Question B.2
For a system with $m=10$ kg, $k=1000$ N/m, its equations of motion have been derived as
$2m\ddot{x_1} + 4kx_1 - kx_2 = f_1(t)$
$m\ddot{x_2} - kx_1 + 4kx_2 = f_2(t)$
(1) Find the natural frequencies of the system.
(2) Determine the associated mode shapes.
(3) Obtain the free vibration response with the following initial conditions:
$x_1(0) = 0$; $x_2(0) = 0$; $\dot{x_1}(0) = 10$ mm/sec; $\dot{x_2}(0) = 0$
(4) If $f_1(t)=f_2(t) = 10\sin2t$ (N) but $m$, $k$ and the initial conditions specified in Step (3) remain
unchanged, will the natural frequencies, the associated mode shapes and free vibration response
be different from those obtained in Steps (1)-(3)? Explain why (use calculations if necessary).