A uniform beam AOB, O being the mid point of AB, mass m, rests on three identical vertical springs with stiffness constants k 1 , k 2 , and k 3 at A, O and B respectively. The bases of the springs are fixed to a horizontal platform.Determine the compression of the springs and their compressional forces in the case: k 1 = k 3 = k and k 2 = 2 k

Question

A uniform beam AOB, O being the mid point of AB, mass m, rests on three identical vertical springs with stiffness constants       k    1  ,       k    2  ,  and       k    3   at A, O and B respectively. The bases of the springs are fixed to a horizontal platform.Determine the compression of the springs and their compressional forces in the case:      k    1    =      k    3    =  k and       k    2    =  2  k

Winston Cooper · Accepted Answer

Because the stiffness       k    1    =      k    3    =  k of the springs off-center are equal, and also their distances to the center, the system is balanced and the beam keeps horizontal. The displacement   Δ  y is the same for all.If       F    e   is the force applied to each of the springs off-center, and       F    m   to the spring at the center:       F    e    =  k  Δ  y and       F    m    =  2  k  Δ  yAs we know that   2      F    e    +      F    m    =  M  g, we have 3 equations and 3 unknown (      F    e    ,      F    m    ,  Δ  y  )  M  g  −  2      F    e    =  2  k  Δ  y  M  g  −  2      F    e    =  2  k            F      e        k      ⟹        F    e    =      1    4    M  gResulting that       F    m    =      1    2    M  g and   Δ  y  =      1          4      k        M  g

Determine the compression of the springs, their compressional forces in the case: k_1=k_3=k, k_2=2k

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