Question 13.4.4: Evaluate the mode shapes and mode frequencies of the vehicle......
Evaluate the mode shapes and mode frequencies of the vehicle shown in Figure 13.4.2 for the case: k_1 = 1.6 × 10^4 N/m, k_2 = 2.5 × 10^4 N/m, L_1 = 1.5 m, L_2 = 1.1 m, m = 730 kg, and I_G = 1350 kg · m².
Dividing the characteristic equation (13.4.8) by mI_G and using the given values for the constants, we obtain
m I_G s^4+\left[m\left(k_1 L_1^2+k_2 L_2^2\right)+I_G\left(k_1+k_2\right)\right] s^2+k_1 k_2\left(L_1+L_2\right)^2=0 (13.4.8)
s^4+105.24 s^2+2744=0The quadratic formula gives the roots s² = −57.61, −47.63. Thus, the four characteristic roots are
s = ±7.59 j, ±6.901 j
These correspond to frequencies of 1.21 Hz and 1.1 Hz.
The mode ratio can be found from equation (1) of Example 13.4.3.
\frac{A_1}{A_2}=\frac{x}{\theta}=\frac{-3500}{730 s^2 + 4.1 \times 10^4} (1)
For mode 1 (s² = −57.61),
\frac{x}{\theta}=\frac{-3500}{730(-57.61) + 4.1 \times 10^4}=3.32 mThus the node is located 3.32 m behind the mass center. Because this node is so far from the mass center, the motion in this mode is predominantly a bounce motion, and this node is called the “bounce center”(see Figure 13.4.3b).
For mode 2 (s² = −47.63),
This node is located 0.562 m ahead of the mass center (because x/θ < 0). Because this node is close to the mass center, the motion in this mode is predominantly a pitching motion, and this node is called the “pitch center” (see Figure 13.4.3c).
