Question

Structural engineers use wireless sensor networks to monitor the condition of dams and bridges. An article 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 Ratio	Frequency (Hz)	Damping Ratio	Frequency (Hz)
0.3	2.72	0.5	1.59
0.3	2.84	0.6	0.77
0.3	3.77	0.6	1.26
0.4	2.07	0.6	1.66
0.4	2.2	0.7	0.89
0.4	2.34	0.7	1
0.4	2.61	0.7	0.66
0.5	1.8	0.8	1.13
0.5	1.93	0.8	0.37

a) Compute the least-squares line for predicting frequency from damping ratio. Round the answers to three decimal places.

b) If two modes differ in damping ratio by 0.2, by how much would you predict their frequencies to differ? Round the answer to three decimal places.

c) Predict the frequency for modes with damping ratio 0.75. Round the answer to three decimal places.

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

e) For what damping ratio would you predict a frequency of 2.0? Round the answer to three decimal places.

Answer 1

Let us construct the scatter plot of frequency (Y) versus Damping Ratio (X) by using the EXCEL.

b)

The least square line equation is

c)

If two modes differ in damping ratio by 0.2, their frequencies to differ by 4.933 times 0.2 is equal to -0.987

d)

Predict the frequency for modes with damping ratio 0.75. as follows

e)

For what damping ratio would you predict a frequency of 2.0, we predict it as follows