Question 13.4.5: Sensitivity of Node Location For the vehicle in Example 13.4...
Sensitivity of Node Location
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}{θ} = \frac{k_{1} L_{1} − k_{2} L_{2}}{m} \frac{1}{ω^{2}_{1} − ω^{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}:
ω_{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} .