User:Egm4313.s12.team13.rlh.r1

Problem Statement
For the spring-dashpot-mass system on p.1-4 (found in notes) draw the FBDs and derive the equation of motion.

A free body diagram of the spring-mass-dashpot found in Sec 1 of the notes must be broken up into three different parts. The spring, the dashpot, and the mass.

The positive y direction equals $$ \overrightarrow{y+} \! $$

The spring's free body diagram is: $$ F_k \longleftarrow\! $$ $$ \longrightarrow\! F_k $$

The dashpot's free body diagram is: $$ F_k \longleftarrow\! $$ $$ \longrightarrow\! F_c $$

The free body diagram of the mass is: $$ F_c \longleftarrow\! $$ $$ \longrightarrow\! F(t) $$

Since there are only two opposing forces on the dashpot those two forces must be equal. :{| style="width:100%" border="0" $$  \displaystyle F_c = F_k = F_i $$     (3.0)
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By definition
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$$  \displaystyle ma=my'' $$     (3.1)
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Using Newtons second law :
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$$  \displaystyle \sum F = ma $$ (3.2)
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using substitution and equations (3.2), (3.1), and (3.0) we get
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$$  \displaystyle \sum F_y = F_(t) - F_i = my'' $$     (3.3)
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Rearranging equation (3.3) gives us the solution

$$ F(t) = F_i + my''$$