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

Authors: Jia-Jang Wu
Journal: Mathematical Problems in Engineering
 (2013)

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## 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[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.


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## Body

## 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 thegeneral 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 A(x) is to take the form of A(x)=Aℓ(x/Lℓ)2 for the general conic shaft and A(x)=(ax+b)n for the specific conic shaft. In the last expressions for A(x), x denotes the longitudinal coordinate of the conic shaft with its origin at the tip (left) end of the general conic shaft, Aℓ denotes the cross-sectional area at x=Lℓ with the subscript ℓ denoting the larger end of the shaft, while a, b, and n denote constants (with b≠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 [2–4], 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 A(x)=(a+bx)n. 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 thepositive-taper conic shaft if its cross-sectional diameter d(x) increases with increasing the longitudinal coordinates x and is called the negative-taper conic shaft if its diameter d(x) decreases with increasing x. 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 i connecting the conic shaft segments (i) and (i+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, [H]{C}=0, is obtained, in which, {C} 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 |H|=0 by using the bisectional method [6] and to obtain the associated integration constants from the equation [H]{C}=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 q cycles if q 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 x for the positive-taper part of the PN-taper shaft by x-=Lℓ+L-ℓ-x for the negative-taper part of the PN-taper shaft (cf. Figure 3) and the parameters (ds, dℓ, Ls, and Lℓ) for the positive-taper part by those (d-s, d-ℓ, L-s, and L-ℓ) for the negative-taper part, where ds and dℓ denote the diameters at smaller and larger ends of the positive-taper part, d-s and d-ℓ denote the corresponding ones of the negative-taper part, Ls and Lℓ denote the coordinates of smaller and larger ends of the positive-taper part with origin o at the tip of the complete positive-taper cone, while L-s and L-ℓ denote those of the negative-taper part with origin o- at the tip of the complete negative-taper cone (cf. Figure 3).Figure 1
The coordinate system for a P-taper conic shaft with diameterdℓ at its larger end, diameter ds at its smaller end, and length Lshaft=L=Lℓ-Ls.Figure 2
A free-free P-taper conic shaft composed ofp conic shaft segments (denoted by (1),(2),…,(i),…,(p)) and carrying one rigid disk with polar mass moment of inertia J^i and one torsional spring with stiffness k^t,i at each node i (i=0,1,2,…,p).Figure 3
The coordinate system for PN-taper conic shaft (ABC¯) composed of both a positive-taper part AB¯ and a negative-taper part BC¯.To confirm the reliability of the presented exact method, the influence of taper ratios (α-) 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., α-≈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.


## 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]:
(1)∂∂x[GIP(x)∂θ(x,t)∂x]=ρIP(x)∂2θ(x,t)∂t2,
where  ρ  and G are mass density and shear modulus of the shaft material, respectively, Ip(x) is the polar moment of inertia for the cross-sectional area A(x) of the shaft at position x, and θ(x,t) is the torsional angle of the shaft at position x and time t. For the P-taper conic shaft as shown in Figure 1, x is the axial coordinate with its origin o at the tip end of the complete conic shaft. It is evident that the revolution of the inclined straight line AB¯ about the horizontal x-axis will generate the longitudinal (lateral) surface of the shaft.Ifdℓ 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 x is given by
(2)dx=(xLℓ)dℓ,
where Lℓ is the total length of the complete conic shaft, and the polar moment of inertia Ip(x) of the cross-sectional area is given by
(3)Ip(x)=πdx432=(xLℓ)4Ip,ℓ,
where Ip,ℓ is the polar moment of inertia for the cross-sectional area at larger end of the conic shaft given by
(4)Ip,ℓ=πdℓ432.
For free vibrations, one has
(5)θ(x,t)=Θ(x)ejωt,
where Θ(x) denotes the amplitude of θ(x,t), and ω denotes the angular natural frequency of conic shaft and j=-1.Substitution of (3) and (5) into (1) gives
(6)ddx[x4dΘ(x)dx]+β2x4Θ(x)=0,
where
(7)β2=ρω2G.
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:
(8)ddχ[χmdydχ]+cχny=0
is given by
(9)y=χν/α[C1Jν(z)+C2Yν(z)],
where Jv(z) and Yv(z) are 1st and 2nd kind Bessel functions of order v,  and the other parameters are given by(10a)α=2n-m+2,(10b)ν=1-mn-m+2,(10c)z=χ1/ααc.Comparing (8) with (6), one finds that
(11)χ=x,y=Θ,m=4,n=4,c=β2.
Substituting (11) into (10a), (10b), and (10c), one obtains,(12a)α=1,(12b)ν=-32,(13)z=χ1/ααc=xβ.
Substituting the last parameters into (9) and using the relationship Yv/2(z)∝J-v/2(z) [8], one has
(14)Θτ(zτ)=βτ3/2zτ-3/2[C1(τ)J3/2(zτ)+C2(τ)J-3/2(zτ)],
where(15a)zτ=βτx,(15b)βτ2=ρωτ2G.In (14), (15a), and (15b), the subscript (or superscript) τ refers to the τth vibrating mode of the conic shaft, while C1(τ) and C2(τ) denote the two corresponding integration constants determined by the associated boundary conditions of the entire conic shaft.


## 4. Boundary Conditions

Figure2 shows a free-free P-taper conic shaft composed of p conic shaft segments (denoted by (1),(2),…,(i),…,(p)) and carrying one rigid disk with polar mass moment of inertia J^i and one torsional spring with stiffness k^t,i at each node i (i=0,1,2,…,p). The compatibility of twisting angles and equilibrium of torques at the arbitrary intermediate node i located at x=xi require that(16a)Θτ,i(zτ,i)=Θτ,i+1(zτ,i),(16b)GIp,iΘτ,i′(zτ,i)=GIp,iΘτ,i+1′(zτ,i)+(ωτ2J^i-k^t,i)Θτ,i(zτ,i),where(17a)zτ,i=βτxi,(17b)Ip,i=(xiLℓ)4Ip,ℓ.The boundary condition for the left end of the entire conic shaft is given by(18)GIp,0Θτ,1′(zτ,0)+(ωτ2J^0-k^t,0)Θτ,1(zτ,0)=0,
where(19a)zτ,0=βτx0=βτLs,(19b)Ip,0=(x0Lℓ)4Ip,ℓ=(LsLℓ)4Ip,ℓ.Similarly, the boundary condition for the right end of the entire conic shaft is(20)GIp,pΘτ,p′(zτ,p)-(ωτ2J^p-k^t,p)Θτ,p(zτ,p)=0,
where(21a)zτ,p=βτxp=βτLℓ,(21b)Ip,p=(xpLℓ)4Ip,ℓ=(LℓLℓ)4Ip,ℓ=Ip,ℓ.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(22a)Θτ,1′(zτ,0)=0,(22b)Θτ,p′(zτ,0)=0(forF-Fshaft),(23a)
Θ
τ
,
1
(
z
τ
,
0
)
=
0
,
(23b)
Θ
τ
,
p
(
z
τ
,
0
)
=
0
(
for
C-C
shaft
)
,(24a)
Θ
τ
,
1
(
z
τ
,
0
)
=
0
,
(24b)
Θ
τ
,
p
′
(
z
τ
,
0
)
=
0
(
for
C-F
shaft
)
,(25a)
Θ
τ
,
1
′
(
z
τ
,
0
)
=
0
,
(25b)
Θ
τ
,
p
(
z
τ
,
0
)
=
0
(
for
F-C
shaft
)
.In (22a)–(25b), the capital letters F and C denote the free and clamped ends of the entire shaft, respectively. Besides, the symbols Θτ,1(zτ,0) and Θτ,1′(zτ,0) denote the twisting angle and its first derivative of the first shaft segment at (left end) node o (cf. Figure 2), respectively. Similarly, Θτ,p(zτ,p) and Θτ,p′(zτ,p) denote those of the final (pth) shaft segment at (right end) node p, respectively.


## 5. Determination of Exact Natural Frequencies and Mode Shapes

From (14), the displacement function of the ith conic shaft segment takes the form(26a)Θτ,i(zτ,i)=βτ3/2zτ,i-3/2[C1,i(τ)J3/2(zτ,i)+C2,i(τ)J-3/2(zτ,i)].
Therefore,
(26b)Θτ,i′(zτ,i)=dΘτ,i(zτ,i)dx=βτ5/2zτ,i-3/2{-32zτ,i-1[C1,i(τ)J3/2(zτ,i)+C2,i(τ)J-3/2(zτ,i)]=βτ5/2zτ,i-3/2+[C1,i(τ)J3/2′(zτ,i)+C2,i(τ)J-3/2′(zτ,i)]32},in which [11]
(27)J3/2(z)=(2πz)1/2(-cosz+sinzz),(28)J-3/2(z)=(2πz)1/2(-sinz-coszz),(29)Jv′(z)=dJvdz(v=32or-32).
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 Figure2, the substitutions of (26a) and (26b) into (18) lead to
(30)C1,1(τ){[-32βτzτ,0-1+fτ,0]J3/2(zτ,0)+βτJ3/2′(zτ,0)}+C2,1(τ){[-32βτzτ,0-1+fτ,0]J-3/2(zτ,0)+βτJ-3/2′(zτ,0)}=0,
where
(31)fτ,0=(ωτ2J^0-k^t,0)(GIp,0).Similarly, substituting (26a) and (26b) into (16a) and (16b), respectively, one obtains
(32)C1,i(τ)J3/2(zτ,i)+C2,i(τ)J-3/2(zτ,i)-C1,i+1(τ)J3/2(zτ,i)-C2,i+1(τ)J-3/2(zτ,i)=0,(33)C1,i(τ){[-32βτzτ,i-1-fτ,i]J3/2(zτ,i)+βτJ3/2′(zτ,i)}+C2,i(τ){[-32βτzτ,i-1-fτ,i]J-3/2(zτ,i)+βτJ-3/2′(zτ,i)}+C1,i+1(τ)βτ[32zτ,i-1J3/2(zτ,i)-J3/2′(zτ,i)]+C2,i+1(τ)βτ[32zτ,i-1J-3/2(zτ,i)-J-3/2′(zτ,i)]=0,
where
(34)fτ,i=(ωτ2J^i-k^t,i)(GIp,i).
Finally, the substitution of (26a) and (26b) into (20) leads to
(35)C1,p(τ){[-32βτzτ,p-1-fτ,p]J3/2(zτ,p)+βτJ3/2′(zτ,p)}+C2,p(τ){[-32βτzτ,p-1-fτ,p]J-3/2(zτ,p)+βτJ-3/2′(zτ,p)}=0,
where
(36)fτ,p=(ωτ2J^p-k^t,p)(GIp,p).Based on (30), (32), (33), and (35), one obtains a characteristic equation:
(37)[H]n-×n-{C}n-×1=0,
where {C}n-×1 is a n-×1 column vector composed of n-=2p integration constants for the τth mode shape of the pth rod segments, C1,i(τ) and C2,i(τ) (i=1,…,i,…,p), that is,
(38){C}n-×1=[C1,1(τ)C2,1(τ)⋯C1,i(τ)C2,i(τ)⋯C1,p(τ)C2,p(τ)]T,
and [H]n-×n- is a n-×n- (with n-=2p) square matrix with its nonzero coefficients given by(39a)H1,1=[-32βτzτ,0-1+fτ,0]J3/2(zτ,0)+βτJ3/2′(zτ,0),(39b)H1,2=[-32βτzτ,0-1+fτ,0]J-3/2(zτ,0)+βτJ-3/2′(zτ,0),(40a)
H
2
i
,
2
(
i
-
1
)
+
1
=
J
3
/
2
(
z
τ
,
i
)
,
(40b)
H
2
i
,
2
(
i
-
1
)
+
2
=
J
-
3
/
2
(
z
τ
,
i
)
,
(40c)
H
2
i
,
2
(
i
-
1
)
+
3
=
-
J
3
/
2
(
z
τ
,
i
)
,
(40d)
H
2
i
,
2
(
i
-
1
)
+
4
=
-
J
-
3
/
2
(
z
τ
,
i
)
,(41a)
H
2
i
+
1,2
(
i
-
1
)
+
1
=
[
-
3
2
β
τ
z
τ
,
i
-
1
-
f
τ
,
i
]
J
3
/
2
(
z
τ
,
i
)
+
β
τ
J
3
/
2
′
(
z
τ
,
i
)
,
(41b)
H
2
i
+
1,2
(
i
-
1
)
+
2
=
[
-
3
2
β
τ
z
τ
,
i
-
1
-
f
τ
,
i
]
J
-
3
/
2
(
z
τ
,
i
)
+
β
τ
J
-
3
/
2
′
(
z
τ
,
i
)
,
(41c)
H
2
i
+
1,2
(
i
-
1
)
+
3
=
β
τ
[
3
2
z
τ
,
i
-
1
J
3
/
2
(
z
τ
,
i
)
-
J
3
/
2
′
(
z
τ
,
i
)
]
,
(41d)
H
2
i
+
1,2
(
i
-
1
)
+
4
=
β
τ
[
3
2
z
τ
,
i
-
1
J
-
3
/
2
(
z
τ
,
i
)
-
J
-
3
/
2
′
(
z
τ
,
i
)
]
,(42a)
H
n
-
,
n
-
-
1
=
[
-
3
2
β
τ
z
τ
,
p
-
1
-
f
τ
,
p
]
J
3
/
2
(
z
τ
,
p
)
+
β
τ
J
3
/
2
′
(
z
τ
,
p
)
,
(42b)
H
n
-
,
n
-
=
[
-
3
2
β
τ
z
τ
,
p
-
1
-
f
τ
,
p
]
J
-
3
/
2
(
z
τ
,
p
)
+
β
τ
J
-
3
/
2
′
(
z
τ
,
p
)
.For the case of a P-taper shaft consisting of one shaft segment (i.e.,p=1), 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., p>1) with i being less than or equal to p-1 (i.e., i≤(p-1)).Nontrivial solution of (37) requires that
(43)|H|=0.
Equation (43) is the frequency equation for the free-free P-taper shaft with each node i carrying one rigid disk J^i and one torsional spring k^t,i (i=0,1,2,…,p). From (43), one may find the natural frequencies of the vibrating system, ωτ (τ=1,2,3,…), and then, with respect to each natural frequency  ωτ, one may determine the values of C1,i(τ) and C2,i(τ) (i=1,2,…,p) from (37). Finally, the substitution of the last integration constants into (26a) will determine the corresponding natural mode shape of the entire conic shaft Θτ(x). Since the values of rotary inertia J^i and rotational stiffness k^t,i 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.,p=1), (37) reduces to
(44)[H]2×2{C}2×1=0,
where(45a){C}2×1=[C1,1(τ)C2,1(τ)]T,(45b)[H]2×2=[H1,1H1,2H2,1H2,2].In (45b), the coefficients H1,1 and H1,2 are the same as those given by (39a) and (39b), while those H2,1 and H2,2 may be obtained from (42a) and (42b) by letting p=1 and n-=2p=2. The results are(46a)H2,1=[-32βτzτ,1-1-fτ,1]J3/2(zτ,1)+βτJ3/2′(zτ,1),(46b)H2,2=[-32βτzτ,1-1-fτ,1]J-3/2(zτ,1)+βτJ-3/2′(zτ,1),where(47)zτ,1=βτxp=βτLℓ.


### 5.2. Clamped-Clamped P-Taper Conic Shaft

If both left and right ends of the P-taper conic shaft shown in Figure2 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(48a)C1,1(τ)J3/2(zτ,0)+C2,1(τ)J-3/2(zτ,0)=0,(48b)C1,p(τ)J3/2(zτ,p)+C2,p(τ)J-3/2(zτ,p)=0.Therefore, the coefficients relating to the boundary conditions, given by (39a), (39b), (42a), and (42b), must be, respectively, replaced by(49a)H1,1=J3/2(zτ,0),(49b)H1,2=J-3/2(zτ,0),(50a)
H
n
-
,
n
-
-
1
=
J
3
/
2
(
z
τ
,
p
)
,
(50b)
H
n
-
,
n
-
=
J
-
3
/
2
(
z
τ
,
p
)
.


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

The formulation presented in Section5.1 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, J^0=k^t,0=0 and J^p=k^t,p=0.


## 5.1. Free-Free P-Taper Conic Shaft

For the free-free P-taper conic shaft shown in Figure2, the substitutions of (26a) and (26b) into (18) lead to
(30)C1,1(τ){[-32βτzτ,0-1+fτ,0]J3/2(zτ,0)+βτJ3/2′(zτ,0)}+C2,1(τ){[-32βτzτ,0-1+fτ,0]J-3/2(zτ,0)+βτJ-3/2′(zτ,0)}=0,
where
(31)fτ,0=(ωτ2J^0-k^t,0)(GIp,0).Similarly, substituting (26a) and (26b) into (16a) and (16b), respectively, one obtains
(32)C1,i(τ)J3/2(zτ,i)+C2,i(τ)J-3/2(zτ,i)-C1,i+1(τ)J3/2(zτ,i)-C2,i+1(τ)J-3/2(zτ,i)=0,(33)C1,i(τ){[-32βτzτ,i-1-fτ,i]J3/2(zτ,i)+βτJ3/2′(zτ,i)}+C2,i(τ){[-32βτzτ,i-1-fτ,i]J-3/2(zτ,i)+βτJ-3/2′(zτ,i)}+C1,i+1(τ)βτ[32zτ,i-1J3/2(zτ,i)-J3/2′(zτ,i)]+C2,i+1(τ)βτ[32zτ,i-1J-3/2(zτ,i)-J-3/2′(zτ,i)]=0,
where
(34)fτ,i=(ωτ2J^i-k^t,i)(GIp,i).
Finally, the substitution of (26a) and (26b) into (20) leads to
(35)C1,p(τ){[-32βτzτ,p-1-fτ,p]J3/2(zτ,p)+βτJ3/2′(zτ,p)}+C2,p(τ){[-32βτzτ,p-1-fτ,p]J-3/2(zτ,p)+βτJ-3/2′(zτ,p)}=0,
where
(36)fτ,p=(ωτ2J^p-k^t,p)(GIp,p).Based on (30), (32), (33), and (35), one obtains a characteristic equation:
(37)[H]n-×n-{C}n-×1=0,
where {C}n-×1 is a n-×1 column vector composed of n-=2p integration constants for the τth mode shape of the pth rod segments, C1,i(τ) and C2,i(τ) (i=1,…,i,…,p), that is,
(38){C}n-×1=[C1,1(τ)C2,1(τ)⋯C1,i(τ)C2,i(τ)⋯C1,p(τ)C2,p(τ)]T,
and [H]n-×n- is a n-×n- (with n-=2p) square matrix with its nonzero coefficients given by(39a)H1,1=[-32βτzτ,0-1+fτ,0]J3/2(zτ,0)+βτJ3/2′(zτ,0),(39b)H1,2=[-32βτzτ,0-1+fτ,0]J-3/2(zτ,0)+βτJ-3/2′(zτ,0),(40a)
H
2
i
,
2
(
i
-
1
)
+
1
=
J
3
/
2
(
z
τ
,
i
)
,
(40b)
H
2
i
,
2
(
i
-
1
)
+
2
=
J
-
3
/
2
(
z
τ
,
i
)
,
(40c)
H
2
i
,
2
(
i
-
1
)
+
3
=
-
J
3
/
2
(
z
τ
,
i
)
,
(40d)
H
2
i
,
2
(
i
-
1
)
+
4
=
-
J
-
3
/
2
(
z
τ
,
i
)
,(41a)
H
2
i
+
1,2
(
i
-
1
)
+
1
=
[
-
3
2
β
τ
z
τ
,
i
-
1
-
f
τ
,
i
]
J
3
/
2
(
z
τ
,
i
)
+
β
τ
J
3
/
2
′
(
z
τ
,
i
)
,
(41b)
H
2
i
+
1,2
(
i
-
1
)
+
2
=
[
-
3
2
β
τ
z
τ
,
i
-
1
-
f
τ
,
i
]
J
-
3
/
2
(
z
τ
,
i
)
+
β
τ
J
-
3
/
2
′
(
z
τ
,
i
)
,
(41c)
H
2
i
+
1,2
(
i
-
1
)
+
3
=
β
τ
[
3
2
z
τ
,
i
-
1
J
3
/
2
(
z
τ
,
i
)
-
J
3
/
2
′
(
z
τ
,
i
)
]
,
(41d)
H
2
i
+
1,2
(
i
-
1
)
+
4
=
β
τ
[
3
2
z
τ
,
i
-
1
J
-
3
/
2
(
z
τ
,
i
)
-
J
-
3
/
2
′
(
z
τ
,
i
)
]
,(42a)
H
n
-
,
n
-
-
1
=
[
-
3
2
β
τ
z
τ
,
p
-
1
-
f
τ
,
p
]
J
3
/
2
(
z
τ
,
p
)
+
β
τ
J
3
/
2
′
(
z
τ
,
p
)
,
(42b)
H
n
-
,
n
-
=
[
-
3
2
β
τ
z
τ
,
p
-
1
-
f
τ
,
p
]
J
-
3
/
2
(
z
τ
,
p
)
+
β
τ
J
-
3
/
2
′
(
z
τ
,
p
)
.For the case of a P-taper shaft consisting of one shaft segment (i.e.,p=1), 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., p>1) with i being less than or equal to p-1 (i.e., i≤(p-1)).Nontrivial solution of (37) requires that
(43)|H|=0.
Equation (43) is the frequency equation for the free-free P-taper shaft with each node i carrying one rigid disk J^i and one torsional spring k^t,i (i=0,1,2,…,p). From (43), one may find the natural frequencies of the vibrating system, ωτ (τ=1,2,3,…), and then, with respect to each natural frequency  ωτ, one may determine the values of C1,i(τ) and C2,i(τ) (i=1,2,…,p) from (37). Finally, the substitution of the last integration constants into (26a) will determine the corresponding natural mode shape of the entire conic shaft Θτ(x). Since the values of rotary inertia J^i and rotational stiffness k^t,i 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.,p=1), (37) reduces to
(44)[H]2×2{C}2×1=0,
where(45a){C}2×1=[C1,1(τ)C2,1(τ)]T,(45b)[H]2×2=[H1,1H1,2H2,1H2,2].In (45b), the coefficients H1,1 and H1,2 are the same as those given by (39a) and (39b), while those H2,1 and H2,2 may be obtained from (42a) and (42b) by letting p=1 and n-=2p=2. The results are(46a)H2,1=[-32βτzτ,1-1-fτ,1]J3/2(zτ,1)+βτJ3/2′(zτ,1),(46b)H2,2=[-32βτzτ,1-1-fτ,1]J-3/2(zτ,1)+βτJ-3/2′(zτ,1),where(47)zτ,1=βτxp=βτLℓ.


## 5.2. Clamped-Clamped P-Taper Conic Shaft

If both left and right ends of the P-taper conic shaft shown in Figure2 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(48a)C1,1(τ)J3/2(zτ,0)+C2,1(τ)J-3/2(zτ,0)=0,(48b)C1,p(τ)J3/2(zτ,p)+C2,p(τ)J-3/2(zτ,p)=0.Therefore, the coefficients relating to the boundary conditions, given by (39a), (39b), (42a), and (42b), must be, respectively, replaced by(49a)H1,1=J3/2(zτ,0),(49b)H1,2=J-3/2(zτ,0),(50a)
H
n
-
,
n
-
-
1
=
J
3
/
2
(
z
τ
,
p
)
,
(50b)
H
n
-
,
n
-
=
J
-
3
/
2
(
z
τ
,
p
)
.


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

The formulation presented in Section5.1 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, J^0=k^t,0=0 and J^p=k^t,p=0.


## 6. Formulation for a PN-Taper Conic Shaft

Figure3 shows the coordinate system of a conic shaft composed of a shaft segment AB¯ with positive taper and a shaft segment BC¯ with negative taper, which is called PN-taper conic shaft in this paper. The key parameters for the positive-taper part AB¯, ds, dℓ, Ls and Lℓ, 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 BC¯ are represented by d-s, d-ℓ, L-s, and L-ℓ, respectively. Since the origin o- for the coordinate system of the negative-taper part BC¯ is located at  x=Lℓ+L-ℓ, the relationship between the axial coordinates  x-  and x is given by
(51)x-=(Lℓ+L-ℓ)-x(forLℓ≤x≤(Lℓ+L-ℓ-L-s)).
If the intermediate node i is located at the junction of positive-taper part AB¯ and negative-taper part BC¯ of the PN-taper conic shaft ABC¯, that is,  x=xi=Lℓ, then (16a), (16b), (33), (41c), and (41d) must be, respectively, replaced by (52a)Θτ,i(zτ,i)=Θτ,i+1(z-τ,i),(52b)GIp,iΘτ,i′(zτ,i)=-GI-p,i+1Θτ,i+1′(z-τ,i)+(ωτ2J^i-k^t,i)Θτ,i(zτ,i),(53)C1,i(τ){[-32βτzτ,i-1-fτ,i]J3/2(zτ,i)+βτJ3/2′(zτ,i)}+C2,i(τ){[-32βτzτ,i-1-fτ,i]J-3/2(zτ,i)+βτJ-3/2′(zτ,i)}-C1,i+1(τ)RIβτ[32z-τ,i-1J3/2(z-τ,i)-J3/2′(z-τ,i)]-C2,i+1(τ)RIβτ[32z-τ,i-1J-3/2(z-τ,i)-J-3/2′(z-τ,i)]=0,(54a)H2i+1,2(i-1)+3=-RIβτ[32z-τ,i-1J3/2(z-τ,i)-J3/2′(z-τ,i)],(54b)H2i+1,2(i-1)+4=-RIβτ[32z-τ,i-1J-3/2(z-τ,i)-J-3/2′(z-τ,i)],where(55a)x-i=(L1+L-1)-xi,(55b)zτ,i=βτxi,(55c)z-τ,i=βτx-i,(55d)Ip,i=(xiLℓ)4Ip,ℓ,(55e)I-p,i+1=(x-iL-ℓ)4I-p,ℓ,(55f)RI=I-p,i+1Ip,i.Furthermore, if the intermediate nodei is located at the negative-taper part BC¯ of the PN-taper conic shaft ABC¯, that is,  x=xi>Lℓ, then (34) and (36) must be, respectively, replaced by
(56)fτ,i=-(ωτ2J^i-k^t,i)(GIp,i),(57)fτ,p=-(ωτ2J^p-k^t,p)(GIp,p).
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 BC¯ of the PN-taper shaft is opposite to that of the positive-taper part AB¯ (cf. Figure 3), that is, d/dx-=-d/dx 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 partAB¯  of the current PN-taper conic shaft ABC¯, and so is for the negative-taper part BC¯, if one obtains the axial coordinate x- from (51) and sets(58a)x=x-,(58b)ds=d-s,(58c)dℓ=d-ℓ,(58d)Ls=L-s,(58e)Lℓ=L-ℓfor the intermediate nodei located in the negative-taper part BC¯, that is, for the case of Lℓ<x=xi≤(Lℓ+L-ℓ-L-s).


## 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 Figure3 must be replaced by a stepped one composed of ne uniform circular shaft elements shown in Figure 4(a). The average diameter of the jth uniform shaft segment, d-j, is determined in the following.(a) The PN-taper conic shaft (ABC¯) is replaced by ne uniform circular shaft     elements. (b) The average diameter d~j (j=1,2,…,ne) for the jth uniform circular shaft element in the positive-taper part AB¯ is determined by d~j=(dj+12xj+1-dj2xj)/(3Δxj).
(a)
(b)From Figure4(b), one sees that the diameters for the cross-sections of the original conic shaft located at the two ends of the jth uniform rod element, dk and dk+1, are given by(59a)dj=(xjLℓ)dℓ,(59b)dj+1=(xj+1Lℓ)dℓ,Based on the requirement that the mass of thejth uniform shaft element (with average diameter d~j) is equal to that of its original jth conic shaft element, one obtains
(60)ρ(πd~j24)ℓj=13ρπ(dj+12xj+1-dj2xj)4.From the last expression, one obtains average diameter of thejth uniform shaft element
(61)d~j=(dj+12xj+1-dj2xj)(3ℓj),
where
(62)ℓj=Δxj=Lshaftne.
Therefore, the average polar moment of inertia for the cross-sectional area of the jth uniform shaft element is given by
(63)I~p,j=πd~j432,
and the corresponding polar mass moment of inertia Jj of the jth uniform shaft element is determined by
(64)Jj=ρℓjI~p,j.Equations (59a)–(61) are for the case of the jth shaft element being located in the positive-taper part AB¯ of the PN-taper conic shaft ABC¯. If the jth shaft element is located in the negative-taper part BC¯, then (59a)–(61) must be, respectively, replaced by(59a)′d-j=(x-jL-ℓ)d-ℓ,(59b)′d-j+1=(x-j+1L-ℓ)d-ℓ,(
60
)
′
ρ
(
π
d
~
j
2
4
)
ℓ
j
=
1
3
ρ
π
(
d
-
j
2
x
-
j
-
d
-
j
+
1
2
x
-
j
+
1
)
4
,
(
61
)
′
d
~
j
=
(
d
-
j
2
x
-
j
-
d
-
j
+
1
2
x
-
j
+
1
)
(
3
ℓ
j
)
.In the last expressions, the values ofx-j and x-j+1 are determined by (51).Based on the foregoing information for thejth uniform rod element (j=1,2,3,…,ne) and shear modulus G of the shaft material, one may obtain the mass matrix and stiffness matrix of each uniform shaft element from [12]:(65a)[m]j=Jj[13161613],(65b)[k]j=(GI~p,jℓj)[1-1-11].Assembly of the elemental mass and stiffness matrices for each of the uniform shaft elements yields the overall mass matrix[m] and overall stiffness matrix [k] of the entire conic shaft. If there exist a rigid disk with rotary inertia J^j and a torsional spring with stiffness k^t,j at x=xj, then J^j and k^t,j must be added to the jth diagonal coefficient of the overall mass matrix [m] and that of the overall stiffness matrix [k], 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 Tables2 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 ρ=7850 kg/m3, Young’s modulus E= 2.068 × 1011 N/m2, shear modulus G=E/[2(1+v)]=  0.7953846 ×  1011 N/m2, Poison’s ratio v=0.3, smaller-end diameter ds=0.03 m, larger-end diameter dℓ=0.05 m, and distance from origin o to the smaller end Ls=3.0 m and that to the larger end Lℓ=5.0 m. The parameters for the negative-taper part BC¯ of the PN-taper shaft ABC¯ (cf. Figure 3) are the same as the corresponding ones of positive-taper part AB¯, that is, d-s=ds, d-ℓ=dℓ, L-s=Ls, and L-ℓ=Lℓ. For convenience, two reference parameters are introduced:(66a)J*=ρI~PLshaft,(66b)kt*=GI~PLshaft,Table 1
The dimensions of a general conic shaft with five taper ratiosα-=dℓ/Lℓ, larger-end diameter dℓ=0.05 m, and shaft length Lshaft=L=2.0 m kept unchanged.
Case
Taper ratioα-=dℓ/Lℓ
Complete shaft lengthLℓ (m)
Length truncatedLs (m)
Smaller-end diameterds (m)
Shaft lengthLshaft=L (m)
Larger-end diameterdℓ (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
0.05Table 2
Influence of taper ratio (α-=dℓ/Lℓ) on the lowest five natural frequencies ωτ (rad/sec) of the P-taper conic shaft (cf. Figure 5) with larger-end diameter dℓ=0.05 m and shaft length Lshaft=L=2.0 m kept unchanged: (a) F-F, (b) C-C, (c) C-F, and (d) F-C BCs.
(a)
Case
Taper ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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 (ne=50).
*
*The exact natural frequencies of uniform shaft obtained from formulas in the appendix.
(b)
Case
Taper ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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 ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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 ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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
22500.1872Table 3
Influence of taper ratio (α-=dℓ/Lℓ) on the lowest five natural frequencies ωτ (rad/sec) of the PN-taper shaft ABC- (cf. Figure 3 or Figure 4(a)) with dimensions of its negative-taper part BC- identical to the corresponding ones of its positive-taper part AB-, and the larger-end diameter dℓ=0.05 m and total shaft length Lshaft=L=4.0 m kept unchanged: (a) F-F, (b) C-C, and (c) C-F (or F-C) BCs.
(a)
Case
Taper ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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 (P=2).
*Natural frequencies obtained from finite element method using 100 shaft elements (ne=100).
**The exact natural frequencies of uniform shaft obtained from formulas in the appendix.
(b)
Case
Taper ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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 ratiosα-
Method
Natural frequencies,ωτ (rad/sec)
ω
1
ω
2
ω
3
ω
4
ω
5
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
11250.0936whereJ* denotes the reference polar mass moment of inertia for rigid disks and kt* denotes the reference stiffness for torsional springs, Lshaft denotes the length of the entire shaft and is given by Lshaft=Lℓ-Ls for the P-taper shaft and Lshaft=(Lℓ+L-ℓ)-(Ls+L-s) for PN-taper shaft; besides, I~p denotes the average polar moment of inertia for the entire taper shaft, that is,(67a)I~p=(Ip,s+Ip,ℓ)2(fortheP-tapershaft),(67b)I~p=(Ip,s+Ip,ℓ+I-p,s+I-p,ℓ)4(forthePN-tapershaft),whereIp,s and Ip,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 AB¯ of the PN-taper shaft ABC¯), respectively, while I-p,s and I-p,ℓ denote the corresponding ones of the negative-taper part BC¯ of the PN-taper shaft. Based on the foregoing given data, one obtains the reference polar mass moment of inertia J*=5.4409436902×10-3 kg-m2 and reference stiffness kt*=1.378230198×104 N·m/rad for the P-taper shaft. Because the dimensions of the positive-taper part AB¯ of the PN-taper shaft ABC¯ are identical to those of the P-taper shaft and the dimensions of the negative-taper part BC¯ of the PN-taper shaft ABC¯ are the same as the positive-taper part AB¯, it is evident that the reference polar mass moment of inertia J*=2×5.4409436902×10-3 kg·m2 and reference stiffness kt*=(1.378230198×104)/2 N·m/rad for the PN-taper shaft.


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

In this paper, the taper ratioα- of the conic shaft is defined by α-=dℓ/Lℓ=ds/Ls, 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 α-=0.02,0.01,0.005,0.0025, and 0.001 is shown in Figure 5 with diameter of the larger end (dℓ=0.05 m) and length of the shaft (Lshaft=L=2.0 m) being kept unchanged. The influence of taper ratio α- on the lowest five natural frequencies of the P-taper shaft, ωτ (τ=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 (ωτ, τ=1 to 5) of the P-taper shaft obtained from either exact method (using single shaft segment, i.e., p=1) or FEM (using 50 shaft elements, i.e., ne=50) converge to the corresponding ones of the uniform circular shaft obtained from the exact formulas given in the appendix, when the taper ratio α- decreases and approaches zero.(ii)
For various BCs, the lowest five natural frequencies of the P-taper shaft obtained from the exact method (usingp=1) are very close to those obtained from FEM (using ne=50).(iii)
For the F-F, C-C, or F-C BCs, the lowest five natural frequencies (ωτ, τ=1 to 5) of the P-taper shaft decrease with the decrease of taper ratio α- as one may see from Tables 2(a), 2(b), and 2(d). This is a reasonable result because the influence of taper ratio α- 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 α-.(iv)
For the C-F BCs (cf. Table2(c)), the lowest two to five natural frequencies (ω2 to ω5) of the P-taper shaft also decrease with the decrease of taper ratio α-. However, the first natural frequency ω1 increases significantly with the decrease of the taper ratio α-. 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 α-.Figure 5
The profiles for longitudinal cross-sections of the conic shaft with five different taper ratios (cf. Table1).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 shaftABC¯ as shown in Figures 3 or 4(a), if the dimensions of its positive-taper part AB¯ 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 BC¯ are identical to the corresponding ones of the positive-taper part AB¯, then the influence of taper ratio α- on the lowest five natural frequencies (ωτ, τ=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 (ωτ, τ= 1 to 5) of the PN-taper shaft obtained from either exact method (using two shaft segments, i.e., p=2) or FEM (using 100 shaft elements, i.e., ne=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 α- decreases and approaches zero.(ii)
For various BCs, the lowest five natural frequencies of the PN-taper shaft obtained from the exact method (usingp=2) are very close to those obtained from FEM (using ne=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 Table3(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 AB¯ of the PN-taper shaft ABC¯ 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. Figure6) are determined in this subsection. The taper ratio of the P-taper shaft is α-=dℓ/Lℓ=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 ds=0.03 m, diameter at larger end dℓ= 0.05 m, coordinate at smaller end x0=Ls=3.0 m, and coordinate at larger end xp=Lℓ=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 J^i=J*/3  (i=0,1,2) with J*=5.4409436902×10-3 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 k^t,i=kt*/3  (i=0,1,2) with kt*=1.378230198×104 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 x0=3.0 m, x1=4.0 m, and x2=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 NJ^ and that of torsional springs by Nk^t in the same table.Table 4
Influence of rigid disks (each with rotary inertiaJ^i=J*/3) or/and torsional springs (each with stiffness k^t,i=kt*/3) 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 J*=5.4409436902×10-3 kg-m2 and reference torsional stiffness kt*=1.378230198×104 Nm/rad.
Case
N
J
^
N
k
^
t
J
^
i
k
^
t
,
i
Method
Natural frequencies,ωτ (rad/sec)
ω
0
#
ω
1
ω
2
ω
3
ω
4
1
3
0
1
3
J
*
0
Exact
—
2475.3743
5026.7728
10924.0242
12894.1665
FEM
—
2475.4996
5027.4548
10931.3000
12904.4532
2
0
3
0
1
3
k
t
*
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
1
3
J
*
1
3
k
t
*
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.The P-taper shaft (cf. case 2 of Table1) carrying: (a) 3 rigid disks each with J^i=J*/3 (i=0,1,2); (b) 3 torsional springs each with k^t,i=kt*/3 (i=0,1,2); and (c) 3 rigid disks and 3 torsional springs each with J^i=J*/3 and k^t,i=kt*/3 (i=0,1,2). The locations of concentrated elements are x0=3.0 m, x1=4.0 m, and x2=5.0 m. Besides, J*=5.4409436902×10-3 kg-m2 and kt*=1.378230198×104 Nm/rad.
(a)
(b)
(c)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 Table2(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 J^i=J*/3 (i=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^t,i=kt*/3 (i=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 of the lowest four natural frequencies of the shaft have the next relationship: ωi,(a)<ωi,(c)<ωi,(b), where ωi,(a), ωi,(b), and ωi,(c) represent the ith 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 Figures6(b) or 6(c), there exists a quasi rigid-body natural frequency ω0 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 ω0 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 ω-0=kt/J; thus, the true rigid body natural frequency for the P-taper shaft shown in Figure 6(b) is given by ω-0,(b)=kt*/J*=1.378230198×104/5.44094369×10-3=1591.5627 rad/sec, and that shown in Figure 6(c) by ω-0,(c)=kt*/(2J*)=1.378230198×104/(2×5.44094369×10-3)=1125.4048 rad/sec, where 2J* is the summation of polar mass moment of inertia of the entire shaft itself (J*) and that of the 3 rigid disks (J*). Comparing with cases 2 and 3 of Table 4, one sees that quasi rigid-body natural frequencies, ω0,(b)=1684.6341 rad/sec and ω0,(c)=1168.6767 rad/sec, are slightly greater than the foregoing corresponding true ones, ω-0,(b)=1591.5627 rad/sec and ω-0,(c)= 1125.4048 rad/sec, respectively.The lowest five mode shapes of the free-free P-taper shaft with taper ratioα-=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)).
(a)
(b)The lowest five mode shapes (including the quasi rigid-body mode) of the “loaded” free-free P-taper shaft shown in Figure6(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., ne=50) are very close to the corresponding ones obtained from the presented exact method using 2 shaft segments (i.e., p=2).In Figure7, 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 Figure8 by using the presented exact method (with 4 shaft segments, i.e., p=4) and the conventional FEM (with 100 shaft elements, i.e., ne=100). The parameters for the positive-taper part  AB¯ of the current PN-taper shaft ABC¯ are the same as those of the P-taper shaft studied in the last subsection, and the parameters for the negative-taper part BC¯ are identical to the positive-taper part AB¯, that is, d-s=ds=0.03 m, d-ℓ=dℓ= 0.05 m, L-s=Ls=3.0 m, and L-ℓ=Lℓ=5.0 m. The polar mass moment of inertia for each of the 5 identical rigid disks is given by J^i=J*/5 (i=0,1,2,3,4) with J*=1.0881887380×10-2 kg-m2 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 k^t,i=kt*/5 with kt*=6.891150988×103 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 ΔLi=Lshaft/p=[(Lℓ+L-ℓ)-(Ls+L-s)]/4=[(5+5)-(3+3)]/4=1.0 m. Since the concentrated elements are located at the ends of the shaft segments, their coordinates with respect to the origin o of the positive-taper shaft AB¯ are given by x0=3.0 m, x1=4.0 m, x2=5.0 m, x3=6.0 m, and x4=7.0 m, respectively, as one may see from Figures 8(a)–8(c).The free-free PN-taper conic shaft carrying: (a) 5 rigid disks each withJ^i=J*/5 (i=0,1,2,3,4); (b) 5 torsional springs each with k^t,i=kt*/5 (i=0,1,2,3,4); and (c) 5 rigid disks and 5 torsional springs each with J^i=J*/5 and k^t,i=kt*/5 (i=0,1,2,3,4). The locations of the concentrated elements are: xi=3.0,4.0,5.0,6.0, and 7.0 m, respectively. Besides, J*=1.0881887380×10-2 kg-m2  and kt*=6.891150988×103 N-m/rad.
(a)
(b)
(c)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 Table5 (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 inertiaJ^i=J*/5) or/and torsional springs (each with stiffness k^t,i=kt*/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 kt*=6.891150988×103N-m/rad.
Case
N
J
^
N
k
^
t
J
^
i
k
^
t
,
i
Method
Natural frequencies,ωτ (rad/sec)
ω
0
#
ω
1
ω
2
ω
3
ω
4
1
5
0
1
5
J
*
0
Exact
—
1688.3916
2388.5998
3733.3469
5098.3850
FEM
—
1688.3959
2388.7487
3733.5282
5099.2798
2
0
5
0
1
5
k
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
1
5
J
*
1
5
k
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.The lowest five mode shapes of the free-free PN-taper shaft with taper ratioα-=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).
(a)
(b)


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

In this paper, the taper ratioα- of the conic shaft is defined by α-=dℓ/Lℓ=ds/Ls, 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 α-=0.02,0.01,0.005,0.0025, and 0.001 is shown in Figure 5 with diameter of the larger end (dℓ=0.05 m) and length of the shaft (Lshaft=L=2.0 m) being kept unchanged. The influence of taper ratio α- on the lowest five natural frequencies of the P-taper shaft, ωτ (τ=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 (ωτ, τ=1 to 5) of the P-taper shaft obtained from either exact method (using single shaft segment, i.e., p=1) or FEM (using 50 shaft elements, i.e., ne=50) converge to the corresponding ones of the uniform circular shaft obtained from the exact formulas given in the appendix, when the taper ratio α- decreases and approaches zero.(ii)
For various BCs, the lowest five natural frequencies of the P-taper shaft obtained from the exact method (usingp=1) are very close to those obtained from FEM (using ne=50).(iii)
For the F-F, C-C, or F-C BCs, the lowest five natural frequencies (ωτ, τ=1 to 5) of the P-taper shaft decrease with the decrease of taper ratio α- as one may see from Tables 2(a), 2(b), and 2(d). This is a reasonable result because the influence of taper ratio α- 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 α-.(iv)
For the C-F BCs (cf. Table2(c)), the lowest two to five natural frequencies (ω2 to ω5) of the P-taper shaft also decrease with the decrease of taper ratio α-. However, the first natural frequency ω1 increases significantly with the decrease of the taper ratio α-. 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 α-.Figure 5
The profiles for longitudinal cross-sections of the conic shaft with five different taper ratios (cf. Table1).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 shaftABC¯ as shown in Figures 3 or 4(a), if the dimensions of its positive-taper part AB¯ 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 BC¯ are identical to the corresponding ones of the positive-taper part AB¯, then the influence of taper ratio α- on the lowest five natural frequencies (ωτ, τ=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 (ωτ, τ= 1 to 5) of the PN-taper shaft obtained from either exact method (using two shaft segments, i.e., p=2) or FEM (using 100 shaft elements, i.e., ne=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 α- decreases and approaches zero.(ii)
For various BCs, the lowest five natural frequencies of the PN-taper shaft obtained from the exact method (usingp=2) are very close to those obtained from FEM (using ne=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 Table3(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 AB¯ of the PN-taper shaft ABC¯ 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. Figure6) are determined in this subsection. The taper ratio of the P-taper shaft is α-=dℓ/Lℓ=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 ds=0.03 m, diameter at larger end dℓ= 0.05 m, coordinate at smaller end x0=Ls=3.0 m, and coordinate at larger end xp=Lℓ=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 J^i=J*/3  (i=0,1,2) with J*=5.4409436902×10-3 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 k^t,i=kt*/3  (i=0,1,2) with kt*=1.378230198×104 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 x0=3.0 m, x1=4.0 m, and x2=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 NJ^ and that of torsional springs by Nk^t in the same table.Table 4
Influence of rigid disks (each with rotary inertiaJ^i=J*/3) or/and torsional springs (each with stiffness k^t,i=kt*/3) 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 J*=5.4409436902×10-3 kg-m2 and reference torsional stiffness kt*=1.378230198×104 Nm/rad.
Case
N
J
^
N
k
^
t
J
^
i
k
^
t
,
i
Method
Natural frequencies,ωτ (rad/sec)
ω
0
#
ω
1
ω
2
ω
3
ω
4
1
3
0
1
3
J
*
0
Exact
—
2475.3743
5026.7728
10924.0242
12894.1665
FEM
—
2475.4996
5027.4548
10931.3000
12904.4532
2
0
3
0
1
3
k
t
*
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
1
3
J
*
1
3
k
t
*
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.The P-taper shaft (cf. case 2 of Table1) carrying: (a) 3 rigid disks each with J^i=J*/3 (i=0,1,2); (b) 3 torsional springs each with k^t,i=kt*/3 (i=0,1,2); and (c) 3 rigid disks and 3 torsional springs each with J^i=J*/3 and k^t,i=kt*/3 (i=0,1,2). The locations of concentrated elements are x0=3.0 m, x1=4.0 m, and x2=5.0 m. Besides, J*=5.4409436902×10-3 kg-m2 and kt*=1.378230198×104 Nm/rad.
(a)
(b)
(c)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 Table2(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 J^i=J*/3 (i=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^t,i=kt*/3 (i=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 of the lowest four natural frequencies of the shaft have the next relationship: ωi,(a)<ωi,(c)<ωi,(b), where ωi,(a), ωi,(b), and ωi,(c) represent the ith 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 Figures6(b) or 6(c), there exists a quasi rigid-body natural frequency ω0 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 ω0 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 ω-0=kt/J; thus, the true rigid body natural frequency for the P-taper shaft shown in Figure 6(b) is given by ω-0,(b)=kt*/J*=1.378230198×104/5.44094369×10-3=1591.5627 rad/sec, and that shown in Figure 6(c) by ω-0,(c)=kt*/(2J*)=1.378230198×104/(2×5.44094369×10-3)=1125.4048 rad/sec, where 2J* is the summation of polar mass moment of inertia of the entire shaft itself (J*) and that of the 3 rigid disks (J*). Comparing with cases 2 and 3 of Table 4, one sees that quasi rigid-body natural frequencies, ω0,(b)=1684.6341 rad/sec and ω0,(c)=1168.6767 rad/sec, are slightly greater than the foregoing corresponding true ones, ω-0,(b)=1591.5627 rad/sec and ω-0,(c)= 1125.4048 rad/sec, respectively.The lowest five mode shapes of the free-free P-taper shaft with taper ratioα-=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)).
(a)
(b)The lowest five mode shapes (including the quasi rigid-body mode) of the “loaded” free-free P-taper shaft shown in Figure6(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., ne=50) are very close to the corresponding ones obtained from the presented exact method using 2 shaft segments (i.e., p=2).In Figure7, 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 Figure8 by using the presented exact method (with 4 shaft segments, i.e., p=4) and the conventional FEM (with 100 shaft elements, i.e., ne=100). The parameters for the positive-taper part  AB¯ of the current PN-taper shaft ABC¯ are the same as those of the P-taper shaft studied in the last subsection, and the parameters for the negative-taper part BC¯ are identical to the positive-taper part AB¯, that is, d-s=ds=0.03 m, d-ℓ=dℓ= 0.05 m, L-s=Ls=3.0 m, and L-ℓ=Lℓ=5.0 m. The polar mass moment of inertia for each of the 5 identical rigid disks is given by J^i=J*/5 (i=0,1,2,3,4) with J*=1.0881887380×10-2 kg-m2 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 k^t,i=kt*/5 with kt*=6.891150988×103 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 ΔLi=Lshaft/p=[(Lℓ+L-ℓ)-(Ls+L-s)]/4=[(5+5)-(3+3)]/4=1.0 m. Since the concentrated elements are located at the ends of the shaft segments, their coordinates with respect to the origin o of the positive-taper shaft AB¯ are given by x0=3.0 m, x1=4.0 m, x2=5.0 m, x3=6.0 m, and x4=7.0 m, respectively, as one may see from Figures 8(a)–8(c).The free-free PN-taper conic shaft carrying: (a) 5 rigid disks each withJ^i=J*/5 (i=0,1,2,3,4); (b) 5 torsional springs each with k^t,i=kt*/5 (i=0,1,2,3,4); and (c) 5 rigid disks and 5 torsional springs each with J^i=J*/5 and k^t,i=kt*/5 (i=0,1,2,3,4). The locations of the concentrated elements are: xi=3.0,4.0,5.0,6.0, and 7.0 m, respectively. Besides, J*=1.0881887380×10-2 kg-m2  and kt*=6.891150988×103 N-m/rad.
(a)
(b)
(c)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 Table5 (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 inertiaJ^i=J*/5) or/and torsional springs (each with stiffness k^t,i=kt*/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 kt*=6.891150988×103N-m/rad.
Case
N
J
^
N
k
^
t
J
^
i
k
^
t
,
i
Method
Natural frequencies,ωτ (rad/sec)
ω
0
#
ω
1
ω
2
ω
3
ω
4
1
5
0
1
5
J
*
0
Exact
—
1688.3916
2388.5998
3733.3469
5098.3850
FEM
—
1688.3959
2388.7487
3733.5282
5099.2798
2
0
5
0
1
5
k
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
1
5
J
*
1
5
k
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.The lowest five mode shapes of the free-free PN-taper shaft with taper ratioα-=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).
(a)
(b)


## 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 diameterdℓ and total length L kept constant (cf. Figure 5), its lowest five natural frequencies (ωτ, τ=1 to 5) in the F-F, C-C, or F-C BCs decrease with the decrease of its taper ratio α- because the polar mass moment of inertia of the entire shaft increases with the decrease of α- and the influence of α- on the supporting stiffness is much smaller. The last trend is also true for the 2nd to 4th  natural frequencies (ω2 to ω4) of the shaft with C-F BCs, but the 1st  natural frequency (ω1) increases significantly with the decrease of taper ratio α- 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 α-.
(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:ωi,J<ωi,Jkt<ωi,kt, where ωi,J, ωi,kt, and ωi,Jktrepresent the ith 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 frequencyω0. The latter is slightly higher than the true rigid-body natural frequency defined by ω-0=∑k^t,i/(Jshaft+∑J^i), where Jshaft denotes the polar mass moment of inertia of the entire shaft itself, while ∑k^t,i and ∑J^i 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.



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*Source: 491062-2013-04-11.xml*