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

Table 5

Influence of rigid disks (each with rotary inertia 𝐽_𝑖̂=𝐽^∗/5) or/and torsional springs (each with stiffness 𝑘_𝑡,𝑖=𝑘^∗_𝑡/5) on the lowest five natural frequencies of the free-free PN-taper conic shaft (cf. Figure 8 and case 2 of Table 1) with reference polar mass moment of inertia 𝐽^∗=1.0881887380×10⁻² kg-m²and reference stiffness of torsional springs 𝑘^∗_𝑡=6.891150988×10³N-m/rad.

Case	𝑁̂𝐽	𝑁𝑘𝑡	𝐽𝑖̂	𝑘𝑡,𝑖	Method	Natural frequencies, 𝜔𝜏 (rad/sec)
						𝜔0^#	𝜔1	𝜔2	𝜔3	𝜔4
1	5	0	15𝐽^∗	0	Exact	—	1688.3916	2388.5998	3733.3469	5098.3850
					FEM	—	1688.3959	2388.7487	3733.5282	5099.2798
2	0	5	0	15𝑘^∗𝑡	Exact	862.2360	3832.1008	5552.7121	8137.9636	10337.4211
					FEM	862.2638	3831.8884	5553.3272	8139.9447	10343.6571
3	5	5	15𝐽^∗	15𝑘^∗𝑡	Exact	587.7894	1836.1973	2489.3670	3791.2496	5136.1437
					FEM	587.7957	1836.1865	2489.5046	3791.4298	5137.0417
Bare F-F PN-taper shaft*					Exact	—	3610.0689	5374.8958	8006.2117	10198.6873

∗The exact values for the lowest four natural frequencies of the “bare” free-free PN-taper shaft taken from case 2 of Table 3(a).
^#Natural frequencies of quasi rigid-body modes.