Question 13.4.5: For the vehicle in Example 13.4.4, investigate the sensitivi......
For the vehicle in Example 13.4.4, investigate the sensitivity of the bounce and pitch node location to uncertainty in the parameter values.
Rearrange the mode ratio expression to obtain
\frac{A_1}{A_2}=\frac{x}{\theta}=\frac{k_1 L_1 – k_2 L_2}{m} \frac{1}{\omega_1^2 – \omega^2}where we have defined ω² = −s², and ω_1 as the natural frequency of a concentrated mass m attached to two springs k_1 and k_2:
\omega_1=\sqrt{\frac{k_1 + k_2}{m}}The value of ω² = −s² is obtained from the characteristic equation. If ω² is close to ω^2_1, then the denominator of x/θ will be very small, and x/θ will be very large. In addition, any slight change in the values of ω^2_1 or ω² due to changes in the parameters k_1, k_2, L_1, L_2, m, or I_G will cause large changes in the value of x/θ and thus large changes in the node location. In Example 13.4.4, ω^2_1 = 4.1 × 10^4/730 = 56.16. One of the roots gives s² = −57.61, which is close to −56.16. Thus we can expect large changes in the node location if slightly different parameter values are used. The other root gives s² = −47.63, which is not as close to −56.16; thus, the corresponding node location will not be as sensitive.
If it is important to have a reliable prediction for the node location, the parameters must be selected—usually by trial and error—so that ω² is not close to ω^2_1.