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

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 [H]{C} = 0 is obtained.Then, the natural frequencies of the torsional shaft are determined from the determinant equation |H| = 0, and, corresponding to each natural frequency, the column vector for the integration constants, {C}, is obtained from the equation [H]{C} = 0. 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.


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 crosssection area () is to take the form of () =  ℓ (/ ℓ ) 2 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,  ℓ denotes the cross-sectional area at  =  ℓ with the subscript ℓ denoting the larger end of the shaft, while , , and  denote constants (with  ̸ = 0).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][3][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 negativetaper 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 ( + 1), 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, []{} = 0, 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 || = 0 by using the bisectional method [6] and to obtain the associated integration constants from the equation []{} = 0.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  =  ℓ +  ℓ −  for the negative-taper part of the PN-taper shaft (cf. Figure 3) and the parameters (  ,  ℓ ,   , and  ℓ ) for the positive-taper part by those (  ,  ℓ ,   , and  ℓ ) for the negative-taper part, where   and  ℓ denote the diameters at smaller and larger ends of the positive-taper part,   and  ℓ denote the corresponding ones of the negative-taper part,   and  ℓ 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  ℓ denote those of the negative-taper part with origin  at the tip of the complete negative-taper cone (cf. Figure 3).
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.,  ≈ 0).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.

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  ℓ 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  ℓ is the total length of the complete conic shaft, and the polar moment of inertia   () of the cross-sectional area is given by where  , ℓ is the polar moment of inertia for the crosssectional area at larger end of the conic shaft given by For free vibrations, one has where Θ() denotes the amplitude of (, ), and  denotes the angular natural frequency of conic shaft and  = √ −1.Substitution of (3) and ( 5) into (1) gives where Equation ( 6) is a Bessel equation with its solution composed of the Bessel functions.

Displacement Function for the Conic Shaft
From [8][9][10], it can be found that the solution for the next differential equation: is given by where  V () and  V () are 1st and 2nd kind Bessel functions of order V, 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  V/2 () ∝  −V/2 () [8], one has where In ( 14), (15a), and (15b), the subscript (or superscript)  refers to the th vibrating mode of the conic shaft, while  () 1 and  () 2 denote the two corresponding integration constants determined by the associated boundary conditions of the entire conic shaft.

Boundary Conditions
Figure 2 shows a free-free P-taper conic shaft composed of  conic shaft segments (denoted by (1), (2), . . ., (), . . ., ()) and carrying one rigid disk with polar mass moment of inertia Ĵ and one torsional spring with stiffness k, at each node  ( = 0, 1, 2, . . ., ).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 Θ ,1 ( ,0 ) = 0, (23a) In (22a)-(25b), the capital letters F and C denote the free and clamped ends of the entire shaft, respectively.Besides, the symbols Θ ,1 ( ,0 ) and Θ  ,1 ( ,0 ) denote the twisting angle and its first derivative of the first shaft segment at (left end) node  (cf. Figure 2), respectively.Similarly, Θ , ( , ) and Θ  , ( , ) denote those of the final (th) shaft segment at (right end) node , respectively.

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.
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 Ĵ and one torsional spring k, ( = 0, 1, 2, . . ., ).From (43), one may find the natural frequencies of the vibrating system,   ( = 1, 2, 3, . ..), and then, with respect to each natural frequency   , one may determine the values of  () 1, and  () 2, ( = 1, 2, . . ., ) from (37).Finally, the substitution of the last integration constants into (26a) will determine the corresponding natural mode shape of the entire conic shaft Θ  ().Since the values of rotary inertia Ĵ and rotational stiffness k, 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.,  = 1), (37) reduces to where In (45b), the coefficients  1,1 and  1,2 are the same as those given by (39a) and (39b), while those  2,1 and  2,2 may be obtained from (42a) and (42b) by letting  = 1 and  = 2 = 2.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 , =  −3/2 ( , ) .

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 freeclamped (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, Ĵ0 = k,0 = 0 and Ĵ = k, = 0.

Formulation for a PN-Taper Conic Shaft
Figure 3    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  +1 are determined by (51).
Based on the foregoing information for the th uniform rod element ( = 1, 2, 3, . . .,   ) 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 Ĵ and a torsional spring with stiffness k, at  =   , then Ĵ and k, 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.

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  = 7850 kg/m 3 , Young's modulus  = 2.068 × 10 11 N/m 2 , shear modulus  = /[2(1 + V)] = 0.7953846 × 10 11 N/m 2 , Poison's ratio V = 0.3, smaller-end diameter   = 0.03 m, largerend diameter  ℓ = 0.05 m, and distance from origin  to the smaller end   = 3.0 m and that to the larger end  ℓ = 5.0 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,   =   ,  ℓ =  ℓ ,   =   , and  ℓ =  ℓ .For convenience, two reference parameters are introduced: where  * denotes the reference polar mass moment of inertia for rigid disks and  *  denotes the reference stiffness for torsional springs,  shaft denotes the length of the entire shaft and is given by  shaft =  ℓ −   for the P-taper shaft and  shaft = ( ℓ +  ℓ ) − (  +   ) for PN-taper shaft; besides, Ĩ denotes the average polar moment of inertia for the entire taper shaft, that is,

Influence of Taper Ratios on the P-Taper Shaft.
In this paper, the taper ratio  of the conic shaft is defined by  =  ℓ / ℓ =   /  , 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  = 0.02, 0.01, 0.005, 0.0025, and 0.001 is shown in Figure 5 with diameter of the larger end ( ℓ = 0.05 m) and length of the shaft ( shaft =  = 2.0 m) being kept unchanged.The influence of taper ratio  on the lowest five natural frequencies of the P-taper shaft,   ( = 1 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 (  ,  = 1 to 5) of the P-taper shaft obtained from either exact method (using single shaft segment, i.e.,  = 1) or FEM (using 50 shaft elements, i.e.,   = 50) 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  = 1) are very close to those obtained from FEM (using   = 50).(iii) For the F-F, C-C, or F-C BCs, the lowest five natural frequencies (  ,  = 1 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 ( 2 to  5 ) of the P-taper shaft also decrease with the decrease of taper ratio .However, the first natural frequency  1 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 .
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.The profiles for longitudinal cross-sections of the conic shaft with five different taper ratios (cf.Table 1).

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 positivetaper part , then the influence of taper ratio  on the lowest five natural frequencies (  ,  = 1 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.,  = 2) or FEM (using 100 shaft elements, i.e.,   = 100) 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  = 2) are very close to those obtained from FEM (using   = 100).
(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 PNtaper 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.

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  =  ℓ / ℓ = 0.01.From case 2 of Table 1, one sees that the pertinent parameters for the current P-taper shaft are: diameter at smaller end   = 0.03 m, diameter at larger end  ℓ = 0.05 m, coordinate at smaller end  0 =   = 3.0 m, and coordinate at larger end   =  ℓ = 5.0 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 Ĵ =  * /3 ( = 0, 1, 2) with  * = 5.4409436902×10 −3 kg-m 2 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 k, =  *  /3 ( = 0, 1, 2) with  *  = 1.378230198 × 10 4 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  0 = 3.0 m,  1 = 4.0 m, and  2 = 5.0 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  Ĵ and that of torsional springs by  k in the same table.
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 Ĵ =  * /3 ( = 0, 1, 2) 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 k, =  *  /3 ( = 0, 1, 2) 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 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  0 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  0 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  0 = √  /; thus, the true rigid body natural frequency for the P-taper shaft shown in Figure 6 4, one sees that quasi rigid-body natural frequencies,  0,() = 1684.6341rad/sec and  0,() = 1168.6767rad/sec, are slightly greater than the foregoing corresponding true ones,  0,() = 1591.5627rad/sec and  0,() = 1125.4048rad/sec, respectively.
The lowest five mode shapes (including the quasi rigidbody 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.,   = 50) are very close to the corresponding ones obtained from the presented exact method using 2 shaft segments (i.e.,  = 2).
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 lowest 5 mode shapes of the free-free "P-taper" shaft carrying 3 rigid disks and 3 torsional springs 1st mode from "exact" method 1st mode from conventional FEM 2nd mode from "exact" method 2nd mode from conventional FEM 3rd mode from conventional FEM 4th mode from conventional FEM 3rd mode from "exact" method 4th mode from "exact" method Axial coordinates,  (m) Quasi rigid-body mode from "exact" method Quasi rigid-body mode from conventional FEM Modal twisting angles, Figure 7: The lowest five mode shapes of the free-free P-taper shaft with taper ratio  = 0.01 (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 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).

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.,  = 4) and the conventional FEM (with 100 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 boldfaced 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  The lowest 5 mode shapes of the free-free "PN-taper" shaft carrying 5 rigid disks and 5 torsional springs 1st mode from "exact" method 1st mode from conventional FEM 2nd mode from "exact" method 2nd mode from conventional FEM 3rd mode from conventional FEM 4th mode from conventional FEM 3rd mode from "exact" method 4th mode from "exact" method Quasi rigid-body mode from "exact" method Quasi rigid-body mode from conventional FEM Modal twisting angles, Figure 9: The lowest five mode shapes of the free-free PN-taper shaft with taper ratio  = 0.01 (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).
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.

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) largerend diameter  ℓ and total length  kept constant (cf. Figure 5), its lowest five natural frequencies (  ,  = 1 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 ( 2 to  4 ) of the shaft with C-F BCs, but the 1st natural frequency ( 1 ) 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  0 .The latter is slightly higher than the true rigid-body natural frequency defined by  0 = √ ∑ k, /( shaft + ∑ Ĵ ), where  shaft denotes the polar mass moment of inertia of the entire shaft itself, while ∑ k, and ∑ Ĵ 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.
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  ℓ = 0.05 m and total shaft length  shaft =  = 4.0 m kept unchanged: (a) F-F, (b) C-C, and (c) C-F (or F-C) BCs.
Ĩ = ( , +  ,ℓ ) 2 (for the P-taper shaft) , (67a) Ĩ = ( , +  ,ℓ +  , +  ,ℓ ) 4 (for the PN-taper shaft) , (67b) where  , and  ,ℓ 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 PNtaper shaft ), respectively, while  , and  ,ℓ 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  * = 5.4409436902 × 10 −3 kg-m 2 and reference stiffness  *  = 1.378230198 × 10 4 N⋅m/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  * = 2 × 5.4409436902 × 10 −3 kg⋅m 2 and reference stiffness  *  = (1.378230198× 10 4 )/2 N⋅m/rad for the PN-taper shaft.

Figure 5 :
Figure5: The profiles for longitudinal cross-sections of the conic shaft with five different taper ratios (cf.Table1).
) 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

1
Axial coordinates,  (m)The lowest 5 mode shapes of the bare free-free "PN-taper" shaft obtained from "exact" Axial coordinates,  (m)

Table 1 :
The dimensions of a general conic shaft with five taper ratios  =  ℓ / ℓ , larger-end diameter  ℓ = 0.05 m, and shaft length  shaft =  = 2.0 m kept unchanged.

Table 2 :
Influence of taper ratio ( =  ℓ / ℓ ) on the lowest five natural frequencies   (rad/sec) of the P-taper conic shaft (cf.Figure5) with larger-end diameter  ℓ = 0.05 m and shaft length  shaft =  = 2.0 m kept unchanged: (a) F-F, (b) C-C, (c) C-F, and (d) F-C BCs.The coordinate system for a P-taper conic shaft with diameter  ℓ at its larger end, diameter   at its smaller end, and length  shaft =  =  ℓ −   .
# Natural frequencies obtained from presented exact method using single shaft segment ( = 1).* Natural frequencies obtained from finite element method using 50 shaft elements (  = 50).* * The exact natural frequencies of uniform shaft obtained from formulas in the appendix.

Table 3 :
Influence of taper ratio ( =  ℓ / ℓ ) on the lowest five natural frequencies   (rad/sec) of the PN-taper shaft  (cf.

Table 4 :
Influence of rigid disks (each with rotary inertia Ĵ =  * /3) or/and torsional springs (each with stiffness k, =  *  /3) on the lowest five natural frequencies of the P-taper conic shaft (cf.Figures 6 and case 2 of Table1) with reference polar mass moment of inertia  * = 5.4409436902 × 10 −3 kg-m 2 and reference torsional stiffness  *  = 1.378230198 × 10 4 Nm/rad.The exact values for the lowest four natural frequencies of the "bare" free-free P-taper shaft taken from case 2 of Table2(a).# Natural frequencies of quasi rigid-body modes.
*The exact values for the lowest four natural frequencies of the "bare" free-free PN-taper shaft taken from case 2 of Table3(a).# Natural frequencies of quasi rigid-body modes.