Consider the example of a $2$DOF lumped

parameter system from class. Use m$1=100$ kg$,$ m$2=100$ kg$,$ k$1=4000$ N$/$m$,$ k $2=1000$

N$/$m$.$ Perform all calculations by hand. Please state your conclusions$/$observations for

each response. Show every step, and write neatly!

a$.$ What is the eigenstructure of this system? Quote your frequencies in $[$Hz$].$

Physically describe what each mode shape implies.

b$.$ Compute the initial condition response for initial conditions q $1(0)=50$ mm$,$ and

mm$/$s$.$ Plot the response using MATLAB.

c$.$ Compute the initial condition response for q$1(0)=50$ mm$,$ and q $2(0)=-50$ mm$.$

Plot the response using MATLAB.

d$.$ Suppose that $\{$f$\}=\{$f$1($t$)$ f $2($t$)\}$ where f $1($t$)=$ sin t and f $2($t$)=0,$ and the same

initial conditions from part $($b$)$ are used. What is the total response? Plot the

response using MATLAB.

e$.$ What initial condition would you choose to ensure that the vibration occurs only

at the second modal frequency? What is the initial condition response in this case.

Plot the response using MATLAB.