Understanding the Equivalence of Two Mechanical Systems

• mech-eng
In summary: What you are asking is what happens to the force on the mass if the springs are stretched. To answer that, you need to solve the equation for the force on the mass.In summary, the picture shows how two systems can be equivalent if they have the same force acting on them. The equation for the force on the mass is given by the sum of the forces on the springs.
mech-eng
I cannot understand the mechanical systems in the picture. In the picture a) how can f is scattered equally to M, K1, K2, B1 and B2 ? Where do you know this? How can two systems be equivalent of each other?

In the picture (b) if f is applied, how can the force on M is Ma, a is acceleration.

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View attachment 111153

Source: Automatic Control Engineering by Francis H.Raven

Thank you.

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Text is perfectly clear. But I concede it's a bit weird at first: after all the various elements seem to be in parallel.

Perhaps it becomes clearer to you if you write the equations for a simpler case.

E.g. just the two springs: ##\ f = (K_1 + K_2) x\ .\ ## And you see that ##\ Z_{\rm tot} = Z_1 + Z_2 \ ## -- that's why they call it series.

Now add the mass and write the equation linking ##\ f, \ M, \ K_1, \ K_2 \ ## in a similar form.

Once you get that, the ##C_1p## and ##C_2p## are a piece of cake.

Ah, Berk has a copy too ! Or did he just pimp yours ? Mine is a first student edition ( 1961 ) where it's page 15 with exactly the same text .

berkeman
I suspect that the beam-like appearance of the mass M is part of what is throwing you off. Instead, think of M as more of a block with two springs and two dashpots supporting it. The external force F acts on the block, as well as forces from each spring and dashpot. All that is of interest here is vertical motion, so ignore the horizontal distribution of elements.

Your question about (b) is not grammatically correct English. Remember what Newton's Second Law says: The SUM of forces on the mass is M*a, not that any single force is M*a.

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