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DYNAMICS - EXAMPLE


Spring-Mass Motion
  Example

 

What is the natural frequency of the system shown if k1 = k2 = k3 = 10 lb/ft?

   
    Solution


Spring Forces

 

Stiffness constant of two springs in parallel,
     F1 + F2 = F
     x k1 + x k2 = x k12
     k1 + k2 = k12

Stiffness constant of two springs in series,
     x1 + x2 = x
     F/k1 + F/k2 = F/k12
     1/k1 + 1/k2 = 1/k12

For the current spring configuration, combine k1 and k2 (series) first,
     1/k12 = 1/k1 + 1/k2
             = 1/10 + 1/10
     k12 = 5

The combined k12 and k3 (parallel),
     k122 = k12 + k3
            = 5 + 10 =15

The natural frequency of the combined system is,
     ω = (k122/m)0.5 = (15/2)0.5

ω = 2.739 rad/s

     
   
 
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