User:Egm4313.s12.team2.ladin

Hello my name is Jonathon.

= Problem R1.3 =

Position Function
$$ y(t)=y_k + y_c\ $$

$$ y'(t)=y'_k + y'_c\ $$

$$ y(t)=y_k + y''_c\ $$

$$ f(w)=f(k)=f(c)=:\ f(I) $$

$$ \overline{f}(t)-\overline{f}(I)=my'' $$

$$ \overline{f}(t)=\overline{f}(I)+my'' $$

= Problem R2.6 =

Problem Statement
Realize spring-dashpot-mass systems in series, as shown in Figure 2.6, with a double root $$ \lambda = -3 $$. Find the values for the parameters k, c, m.

Figure 2.6

Solution
The equation of motion for spring-dashpot-mass system in series is: $$ m(y''_k+\frac{k}{c}y'_k)+ky_k=f(t)\ $$ Distribute m: $$ my''_k+\frac{mk}{c}y'_k+ky_k=f(t)\ $$ The standard L2-ODE-CC with excitation is: $$ Ay''+By'+Cy=r(x)\ $$ The homogeneous equation is: $$ Ay''_h+By'_h+Cy_h=0\ $$ Given the double root: $$ \lambda=-3\ $$

The characteristic equation is: $$ [\lambda-(-3)]^2=(\lambda+3)^2=\lambda^2+6\lambda+9=0\ $$ The homogeneous L2-ODE-CC for this characteristic equation is: $$ y''+6y'+9y=0\ $$ Compare coefficients with: $$ my''_k+\frac{mk}{c}y'_k+ky_k=f(t)\ $$ The result is: $$ m=1\ $$

$$ k=9\ $$

$$ 6=\frac{mk}{c}\ $$

$$ 6=\frac{mk}{c}\rightarrow c=\frac{mk}{6}=\frac{(1)(9)}{6}\ $$

$$ c=\frac{3}{2}\ $$