Chapter 6.4, Problem 12P

Consider the two-mass, three-spring system of Example 2.Instead of solving the system of four first order equations, we indicate here how to proceed directly from the system of two second order equations given in Eq.(18) of section 6.1.

(a) Show that using the parameter values of Example 2, $m_1=2,m_2=\frac{9}{4},k_1=1,k_2=3,k_3=\frac{15}{4}$ and assuming that $F_1\left(t\right)=0$ and $F_2\left(t\right)=0$, Eqs.(18) in section 6.1 can be written in the form

$\text{y}\text{'}\text{'}=\left(\begin{array}{cc}-2& 3/2\\ 4/3& -3\end{array}\right)\text{y=By}\text{.}$

(b) Assume that $\text{y}=e^{\omega t}\text{u }$ and show that

$\left(\text{B}-{\omega }^2\text{I}\right)\text{u}=0$.

Note that ${\omega }^2$ is an eigenvalue of $\text{B}$ corresponding to the eigenvector $\text{u}$.

(c) Find the eigenvalues and eigenvectors of $\text{B}$.

(d) Write down the expressions for $y_1$ and $y_2$. There should be four arbitrary constants in these expressions.

(e) By differentiating the result from part (d), write down expressions for $y{\text{'}}_1$ and $y{\text{'}}_2$. Your results from parts (d) and € should agree with Eq.(13) in the text.