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Journal of Sound and Vibration 331 (2012) 107–116
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi
Longitudinal vibrations in circular rods: A systematic approach
N.G. Stephen a,n, K.F. Lai b, K. Young b, K.T. Chan c
a
Faculty of Engineering and the Environment, Mechanical Engineering, University of Southampton, Highfield, Southampton SO17 1BJ, UK
Department of Physics, The Chinese University of Hong Kong, Hong Kong, China
c
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China
b
a r t i c l e i n f o
abstract
Article history:
Received 15 July 2011
Accepted 22 August 2011
Handling Editor: S. Ilanko
Available online 23 September 2011
A systematic method is developed for expressing the frequency squared o2 and the
corresponding displacement fields of harmonic waves in a long thin rod as an even
power series in qa, where q is the wavenumber along the rod and a is a representative
transverse dimension. For longitudinal waves in a circular rod, the evaluation is reduced
to algebraic recursion, giving coefficients analytically in terms of Poisson’s ratio v, to
many orders. The second nontrivial coefficient, corresponding to Rayleigh–Love theory
in the present longitudinal case and Timoshenko theory in the flexural case, is thus put
on a firm footing without reliance on ad hoc physical assumptions. The results are
compared to available exact predictions, and shown to be accurate for moderate values
of qa (5% accuracy for qa r 1.5) with just two terms. Improvements based on the
Rayleigh quotient guarantee positivity and the correct asymptotic power, and the
variational principle further ensures that the accuracy improves monotonically with the
order of approximation. With these features, accurate results are obtained for larger qa
(5% accuracy for qa r 3), so that results are valid for rods that are by no means thin.
Application of these methods to the flexural case has been presented separately.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The vibration of thin beams and rods is a familiar topic; see for example Ref. [1, Chapter 8]. When some characteristic
transverse dimension a is small compared to the length L, a/L51 provides a small parameter in terms of which simple but
accurate approximations are possible. The key idea is that variation of the strain in the direction transverse to the axis is
negligible, and the vibration can be described by a single variable, for example the transverse displacement of the centerline in the Euler–Bernoulli theory of flexural vibrations, or the axial displacement for longitudinal vibrations. Such theories
are often termed classic, or first-order (although in the notation of the present paper, they would be fourth and second
order, respectively, in (a/L)).
For flexural vibrations, Timoshenko beam theory (see Ref. [1, Section 8.5.8]) is an improvement to the Euler–Bernoulli
theory and has been termed second-order [2] (again in the notation of the present paper it is sixth order in (a/L)). The
improvement comes about through the inclusion of rotary inertia, and the introduction of the shear angle as a second
center-line variable; but since shear strain must vary in the transverse direction, it also includes a shear coefficient k
which attempts to account for that variation. For longitudinal vibrations, the analogous higher-order approximation is
known as Rayleigh–Love (see Ref. [3, article 278]). These approaches, based on powerful physical insight, are practically
n
Corresponding author.
E-mail address: ngs@soton.ac.uk (N.G. Stephen).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jsv.2011.08.021
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N.G. Stephen et al. / Journal of Sound and Vibration 331 (2012) 107–116
t, T
time, normalizing integral related to
kinetic energy
u, v, w displacement components
U, W
radial
and
longitudinal
displacement
functions
vph, V
phase velocity, strain energy
x, y, z
Cartesian coordinates
Y
v2ph in suitable units
a, b, g coefficients in expansion
d
Kronecker delta
k
shear coefficient
l
Lamé constant
n
Poisson’s ratio
n, xj
displacement vector and its components
r
density
t
non-dimensional radius, r/a
o, o0, os radian frequency, characteristic frequencies
Nomenclature
a
c, C
e, E
f
G
i
i, j, k, l
L
m, M
n, N
n, nj
q
r
S
radius of rod
parameter, stiffness constants
unit vector, Young’s modulus
surface traction
shear
pffiffiffiffiffiffiffi modulus
1
indices, powers of qa
length of rod, half-wavelength
mode order, longitudinal modulus
order in powers of qa, maximum value of
order n, integer
unit normal vector and its components
wavenumber for a traveling wave, mp/L for
standing wave
radial coordinate
series, Eq. (2)
useful and can be very accurate for thin beams; nevertheless, the mathematical foundation is open to some questions.
First, different forms have been proposed for k [4,5]; although the one proposed by Stephen and Levinson [2,6] and more
recently Hutchinson [7,8] is the most generally accepted, the differences continue to be debated [9,10]. Second, even the
correct k at best represents an effective theory that incorporates two degrees of freedom (compared to one degree of
freedom in the Euler–Bernoulli approximation) out of infinitely many, and cannot be exact. Therefore one should try to
characterize – and in simple cases determine – the corrections to higher orders.
The reference to different orders foreshadows our approach. This and a companion paper, Chan et al. [11], present a
systematic formulation of vibrations in long thin rods, through a power-series expansion in (a/L), or equivalently qa, where
q¼ mp/L is the wavenumber, with the integer m being the mode number; quantities such as the frequency will only
involve even powers of qa. The systematic expansion replaces reliance on any physical insight or effective theory. The
lowest nontrivial order will reproduce classic results such as Euler–Bernoulli, while the next order in (qa)2 will reproduce
results such as those of Timoshenko with a precise determination of k, for which any remaining doubts are unequivocally
settled. This program will also demonstrate that the corrections to Timoshenko theory lie in the next power of (qa)2.
Flexural vibration is treated in the companion paper [11], while the present paper deals with the case of the
longitudinal vibrations of a circular rod, in which the symmetry allows the power series to be developed to many orders
(Sections 2 and 3). The expansion, checked against an available exact solution [12], establishes the mathematical validity
of the formalism and reveals some of its general properties, laying the foundation for other applications. The second part of
this paper (Sections 4 and 5) develops methods to extend the results to higher values of qa (thicker rods).
Consider an infinitely long rod along the z-direction, NozoN, with uniform cross-section in the xy-plane, in the
present case the disk r2 ¼x2 þy2 ra2. Attention is focused on sinusoidal oscillations pcos ot with a radian frequency o.
For a uniform rod, the variation along z can be expressed as plane waves pexp(iqz) with wavenumber q; vibrations with
different values of q are independent. The complex notation allows the replacement @/@z-iq and it is always understood
that either the real part or the imaginary part is taken. For example, if the imaginary part is taken, exp(iqz)-sin(qz), and a
finite rod of length L, say with fixed ends, is readily described by restricting to q¼mp/L, m¼1, 2, 3, .y However, it is
convenient to regard q as a continuous variable, in terms of which a power series expansion is sought. The formalism is
developed in Section 2, but one can anticipate the general form of the result. First, express o in terms of a characteristic
frequency o0
o20 ¼
E
ra2
,
(1)
chosen so that the leading coefficient is unity, where E is Young’s modulus and r is the density; the ratio o2 =o20 is then
expanded in (qa). Only even powers appear because left and right propagating waves (z- z , q- q) have the same
frequencies. Thus one can write the general expression
o2 X
¼
a ðqaÞ2n S,
o20 n ¼ 1 2n
(2)
and the task is to determine a2n. The leading term is a2 in the present longitudinal case but a4 in the flexural case [11],
since longitudinal stresses are O(1) in the limit a-0, while bending stresses are O(a2).
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109
The dimensionless coefficients a2n can only depend on the Poisson ratio n in the case of an isotropic material, and on
dimensionless shape parameters characterising the cross-section (e.g. the aspect ratio of a rectangular cross-section). For a
circular rod, there is no shape parameter, and a2n ¼ a2n(n). The present paper first establishes a formalism that allows a2n to
be calculated order by order; significantly the calculation is reduced to algebraic recursion, which has been implemented
to 20 orders in qa as an algebraic function of n.
2. Formulation of problem
2.1. Equations of motion
Consider an isotropic material with stiffness constants
Cij,kl ¼ ldij dkl þ Gðdik djl þ dil djk Þ,
(3)
n ¼ uex þ vey þ wez
(4)
in which there is a displacement field
where u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) are the displacement components along the x-, y- and z-directions, respectively, and
t is the time. The equations of motion are as follows:
r@2t u ¼ M@2x u þGð@2y þ @2z Þu þ ðG þ lÞð@x @z w þ @x @y vÞ,
(5a)
r@2t v ¼ M@2y v þ Gð@2x þ @2z Þv þ ðG þ lÞð@y @z w þ@x @y uÞ,
(5b)
r@2t w ¼ M@2z wþ Gð@2x þ @2y Þw þ ðG þ lÞð@x @z u þ@y @z vÞ,
(5c)
where the shear modulus G, the longitudinal modulus M and the Lamé constant l can be expressed in terms of Young’s
modulus E by
G¼
E
,
2ð1 þ nÞ
M¼
ð1nÞE
,
ð1 þ nÞð12nÞ
l¼
nE
:
ð1 þ nÞð12nÞ
(6)
Restrict to plane waves pexp(iqz) and make use of cylindrical symmetry; then the displacements are reduced to two
functions U(r) and W(r)
u ¼ iðx=aÞUðrÞexpðiqzÞ
(7a)
v ¼ iðy=aÞUðrÞexpðiqzÞ,
(7b)
w ¼ WðrÞexpðiqzÞ
(7c)
Factors of i in Eqs. (7a) and (7b) anticipate the quarter-cycle phase difference between the transverse and longitudinal
displacements (e.g. wpsin qz, up cos qz), so that U and W as defined are both real; the factor a 1 in Eqs. (7a) and (7b)
renders U conveniently dimensionless. A purely real version is sketched in Appendix A.
The equations of motion then reduce to the following, where henceforth prime denotes d/dr:
ro2 U ¼ MðU 00 þ3U 0 =rÞ þ ðqaÞðMGÞW 0 =rq2 GU,
(8a)
ro2 W ¼ q2 MW þGðW 00 þW 0 =rÞðq=aÞðMGÞðrU 0 þ2UÞ:
(8b)
2.2. Boundary conditions
The absence of surface traction is given by
fi ¼ nj Cij,kl @k xl ¼ 0
(9)
evaluated at r ¼a, where nj is a component of the unit normal. Straightforward evaluation then gives
W 0 ðaÞqUðaÞ ¼ 0,
(10a)
ðl þ 2GÞ½aU 0 ðaÞ þ UðaÞ þ lUðaÞ þ ðqaÞlWðaÞ ¼ 0,
(10b)
where the two equations, respectively, refer to the z- and r-components of Eq. (9). The task is therefore reduced to solving
two ordinary differential equations (8a,b) subject to these boundary conditions.
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2.3. Expansion for thin rods
For rods that are thin (compared to the wavelength or L), expand all the wave functions and the eigenvalue in powers of
qa 51. The zero-order solution involves only W, and inspection reveals that W and U contain only even and odd powers of
qa, respectively. So put
WðrÞ ¼ W0 ðtÞ þðqaÞ2 W2 ðtÞ þ ðqaÞ4 W4 ðtÞ þ ,
(11a)
UðrÞ ¼ ðqaÞ U1 ðtÞ þðqaÞ3 U3 ðtÞ þ :
(11b)
The functions W2n , U2n þ 1 depend on the non-dimensional radius
t ¼ r=a:
(12)
When these are put into Eqs. (8) and (10), the problem can be solved order-by-order. At each order the second-order
ordinary differential equations can be integrated, because the relevant combinations are, respectively
U 00 þ3U 0 =r ¼
1 d 3 0
ðr U Þ,
r 3 dr
W 00 þW 0 =r ¼
1d
ðrW 0 Þ:
r dr
(13)
Straightforward computation then leads to the following, where the first line is just a normalization convention:
W0 ¼ 1,
(14a)
U1 ¼ n,
(14b)
W2 ¼ nt2 =2,
(14c)
a2 ¼ 1:
(14d)
A possible constant term in W2 is removed because the solution is normalized to W(0) ¼1 for all qa, which means, in
particular
W0 ð0Þ ¼ 1,
W2 ð0Þ ¼ W4 ð0Þ ¼ ¼ 0:
(15)
This calculation can in principle be extended to any order in qa without difficulties; we have done so up to fourth order.
The higher-order results can be read off from Tables 1–3 discussed below, and will not be separately exhibited. It can be
seen that W2n , U2n þ 1 are polynomials of order 2n in t. This property will be important in Section 5.
2.4. Reduction to algebraic recursion
Because of the general properties alluded to above, the functions can be written as
W2n ¼
n
X
2j
b2j
U2n þ 1 ¼
2n t ,
j¼0
n
X
2j
g2j
2n þ 1 t :
(16)
j¼0
2j
in terms of the coefficients a2n ðnÞ , b2n ðnÞ and g2j
2n þ 1 ðnÞ. In some of the formulas below, it is convenient to go beyond the
ranges of indices implied by Eqs. (11) and (16), and understand the coefficients to be zero if no0, jo 0, or j 4n. Putting Eq.
(16) into Eqs. (8) and (10) yields algebraic recursion relations for these coefficients
g2k
2n þ 1 ¼
n
12n
1
ð1 þ nÞð12nÞ X
g2k2
b2k
a g2k2 ,
2n 2n1 2ð1nÞð2kþ 2Þ
ð1nÞð2kÞð2k þ 2Þ j ¼ 1 2j 2n2j þ 1
2ð1nÞð2kÞð2k þ1Þ
g02n þ 1 ¼ n
X
½1þ ð1nÞkg2k
2n þ 1 n
k¼1
b2k
2n ¼
n
X
b2k
2n ,
(17b)
k¼0
n1
1n
1
1þn X
2k2
2k2
b
g
a2j b2k2
2n2
2n2j ,
2n1
2ð12nÞk
2ð12nÞk2
2k2 j ¼ 1
b22n ¼ (17a)
n
X
2k
kb2n þ
k¼2
n1
1X
g2k ,
2 k ¼ 0 2n1
(17c)
(17d)
which come, respectively, from Eqs. (8a), (10b), (8b) and (10a).
2k
Recursion relation (17c) determines b2n for all k, with two exceptions. First, for k¼0, the right-hand-side involves the
negative index 2k 2¼ 2, and therefore vanishes. Thus
b02n ¼ 0,
(18)
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N.G. Stephen et al. / Journal of Sound and Vibration 331 (2012) 107–116
111
consistent with Eq. (15). Second, for k¼1, Eq. (17c) reduces to
b22n ¼
n
1n
1
1þn X
b02n2 þ
g02n1 a b0 :
2ð12nÞ
2ð12nÞ
2 j ¼ 1 2j 2n2j
(19)
But on the right-hand-side, the first term vanishes unless 2n 2¼0, on account of Eq. (15). For the same reason, the terms
in the sum are all zero unless 2n 2j ¼0. Thus Eq. (19) simplifies to
b22n ¼
1n
1
1þn
dn,1 þ
g0 a2n :
2ð12nÞ 2n1
2ð12nÞ
2
(20)
When this is compared with Eq. (17d), the coefficients a2n are determined
n
n1
X
1þ n
1X
1
1n
2k
a2n ¼
kb2n g2k
g02n1 þ
d :
2n1 þ
2
2ð12
2
n
Þ
2ð12
nÞ n,1
k¼2
k¼0
(21)
3. Results and discussion
3.1. Results of recursion
0
Starting from the seed b0 ¼ 1, the coefficients are determined recursively as functions of n, using MATHEMATICA, up to
20 orders in qa. Cancellation of the singularity at n ¼ 1/2 provides a useful check. Tables 1a and 1b show a2n analytically up
to 12 orders in qa (higher orders being too cumbersome to display) and numerically to 20 orders; all numerical results in
2j
this paper refer to n ¼0.30. Some results for b2n , g2j
2n þ 1 are illustrated in Tables 2 and 3. To be explicit
2
E
n
n2
6
2
3
4
ð7
þ
4
o2 ¼ 2 ðqaÞ2 ðqaÞ4 þ
n
þ
32
n
4
n
24
n
ÞðqaÞ
þ
:
(22)
2
ra
48ð1n2 Þ
3.2. Comparison with exact theory
The exact frequency equation for the solid circular rod is presented as a 2 2 determinant in article 201 of Love [3],
wherein each element involves a Bessel function of order zero or one, and/or its derivatives. The theory was developed by
Armenakas et al. [12] for the hollow circle of wall-thickness H (equal to the radius a for the solid
pffiffiffiffiffiffiffiffifficircular rod) and exact
numerical results are presented as tables of O ¼ o/os, versus H/L¼qa/p, where os ¼ ðp=HÞ G=r is the lowest simple
Table 1a
Values of a2n as functions of n. The leading coefficient is a2 ¼1, while for 2nZ 4, it is convenient to put a2n ¼ N 1 n2 ð1 þ nÞ1 ð1nÞ2n f ðnÞ, where N is an
integer, and f(n) is a polynomial of order 3n 5 whose coefficients (all integers) are listed in ascending order. For example, the second row in the table
means a6 ¼ ð1=48Þn2 ð1 þ nÞ1 ð1nÞ1 ð7þ 4n þ 32n2 4n3 24n4 Þ. Note also that the first row can be simplified to a4 ¼ n2/2. Results have been obtained
up to 2n¼20, but the higher orders are too cumbersome to display.
2n
N
Coefficients of f(n) in ascending order
4
6
8
10
12
2
48
768
23040
2211840
1, þ 1
7, 4, 32, 4, 24
33, 69, 282, 536, 580, 956, 320, 480
292, 1052, 3547, 15063, 5310, 55442, 16964, 71964, 39480, 30240, 20160
8269, 42283, 127415, 925043, 254438, 5139760, 5727184, 9885104, 16962752,
4980512, 18407424, 3575040, 6773760, 2903040
Table 1b
Values of a2n for n ¼ 0.30.
2n
a2n
2
4
6
8
10
12
14
16
18
20
1.000
4.500 10 2
6.640 10 3
6.694 10 4
2.456 10 5
3.030 10 5
7.318 10 6
7.338 10 7
1.311 10 7
7.482 10 8
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N.G. Stephen et al. / Journal of Sound and Vibration 331 (2012) 107–116
Table 2
2k
2k
Values of b2n as functions of n, expressed as b2n ¼ N 1 nð1nÞ1n f ðnÞ, where N is an integer, and f(n)
is a polynomial whose coefficients (all integers) are listed in ascending order. For example, the last
6
row in the table means b6 ¼ ð1=1152Þnð1nÞ2 ð13n þ 2n2 þ 8n3 4n4 Þ.
2n, 2k
N
Coefficients of f(n) in ascending order
0,
2,
2,
4,
4,
4,
6,
6,
6,
6,
1
1
2
1
16
32
1
192
256
1152
0
0
1
0
3, 4, 2, 4
1, 2, 2
0
11, 39, 10, 104, 52, 64, 48
3, 1, 30, 16, 24, 16
1, 3, 2, 8, 4
0
0
2
0
2
4
0
2
4
6
Table 3
2k
1
nð1nÞn f ðnÞ, where N is an integer, and
Values of g2k
2n þ 1 as functions of n, expressed as g2n þ 1 ¼ N
f(n) is a polynomial whose coefficients (all integers) are listed in ascending order. For example, the
second row in the table means g03 ¼ ð1=8Þnð1nÞ1 ð3 þ 6n þ 2n2 4n3 Þ.
2nþ 1, 2k
N
Coefficients of f(n) in ascending order.
1,
3,
3,
5,
5,
5,
1
8
8
96
64
192
1
3, 6, 2, 4
1, 0, 2
11, 47, 18, 112, 76, 64, 48
3, 5, 12, 24, 8, 16
1, 1, 4, 0, 4
0
0
2
0
2
4
thickness-shear frequency
offfi an infinite plate also of thickness H. For comparison with the series S on the right-hand-side
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
of Eq. (2), o=os ¼ 2ð1 þ nÞS=p. Instead of the frequency, we consider
Y¼
ðo=os Þ2
ðqaÞ2
¼
O2
pðH=LÞ2
¼
v2ph
ðos aÞ2
,
(23)
where vph ¼ o/q is the phase velocity; Y is convenient because it stays within a relatively small range.
Table 4 and Fig. 1 show the exact results for Y together with those obtained from the series (2) truncated at 2n¼2N.
The N ¼2 truncation (Rayleigh–Love) is accurate to 5% for H/Lr0.5 (or qa r0.5p), while the N ¼3 truncation is accurate for
H/Lr0.7 (or qa r0.7p). Higher powers do not significantly increase the range of validity, for reasons explained below. In
any event, one does not expect higher powers to be available in other situations to which these methods are intended to be
transplanted.
3.3. Possible application to thicker rods
Attempts to use the power series for larger qa run into three difficulties. (a) The coefficients in Table 1b seem to suggest
a radius of convergence of (qa)2 E3, consistent with the limit of accuracy shown in Table 4. (b) There is no guarantee of
positivity. For example, the polynomial with two terms a2(qa)2 þ a4(qa)4 changes sign at (qa)2 ¼ a2/a4 ¼ 2/n2 ( E22 for
n ¼0.3). (c) For q-N, the polynomial predicts o2pq2N, whereas both physical intuition and the exact solution require a
finite phase velocity, i.e. o2pq2. The next few sections describe attempts to guarantee positivity and the correct
asymptotic power, which would enable the range of accuracy to be extended.
4. Rayleigh quotient
4.1. General formulation of Rayleigh quotient
In many circumstances, a better approximation is provided by the Rayleigh quotient [1], closely related to the
variational principle and automatically ensuring positivity. In general, the eigenvalue can be expressed as the minimum
V
T
o2 ¼ min ,
(24)
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N.G. Stephen et al. / Journal of Sound and Vibration 331 (2012) 107–116
113
Table 4
Values of Y (the phase velocity squared in suitable units, as defined in Eq. (23)) as a function of H/L¼ qa/p. The tabulation
shows the exact solution; the power series truncated at (qa)2N, N ¼1 (classic approximation), N ¼2 (Rayleigh–Love
approximation), and N ¼ 3; the Rayleigh quotient (RQ) with eigenfunctions expanded up (qa)N, with N ¼ 3, and the
variational approximation (VA), using a polynomial up to rN, with N ¼ 3. All data refer to n ¼ 0.30. Entries which differ from
the exact solution by more than 5% are shown in bold.
H
L
0.01
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.30
1.50
1.80
2.00
3.00
4.00
5.00
6.00
Y
qa
H
¼p
L
0.0314
0.157
0.314
0.628
0.943
1.26
1.57
1.89
2.20
2.51
2.83
3.14
4.08
4.71
5.65
6.28
9.42
12.6
15.7
18.8
Exact
N ¼1
Classic
N ¼2
Rayleigh–Love
N¼3
N¼ 3
RQ
N ¼3
VA
0.263
0.263
0.262
0.259
0.251
0.240
0.222
0.199
0.175
0.154
0.137
0.125
0.104
0.0974
0.0921
0.0902
0.0870
0.0865
0.0864
0.0865
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.263
0.262
0.259
0.253
0.245
0.234
0.221
0.206
0.189
0.169
0.146
0.0657
0.000
0.116
0.205
0.790
1.61
2.66
3.95
0.263
0.263
0.262
0.259
0.252
0.240
0.224
0.199
0.165
0.119
0.057
0.024
0.421
0.862
1.90
2.93
14.6
45.2
109
225
0.263
0.263
0.262
0.259
0.252
0.240
0.224
0.203
0.180
0.161
0.150
0.146
0.148
0.149
0.148
0.146
0.132
0.122
0.116
0.111
0.263
0.263
0.262
0.259
0.252
0.240
0.223
0.201
0.177
0.155
0.138
0.125
0.110
0.109
0.112
0.114
0.120
0.120
0.117
0.114
0.30
0.25
Y
0.20
4
0.15
3
0.10
1
2
0.05
-1.0
-0.5
0.0
0.5
X
Fig. 1. Plot of Y (the phase velocity squared in suitable units, as defined in Eq. (23)) versus X ¼ log10(H/L), for the data in Table 4. Line 1: power series
truncated at (qa)2N , N ¼3; Line 2: exact solution; Line 3: variational method using a polynomial up to rN , N¼ 3; Line 4: Rayleigh quotient with
eigenfunctions expanded to (qa)N , N ¼3.
where V and T are, respectively, related to the strain energy and a normalizing integral related to the kinetic energy
Z
1
3
d xCij,kl ð@i xj Þð@k xl Þ,
V¼
(25a)
2
T¼
1
2
Z
3
d xrxj xj :
(25b)
The integrals are over the entire solid, and the minimization is carried out over all possible xj that satisfy the boundary
conditions. The minimum is attained when xj equals the eigenfunction.
4.2. Application to present problem
Several simplifications are possible for the present problem. (a) All quadratic expressions should be changed by
n
(schematically) xx-xnx. (b) Derivatives can be written as @i ¼ @?
i þiqdi3 when acting on x. (c) All x x terms are uniform in
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z, so the integral is reduced to the two transverse dimensions. With these simplifications and scaling out some constants
o2
V~
¼ min ,
2
o0
T~
Z
T~ ¼ 2
1
(26)
dttðW 2 þ t2 U 2 Þ,
(27a)
0
V~ ¼ V~ l þ V~ G ,
V~ l ¼
V~ G ¼
1
1þn
Z
1
2n
ð1 þ nÞð12nÞ
Z
1
(27b)
dtt½2U þ tðdU=dtÞ þðqaÞW2 ,
(27c)
0
dttf2U 2 þ 2½dðtUÞ=dt2 þ ½ðqaÞtUðdW=dtÞ2 þ 2ðqaÞ2 W 2 g,
(27d)
0
The overall constant is chosen so that T~ -1 for qa-0, while the manifestly positive terms V~ l and V~ G arise from the l and G
terms in the strain energy.
The power series solution for U and W in Eq. (11) are then truncated at a maximum power (qa)N and substituted into
Eq. (27), and the resultant squares evaluated. This approximation has three advantages. (a) By first truncating and then
squaring (rather than the other way round), positivity is guaranteed. (b) The denominator goes up to the power (qa)2N,
while the numerator goes up to two more powers—thus guaranteeing the correct asymptotic power. As an example, the
case of N ¼3 (which requires that the functions U and W be evaluated to (qa)3), leads to the expression
o2 V2 ðqaÞ2 þ V4 ðqaÞ4 þ V6 ðqaÞ6 þ V8 ðqaÞ8
¼
,
o20
T0 þ T2 ðqaÞ2 þ T4 ðqaÞ4 þT6 ðqaÞ6
(28)
where the coefficients are given in Table 5 for n ¼0.3. The resultant values of Y are given in Table 4 and Fig. 1, being
accurate to 5% for H/Lr0.8 (or qa r0.8p), a significant improvement over the power series with two or even three terms.
(c) Although the eigenfunctions are truncated at order N (¼3), the Rayleigh quotient has an accuracy up to order 2N ( ¼6),
as recently demonstrated [13]. With such qualitative advantages for both large and small qa, it should not be surprising
that the method works well over a large range of qa.
4.3. The limit of large qa
It appears difficult to go beyond H/LE1 (or qa E p) by approximating the Rayleigh quotient in the manner described
above, and the reason can be understood by analyzing the limit qa-N. If the eigenfunctions are truncated at an odd
power (qa)N, one would have 9U9b9W9 in this limit, and by considering the leading terms, it can be shown that Eq. (26)
will approach [2(1þ n)] 1(qa)2, corresponding to the bulk shear wave speed
v2ph ¼
G
r
,
Y¼
1
p2
¼ 0:101:
(29a)
On the other hand, truncation at an even N would cause 9U959W9 in this limit, and Eq. (26) will approach
ð1nÞ½ð1 þ nÞð12nÞ1 ðqaÞ2 , corresponding to the bulk longitudinal wave speed
M
2
1n
v2ph ¼ , Y ¼ 2
¼ 0:355:
(29b)
r
p 12n
Neither limit agrees with the exact value 0.0865, though Eq. (29a) is much closer.
It is not surprising that any finite truncation fails to give the qa-N limit correctly. In fact, the exact asymptotic value
0.0865 shows that the eigenfunction is a Rayleigh surface wave [14,15], which is concentrated in a thin layer q 1 near
the surface r ¼a, and only functions that allow such behavior to be captured will succeed in giving an accurate estimate.
The above analysis also shows that if N is successively increased, then at large (and by extension also moderate) values
of qa, the predicted value of Y will alternate between Eqs. (29a) and (29b)—in other words the fourth order will be worse
Table 5
The coefficients in the Rayleigh quotient with displacements evaluated to
third order; for n ¼ 0.3; see Eq. (28) for the definition of the coefficients.
n
Tn
Vn
0
2
4
6
8
1.000
0.105
1.684 10 2
5.386 10 4
–
–
1.000
0.150
1.493 10 2
2.071 10 4
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115
Table 6
Comparison of various approximation methods, corresponding to the last three columns in Table 4. The various attributes are explained in Section 6.
Method
Thin limit
Asymptotic power
Positivity
Monotonicity
Truncated power series in qa (Section 3)
Rayleigh quotient, expanded in powers of qa (Section 4)
Variational method, in powers of r (Section 5)
Yes
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
No
Yes
than the third order shown above. As an example, at H/L¼0.8, the third order, fourth order and exact values are 0.161
(close to Eq. (29a)), 0.328 (close to Eq. (29b)) and 0.154. One is therefore led to look for methods that guarantee monotonic
improvement as the order of approximation is increased.
5. Variational method based on polynomial in r
5.1. General formulation
The exact Rayleigh quotient (i.e. Eqs. (26)–(27)), as a minimization principle, allows another systematic approximation
by exploiting an important feature of the expansion in powers of qa. The solution for W2n and U2n þ 1 are polynomials in r,
whose general term goes as qnrk with k rn. Thus for accuracy to a certain power (qa)n, it suffices to consider only a
polynomial in r up to the same power. For example, to third-order accuracy in qa, one may represent U and W as
U ¼ ðqaÞðg0 þ g2 t2 Þ,
0
(30a)
2
W ¼ b þ b t2 ,
0
2
0
(30b)
2
where b , b , g , g are functions of q; because of the factor of (qa) extracted in Eq. (30a) and dimensional reasons they
depend only on (qa)2. By putting Eq. (16) into Eq. (11) and reversing the order of summation, we see that these new
q-dependent coefficients are related to those in Eq. (16) by
b2j ¼
1
X
n¼j
2n
b2j
g2j ¼
2n ðqaÞ ,
1
X
2n
g2j
2n þ 1 ðqaÞ :
(31)
n¼j
In this approximation, V and T are no longer functionals of U(r) and W(r), but functions of the discrete variables b0, b2, g0,
g2. For example, Eq. (27a) is readily evaluated to be
0
0 2
2
T~ ¼ ðb Þ2 þ b b þð1=3Þðb Þ2 þ ðqaÞ2 ½ð1=2Þðg0 Þ2 þð2=3Þg0 g2 þ ð1=4Þðg2 Þ2 ,
(32)
while Eqs. (27b)–(27d) are likewise quadratics that are only slightly more complicated.
The actual minimization is straightforward. The four variables satisfy two constraints due to the boundary conditions
(10), and there is an arbitrary normalization say b0 ¼1. Thus the ratio (26) depends on only one remaining variable, say g0,
in terms of which the minimum is sought. It is possible to carry this out analytically as a function of n, but for simplicity we
have only done so numerically for n ¼0.3. The results for Y are given in Table 4 and Fig. 1, showing 5% accuracy for H/Lr1.0
(or qa r p), a further improvement over the expansion to the corresponding power of qa (Section 4), and taking the theory
well beyond the domain of long and thin rods. The improvement is guaranteed, since the approximate solution in Section 4
is within the class (30) over which the minimization is carried out. Importantly, as the maximum power of r is increased,
the minimization is carried out over a larger set, so the results must improve monotonically—an important advantage over
the expansion in qa.
Incidentally, since the exact solution in the qa-N limit appears to be a Rayleigh surface wave [14,15] with
displacement fields of the schematic form exp [cq(r a)] with c¼O(1), a variational trial function that allows for such
possibilities may yield accurate results over the whole range of qa.
6. Conclusion
The present problem of longitudinal vibrations in a circular rod has an exact solution [3], so the purpose of this paper is
not to determine the answer, but to develop and evaluate approximation methods that can be transplanted to other
situations. In these terms, we have shown that the classic and Rayleigh–Love theories can be understood as the first two
terms of a systematic expansion in powers of (qa)2, so that ad hoc assumptions (even though they are based on powerful
insights) can be avoided in the derivations of these theories, and the nature of the corrections can be precisely
characterized (namely the next power of (qa)2). These conclusions have been useful in the analogous problem of flexural
vibrations and in particular in placing Timoshenko theory [1] and the shear coefficient [4-10] on an unambiguous
foundation [11].
The present study also highlights the importance of a number of conditions that a good approximation scheme needs to
satisfy: (a) accuracy in the thin limit, i.e. being correct to a few powers of (qa)2; (b) the correct asymptotic power, i.e. the
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phase velocity being constant for qa-N; (c) positivity for o2; (d) monotonicity of the accuracy in the order of
approximations. The various approximation schemes are compared in these attributes in Table 6; the variational method
based on a polynomial in r satisfies all these requirements, and unsurprisingly is accurate over the largest range of qa, so
much so that it should be practically useful for any circular cylinder that can reasonably be called a ‘rod’. These lessons
should be useful when more complicated geometries are discussed.
Acknowledgment
We thank C.Y. Ng for help with the graphics.
Appendix A. Real formulation
Since all the coefficients in the defining equations are real, the real and imaginary parts of a complex solution are
separately solutions. So the imaginary part of Eq. (7) leads to a real solution
u ¼ ðx=aÞUðrÞcos qz,
v ¼ ðy=aÞUðrÞcos qz,
w ¼ WðrÞsin qz,
(A1)
ur ¼ ðx=rÞu þ ðy=rÞv
¼ ðr=aÞUðrÞcos qz:
(A2)
and the first two can be combined into
To all these spatial functions should be appended the same sinusoidal factor in time, say sin ot. The rest of the paper,
from Eq. (8) onwards, then involves purely real quantities.
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