SolSec1_7 - Problems and Solutions Section 1.7(1.82 through...

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Problems and Solutions Section 1.7 (1.82 through 1.89) 1.82 Choose a dashpot's viscous damping value such that when placed in parallel with the spring of Example 1.7.2 reduces the frequency of oscillation to 9 rad/s. Solution: The frequency of oscillation is ! d = ! n 1 " # 2 From example 1.7.2: ! n = 10 rad/s, m = 10 kg, and k = 10 3 N/m So, 9 = 10 1 ! " 2 ! 0.9 = 1 " # 2 ! (0.9) 2 = 1 " # 2 ! = 1 " 0.9 ( ) 2 = 0.436 Then c = 2 m ! n " = 2(10)(10)(0.436) = 87.2 kg/s 1.83 For an underdamped system, x 0 = 0 and v 0 = 10 mm/s. Determine m, c, and k such that the amplitude is less than 1 mm. Solution: Note there are multiple correct solutions. The expression for the amplitude is: A 2 = x 0 2 + ( v o + !" n x o ) 2 " d 2 for x o = 0 # A = v o " d < 0.001 m # " d > v o 0.001 = 0.01 0.001 = 10 So ! d = k m 1 " # 2 ( ) > 10 $ k m 1 " # 2 ( ) > 100, $ k = m 100 1 " # 2 (1) Choose ! = 0.01 " k m > 100.01 (2) Choose m = 1 kg ! k > 100.01 (3) Choose k = 144 N/m >100.01 ! " n = 144 rad s = 12 rad s ! " d = 11.99 rad s ! c = 2 m #" n = 0.24 kg s
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1.84 Repeat problem 1.83 if the mass is restricted to lie between 10 kg < m < 15 kg. Solution: Referring to the above problem, the relationship between m and k is k >1.01x10 -4 m after converting to meters from mm. Choose m =10 kg and repeat the calculation at the end of Problem 1.82 to get
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