Question 13.4.2: Find and interpret the mode ratios for the system shown in F......

For this case, equation (3) in Example 13.4.1 becomes

\left(m_1s^2+k_1+k_2\right)\left(m_2s^2+k_2+k_3\right)-k_2^2=0          (3) in Example 13.4.1

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

\left(m_1s^2+k_1+k_2\right)A_1 – k_2 A_2=0          (1) of Example 13.4.1

for mode 1, and

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.