2 k " sin cos ) 1 2 mg " sin = 0 Using the linear, small angle approximations sin " and
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This note was uploaded on 03/31/2009 for the course MECHANICAL MAE351 taught by Professor J.g.lee during the Spring '09 term at Korea Advanced Institute of Science and Technology.

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SolSec1_8 - Problems and Solutions Section 1.8 (1.90...

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