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Problem R1.2
Derive the equation of motion of the spring-mass-dashpot in fig. 5.3 in K 2011 p 85 with an applied force r(t) on the ball.

In order to derive the equation of motion of the spring-mass-dashpot with an applied force of $$ \displaystyle r(t)$$, the formula of a non-homogenous equation must first be understood.

In the non homogenous equation:
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$$  \displaystyle y'' + p(x)y' + q(x)y = r(x) $$     (2.0) The $$\displaystyle y $$ represents position, $$ \displaystyle y' $$ represents velocity, and $$ \displaystyle y'' $$ represents acceleration.
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Figure 5.3 consist of a mass $$\displaystyle m$$, a spring constant $$\displaystyle c$$, and a damper coefficent $$\displaystyle k$$ These components create 3 internal forces- intertia $$\displaystyle my''$$, damping force $$\displaystyle cy'$$, and restoring force $$\displaystyle ky$$. Along with these 3 forces is also a driving force of $$ \displaystyle r(t)$$.

Solution
Using these 4 components, equation 1 becomes:
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$$\displaystyle my'' + cy' + ky = r(t)$$
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Section 1 Lecture Notes