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

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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 cos ! " 1 yields a) m ! 2 3 "" ! + 2 k ! 2 " mg ! 2 # $ % & ( ! = 0 Since the leading coefficient is positive the sign of the coefficient of θ determines the stability.
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