Question 13.4.2: Nonsymmetric Modes Find and interpret the mode ratios for th...

For this case, equation (3) in Example 13.4.1 becomes
3u² + 13αu + 8α² = 0
where u = s² and α = k/m. From the quadratic formula we obtain u = −0.743α and u = −3.591α. Thus the two modal frequencies are ω_{1} = \sqrt{0.743α} = 0.862 \sqrt{k/m} and ω_{2} = \sqrt{3.591α} = 1.89 \sqrt{k/m} . From equation (1) of Example 13.4.1 the mode ratios are computed as
\frac{A_{1}}{A_{2}} = \frac{2α}{s^{2}  +  3α} = 0.886

for mode 1, and
\frac{A_{1}}{A_{2}} = \frac{2α}{s^{2}  +  3α} = −3.39
for mode 2.
Thus in mode 1 the masses move in the same direction with the amplitude of mass m_{1} equal to 0.886 times the amplitude of mass m_{2}. This oscillation has a frequency of ω_{1} = 0.862 \sqrt{k/m}.
In mode 2, the masses move in the opposite direction with amplitude of mass m_{1} equal to 3.39 times the amplitude of mass m_{2}. This oscillation has a higher frequency of ω_{2} = 1.89 \sqrt{k/m}.
To stimulate the first mode, displace mass m_{1} 0.866 times the initial displacement of mass m_{2}, in the same direction. To stimulate the second mode, displace mass m_{1} 3.39 times the initial displacement of mass m_{2}, but in the opposite direction.