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	1.98
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

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.

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

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

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

Answer 1

	Damping ratio (X)	Frequency (Y)	X2	Y2	XY
1	0.3	2.72	0.09	7.3984	0.816
2	0.3	2.84	0.09	8.0656	0.852
3	0.3	3.77	0.09	14.2129	1.131
4	0.4	2.07	0.16	4.2849	0.828
5	0.4	2.2	0.16	4.84	0.88
6	0.4	2.34	0.16	5.4756	0.936
7	0.4	2.61	0.16	6.8121	1.044
8	0.5	1.8	0.25	3.24	0.9
9	0.5	1.93	0.25	3.7249	0.965
10	0.5	1.98	0.25	3.9204	0.99
11	0.6	0.77	0.36	0.5929	0.462
12	0.6	1.26	0.36	1.5876	0.756
13	0.6	1.66	0.36	2.7556	0.996
14	0.7	0.89	0.49	0.7921	0.623
15	0.7	1	0.49	1	0.7
16	0.7	0.66	0.49	0.4356	0.462
17	0.8	1.13	0.64	1.2769	0.904
18	0.8	0.37	0.64	0.1369	0.296
Total	9.5	32	5.49	70.5524	14.541