Question: Structural engineers use wireless sensor networks to monitor the condition of dams and bridges. The article Statistical Analysis of Vibration Modes of a Suspension Bridge

Structural engineers use wireless sensor networks to monitor the condition of dams and bridges. The article €œStatistical Analysis of Vibration Modes of a Suspension Bridge Using Spatially Dense Wireless Sensor Network€ (S. Pakzad and G. Fenves, Journal of Structural Engineering, 2009:863€“872) describes an experiment in which accelerometers were placed on the Golden Gate Bridge for the purpose of estimating vibration modes. For 18 vertical modes, the system was underdamped (damping ratio< 1). Following are the damping ratios and frequencies for those modes.

Damping Frequency Damping Frequency (Hz) (Hz) Ratio Ratio 1.53 2.72 0.3 0.5 0.3 2.84 0.6 0.77 0.3 1.26 3.77 0.6 0.4 2.07

a. Construct a scatterplot of frequency (y) versus damping ratio (x). Verify that a linear model is appropriate.

b. Compute the least-squares line for predicting frequency from damping ratio.

c. If two modes differ in damping ratio by 0.2, by how much would you predict their frequencies to differ?

d. Predict the frequency for modes with damping ratio 0.75.

e. Should the equation be used to predict the frequency for modes that are overdamped (damping ratio > 1)? Explain why or why not.

f. For what damping ratio would you predict a frequency of 2.0?

Damping Frequency Damping Frequency (Hz) (Hz) Ratio Ratio 1.53 2.72 0.3 0.5 0.3 2.84 0.6 0.77 0.3 1.26 3.77 0.6 0.4 2.07 0.6 1.66 0.4 2.20 0.7 0.89 0.4 2.34 0.7 1.00 0.7 0.4 2.61 0.66 0.8 0.5 1.80 1.13 0.5 1.93 0.8 0.37

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