Torsional Vibrations of a Conic Shaft with Opposite Tapers Carrying Arbitrary Concentrated Elements

Table 4

Influence of rigid disks (each with rotary inertia 𝐽_𝑖̂=𝐽^∗/3) or/and torsional springs (each with stiffness 𝑘_𝑡,𝑖=𝑘^∗_𝑡/3) on the lowest five natural frequencies of the P-taper conic shaft (cf. Figures 6 and case 2 of Table 1) with reference polar mass moment of inertia 𝐽^∗=5.4409436902×10⁻³ kg-m² and reference torsional stiffness 𝑘^∗_𝑡=1.378230198×10⁴ Nm/rad.

Case	𝑁̂𝐽	𝑁𝑘𝑡	𝐽𝑖̂	𝑘𝑡,𝑖	Method	Natural frequencies, 𝜔𝜏 (rad/sec)
						𝜔0^#	𝜔1	𝜔2	𝜔3	𝜔4
1	3	0	13𝐽^∗	0	Exact	—	2475.3743	5026.7728	10924.0242	12894.1665
					FEM	—	2475.4996	5027.4548	10931.3000	12904.4532
2	0	3	0	13𝑘^∗𝑡	Exact	1684.6341	5932.6585	10654.8699	15394.2260	20355.8291
					FEM	1684.7292	5933.5926	10661.9274	15416.7018	20409.3205
3	3	3	13𝐽^∗	13𝑘^∗𝑡	Exact	1168.6767	2853.7625	5191.2778	10939.0990	12923.7224
					FEM	1168.6972	2853.8492	5191.9785	10946.3775	12934.0271
Bare F-F P-taper shaft*					Exact	—	5374.8958	10198.6873	15134.0131	20100.9780

∗The exact values for the lowest four natural frequencies of the “bare” free-free P-taper shaft taken from case 2 of Table 2(a).
^#Natural frequencies of quasi rigid-body modes.