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

Table 2

Influence of taper ratio (𝛼=𝑑_ℓ/𝐿_ℓ) on the lowest five natural frequencies 𝜔_𝜏 (rad/sec) of the P-taper conic shaft (cf. Figure 5) with larger-end diameter 𝑑_ℓ=0.05 m and shaft length 𝐿_shaft=𝐿=2.0 m kept unchanged: (a) F-F, (b) C-C, (c) C-F, and (d) F-C BCs.

(a)

Case	Taper ratios 𝛼	Method	Natural frequencies, 𝜔𝜏 (rad/sec)
			𝜔1	𝜔2	𝜔3	𝜔4	𝜔5
1	0.02	Exact^#	7384.4129	11854.5424	16427.4469	21133.5082	25931.2855
		FEM*	7382.5338	11854.2819	16438.2035	21170.1886	26013.7379
2	0.01	Exact	5374.8958	10198.6873	15134.0131	20100.9780	25081.0046
		FEM	5375.3828	10204.5966	15155.2176	20152.2975	25182.2201
3	0.005	Exact	5074.8956	10037.9356	15025.4133	20019.1472	25015.3982
		FEM	5075.6554	10044.3910	15047.4403	20071.5715	25118.0086
4	0.0025	Exact	5016.8720	10008.5200	15005.7524	20004.3882	25003.5861
		FEM	5017.6806	10015.0732	15027.9268	20057.0107	25106.4467
5	0.001	Exact	5002.5736	10001.3500	15000.9698	20000.8006	25000.7158
		FEM	5003.3940	10007.9269	15023.1800	20053.4711	25103.6372
Uniform shaft**			5000.0416	10000.0832	15000.1248	20000.1664	25000.2080

#Natural frequencies obtained from presented exact method using single shaft segment (𝑃=1).
∗Natural frequencies obtained from finite element method using 50 shaft elements (𝑛_𝑒=50).
∗∗The exact natural frequencies of uniform shaft obtained from formulas in the appendix.

(b)

Case	Taper ratios; 𝛼	Method	Natural frequencies, 𝜔𝜏 (rad/sec)
			𝜔1	𝜔2	𝜔3	𝜔4	𝜔5
1	0.02	Exact^#	5966.9661	10667.3679	15490.2119	20382.6984	25312.3748
		FEM*	5966.2987	10670.9583	15508.0439	20429.6221	25408.1553
2	0.01	Exact	5128.1859	10066.7014	15044.8782	20033.8229	25027.1676
		FEM	5128.8914	10073.0559	15066.7556	20086.0475	25129.5265
3	0.005	Exact	5025.1181	10012.7167	15008.5594	20006.4959	25005.2732
		FEM	5025.9195	10019.2558	15030.7127	20059.0900	25108.0980
4	0.0025	Exact	5005.6582	10002.8966	15002.0014	20001.5744	25001.3349
		FEM	5006.4761	10009.4684	15024.2039	20054.2346	25104.2431
5	0.001	Exact	5005.6582	10002.8966	15002.0014	20001.5744	25001.3349
		FEM	5001.7077	10007.0855	15022.6214	20053.0546	25103.3066
Uniform shaft**			5000.0416	10000.0832	15000.1248	20000.1664	25000.2080

(c)

Case	Taper ratios; 𝛼	Method	Natural frequencies, 𝜔𝜏 (rad/sec)
			𝜔1	𝜔2	𝜔3	𝜔4	𝜔5
1	0.02	Exact^#	447.0176	7811.0650	12760.2959	17706.1543	22667.7637
		FEM*	447.9340	7812.6640	12771.1535	17738.7421	22739.4090
2	0.01	Exact	1579.8439	7313.7896	12390.7453	17422.4856	22439.9285
		FEM	1580.1051	7316.8955	12404.1570	17458.5569	22515.9715
3	0.005	Exact	2068.4024	7379.8464	12428.7547	17449.3316	22460.7130
		FEM	2068.5822	7382.8417	12441.9769	17485.1338	22536.4057
4	0.0025	Exact	2290.9387	7435.7082	12461.7079	17472.7621	22478.9027
		FEM	2291.0813	7438.6027	12474.7612	17508.3261	22554.2862
5	0.001	Exact	2417.9471	7473.5121	12484.2105	17488.8006	22491.3660
		FEM	2418.0663	7476.3373	12497.1478	17524.2010	22566.5373
Uniform shaft**			2500.0208	7500.0624	12500.1040	17500.1456	22500.1872

(d)

Case	Taper ratios; 𝛼	Method	Natural frequencies, 𝜔𝜏 (rad/sec)
			𝜔1	𝜔2	𝜔3	𝜔4	𝜔5
1	0.02	Exact^#	5735.5766	9980.7807	14380.7452	18967.7823	23687.7230
		FEM*	5733.4631	9977.5119	14382.5800	18986.8702	23741.3914
2	0.01	Exact	3610.0689	8006.2117	12816.3662	17728.7711	22678.9066
		FEM	3609.7962	8007.9786	12827.5397	17761.6872	22750.8551
3	0.005	Exact	2969.4822	7681.4675	12610.5906	17579.4008	22561.9393
		FEM	2969.4664	7683.8987	12622.8685	17613.8717	22635.9044
4	0.0025	Exact	2717.6065	7578.0625	12547.2117	17533.8559	22526.4265
		FEM	2717.6603	7580.6924	12559.8220	17568.7953	22600.9996
5	0.001	Exact	2583.3737	7528.6619	12517.3057	17512.4412	22509.7535
		FEM	2583.4588	7531.3848	12530.0716	17547.6000	22584.6113
Uniform shaft**			2500.0208	7500.0624	12500.1040	17500.1456	22500.1872