Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 491062, 19 pages

http://dx.doi.org/10.1155/2013/491062

Research Article

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

Jia-Jang Wu

Department of Marine Engineering, National Kaohsiung Marine University, No. 482, Jhong-Jhou 3rd Road, Cijin District, Kaohsiung City 80543, Taiwan

Received 20 December 2012; Accepted 3 February 2013

Academic Editor: Chang-Hua Lien

Copyright © 2013 Jia-Jang Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to present the exact solution for free torsional vibrations of a linearly tapered circular shaft carrying a number of concentrated elements. First of all, the equation of motion for free torsional vibration of a conic shaft is transformed into a Bessel equation, and, based on which, the exact displacement function in terms of Bessel functions is obtained. Next, the equations for compatibility of deformations and equilibrium of torsional moments at each attaching point (including the shaft ends) between the concentrated elements and the conic shaft with positive and negative tapers are derived. From the last equations, a characteristic equation of the form is obtained. Then, the natural frequencies of the torsional shaft are determined from the determinant equation , and, corresponding to each natural frequency, the column vector for the integration constants, , is obtained from the equation . Substitution of the last integration constants into the associated displacement functions gives the corresponding mode shape of the entire conic shaft. To confirm the reliability of the presented theory, all numerical results obtained from the exact method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.

1. Introduction

In this paper, a conic shaft with its longitudinal (lateral) surface generated by revolving an inclined straight line about its longitudinal axis is called the general conic shaft and that generated by revolving an inclined curve about its longitudinal axis is called the specific conic shaft. The main difference between the last two conic shafts is that the variation of cross-section area is to take the form of for the general conic shaft and for the specific conic shaft. In the last expressions for , denotes the longitudinal coordinate of the conic shaft with its origin at the tip (left) end of the general conic shaft, denotes the cross-sectional area at with the subscript denoting the larger end of the shaft, while , , and denote constants (with ). For a general conic shaft, the exact solution for the natural frequencies and mode shapes of free transverse vibrations has been presented in [1], while for a specific conic shaft, that of free longitudinal vibration has been reported in [2, 3], and that of free torsional vibration in [4]. However, for a general conic shaft, the information concerning its exact solution for the natural frequencies and mode shapes of free torsional vibration is not yet obtainable from the existing literature. In [2–4], the exact solution for free longitudinal or free torsional vibration of a specific conic rod is obtained by using appropriate transformations to reduce its equation of motion to the analytically solvable standard form with the specific area variation . In [5], the numerical solution for the lowest several natural frequencies of the free-free specific conic shafts is determined from the so-called target function method. Because the exact solution for the natural frequencies and mode shapes of free torsional vibration of the most practical general conic shafts is not yet presented, this paper tries to provide some information for this topic.

For convenience, in this paper, a linearly tapered beam is called the positive-taper conic shaft if its cross-sectional diameter increases with increasing the longitudinal coordinates and is called the negative-taper conic shaft if its diameter decreases with increasing . Furthermore, a conic shaft consisting of one positive-taper conic shaft segment only is called the P-taper shaft (cf. Figures 1 and 2) and that composed of both a positive-taper and a negative-taper conic shaft segments is called the PN-taper shaft (cf. Figure 3). The exact natural frequencies and mode shapes for the free torsional vibrations of a P-taper shaft with or without carrying any number of concentrated elements (such as rigid disks and/or torsional springs) are determined through the following procedures: (a) to transform the equation of motion of the P-taper shaft into the Bessel equation and to obtain the exact solution for the (torsional) angular displacements in terms of the Bessel functions. (b) To establish the equations of compatibility for deformations and those of equilibrium for torsional moments at a typical intermediate node connecting the conic shaft segments and , and those at the two ends of the entire conic shaft. Based on the last equations and the prescribed boundary conditions, corresponding to each of the trial natural frequencies, a characteristic equation for the entire conic shaft, , is obtained, in which, is a column vector composed of the integration constants of all conic shaft segments and [H] is a square matrix composed of the associated coefficients. (c) To determine the natural frequency of the vibrating system from the determinant equation by using the bisectional method [6] and to obtain the associated integration constants from the equation . The substitution of the latter integration constants into the associated displacement function for each of the conic shaft segments will determine the corresponding mode shape. The last steps (b) and (c) must be repeated cycles if pairs of natural frequency and associated mode shape are required. It is found that the foregoing procedures for obtaining the exact solution of free torsional vibrations of a P-taper shaft are also available for that of a PN-taper shaft carrying arbitrary concentrated elements. The key point is to replace the longitudinal (axial) coordinate for the positive-taper part of the PN-taper shaft by for the negative-taper part of the PN-taper shaft (cf. Figure 3) and the parameters (, , , and ) for the positive-taper part by those (, , , and ) for the negative-taper part, where and denote the diameters at smaller and larger ends of the positive-taper part, and denote the corresponding ones of the negative-taper part, and denote the coordinates of smaller and larger ends of the positive-taper part with origin at the tip of the complete positive-taper cone, while and denote those of the negative-taper part with origin at the tip of the complete negative-taper cone (cf. Figure 3).

Figure 1: The coordinate system for a P-taper conic shaft with diameter at its larger end, diameter at its smaller end, and length .

Figure 2: A free-free P-taper conic shaft composed of conic shaft segments (denoted by ) and carrying one rigid disk with polar mass moment of inertia and one torsional spring with stiffness at each node (.

Figure 3: The coordinate system for PN-taper conic shaft () composed of both a positive-taper part and a negative-taper part .

To confirm the reliability of the presented exact method, the influence of taper ratios () of the P-taper and PN-taper shafts on their lowest four natural frequencies in various boundary conditions (BCs) is studied. It is found that the lowest four natural frequencies of all conic shafts converge to the corresponding ones of the associated uniform shafts in various BCs when the taper ratios of the conic shafts approach zero (i.e., ). Furthermore, all numerical results obtained from the presented exact method are also compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved. For convenience, a conic shaft without any attachments is called the bare shaft and the one carrying any concentrated elements is called the loaded shaft in this paper.

2. Bessel Equation for the Torsional Vibration of a P-Taper Conic Shaft

For a nonuniform shaft performing free torsional vibrations, its equation of motion takes the form [7]:

where and are mass density and shear modulus of the shaft material, respectively, is the polar moment of inertia for the cross-sectional area of the shaft at position , and is the torsional angle of the shaft at position and time . For the P-taper conic shaft as shown in Figure 1, is the axial coordinate with its origin at the tip end of the complete conic shaft. It is evident that the revolution of the inclined straight line about the horizontal -axis will generate the longitudinal (lateral) surface of the shaft.

If denotes the diameter of the larger end of the truncated conic shaft (cf. Figure 1), then the diameter for the cross-section area located at position is given by

where is the total length of the complete conic shaft, and the polar moment of inertia of the cross-sectional area is given by

where is the polar moment of inertia for the cross-sectional area at larger end of the conic shaft given by

For free vibrations, one has

where denotes the amplitude of , and denotes the angular natural frequency of conic shaft and .

Substitution of (3) and (5) into (1) gives

where

Equation (6) is a Bessel equation with its solution composed of the Bessel functions.

3. Displacement Function for the Conic Shaft

From [8–10], it can be found that the solution for the next differential equation:

is given by

where and are 1st and 2nd kind Bessel functions of order , and the other parameters are given by

Comparing (8) with (6), one finds that

Substituting (11) into (10a), (10b), and (10c), one obtains,

Substituting the last parameters into (9) and using the relationship [8], one has

where

In (14), (15a), and (15b), the subscript (or superscript) refers to the th vibrating mode of the conic shaft, while and denote the two corresponding integration constants determined by the associated boundary conditions of the entire conic shaft.

4. Boundary Conditions

Figure 2 shows a free-free P-taper conic shaft composed of conic shaft segments (denoted by ) and carrying one rigid disk with polar mass moment of inertia and one torsional spring with stiffness at each node (. The compatibility of twisting angles and equilibrium of torques at the arbitrary intermediate node located at require that

where

The boundary condition for the left end of the entire conic shaft is given by

where

Similarly, the boundary condition for the right end of the entire conic shaft is

where

Equations (18) and (20) are the nonclassical boundary conditions. The general classical boundary conditions (without any attachments at each end of the shaft) are given by

In (22a)–(25b), the capital letters and denote the free and clamped ends of the entire shaft, respectively. Besides, the symbols and denote the twisting angle and its first derivative of the first shaft segment at (left end) node (cf. Figure 2), respectively. Similarly, and denote those of the final (th) shaft segment at (right end) node , respectively.

5. Determination of Exact Natural Frequencies and Mode Shapes

From (14), the displacement function of the th conic shaft segment takes the form

Therefore,

in which [11]

Now, the exact natural frequencies and the associated mode shapes of a conic shaft carrying arbitrary number of concentrated elements (including rigid disks and/or torsional springs) with various boundary conditions can be determined as follows.

5.1. Free-Free P-Taper Conic Shaft

For the free-free P-taper conic shaft shown in Figure 2, the substitutions of (26a) and (26b) into (18) lead to

where

Similarly, substituting (26a) and (26b) into (16a) and (16b), respectively, one obtains

where

Finally, the substitution of (26a) and (26b) into (20) leads to

where

Based on (30), (32), (33), and (35), one obtains a characteristic equation:

where is a column vector composed of integration constants for the th mode shape of the th rod segments, and (), that is,

and is a (with ) square matrix with its nonzero coefficients given by

For the case of a P-taper shaft consisting of one shaft segment (i.e., ), only (39a), (39b), (42a), and (42b) are required for the determination of natural frequencies and associated mode shapes, and (40a)–(40d) and (41a)–(41d) are further required for the case of total number of shaft segments being greater than 1 (i.e., ) with being less than or equal to (i.e., ).

Nontrivial solution of (37) requires that

Equation (43) is the frequency equation for the free-free P-taper shaft with each node carrying one rigid disk and one torsional spring (). From (43), one may find the natural frequencies of the vibrating system, (), and then, with respect to each natural frequency , one may determine the values of and () from (37). Finally, the substitution of the last integration constants into (26a) will determine the corresponding natural mode shape of the entire conic shaft . Since the values of rotary inertia and rotational stiffness are arbitrary including zero, the foregoing formulation is available for arbitrary cases of the free-free loaded conic shaft including the bare one.

For the special case of a free-free conic shaft consisting of only one shaft segment (i.e., ), (37) reduces to

where

In (45b), the coefficients and are the same as those given by (39a) and (39b), while those and may be obtained from (42a) and (42b) by letting and . The results are

where

5.2. Clamped-Clamped P-Taper Conic Shaft

If both left and right ends of the P-taper conic shaft shown in Figure 2 are clamped, then the effects of all concentrated elements at the last two ends are nil. In such a case, the foregoing nonclassical boundary conditions of the conic shaft are the same as the C-C classical ones given by (23a) and (23b). Substitution of (26a) into (23a) and (23b), respectively, yields

Therefore, the coefficients relating to the boundary conditions, given by (39a), (39b), (42a), and (42b), must be, respectively, replaced by

5.3. Clamped-Free (or Free-Clamped) P-Taper Conic Shaft

The formulation presented in Section 5.1 is also available for the free vibration analysis of the clamped-free (C-F) and free-clamped (F-C) P-taper conic shaft: the BCs given by (49a), (49b), (42a), and (42b) are required for a C-F shaft, and the BCs given by (39a), (39b), (50a), and (50b) are required for a F-C shaft. It is noted that, for the classical boundary conditions, and .

6. Formulation for a PN-Taper Conic Shaft

Figure 3 shows the coordinate system of a conic shaft composed of a shaft segment with positive taper and a shaft segment with negative taper, which is called PN-taper conic shaft in this paper. The key parameters for the positive-taper part , , , and , are the same as those shown in Figures 1 or 2 for the P-taper conic shaft, and the corresponding ones of the negative-taper part are represented by , , , and , respectively. Since the origin for the coordinate system of the negative-taper part is located at , the relationship between the axial coordinates and is given by

If the intermediate node is located at the junction of positive-taper part and negative-taper part of the PN-taper conic shaft , that is, , then (16a), (16b), (33), (41c), and (41d) must be, respectively, replaced by

where

Furthermore, if the intermediate node is located at the negative-taper part of the PN-taper conic shaft , that is, , then (34) and (36) must be, respectively, replaced by

The negative signs (−) in the right hand sides of (52b), (56), and (57) are due to the fact that the sign for the first derivative of the negative-taper part of the PN-taper shaft is opposite to that of the positive-taper part (cf. Figure 3), that is, according to (51).

It is noted that the formulation for the P-taper conic shaft presented in the last section is available for the positive-taper part of the current PN-taper conic shaft , and so is for the negative-taper part , if one obtains the axial coordinate from (51) and sets

for the intermediate node located in the negative-taper part , that is, for the case of .

7. Free Torsional Vibration Analysis of a Conic Shaft by FEM

In order to use the conventional FEM to tackle the examined problem, the conic shaft such as that shown in Figure 3 must be replaced by a stepped one composed of uniform circular shaft elements shown in Figure 4(a). The average diameter of the th uniform shaft segment, , is determined in the following.

(a)

(b)

Figure 4: (a) The PN-taper conic shaft () is replaced by uniform circular shaft elements. (b) The average diameter () for the th uniform circular shaft element in the positive-taper part is determined by .

From Figure 4(b), one sees that the diameters for the cross-sections of the original conic shaft located at the two ends of the th uniform rod element, and , are given by

Based on the requirement that the mass of the th uniform shaft element (with average diameter ) is equal to that of its original th conic shaft element, one obtains

From the last expression, one obtains average diameter of the th uniform shaft element

where

Therefore, the average polar moment of inertia for the cross-sectional area of the th uniform shaft element is given by

and the corresponding polar mass moment of inertia of the th uniform shaft element is determined by

Equations (59a)–(61) are for the case of the th shaft element being located in the positive-taper part of the PN-taper conic shaft . If the th shaft element is located in the negative-taper part , then (59a)–(61) must be, respectively, replaced by

In the last expressions, the values of and are determined by (51).

Based on the foregoing information for the th uniform rod element () and shear modulus of the shaft material, one may obtain the mass matrix and stiffness matrix of each uniform shaft element from [12]:

Assembly of the elemental mass and stiffness matrices for each of the uniform shaft elements yields the overall mass matrix and overall stiffness matrix of the entire conic shaft. If there exist a rigid disk with rotary inertia and a torsional spring with stiffness at , then and must be added to the th diagonal coefficient of the overall mass matrix and that of the overall stiffness matrix , respectively. Finally, imposing the specified boundary conditions of the entire conic shaft and solving the resulting characteristic equation, one will determine the natural frequencies and the corresponding mode shapes of the entire conic shaft.

8. Numerical Results and Discussions

Except Tables 2 and 3, all numerical results for the P-taper conic shaft (cf. Figures 1 or 2) in this paper are based on the following data (cf. case 2 of Table 1): mass density  kg/m³, Young’s modulus 2.068 × 10¹¹ N/m², shear modulus 0.7953846 × 10¹¹ N/m², Poison’s ratio , smaller-end diameter  m, larger-end diameter  m, and distance from origin to the smaller end  m and that to the larger end  m. The parameters for the negative-taper part of the PN-taper shaft (cf. Figure 3) are the same as the corresponding ones of positive-taper part , that is, , , , and . For convenience, two reference parameters are introduced:

Table 1: The dimensions of a general conic shaft with five taper ratios , larger-end diameter  m, and shaft length  m kept unchanged.

Case
Taper ratio

Complete shaft length

(m)
Length truncated

(m)
Smaller-end diameter

(m)
Shaft length

(m)
Larger-end diameter

(m)

1
0.02
2.5
0.5
0.01

2
0.01
5.0
3.0
0.03

3
0.005
10.0
8.0
0.04
2.0
0.05

4
0.0025
20.0
18.0
0.045

5
0.001
50.0
48.0
0.048

Uniform rod
—
—
0.05
2.0
0.05

Table 2: Influence of taper ratio () on the lowest five natural frequencies (rad/sec) of the P-taper conic shaft (cf. Figure 5) with larger-end diameter  m and shaft length  m kept unchanged: (a) F-F, (b) C-C, (c) C-F, and (d) F-C BCs.

(a)

Case
Taper ratios
Method

Natural frequencies, (rad/sec)

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 ().

Natural frequencies obtained from finite element method using 50 shaft elements ().

The exact natural frequencies of uniform shaft obtained from formulas in the appendix.

(b)

Case
Taper ratios

Method

Natural frequencies, (rad/sec)

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

Method

Natural frequencies, (rad/sec)

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

Method

Natural frequencies, (rad/sec)

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

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)

Case
Taper ratios

Method

Natural frequencies, (rad/sec)

1
0.02
Exact#
5735.5766
7384.4129
9980.7807
11854.5424
14380.7452

FEM*
5733.4631
7382.5338
9977.5119
11854.2819
14382.5800

2
0.01
Exact
3610.0689
5374.8958
8006.2117
10198.6873
12816.3662

FEM
3609.7962
5375.3828
8007.9786
10204.5966
12827.5397

3
0.005
Exact
2969.4822
5074.8956
7681.4675
10037.9356
12610.5906

FEM
2969.4664
5075.6554
7683.8987
10044.3910
12622.8685

4
0.0025
Exact
2717.6065
5016.8720
7578.0625
10008.5200
12547.2117

FEM
2717.6603
5017.6806
7580.6924
10015.0732
12559.8220

5
0.001
Exact
2583.3737
5002.5736
7528.6619
10001.3500
12517.3057

FEM
2583.4588
5003.3940
7531.3848
10007.9269
12530.0716

Uniform shaft**
2500.0208
5000.0416
7500.0624
10000.0832
12500.1040

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)

Case
Taper ratios

Method

Natural frequencies, (rad/sec)

1
0.02
Exact#
447.0176
5966.9661
7811.0650
10667.3679
12760.2959

FEM*
447.9340
5966.2987
7812.6640
10670.9583
12771.1535

2
0.01
Exact
1579.8439
5128.1859
7313.7896
10066.7014
12390.7453

FEM
1580.1051
5128.8914
7316.8955
10073.0559
12404.1570

3
0.005
Exact
2068.4024
5025.1181
7379.8464
10012.7167
12428.7547

FEM
2068.5822
5025.9195
7382.8417
10019.2558
12441.9769

4
0.0025
Exact
2290.9387
5005.6582
7435.7082
10002.8966
12461.7079

FEM
2291.0813
5006.4761
7438.6027
10009.4684
12474.7612

5
0.001
Exact
2417.9471
5000.8859
7473.5121
10000.5058
12484.2105

FEM
2418.0663
5001.7077
7476.3373
10007.0855
12497.1478

Uniform shaft**
2500.0208
5000.0416
7500.0624
10000.0832
12500.1040

(c)

Case
Taper ratios

Method

Natural frequencies, (rad/sec)

1
0.02
Exact#
314.6200
5841.7915
7596.9631
10286.6906
12314.5569

FEM*
315.2569
5840.2785
7596.7480
10286.1690
12319.6401

2
0.01
Exact
1016.5493
4182.8341
6408.1802
8927.6668
11352.3196

FEM
1016.6576
4182.8740
6409.6737
8931.5092
11361.3636

3
0.005
Exact
1197.3804
3842.8258
6280.1632
8784.1136
11269.3945

FEM
1197.4156
3843.1091
6281.7514
8788.4149
11278.7052

4
0.0025
Exact
1237.8483
3771.2365
6256.7605
8757.6678
11254.3883

FEM
1237.8664
3771.5690
6258.3630
8762.0525
11263.7458

5
0.001
Exact
1248.1680
3753.2346
6251.0591
8751.2135
11250.7384

FEM
1248.1817
3753.5794
6252.6651
8755.6185
11260.1073

Uniform shaft**
1250.0104
3750.0312
6250.0520
8750.0728
11250.0936

where denotes the reference polar mass moment of inertia for rigid disks and denotes the reference stiffness for torsional springs, denotes the length of the entire shaft and is given by for the P-taper shaft and for PN-taper shaft; besides, denotes the average polar moment of inertia for the entire taper shaft, that is,

where and denote the polar moment of inertia of the smaller-end area and that of the larger-end area of the P-taper shaft (or the positive-taper part of the PN-taper shaft ), respectively, while and denote the corresponding ones of the negative-taper part of the PN-taper shaft. Based on the foregoing given data, one obtains the reference polar mass moment of inertia  kg-m² and reference stiffness  Nm/rad for the P-taper shaft. Because the dimensions of the positive-taper part of the PN-taper shaft are identical to those of the P-taper shaft and the dimensions of the negative-taper part of the PN-taper shaft are the same as the positive-taper part , it is evident that the reference polar mass moment of inertia  kgm² and reference stiffness  Nm/rad for the PN-taper shaft.

8.1. Influence of Taper Ratios on the P-Taper Shaft

In this paper, the taper ratio of the conic shaft is defined by , and the five cases of taper ratio and the associated parameters of the P-taper shaft are shown in Table 1. Furthermore, the sketch for the P-taper shaft with taper ratios , and 0.001 is shown in Figure 5 with diameter of the larger end ( m) and length of the shaft ( m) being kept unchanged. The influence of taper ratio on the lowest five natural frequencies of the P-taper shaft, ( to 5), is shown in Tables 2(a), 2(b), 2(c), and 2(d) for the F-F, C-C, C-F, and F-C BCs, respectively. In which, Exact refers to the exact method presented in this paper, FEM refers to the conventional finite element method and the natural frequencies of the uniform shaft listed in the final row of each table (and denoted by the bold-faced digits) are obtained from the formulas given in the appendix at the end of this paper. From Tables 2(a)–2(d), one finds that:

(i)For various BCs, the lowest five natural frequencies (, to 5) of the P-taper shaft obtained from either exact method (using single shaft segment, i.e., ) or FEM (using 50 shaft elements, i.e., ) converge to the corresponding ones of the uniform circular shaft obtained from the exact formulas given in the appendix, when the taper ratio decreases and approaches zero.(ii)For various BCs, the lowest five natural frequencies of the P-taper shaft obtained from the exact method (using ) are very close to those obtained from FEM (using ).(iii)For the F-F, C-C, or F-C BCs, the lowest five natural frequencies (, to 5) of the P-taper shaft decrease with the decrease of taper ratio as one may see from Tables 2(a), 2(b), and 2(d). This is a reasonable result because the influence of taper ratio on the supporting stiffness is very small for the above-mentioned BCs (with right-end clamped and diameter being kept unchanged) and the polar mass moment of inertia of the entire shaft increases with the decrease of taper ratio .(iv)For the C-F BCs (cf. Table 2(c)), the lowest two to five natural frequencies ( to ) of the P-taper shaft also decrease with the decrease of taper ratio . However, the first natural frequency increases significantly with the decrease of the taper ratio . This is also a reasonable result because, near the (left) clamped end, the increasing effect of supporting stiffness is much greater than that of polar mass moment of inertia due to the decrease of taper ratio .

Figure 5: The profiles for longitudinal cross-sections of the conic shaft with five different taper ratios (cf. Table 1).

Based on the foregoing reasonable results, it is believed that the theory and computer programs developed for the presented exact method should be reliable. It is worthy of mention that because the exact solution of the current research is determined based on the bar theory [7], the beam element theory [12] is used in FEM. The fundamental theory of solid element is different from the bar theory. Therefore, the solid element is not adopted in FEM.

8.2. Influence of Taper Ratios on the PN-Taper Shaft

For the PN-taper shaft as shown in Figures 3 or 4(a), if the dimensions of its positive-taper part are the same as those of the P-taper shaft (cf. Figure 5) studied in the last (Section 8.1), and the dimensions of the negative-taper part are identical to the corresponding ones of the positive-taper part , then the influence of taper ratio on the lowest five natural frequencies (, to 5) of the PN-taper shaft is shown in Tables 3(a), 3(b), and 3(c) for the F-F, C-C, and C-F (or F-C) BCs, respectively. From Table 3, one sees that:

(i)for various BCs, the lowest five natural frequencies (, 1 to 5) of the PN-taper shaft obtained from either exact method (using two shaft segments, i.e., ) or FEM (using 100 shaft elements, i.e., ) converge to the corresponding ones of the uniform circular shaft (cf. bold-faced digits in the final row of each table) obtained from the exact formulas given in the appendix, when the taper ratio decreases and approaches zero.(ii)For various BCs, the lowest five natural frequencies of the PN-taper shaft obtained from the exact method (using ) are very close to those obtained from FEM (using ).(iii)Because of symmetry in geometrical structures, the lowest five natural frequencies of the C-F PN-taper shaft are identical to those of F-C PN-taper shaft.(iv)Comparing Table 3(b) with Table 2(c), one sees that the 1st, 3rd, and 5th natural frequencies of the PN-taper shaft (with C-C BCs) are exactly equal to the 1st, 2nd, and 3rd natural frequencies of the P-taper shaft (with C-F BCs), respectively. This is because the dimensions and left-end boundary conditions of the positive-taper part of the PN-taper shaft are the same as those of the P-taper shaft, and the 1st, 3rd, and 5th mode shapes of the former are the same as the 1st, 2nd, and 3rd mode shapes of the latter, respectively.

In view of the foregoing reasonable results, one believes that the theory and computer programs developed for the PN-taper shaft should be reliable.

8.3. A Free-Free P-Taper Shaft Carrying Multiple Concentrated Elements

To show the ability of the presented exact method for obtaining the exact solution of free torsional vibration of the conic shaft carrying arbitrary concentrated elements, the lowest five natural frequencies and some associated mode shapes of a free-free P-taper shaft carrying 3 identical rigid disks and 3 identical torsional springs (cf. Figure 6) are determined in this subsection. The taper ratio of the P-taper shaft is . From case 2 of Table 1, one sees that the pertinent parameters for the current P-taper shaft are: diameter at smaller end  m, diameter at larger end 0.05 m, coordinate at smaller end  m, and coordinate at larger end  m. It is noted that the origin of the last coordinates is at the tip of the conic shaft (cf. Figure 6). In Figure 6(a), the P-taper shaft carries 3 identical rigid disks, where the polar mass moment of inertia of each disk is given by () with  kg-m² denoting the reference polar mass moment of inertia mentioned at the beginning of the current section. In Figure 6(b), the P-taper shaft carries 3 identical torsional springs, where the stiffness of each torsional spring is given by () with  Nm/rad denoting the reference stiffness mentioned at the beginning of the current section. In Figure 6(c), the 3 identical rigid disks shown in Figure 6(a) and the 3 identical torsional springs shown in Figure 6(b) are attached to the same P-taper shaft. The locations of the concentrated elements (rigid disks or torsional springs or both rigid disks and torsional springs) are  m,  m, and  m, as one may see from Figure 6. For convenience, in Table 4, the loaded P-taper shaft shown in Figures 6(a), 6(b), and 6(c) are called cases 1, 2, and 3, respectively. Besides, the total number of rigid disks is denoted by and that of torsional springs by in the same table.

Table 4: Influence of rigid disks (each with rotary inertia ) or/and torsional springs (each with stiffness ) 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  kg-m² and reference torsional stiffness  Nm/rad.

Case

Method
Natural frequencies, (rad/sec)

1
3
0

0
Exact
—
2475.3743
5026.7728
10924.0242
12894.1665

FEM
—
2475.4996
5027.4548
10931.3000
12904.4532

2
0
3
0

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

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.

(a)

(b)

(c)

Figure 6: The P-taper shaft (cf. case 2 of Table 1) carrying: (a) 3 rigid disks each with (); (b) 3 torsional springs each with (); and (c) 3 rigid disks and 3 torsional springs each with and (). The locations of concentrated elements are  m,  m, and  m. Besides,  kg-m² and  Nm/rad.

Comparing with the lowest four natural frequencies of the bare free-free P-taper shaft (without carrying any concentrated elements) given by case 2 of Table 2(a) and listed in the final row of Table 4 (denoted by the bold-faced digits), one sees that the 3 identical rigid disks, each with () as shown in Figure 6(a), reduce the lowest four natural frequencies of the P-taper shaft (cf. case 1 in Table 4). On the contrary, the 3 identical torsional springs, each with () as shown in Figure 6(b), raise the lowest four natural frequencies of the P-taper shaft (cf. case 2 in Table 4). When the last concentrated elements are simultaneously attached to the P-taper shaft as shown in Figure 6(c), it is under expectation that the numerical values of the lowest four natural frequencies of the shaft have the next relationship: , where , , and represent the th natural frequencies of the P-taper shafts shown in Figures 6(a), 6(b), and 6(c), respectively.

It is noted that, for a free-free shaft supported by torsional springs such as that shown in Figures 6(b) or 6(c), there exists a quasi rigid-body natural frequency as one may see from Table 4. The quasi straight lines (—— and - -- -) appearing in the upper side of Figure 7(b) are the mode shapes corresponding to the above-mentioned frequencies for the shaft shown in Figure 6(c), obtained from the presented exact method and the conventional FEM, respectively. For a true rigid shaft, its natural frequency is given by ; thus, the true rigid body natural frequency for the P-taper shaft shown in Figure 6(b) is given by  rad/sec, and that shown in Figure 6(c) by  rad/sec, where is the summation of polar mass moment of inertia of the entire shaft itself () and that of the 3 rigid disks (). Comparing with cases 2 and 3 of Table 4, one sees that quasi rigid-body natural frequencies,  rad/sec and  rad/sec, are slightly greater than the foregoing corresponding true ones,  rad/sec and = 1125.4048 rad/sec, respectively.

(a)

(b)

Figure 7: The lowest five mode shapes of the free-free P-taper shaft with taper ratio (cf. case 2 of Table 1): (a) for the bare shaft (without any attachments as shown in Figure 1); (b) for the loaded shaft (carrying 3 rigid disks and 3 torsional springs as shown in Figure 6(c)).

The lowest five mode shapes (including the quasi rigid-body mode) of the “loaded” free-free P-taper shaft shown in Figure 6(c) are plotted in Figure 7(b). Comparing with the corresponding ones of the “bare” free-free P-taper shaft shown in Figure 7(a), one sees that the lowest four (elastic) mode shapes of the “loaded” shaft are much different from those of the “bare” shaft. From Table 4, one sees that all the natural frequencies obtained from FEM using 50 shaft elements (i.e., ) are very close to the corresponding ones obtained from the presented exact method using 2 shaft segments (i.e., ).

In Figure 7, the mode shapes obtained from the presented exact method are represented by the solid lines (—) and those obtained from FEM by the dashed lines (- - -). Furthermore, the 1st, 2nd, 3rd, and 4th mode shapes obtained from the exact method are denoted by the symbols •, +, ▲, and ■, respectively, while those from FEM by the symbols , ×, and □, respectively. Because all the natural frequencies obtained from FEM are very close to the corresponding ones obtained from the presented exact method as one may see from Table 4, so are the associated mode shapes obtained from the above two methods as shown in Figure 7(b).

8.4. A Free-Free PN-Taper Shaft Carrying Multiple Concentrated Elements

This subsection studies the free vibration characteristics of a free-free PN-taper shaft carrying 5 identical rigid disks and 5 identical torsional springs as shown in Figure 8 by using the presented exact method (with 4 shaft segments, i.e., ) and the conventional FEM (with 100 shaft elements, i.e., ). The parameters for the positive-taper part  of the current PN-taper shaft are the same as those of the P-taper shaft studied in the last subsection, and the parameters for the negative-taper part are identical to the positive-taper part , that is,  m, 0.05 m,  m, and  m. The polar mass moment of inertia for each of the 5 identical rigid disks is given by () with   denoting the reference polar mass moment of inertia mentioned at the beginning of this section and the stiffness for each of the torsional springs is given with  Nm/rad denoting the reference stiffness. From Figure 8, one sees that the entire PN-taper shaft is composed of four shaft segments and the length of each shaft segment is given by  m. Since the concentrated elements are located at the ends of the shaft segments, their coordinates with respect to the origin of the positive-taper shaft are given by  m,  m,  m,  m, and  m, respectively, as one may see from Figures 8(a)–8(c).

(a)

(b)

(c)

Figure 8: The free-free PN-taper conic shaft carrying: (a) 5 rigid disks each with (); (b) 5 torsional springs each with (); and (c) 5 rigid disks and 5 torsional springs each with and (). The locations of the concentrated elements are: , and 7.0 m, respectively. Besides,  kg-m² and  N-m/rad.

For convenience of comparison, the lowest four natural frequencies and the five mode shapes of the “bare” free-free PN-taper shaft (without carrying any concentrated elements) are shown in the final row of Table 5 (denoted by bold-faced digits) and Figure 9(a), respectively. Comparing Table 5 for the current free-free PN-taper shaft with Table 4 for the free-free P-taper shaft, one sees that the conclusions drawn from Table 4 are also correct for Table 5. However, there exists a significant difference between Figures 9(b) and 7(b) that the lowest five mode shapes (including the quasi rigid-body mode) of the loaded PN-taper shaft shown in Figure 9(b) are symmetrical (or antisymmetrical) with respect to its middle cross-section due to the fact that both the structural dimensions and the loading conditions are symmetrical with respect to its middle cross-section, but this is not true for those of the loaded P-taper shaft shown in Figure 7(b). Of course, the lowest five mode shapes of the bare PN-taper shaft as shown in Figure 9(a) are also symmetrical (or antisymmetrical) with respect to its middle cross-section due to symmetry of the structure. The legends of Figure 9 are the same as those of Figure 7. It is seen that all mode shapes obtained from FEM are very close to the corresponding ones obtained from the exact method as one may see from Figure 9(b). This is under expectation because the associated natural frequencies shown in Table 5 are very close to each other.

Table 5: Influence of rigid disks (each with rotary inertia ) or/and torsional springs (each with stiffness ) 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  kg-m²and reference stiffness of torsional springs N-m/rad.

Case

Method
Natural frequencies, (rad/sec)

1
5
0

0
Exact
—
1688.3916
2388.5998
3733.3469
5098.3850

FEM
—
1688.3959
2388.7487
3733.5282
5099.2798

2
0
5
0

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

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.

(a)

(b)

Figure 9: The lowest five mode shapes of the free-free PN-taper shaft with taper ratio (cf. case 2 of Table 1): (a) for the bare shaft (without any attachment, cf. Figure 3); (b) for the loaded shaft (carrying 5 rigid disks and 5 torsional springs as shown in Figure 8).

9. Conclusions

(1)Since the exact solution for free torsional vibrations of the general conic shafts is not yet obtainable from the existing literature, the presented exact method is significant in this aspect. Besides, the exact natural frequencies and associated mode shapes for free torsional vibrations of the general conic shafts carrying any number of rigid disks or/and torsional springs obtained from the presented exact method are also significant for evaluating the accuracy of various existing approximate methods concerned (such as FEM).(2)For a conic shaft composed of two shaft segments with opposite tapers, it is difficult to obtain its exact natural frequencies and associated mode shapes of free torsional vibrations. The simple theory presented in this paper is helpful for solving this kind of problem.(3)The formulation of this paper is very flexible because (i) the locations of the concentrated elements on the shaft are arbitrary; (ii) the total number of concentrated elements attached to the shaft is also arbitrary including zero; and (iii) each node may be attached by a rigid disk or a torsional spring or both a rigid disk and a torsional spring.(4)For a P-taper conic shaft with its (right) larger-end diameter and total length kept constant (cf. Figure 5), its lowest five natural frequencies (, to 5) in the F-F, C-C, or F-C BCs decrease with the decrease of its taper ratio because the polar mass moment of inertia of the entire shaft increases with the decrease of and the influence of on the supporting stiffness is much smaller. The last trend is also true for the 2nd to 4th natural frequencies ( to ) of the shaft with C-F BCs, but the 1st natural frequency () increases significantly with the decrease of taper ratio because near the (left) clamped end, the increasing effect of supporting stiffness is much greater than that of polar mass moment of inertia due to the decrease of taper ratio . (5)For a conic shaft (either P-taper or PN-taper) carrying arbitrary rigid disks and the same number of torsional springs, if the locations of the rigid disks are identical to the corresponding torsional springs, then the lowest five natural frequencies of the shaft decrease due to carrying the rigid disks only and increase due to carrying the torsional springs only. However, the lowest five natural frequencies of the shaft carrying both the rigid disks and the torsional springs have the next relationship: , where , , and represent the th natural frequencies of the conic shaft carrying the rigid disks only, the torsional springs only, and both the rigid disks and the torsional springs, respectively.(6)For a conic shaft (either P-taper or PN-taper) carrying torsional springs or both rigid disks and torsional springs, there exists a quasi rigid-body natural frequency . The latter is slightly higher than the true rigid-body natural frequency defined by , where denotes the polar mass moment of inertia of the entire shaft itself, while and denote the summation of stiffness of all torsional springs and that of polar mass moment of inertia of all rigid disks attached to the shaft, respectively.

Appendix

Exact Natural Frequencies and Mode Shapes of a Uniform Shaft

The characteristic equation for free torsional vibration of a uniform shaft is given by [7]

where

The last expressions are similar to those of the same uniform shaft performing free longitudinal vibration [7]; thus, the exact solution for natural frequencies and mode shapes of free longitudinal vibration of a uniform shaft is available for those of free torsional vibration of the same shaft, if the Young’s modulus for longitudinal vibrations is replaced by the shear modulus for torsional vibrations. The results for various boundary conditions are listed in the following.

F-F Shaft and C-C Shaft. The exact natural frequencies for the torsional vibrations of a uniform shaft with free-free (F-F) boundary conditions are the same as those with clamped-clamped (C-C) ones. They are given by

However, the associated mode shapes are different from each other and given by

Since represents the rigid-body mode, the formulas given by (A.3) with are correct only for the F-F shaft.

(ii) C-F Shaft. The exact natural frequencies and mode shapes of a uniform shaft with clamped-free (C-F) boundary conditions are given by

(iii) F-C Shaft. The exact natural frequencies and mode shapes for a uniform shaft with free-clamped (F-C) boundary conditions are given by

It is noted that the frequency parameters given by (A.9) are the same as those given by (A.6), but the corresponding mode shapes defined by (A.11) are different from those defined by (A.8).

Acknowledgment

This work was supported by the National Science Council, the Republic of China, under Contract no. NSC 101-2221-E-022-002.

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