Mathematical Problems in Engineering

Research Article

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

Table 3

Influence of taper ratio ( ) on the lowest five natural frequencies (rad/sec) of the PN-taper shaft (cf. Figure 3 or Figure 4(a)) with dimensions of its negative-taper part identical to the corresponding ones of its positive-taper part , and the larger-end diameter  m and total shaft length  m kept unchanged: (a) F-F, (b) C-C, and (c) C-F (or F-C) BCs.
(a)
CaseTaper ratios
Method
Natural frequencies, (rad/sec)
10.02Exact#5735.57667384.41299980.780711854.542414380.7452
FEM*5733.46317382.53389977.511911854.281914382.5800
20.01Exact3610.06895374.89588006.211710198.687312816.3662
FEM3609.79625375.38288007.978610204.596612827.5397
30.005Exact2969.48225074.89567681.467510037.935612610.5906
FEM2969.46645075.65547683.898710044.391012622.8685
40.0025Exact2717.60655016.87207578.062510008.520012547.2117
FEM2717.66035017.68067580.692410015.073212559.8220
50.001Exact2583.37375002.57367528.661910001.350012517.3057
FEM2583.45885003.39407531.384810007.926912530.0716
Uniform shaft**2500.02085000.04167500.062410000.083212500.1040
Natural frequencies obtained from presented exact method using two shaft segments ( ).
*Natural frequencies obtained from finite element method using 100 shaft elements ( ).
**The exact natural frequencies of uniform shaft obtained from formulas in the appendix.
(b)
CaseTaper ratios
Method
Natural frequencies, (rad/sec)
10.02Exact#447.01765966.96617811.065010667.367912760.2959
FEM*447.93405966.29877812.664010670.958312771.1535
20.01Exact1579.84395128.18597313.789610066.701412390.7453
FEM1580.10515128.89147316.895510073.055912404.1570
30.005Exact2068.40245025.11817379.846410012.716712428.7547
FEM2068.58225025.91957382.841710019.255812441.9769
40.0025Exact2290.93875005.65827435.708210002.896612461.7079
FEM2291.08135006.47617438.602710009.468412474.7612
50.001Exact2417.94715000.88597473.512110000.505812484.2105
FEM2418.06635001.70777476.337310007.085512497.1478
Uniform shaft**2500.02085000.04167500.062410000.083212500.1040
(c)
CaseTaper ratios
Method
Natural frequencies, (rad/sec)
10.02Exact#314.62005841.79157596.963110286.690612314.5569
FEM*315.25695840.27857596.748010286.169012319.6401
20.01Exact1016.54934182.83416408.18028927.666811352.3196
FEM1016.65764182.87406409.67378931.509211361.3636
30.005Exact1197.38043842.82586280.16328784.113611269.3945
FEM1197.41563843.10916281.75148788.414911278.7052
40.0025Exact1237.84833771.23656256.76058757.667811254.3883
FEM1237.86643771.56906258.36308762.052511263.7458
50.001Exact1248.16803753.23466251.05918751.213511250.7384
FEM1248.18173753.57946252.66518755.618511260.1073
Uniform shaft**1250.01043750.03126250.05208750.072811250.0936