User:Egm4313.sp12.team4.diroma/report1

= Problem 1 =

=Problem Statement=

Derive the equation of motion of a spring-dashpot system in parallel with a mass and applied force, $$   f(t)$$



Force Balance
Using a force balance, the following equation can be derived:

$$ f(t) = my'' + f_I $$ (1) $$\Rightarrow $$ where$$(f_I = f_c = f_k) $$

Adding the displacements due to the spring $$ (y_k)$$ and the dashpot $$ (y_c)$$ we can write the following equation:

$$ y=y_k+y_c $$ (2)

Taking two time derivatives of (2) gives:

$$ y = y_k + y''_c $$                                              (3)

Since $$f_k=f_c $$ ; We can write:

$$ ky_k = cy'_c $$ (4)

Now, solving for $$ y'_c $$ in (4), we find:

$$ y'_c = \frac{k}{c}y_k $$

We can substitute our new value for $$ y'_c $$ into (3) to obtain:

$$ y = y_k + (y'_c)' $$ ==> $$ y = y_k + (\frac{k}{c}y_k)' $$ ==> $$ y = y_k + \frac{k}{c}y'_k $$

We can now substitute for $$ y'' $$ and $$ f(I) $$ in equation (1) to obtain an equation in terms of $$ y_k $$ :

$$ m(y''_k + \frac{k}{c}y'_k) + ky_k = f(t) $$