Structural Damping Effects on Dynamic Instability of Subtangentially Loaded and Shear Deformable Beck’s Columns

Table 6

Divergence and damped flutter loads of Beck’s columns for , 0.005, and 0.02 with variation of .

Subtangentiality, α	0.0	0.1	0.2	0.3	0.321	0.354	0.4	0.439	0.5	0.6	0.7	0.8	0.9	1.0
					0.317^†	0.352^†		0.433^†	0.488^†
					0.306^‡	0.345^‡		0.413^‡	0.456^‡
Undamped second flutter load,	—	—	—	—	19.42	24.13	28.78	33.34	49.35	—	—	—	—	—
					17.68^†	22.41^†	27.35^†	31.78^†	—
					13.96^‡	18.81^‡	—	—	—
Undamped first flutter load, (= externally damped flutter load, )	—	—	—	—	19.42	17.08	16.40	16.16	16.05	16.26	16.79	17.59	18.67	20.05
					17.68^†	15.39^†	14.73^†	14.52^†	14.37^†	14.47^†	14.84^†	15.43^†	16.20^†	17.16^†
					13.96^‡	11.89^‡	11.29^‡	11.21^‡	11.03^‡	10.94^‡	11.08^‡	11.35^‡	11.70^‡	12.12^‡
2% damped flutter load,	—	—	—	—	19.22	17.08	15.86	14.55	14.20	14.12	14.40	14.98	15.85	17.03
					17.48^†	15.39^†	14.22^†	13.17^†	12.85^†	12.71^†	12.92^†	13.36^†	14.01^†	14.87^†
					13.80^‡	11.89^‡	10.74^‡	10.38^‡	10.12^‡	9.857^‡	9.946^‡	10.17^‡	10.49^‡	10.90^‡
Internally damped flutter load,	—	—	—	—	19.42	17.08	15.86	14.55	9.870	9.379	9.305	9.529	10.05	10.94
					17.48^†	15.39^†	14.22^†	13.17^†	9.237^†	8.719^†	8.657^†	8.855^†	9.303^†	9.983^†
					13.96^‡	11.89^‡	10.74^‡	10.35^‡	7.797^‡	7.231^‡	7.194^‡	7.338^‡	7.637^‡	8.083^‡
2nd divergence load,	22.21	21.17	19.89	18.23	17.81	17.08	15.86	14.55	9.870	—	—	—	—	—
	19.99^†	19.05^†	17.90^†	16.40^†	16.09^†	15.39^†	14.22^†	13.17^†	9.237^†
	15.38^‡	14.66^‡	13.77^‡	12.59^‡	12.51^‡	11.89^‡	10.74^‡	10.35^‡	7.797^‡
1st divergence load,	2.467	2.830	3.325	4.055	4.257	4.626	5.292	6.099	9.870	—	—	—	—	—
	2.437^†	2.795^†	3.286^†	4.012^†	4.174^†	4.560^†	5.263^†	5.952^†	9.237^†
	2.351^‡	2.697^‡	3.174^‡	3.891^‡	3.947^‡	4.361^‡	5.204^‡	5.509^‡	7.797^‡

The unmarked, , and marked values denote critical loads corresponding to , 0.005, and 0.02, respectively.