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

analianopolisca 2022-08-13 Answered
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
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Answers (1)

Winston Cooper
Answered 2022-08-14 Author has 12 answers
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 y
As 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 g
Resulting that F m = 1 2 M g and Δ y = 1 4 k M g
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