Nonlinear dynamics of straight fluid-conveying pipes with general
boundary conditions and additional springs and masses
T. Zhang 1,2,*, H. Ouyang 2, Y.O. Zhang 1, B.L. Lv 2,3
1. School of Naval Architecture and Ocean Engineering, Huazhong University of Science & Technology, Wuhan 430074, China
2. School of Engineering, University of Liverpool, The Quadrangle, Liverpool L69 3GH, UK
3. College of Power and Energy Engineering, Harbin Engineering University, Harbin, 150001, P. R. China
Abstract
In this paper, nonlinear equations of three-dimensional motion are established for straight
fluid-conveying pipes with general boundary conditions. Firstly ten springs at the two ends of the
pipes are used to represent the general boundary conditions, and the displacements are expressed as
the superposition of a Fourier cosine series and four supplementary functions to satisfy the
boundary conditions. Then in the Lagrangian framework, three coupled equations of motion in
compact matrix form are derived by using the extended Hamilton’s principle, and the effect of the
extra linear springs and lumped masses at arbitrary positions along the pipe are also considered in
these equations. The natural frequencies of pipes with several different boundary conditions are
computed by the linearized dynamic equations in order to validate the method through comparisons
with results from other reliable sources. Nonlinear dynamic behaviour of the pipes with different
boundary conditions is analysed by solving the nonlinear equations using the Runge-Kutta scheme
at last. It is found that the method presented in the paper could conveniently and effectively solve
the nonlinear vibration problem of fluid-conveying pipes with general boundary conditions and
additional springs and masses.
Keywords: Nonlinear, Dynamics, Fluid-conveying pipes, General boundary conditions,
Stability
*
Corresponding author. Present address: School of Engineering, University of Liverpool, The Quadrangle, Liverpool L69 3GH,
UK. Tel.:+86 13995559242;
E-mail address: [email protected] (T. Zhang); [email protected] (H. Ouyang); [email protected]
(Y.O. Zhang)
1
1. Introduction
Fluid-conveying pipes are one of the most common and important structures widely used in
engineering and the dynamics of these pipes has been a popular research topic. A significant
amount of research has been carried out on dynamics of fluid-conveying pipes. From the literature
in the past 50 years, various numerical or approximate analytical methods have been applied to pipe
dynamics [1-4]. The research into dynamics of straight pipes can be roughly divided into two
categories according to the dynamic characteristics of the pipes: those with rigid boundaries so that
the non-conservative flow-induced force has no effect on the system stability and those with nonrigid boundaries so that the non-conservative flow-induced force has a strong influence on the
system stability.
A pipe whose translations at both ends are fully constrained is a conservative system if
dissipative damping is ignored, and its equations of motion are typically described in the
Lagrangian framework and derived from Hamilton’s principle or Newton’s laws of motion. The
equations can be discretized and solved by many methods. The modal superposition method [5-7]
and the Galerkin method [8-10] are conveniently used for pipelines modelled as beams. There are
other similar approaches such as Differential Quadrature Method (DQM) [11, 12], Differential
Transformation Method (DTM) [13, 14] and Finite Difference Method (FDM) [15]. A pipe can
even be taken as a cylindrical shell [16]. The finite element method (FEM) is more suitable for
complex pipe systems [17, 18]. Moreover, the interaction between pipes and fluids can be studied in
more details by the FEM, or by the Spectral Element Method (SEM) [19, 20]. In addition, the wave
approach [21] and the transfer matrix method (TMM) [22, 23] are also well-suited to analysis of
periodic or multi-span pipelines. It seems that most of the afore-mentioned papers are more
concerned with linear dynamics of pipes and the influence on the structure of fluids than with the
nonlinear dynamic behaviour of pipes [24].
Being an inherently non-conservative system, the dynamic behaviour of a cantilever fluidconveying pipe is more interesting and complicated than a simply-supported or clamped pipe (either
2
of which is a conservative system when damping is absent), especially at very high fluid velocity.
The motion of the pipe exhibits complex bifurcations and even chaos usually associated with strong
nonlinearities [25-30]. It should be noted that Païdoussis and his co-workers set up a set of
nonlinear equations of motion in the Eulerian framework for large deflection. Only the lateral
motion of the pipe as a function of the curvilinear coordinate along the centreline of the pipe needs
to be studied because the longitudinal displacement can be expressed by the lateral displacement
with the inextensibility condition [3, 26, 27]. Vertical cantilever pipes with spring supports and
concentrated masses in 2D vibration [31-33] and 3D vibration [34-38] were studied, and these
works revealed that the vibration involved bifurcation and was really complex for non-conservative
pipe systems. On the other hand, it is also very important to get convergent and accurate results at
lowest possible cost [39]. In contrast with the Eulerian approach mentioned above, a novel extended
Lagrange version of the non-linear equations for cantilever pipes was also presented [40] and the
numerical results agreed with the results of [2]. Moreover, it is worth noting that a threedimensional, geometrically exact theory for flexible fluid-conveying tubes with variable cross
sections was derived in [41]. The dynamic change of the cross section showed significant effects on
both the linear and nonlinear solutions.
In this paper, new 3-dimensional nonlinear equations of motion are derived from the extended
Hamilton’s principle for a fluid-conveying pipe with general boundary conditions. In contrast with
other published works, the displacements in the three perpendicular directions are expressed as
special Fourier series [42], which is also called improved Fourier series method in some papers [43,
44]. One major advantage of the method is that all kinds of boundary conditions can be covered by
suitable springs at both ends of the pipe. The unknown coefficients of the special Fourier series can
be viewed as generalized coordinates of displacements. Another advantage is that energy variation
is with respect to each generalized coordinates but not directly to displacements. Then by means of
the extended Hamilton’s principle [1, 3], the equations of motion with square and cubic nonlinear
stiffness terms can be conveniently obtained for the fluid-conveying pipe. Moreover, linear springs
3
and lumped masses at arbitrary positions along the pipe can also be included in the dynamic system
in order to widen the practical application of this method. Its validity and accuracy are confirmed by
results of simulated numerical examples compared with the results from the published literature,
and the linear and nonlinear dynamic behaviour is analysed in several cases with different boundary
conditions at last.
2. Problem formulation
To simplify the analysis, a uniform Euler-Bernoulli beam is used to model a typical slender
fluid-conveying pipe. Some devices and elastic supports on the pipe are included as lumped masses
and linear springs, as shown in Fig. 1. A Cartesian coordinate system is also defined.
Fig. 1 Schematics of a pipe conveying fluid with multiple elastic supports and lumped masses.
To represent general boundary conditions at both ends of the pipe, there are respectively five
springs (two rotational springs about the y and z axes and three linear springs in three perpendicular
directions) [42]. In addition, there are ns elastic supports modelled as linear grounded springs and
nm devices modelled as lumped masses along the pipe, with the location of the i-th set of springs k xxi ,
k xyi and k xzi marked by xi  i  1, 2,...ns  and the location of the j-th lumped mass mx j marked by
4
x j  j  1, 2,...nm  , where the superscripts indicate the directions of the springs. Taking into account
the springs at the two ends of the pipe, and x0 and xl added into the sequence xi  , then the
stiffness of linear springs at the two ends are denoted by kxx0 , kxy0 , kxz0 , k xxl , k xyl and k xzl . Similarly
the stiffness values of the rotational springs are denoted as K xy0 , K xz0 , K xyl and K xzl . There is always
fluid flow inside the pipe at a constant velocity V , and a balanced tensile force T0 is applied at both
ends (tension is positive and compression is negative).
3. Equations of Motion
For a three-dimensional Euler-Bernoulli beam, respectively the longitudinal, transverse and
vertical displacements of a point on the pipe in the three directions are given by:
v  x, t 
w  x, t 

z
U x  x, y, z , t   u  x, t   y
x
x

U  x, y, z , t   v  x, t 
 y
U z  x, y, z , t   w  x, t 

(1)
where x , y and z are the coordinates of a point of the pipe in the Lagrangian coordinate system,
and u , v and w are the longitudinal, transverse and vertical displacements of the neutral axis of
the pipe, respectively.
Using a nonlinear strain-displacement relation and neglecting the shear strain, the normal
strain of a point in the pipe can be expressed as:
2
2
U x 1  U x  1  U y  1  U z 
x 
 
  

  
x 2  x  2  x  2  x 
2
(2)
Adding the longitudinal normal strain caused by the axial force T0 acting at the ends of the
pipe, the x-direction normal strain becomes:
x 
T0
1
1
2
 u  u2  v2  w2   1  u  yv  zw    yv  zw 
EAp
2
2
5
(3)
where the primes denote the differentiation with respect to the longitudinal coordinate x , and Ap
denotes the cross-sectional area of the pipe, and E is the Young's modulus of the pipe’s material.
Only considering the longitudinal normal strain in this study, the strain energy U p of the pipe
is:
Up 
2
1 l
E



x dAdx
2 0 Ap
(4)
Substituting Eq. (3) into Eq. (4), and considering that the longitudinal displacement of the pipe
is a one-order lower quantity than the transverse and vertical displacements, the following equation
is obtained:
Up 
1 2
 2
2
2
2
2
2 2










u

2

u


u



u
v

w

v

w





 dx
0
0
0
0
2 0 
4
EAp
l
(5)
EI

2
1 2

2 
5
0 1   0  3u  2  v  w  dx  Θ( )
l
where  0  T0 / EAp , and I is the second moment of Ap . Θ(5 ) represents terms in the fifth order
of magnitude or above and thus are subsequently ignored.
The displacement vector R of an arbitrary point  x, 0, 0  on the centre line of the pipe can be
written as:
R   x  u  i + vj + wk
(6)
where i , j and k are the three orthogonal basis vectors in the Cartesian coordinate system,
respectively.
The velocity vector Vp of the same centre point is:
Vp  ui  vj  wk
in which, the overhead dot represents differentiation with respect to time t as usual.
The kinetic energy T p of the pipe can be represented by:
6
(7)
Tp 
mp
2
 u
l
2
0
 v 2  w2 dx
(8)
where m p is the mass per unit length of the pipe. It is assumed that the fluid velocity is uniform
over the entire cross-section of the pipe.
Because of the deformation of the pipe, the relative velocity vector Vr of the fluid at the centre
point in the pipe cross section is given by [1]
Vr  V 1  u   i  vi  wk 
(9)
Taking the motion of the structure into account, the absolute fluid velocity vector Va at the
same point in the Cartesian coordinate system can also be written as follows:

Va  Vp  Vr  u  V 1  u  i   v  Vv  i +  w  Vw  k

(10)
When the rotary inertia effect and the secondary flow effect are neglected, the kinetic energy
T f of the fluid inside the pipe can be described as:
Tf 
mf
2
 u
l
0
2
 v 2  w2   2V u 1  u   vv  ww

(11)
V 2 1  u   v2  w2  dx


2
where m f is the mass per unit length of the fluid.
Considering the elastic supports at the two ends of the pipe and ns intermediate linear springs,
the total potential energy of all springs may be expressed as:
Uk 
1 ns 1 x 2
1
 k xi u  xi   k xyi v 2  xi   k xzi w2  xi    K xy0 v2  x0 

2 i 0
2
1
1
1
 K xy0 v2  xl   K xzl w2  x0   K xzl w2  xl 
2
2
2
(12)
Similarly, for the nm lumped masses on the pipe, the total kinetic energy of them can be
written as:
7
Tm 
1 nm
mx u 2  x j   v 2  x j   w2 ( x j ) 
2 j 1 j 
(13)
Suppose an external concentrated force vector F acts at x f on the pipe. Then the virtual work
done by all external forces may be expressed as:
δW  T0 δu  xl   δu  x0    Fxxf  t  δu  x f   Fxyf  t  δv  x f   Fxzf t  δw  x f 
(14)
where Fxxf , Fxyf and Fxzf are the components of force F projected onto the x, y and z axes,
respectively; δ is the variational operator.
For the above pipe, implementation of the Hamilton’s principle leads to equation below:

t2
t1
t2
δ(Tp  Tf  Tm  U p  U k )dt   δWdt
t1
   m f V  R  V   R  dt  0
t1
0
(15)
l
t2
where the last term of Eq. (15) on the left-hand side is the variations of the virtual momentum at the
two ends. This term becomes zero when both ends of the pipe are constrained.
 is the outward normal vector at the two ends of the pipe whose expression is respectively:
 0   1  u  x0   i  v  x0  j  w  x0  k  ;
 l  1  u  xl   i  v  xl  j  w  xl  k  ;
(16)
The improved Fourier series method [43] is used to transform the equations of motion in the
physical coordinates into those in the generalised coordinates and is described below.
The three displacements can be expanded as:
 u ( x, t )   Tu ( x)q u (t )

T
 v( x, t )   v ( x)q v (t )
 w( x, t )   T ( x)q (t )
w
w

(17)
where  u ,  v ,  w , q u , q v and q w are vector functions.
In order to simplify the equations, the following dimensionless quantities are introduced:
8
 EI
x
u
v
w
  ,   ,   ,   ,  
m m
l
l
l
l
p
 f
1/2



1/2
mf
Mf 
t
,  
,
 Vl ,  
2
m f  mp
l
 EI 
Ap l 2
k xxi l 3
k xyi l 3
k xzi l 3
K xy0 l
T0l 2
η
ς
ζ
ς


,
,
,
,
,
,
 i 
 i 
 i 
0 

I
EI
EI
EI
EI
EI
mx j
Fxxf l 2
K xzl l  
η
j
 
 
 
 m f  m p  l ,  f  EI ,
EI ,
EI ,
EI ,
ζ
0
K xz0 l
ς
1
K xyl l
ζ
1
(18)
ς 
f
y 2
xf
F l
EI ,
ζ 
f
z 2
xf
F l
EI
Following the improved Fourier series method, these vector functions described above now
become:
    1, cos   , cos  2  , , cos  n  , f ( ), f ( )T
1
2
 η
        1, cos   , cos  2  , , cos  n  , f ( ), f ( ), f ( ), f ( )T
ζ
1
2
3
4
 ς
T

η
η
η
η
η
q η    a0 ( ), a1 ( ), , an ( ), c1 ( ), c2 ( )

T
q ς    a0ς   , a1ς   , , anς   , c1ς   , c2ς   , c3ς ( ), c4ς ( )

q    a ζ   , a ζ   , , a ζ   , c ζ   , c ζ   , c ζ ( ), c ζ ( )T
0
1
n
1
2
3
4
 ζ
(19)
where the elements of vectors q η , q ς and q ζ are functions of only dimensionless time  . Functions
fi   (i  1, 2,3, 4) are defined by:

27    1
 3 
 f1    28 sin  2   28 sin  2 
 




27
   1
 3 
 f 2     cos    cos 

28

 2  28
 2 

 f    3 sin     1 sin  3 
 


 3
4  2  4  2 

 f     3 cos     1 cos  3 
 


 4
4
 2  4
 2 
9
(20)
Actually vector functions  η ,  ς and  ζ in the Fourier series are all truncated to a finite
number of terms N in this study, and vectors q η , q ς and q ζ can be considered sets of generalised
coordinates of the three displacements.
Substituting Eqs. (5), (8), (11), (12), (13) and (17) into Eq. (15), a procedure of variational
operations with respect to all these generalised coordinates is executed, and then the following
equations governing the motion of the fluid-conveying pipe are obtained:
 q Tς N1q ς   q Tζ N1q ζ 
 T
 

q ς N 2q ς  q Tζ N 2q ζ 

M η q   C ηq   K ηq η 

 ξηf f η , n  N  3

 

 T
  T

q ς N nq ς  q ζ N nq ζ 
 q Tη N1q ς 
T
 T
   q ς 11q ς
q
N
q


M ς q ς  Cς q ς  K ς q ς   η 2 ς    


 T
  q Tς n1q ς
q
N
q
 η n ς 
 qTζ 11q ζ


qTζ n1q ζ

q Tς 1nq ς 


q Tς nnq ς 
(21.b)
q ζT 1nq ζ  

ς
  q ς  ξf fς , n  N  5
q Tζ nnq ζ  
 q Tη N1q ζ 
T
 T
   q ζ 11q ζ
q Nq

M ζ q ζ  Cζ q ζ  K ζ q ζ   η 2 ζ    


 T
  q Tζ n1q ζ
q η N nq ζ 
 q Tς 11q ς


q Tς n1q ς

(21.a)
q Tζ 1nq ζ 


T
q ζ nnq ζ 
(21.c)
q ςT 1nq ς  

ζ
  qζ  ξf fζ , n  N  5
q Tς nnq ς  
in which, N i and ij are symmetric matrices, and N i are asymmetrical matrices and they are
related to the nonlinear terms in Appendix A. M η , M ς and M ζ are mass matrices, and the
10
dimension is  N  5   N  5 for M ς and M ζ , and is
 N  3   N  3 for M η ; f η , f ς and f ζ
are
vectors due to the external load. In addition,
nm

ˆ
M

M

M


 ς
ζ
i M j
j 1

 M  M (1: N  3;1: N  3)
ς
 η
(22)
These mass matrices are all symmetrical, and their elements are given in Appendix A. For the
axial motion, mass matrix M η is formed from the first N  3 columns and the first N  3 rows of
matrix M ς ; Cη , C ς and Cζ are damping matrices, each of which consists of two parts if the
Rayleigh damping is ignored. One part is an anti-symmetrical matrix C caused by the Coriolis
force of the fluid inside the pipe, whereas the other part is a symmetrical matrix C caused by the
non-conservative force at both ends of the pipe. These damping matrices can be written as follows:
Cς  Cζ  β   C  C 

Cη  Cς 1: N  3;1: N  3
(23)
The elements in matrices C and C are given in Appendix B, and Cη is also a part of C ς , in
the same way as M η is obtained from M ς . If viscosity of the fluid is considered, a linear damping
matrix C should be added into the right-hand side of Eq. (23) due to internal energy dissipation of
the pipe as a result of the friction between the pipe and the fluid:
C  c0M  c1K
(24)
where c0 and c1 are Rayleigh damping constants, and K is the stiffness matrix of the pipe itself
introduced below.
K η , K ς and K ζ are stiffness matrices in the three perpendicular directions respectively, and
they can be expressed as sums of several matrices:
11
ns



2 ˆ
2
K






K

K


K

K1i  1: N  3;1: N  3

 η 


i 1



ns


2 ˆ
 K ς   2 K  K 2i
K ς  1   /   K       K
i 1

ns

ˆ  K   2K  K
K ζ  (1   /  )K      2  K

ζ
3i

i 1
(25)
K is a symmetrical matrix. Matrix K̂ on the right-hand side of Eq. (25) is due to the
centrifugal force caused by the fluid flow and the force acting at both ends of the pipe. Matrix K is
due to the elastic supports at both ends of the pipe. Matrix K is an asymmetrical matrix caused by
the non-conservative force at both ends of the pipe with elastic supports and the last matrix on the
right-hand side of the three expressions is caused by the elastic supports along the pipe.
4. Natural frequencies of fluid-conveying pipes
For modal analysis of the pipe, it is necessary to obtain the linearized equations of motion in
the neighbourhood of equilibrium positions. In this situation when the external force is neglected,
the linear equations of motion are given by:
M ηq η  C ηq η  K ηq η  0

 M ς q ς  Cς q ς  K ς q ς  0
M q  C q  K q  0
ζ ζ
ζ ζ
 ζ ζ
(26)
Actually the above equations which are independent of each other are the free vibration
equations of three displacements of the fluid-conveying pipe respectively, and the natural
frequencies of the system are solved numerically first. The three equations in Eq. (26) are in the
same form, so only the solution process of the vertical displacement is addressed here as an
example. By means of an equation reduction technique [45], the free vibration equation of the
vertical displacement can be rewritten in the form:
AY  BY  0
12
(27)
M 0 
Cζ K ζ 
q ζ 
A ζ
, B
, Y  


 0 
  0 
q ζ 
The complex eigenvalue n of Eq. (27) satisfy:
B  n A  0
(28)
Eq. (27) can be expressed in several different forms, but with the form above matrix A is still
symmetrical and this important feature is essential in overcoming the potential ill-posedness of the
algorithm used in this paper. Solving the eigenvalue problem of Eq. (27), N  5 pairs of

eigenvalues (also called poles) (n , n ) are obtained, where ‘-’ denotes the complex conjugate. It is
known that the imaginary part of the pole is the natural frequency n while the real part is about the
stability of the system.
It should be pointed out that Eq. (28) would be a serious ill-conditioned equation with the
increase of N (for example, N  100 ). For low frequencies, the value of N has a very small effect
on the accuracy of the eigenvalues; but for higher frequencies, the effect becomes greater, and very
high frequencies may be quite wrong. In the present paper, an improved algorithm is proposed for
dealing with the ill-conditioned eigenvalue problem and its details are given in Appendix B.
All the relevant equations are derived for general elastic boundary conditions. It is very easy to
simulate all kinds of classical boundary conditions by setting the spring constants at the two ends to
appropriate values. Table 1 lists values of the spring constants for four ideal boundary conditions of
the pipe.
Table 1 The spring constants to represent the ideal boundary conditions
Boundary
 0η
 0ς
 0ζ
 0ς
 0ζ
1η
 1ς
1ζ
1ς
conditions
Cantilever
0
0
0
0
1010
1010
1010
1010
1010
Pinned-pinned 1010 1010 1010
0
0
0
1010
1010
1010
Pinned0
0
1010
1010
1010
1010
1010
1010
1010
clamped
13
1ζ
0
0
1010
Clampedclamped
1010
1010
1010
1010
1010
1010
1010
1010
1010
1010
In order to explore the convergence and efficiency of this method, the natural frequencies of
the pipe without fluids are computed and compared with those reported in [46], as in Table 2.
Table 2 The natural frequencies of the pipe as N increases    0.1,   0,   0,   8.08e4
Clamped
-clamped
Exact a
a
N
1
3
5
10
50
100
1η
903.9174
893.1090
893.0321
893.0164
893.0137
893.0137
893.0084
2η
1822.8926
1786.9814
1786.1600
1786.0423
1786.0322
1786.0322
1786.0168
3η
3547.5664
2762.0153
2682.1302
2679.0977
2679.0407
2679.0407
2679.0253
1ζ
22.4134
22.3739
22.3733
22.3733
22.3732
22.3730
22.3733
2ζ
61.9520
61.6729
61.6728
61.6729
61.6727
61.6729
61.6728
3ζ
250792.8
123.2994
120.9061
120.9038
120.9042
120.9040
120.9034
results from Blevins (1976).
It is seen that the results of this paper are close to the exact solutions presented by Blevins [46]
with increasing value of N and for the first several frequencies, only a few terms of the Fourier
series could lead to an accurate solution. Table 3 compares the first four natural frequencies of the
fluid-conveying pipe with various stiffness values of both rotational and translational constraints, in
which the Fourier series is truncated to N  50 . By changing the stiffness of both the rotational and
linear springs, the boundary conditions can turn from the pinned-pinned case to the clampedclamped case. The classical pinned-pinned and clamped-pinned cases show good agreement with
[47]. The natural frequencies of the pinned-free pipe with various stiffness of the linear springs are
shown in Table 4. Different cases with changing the stiffness of the linear springs at the pinned end
are computed, and the pinned-free pipe (  0η   0ς   0ζ  1010 , 1η  1ς =1ζ  0 , 1η  1ς  1ζ  0 )
finally turns into a cantilever pipe. In Table 5, the natural frequencies of the pipe with various
stiffness values of the linear springs at both ends are listed. The pipe is assumed with
14
 0η   0ς   0ζ  1 and 1η  1ς =1ζ  1 , and then the stiffness of the linear springs at both ends are
changed at the same time.
Table 3 The natural frequencies of the fluid-conveying pipe with various stiffness of both the
rotational and linear springs ( 1η  1ς =1ζ =1010 ,   0.1 ,   0 ,   0 )
 0η ,  0ς ,  0ζ 0η , 0ς , 0ζ
0.01
0
1
0
100
0
10
Pinned-pinned
10
0
Exacta
1010
0
10
10
10
10
10
100
Clamped-pinned
1010
1010
Exacta
1010
1010
10
10
1010
1010
1010
10
Clamped-clamped
10
1010
a
results from Thomson (1988)
1η , 1ς , 1ζ
0
0
0
0
0
0
0
0
0
10
100
1010
1
0.1726
1.7156
8.9320
9.8696
9.8696
13.4296
15.1258
15.4182
15.4182
19.6273
21.9518
22.3733
2
15.4195
15.5487
26.5041
39.4784
39.4784
44.7216
49.0452
49.9649
49.9649
55.5005
60.5461
61.6728
3
49.9653
50.0050
54.5712
88.8265
88.8264
95.0931
102.3850
104.2477
104.2477
110.7090
118.7590
120.9035
4
104.2479
104.2669
106.3011
157.9136
157.9137
164.8569
175.1760
178.2695
178.2697
185.3459
196.4160
199.8592
Table 4 The natural frequencies of the pinned-free fluid-conveying pipe with various stiffness of the
linear springs at the pinned end (  0η   0ς   0ζ  1010 , 1η  1ς =1ζ  0 , 1η  1ς  1ζ  0 ,   0.1 ,
  0,  0)
 , ,
0.01
1
100
Cantilever
1010
Exact
1010
η
0
ς
0
ζ
0
1
0.1734
1.5572
3.4477
3.5159
3.5160
2
15.4277
16.2501
21.6200
22.0345
22.0345
3
49.9749
50.8958
60.5700
61.6979
61.6972
4
104.2577
105.1983
118.7574
120.9020
120.9019
Table 5. The natural frequencies of the pipe with various stiffness of the linear springs at both ends
(  0η   0ς   0ζ  1 , 1η  1ς =1ζ  1 ,   0.1 ,   0 ,   0 )
0η , 0ς , 0ζ
0.01
1
100
1010
1η , 1ς , 1ζ
0.01
1
100
1010
2
1
1.4026 2.4940
1.4058 4.9877
1.4121 9.8859
1.4123 10.0701
15
3
22.5900
25.6353
38.7783
39.5291
4
61.7777
62.2332
87.1790
88.8490
Another verification for the accuracy of the method is provided by the example of a pinnedpinned pipe with various dimensionless fluid velocities. As seen in Table 6, the computed natural
frequencies approach the exact values closely. When    , the first eigenvalue becomes zero, and
this means that the pipe becomes unstable. For the pipes with multiple supports (ζi  1010 ) , it is
obvious that the solutions are also exactly consistent with the FEM and TMM results [22] as shown
in Table 7. Good agreement also can be found between the presented results and those reported in
[14] as shown in Table 8.
Table 6 The natural frequencies of the pined-pined pipe with various fluid velocities

0.5
1.0
2.0
3.0
𝜋
a
Presented
method
Exact a
Error(%)
Presented
Exact a
Error(%)
Presented
Exact a
Error(%)
Presented
Exact a
Error(%)
Presented
Exact a
Error(%)
  0.1,   0


3ζ
4ζ
9.7347
39.3620
88.7146
157.7973
9.7438
-0.09
9.3207
9.3563
-0.38
7.4903
7.6112
-1.58
2.8159
2.9296
-3.88
0.0000
0
----
39.3532
0.02
39.0108
38.9752
0.09
37.5847
37.4250
0.43
35.1114
34.6877
1.25
34.6684
34.1893
1.40
88.7014
0.01
88.3806
88.3250
0.06
87.0333
86.8034
0.26
84.7543
84.2062
0.65
84.3545
83.7463
0.73
157.7886
0.01
157.4726
157.4129
0.04
156.1603
155.9008
0.17
153.9655
153.3477
0.40
153.5802
152.8992
0.45
ζ
1
ζ
2
results from Blevins (1976).
Table 7 The natural frequencies of the pined-pined pipe with six middle supports
EI
1
 0.2629
M p  M f l2
  0.5379,   0,

0.1l 0.1l
1ζ
2ζ
0.1l
3ζ
16
0.4l
0.1l 0.1l 0.1l
4ζ
5ζ
0
a
Presented
113.6072
321.8330
637.3554
986.9637
1042.8165
TMM a
113.6057
321.8237
637.1004
986.0757
1041.9966
FEM a
113.6314
321.8364
637.3842
987.0748
1042.8857
3.00
110.1645
318.4551
633.7701
982.5034
1037.9208
6.00
99.5001
308.2218
622.9415
969.0158
1023.6765
12.00
47.7057
265.3897
578.4943
913.2545
964.6553
13.65
0.0000
247.4455
560.4837
890.5315
940.8224
results from Wu (2001).
Table 8 Critical velocities of different pipe types
Critical Velocities
Pipe Type
Ni
Païdoussis Present
st
Pinned-pinned Pipe (1 mode)
3.1416
π
3.14
st
nd
Pinned-pinned Pipe (1 and 2 combined)
6.3941
6.38
6.40
st
Clamped-clamped Pipe (1 mode)
6.2832
2π
6.28
st
nd
Clamped-clamped Pipe (1 and 2 combined) 9.2946
9.3
9.29
rd
Cantilevered Pipe (3 mode)
9.3224
9.3
9.33
st
Clamped-spring-mass (1 mode)
5.59
5. Stability of the fluid-conveying pipes
The dynamic behaviour of the fluid-conveying pipe is very complicated with various boundary
conditions, and sometimes the stable region and unstable region may intertwine with increasing
fluid velocity. The minimum velocity at which the pipe loses stability is of most concern usually. A
dimensionless critical velocity  c is defined as:
c  min(d , f )
(29)
where d and c are the critical velocity for divergence and flutter instability respectively.
Numerical methods such as the bisection method can be used to determine the minimum
unstable velocity form the curves of the eigenvalues versus fluid velocities. For these three kinds of
ideal boundary conditions (Pinned-Pinned, Clamped-Clamped and Clamped-Pinned), one can see
17
the pipes are a gyroscopic system, and therefore the critical velocity is independent of  shown by
black solid lines in Fig. 2.
Fig. 2 The critical instability velocity c as a function of  with various boundary conditions.
The pipe always loses stability by divergence at first for fully or partly constrained boundaries.
However, for a cantilever pipe, flutter instability may occur first, and the dimensionless critical
velocity increases with the increase of  . Moreover, the curve can be roughly divided into three
segments, and in these different regions instability of the cantilever pipe appears with different
modes. If a linear spring is added at the free end of the pipe, the stiffness of the spring would have a
great influence on the stability of the pipe. When the dimensionless stiffness coefficient 1ζ is
relatively large (for example,  50 ), the critical velocity is independent of  ; but when 1ζ is
relatively small, it increases the critical velocity of pipe when   0.3 . Fig. 3 shows the critical
velocity of the cantilever pipe with a lumped mass and a linear spring at the free end. It is obvious
that the critical velocity decreases with the increase of the lamped mass when   0.7 .
18
Fig. 3 The critical velocity c as a function of  for different lumped mass 1 .
For a pinned-pinned pipe supported by two rotational springs of an identical stiffness value at
the two ends, the critical velocity curves versus stiffness coefficient  0ζ are demonstrated in Fig. 4.
When 0ζ  100 , the critical velocity  c is nearly equal to that of the clamped-clamped pipe. In
addition, the effects of the axial force  are also shown in Fig. 4. It is evident that the axial
compressive force reduces the critical velocity.
19
Fig. 4 The critical velocity c versus the rotational spring stiffness at two ends of the pipe.
6. Nonlinear dynamic responses
When N is large, it will be costly and very difficult to get convergent results if one solves the
original set of time-dependent Eq. (21) directly. Therefor a transformation method will be used to
overcome this drawback of the equations.
For the pipe with zero fluid velocity, one can get the i -th vibration mode

ς
ζ
 i ,  i and  i from
Eq. (26), and then a modal transformation may be introduced as follows:
η
η
η
q η  Q η X η ; Qη   ,   
nη 
 1 2

n
 N  1
(30.a)
ς
ς
ς
q ς  Q ς Xς ; Qς   ,   
nς 
 1 2

n
ς
 N  1
(30.b)
ζ
ζ
ζ
q ζ  Q ζ X ζ ; Q ζ    ,   
nζ 
 1 2

n
 N  1
(30.c)
η
ζ
By substituting Eq. (30) into Eq. (21), and left-multiplying it by QTη , QTς and QTζ respectively,
then the new reduced equations of motion may be expressed as:
20
QTη MηQη Xη  QTη ξηf fη  QTη CηQη Xη  QTη K ηQηXη
  XTς Q Tς N1Q ς Xς   XTζ Q Tζ N1Q ζ X ζ  
 T T
 

X Q ς N 2Q ς Xς   XTζ Q Tζ N 2Q ζ X ζ  
T  ς
Q η 


 

 T T
  T T

  X ς Q ς N n Q ς X ς   X ζ Q ζ N nQ ζ X ζ  





(31.a)
 XTη Q Tη N1Q ς Xς 
 T T

X Q η N 2Q ς X ς 
T
T 
T
T
T  η
Q ς M ς Q ς X ς  Q ς ξf f ς  Q ς CςQ ς X ς  Q ς K ςQ ς X ς  Q ς


 T T

 X η Q η N nQ ς X ς 
  XTς QTς 11Q ς Xς

QTς  
  XT QT  Q X
  ς ς n1 ς ς
 XTζ QTζ 11Qζ Xζ


 XTζ QTζ  n1Qζ Xζ

XTς Q Tς 1nQ ς X ς 


XTς Q Tς nnQ ς X ς 
(31.b)
XTζ QTζ 1nQ ζ Xζ  

  Qς Xς
T T
Xζ Q ζ nnQ ζ Xζ  
 XTη Q Tη N1Q ζ X ζ 
 T T

X Q η N 2Q ζ X ζ 
T
T ζ
T
T
T  η
Q ζ M ζ Q ζ Xζ  Q ζ ξf fζ  Q ζ Cζ Q ζ X ζ  Q ζ K ζQ ζ X ζ  Q ζ


 T T

 X η Q η N nQ ζ Xζ 
  XTζ QTζ 11Qζ Xζ

QTζ  
  XT Q T  Q X
  ζ ζ n1 ζ ζ
 XTς QςT 11Qς Xς


 XTς QςT n1Q ς Xς

XTζ Q ζT 1nQ ζ Xζ 


XTζ Q Tζ nnQ ζ Xζ 
(31.c)
XTς QTς 1nQ ς Xς  

  Q ζ Xζ
T T
Xς Q ς nnQ ς Xς  
For the multi-degree-of-freedom dynamic system with square and cubic stiffness nonlinearities,
although there are many different numerical methods which can be employed to obtain very
accurate numerical solutions [28], a fourth-order auto-adaptive step-size Runge-Kutta scheme is
applied to deal with the time-dependent system more conveniently to demonstrate the effectiveness
21
of this method. In order to reduce the cost of computation and maintain appropriate accuracy of
results, N is generally taken as 5 and nη  3 , nς  nζ  6 .
As an example, the nonlinear dynamic responses of a clamped-clamped fluid-conveying pipe
without viscous damping are investigated firstly with an external harmonic force at the middle of
ζ
the pipe only acting in the vertical direction (0.5
 f0sin f  ) , and the initial conditions employed
here are that displacement and velocity of the pipe are zero. The external harmonic force is very
small and is introduced to tease out the stability behaviour from the dynamic response. This idea
was also used for this purpose in [6].
In order to verify the nonlinear model, the vertical dynamic responses obtained respectively
from the nonlinear and linear equations are compared in Fig. 5. When the fluid velocity is lower
than the critical velocity c  2 , the pipe remains stable under the small disturbances of the
external load. It can be seen that the results of nonlinear and linear models completely agree in the
subcritical range. If the fluid velocity is greater than  c , the results of the linear model exhibit
divergence or flutter instability; however a convergent solution is possible from the nonlinear model.
Fig. 5 also demonstrates that even with a small perturbation the vibration amplitude of the pipe will
suddenly increase by a large amount as long as   c and the displacement will vibrate around a
new equilibrium point (which is not the initial static equilibrium point).
22
Fig. 5 Dynamic responses of the vertical displacement at the middle of a clamped-clamped pipe
from the nonlinear and linear models. (  =0.5,  =2.0e3, f 0 =0.001,  f =24 π )
Proportional viscous damping may have a great influence on the dynamic responses of the
fluid-conveying pipe when its translations at boundaries are fully constrained. The dynamic
response of a clamped-clamped pipe with different amounts of proportional viscous damping is
computed and the vertical displacement at the middle of the pipe is shown in Fig. 6. There are two
different unstable phenomena (divergence and flutter) in the whole range of the fluid velocity (up to
14), and therefore results at two typical unstable fluid velocities (   6.5 and   11.0 ) are
compared to find out the effect of the viscous damping.
23
Fig. 6 Dynamic responses of the vertical displacement at various levels of proportional damping.
(  =0.5,  =2.0e3)
In the divergence region predicted by the linear stability analysis, the vibrations of the pipe end
up in the form of regular periodic motion with low frequencies when nonlinearity is included, and
the pipe may jump between two symmetrical equilibrium positions if the viscous damping is zero.
For the flutter instability predicted by the linear model, the pipe oscillates violently at higher
frequencies than in the divergence region. Moreover, with the increase in viscous damping, the
oscillating amplitude of the pipe reduces evidently. In these two cases, if time is long enough, the
proportional damping would make the motion of pipe decay completely.
For the linear model without any viscous damping causing energy loss, the displacement of the
pipe grows continuously because of the appearance of resonance when the frequency of external
harmonic force is equal to the fundamental natural frequency of the dynamic system. However, the
displacement does not grow with time in the nonlinear model as seen from Fig. 7. Furthermore the
responses of the point at the middle of the pipe with three different boundary conditions are also
24
shown in Fig. 7, and for each boundary condition the frequency of external harmonic forces is equal
to the first natural frequency of the fluid-conveying pipe respectively.
Fig. 7 The resonance of the pipe with different boundary conditions. (  =2.0,  =0.2,  =2.0e3)
The nonlinear dynamics of the cantilever pipe is more complex and interesting . A bifurcation
diagram may be useful to illustrate the overall dynamics of the system. Therefore the results of a
cantilever pipe as a typical case are shown in Fig. 8. The system is stable at low fluid velocity
(   9.3 ), and then a Hopf bifurcation occurs at   9.4 and the pipe would lose stability. The
range of fluid velocity that leads to distinct dynamic behaviour can probably be divided into three
regions based on the nature of the dynamic response. The time histories and the phase portraits of
the tip displacement are also given in Fig. 9 at three flow velocities for the three different regions of
distinct dynamic behaviour respectively. In the region of 9.4    11.8 , the dynamic response is a
periodic oscillation at greater amplitude than that of the response at lower fluid velocities. The pipe
25
appears to experience quasi-periodic oscillations with several frequencies when 11.8    13.2 ,
and the pipe may undergo chaotic motion when   13.2 . Moreover, asymmetrical oscillations in
the y and z directions occur at a fluid velocity   12.8 , and the symmetry is regained shortly after a
small increase in fluid velocity (   12.95 ).
Fig. 8 The bifurcation diagram for a cantilevered pipe.
26
Fig. 9 The time histories and the phase portraits of tip displacement.
In addition, the longitudinal displacement at the free tip is also illustrated in the figure, and it is
clear that its amplitude increase with the fluid velocity, and wherefore the aforementioned
assumption that the longitudinal displacement is a second-order small quantity in comparison with
the other two displacements may not be valid for larger values of fluid velocity. In that case those
2
2
2
quantities including u  (or u 3 , u 4 , u2v2 , u2 w2 , u2v2 , u  w ) should not be ignored in the strain
energy given by Eq. (8). However, it is convenient to derive the new nonlinear equations of motion
by the presented method if these quantities are included.
A case of a cantilever pipe with two linear springs and a lumped mass added at the end of the
pipe is studied further to validate the approach in this paper at last. The bifurcation diagram of the
tip displacement at the free end of the pipe is shown in Fig. 10, and Hopf bifurcations are seen to
develop at about   5.65 for the transverse displacement, which is slightly greater than the
unstable velocity ( c  5.59 ) obtained by the linear analysis as shown in Table 8. However, Hopf
27
bifurcation of the vertical displacement takes place at a smaller fluid velocity   5.55 in spite of a
larger stiffness of the spring than the transverse spring. In the region of 5.6    6.6 , the pipe
oscillates symmetrically in a quasi-periodic form and vibration in the transverse direction diverges
at a higher rate than that in the vertical direction. Moreover chaotic oscillation occurs when   6.7 ,
but quasi-periodic motion can be observed at   7.1 within the chaos-dominated region.
Fig. 10 Bifurcation diagram for a cantilever pipe with tow linear springs and a lumped mass added
at the middle of the pipe.
Furthermore, at three typical flow velocities, the tip displacement projected onto the yz  plane
and its power spectrum density (PSD) are presented in Fig. 11. The periodic motion is evident for
both the transverse and vertical displacements at   5.7 , and the first dimensionless frequencies of
the motions (separately denoted with f ς and f ζ ) are different according to the PSD (the first
dimensionless frequency of the vertical motion is larger than that of the transverse motion) because
of the larger stiffness of the vertical spring. For the chaotic oscillations (observed from the graph of
the projected tip displacement) at   6.8 , there are not significant peaks in the PSD curve. It is
very interesting to see that periodic motions are regained abruptly at   7.1 , and in this case, the
28
first frequency of the vertical motion is half of that of the transverse motion. Moreover, the spacing
between the adjacent PSD peaks is the same for the two motions. Chaotic oscillations would reemerge immediately with a small increase in fluid velocity.
Fig. 11 Tip displacement projected onto yz-plane and PSD diagrams of the tip displacement at
different flow velocities.
The nonlinear dynamical behaviour of fluid-conveying pipes is very interesting and complex,
especially when extra springs and lumped masses are added. Pipe vibration due to unsteady internal
flow can also be studied by the presented method, if these flow characteristics are taken into
account in the kinetic energy when nonlinear equations of motion are derived. This will be one
future research topic.
7. Conclusions
A new set of equations of motion are derived for the linear and nonlinear vibrations of straight
fluid-conveying pipes in this paper. The mathematical expressions of these equations are
29
considerably simple and can easily be programmed. The biggest advantage of the equations is their
convenience in dealing with pipes with general boundary conditions. An analytical expression for
the pipe vibration in the form of a special Fourier cosine series and four supplementary functions is
given.
The comparison of several models in the linear stability analysis serves to demonstrate that the
results of the equations are very accurate even with small numbers of truncated terms in the special
Fourier series, and the eigenvalue analysis of the pipe is able to determine various unstable regions
for the fluid velocity. For the pipe with translations at boundaries fully constrained, the nondimensional unstable velocity is constant at various fluid mass ratio  (mass of the fluid over the
mass of the fluid plus the mass of the pipe) and divergence instability would first take place
generally. However, the unstable velocity would increase with larger  for other boundary
conditions and flutter instability may occur first (before divergence) in these cases.
When becoming unstable, the dynamics of the pipe whose translations are fully constrained at
both ends could typically evolve into a new equilibrium position after a small perturbation, while
the nonlinear oscillations of the pipe are normally asymmetric about the initial equilibrium position.
Moreover, viscous damping would significantly affect the pipe vibration. For pipes with
translations at boundaries allowed at one end, unlike pipes whose translations at boundaries are
fully constrained, both unstable symmetrical and asymmetrical oscillations could occur when
subjected to an external disturbance. With the increase in the fluid velocity, the nonlinear vibration
would gradually evolve into three different forms: periodic motion, quasi-periodic motion and
chaotic motion; and the magnitude of oscillations would increase too. In addition, the longitudinal
displacement can no longer be taken as a lower order quantity than the other displacements when
the fluid velocity is quite large.
Acknowledgements
30
This study was supported by the National Defence Fund of Huazhong University of Science
and Technology (No. 01-18-140019). A large part of this work was carried out at the School of
Engineering, University of Liverpool during the year-long visit of the first and fourth authors.
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35
Appendix A:
Since the special Fourier series expressions of the displacements are composed of two distinct
parts, all square system matrices for linear terms in Eq. (21) can be decomposed into four parts
below, in order to concisely express them.
M
M   aaT
Mac
 ˆ
ˆ   K aa
K
ˆ T
K
ac
M ac 
 C
; C   aa T

Mcc 
 Cac
η
ˆ 
K
 K aa
ac
 ; Kη   η T
ˆ 
K
K ac
cc 
Cac 
C
; C   aaT

Ccc 
Cac
η

 K ςaa
K ac
;
K

 ςT
ς
η 
K cc

K ac
Cac 
 K aa
; K  T
Ccc 
K ac
ς

 K ζaa
K ac
;
K

 ζT
ζ
ς 
K cc

K ac
K ac 
;
K cc 
ζ

0 K ac 
K ac
;K  
;
ζ 
K cc 
0 K cc 
where superscript T denotes the matrix transpose operation, and the dimension of all the partitioned
matrices with subscript aa is ( N  1)  ( N  1) . Moreover, the dimension of all the partitioned
matrices with subscript cc is 4  4 . Then the elements of these partitioned matrices are given by:
mij  1 i  j  0

Maa  [mij ] N 1 N 1 , mij  0.5 i  j  0 ;

mij  0 i  j
Caa  [cij ] N 1 N 1

i 2 j 2 4
k 
K aa  [kij ] N 1 N 1 ,  ij
2
kij  0

i j
cij  0


 1i  j   1  i 2  j 2 
, 
;


cij 
i j
j2  i2


i j
i j
ˆ  [kˆ ]
;K
aa
ij  N 1 N 1
 ˆ ij 2
kij 
2
, 
kˆ  0
 ij
i j
i j
K aaη  [kijη ]( N 1)( N 1) , kijη  10  (1)(i  j ) 11 ; K ςaa  [kijς ]( N 1)( N 1) , kijς   20  (1)(i  j )  21 ;
K ζaa  [kijζ ]( N 1)( N 1) , kijζ   30  (1)(i  j )  31 ; Caa  [cij ] N 1 N 1 , cij  1   1
36
i  j 
;
;
Mac  [mis ] N 14
Cac  [cis ] N 14
Cac  [cis ] N 14
ˆ  [kˆ ]
K
ac
is  N 14
ci1   1i

ci 2  1
, 
;
i
ci 3   1

ci 4  1

120  48i 2
m

 i1
7  16i 4  40i 2  9 


i 1
mi 2   1  mi1
, 
;
12
m 
 i 3  16i 4  40i 2  9 

m   1i 1  m
i3
 i4
2
4

i 63  208i  112i
c


1


 i1
7(16i 4  40i 2  9)

c   1i  c
i2
i1
, 
;
2
4
c  1 i 9  16i  16i
 i 3   16i 4  40i 2  9

i
ci 4   1  ci 3
K ac  [kis ] N 14

120i 2  48i 4  

ˆ
ki1 
7 16i 4  40i 2  9 


i 1
ˆ
ˆ
, ki 2   1  ki1
;
2
ˆ
12i 
 ki 3 
4
16i  40i 2  9

kˆ   1i 1  kˆ
i3
 i4
K ςac  [kisς ] N 14
 kiς1  (1)i  21
 ς
 ki 2   20
,  ς
;
i
 ki 3  (1)  21
 ς
 ki 4   20

27 3i 2
k

 i1
7 16i 4  40i 2  9 


i 1
k  1  ki1
,  i2  
;
4 3

12i 
ki 3  16i 4  40i 2  9

ki 4   1i 1  ki 3
η
K ac
 [kisη ] N 14
K ζac  [kisζ ] N 14
37
 kiη1  (1)i 11
 η
 ki 2  10
,  η
;
i
 ki 3  (1) 11
 η
 ki 4  10
 kiζ1  ( 1)i  31
 ζ
 ki 2   30
,  ζ
;
i
 ki 3  (1)  31
 ζ
 ki 4   30
K ac  [kis ] N 14
3

ki1  7

( i 1)

, ki 2   1 ki1 ;
k  0
 i3

 ki 4  0
where i, j  0,1, 2  N and s  1, 2,3, 4 .
 365
 784

  253
 294
M cc  
 41
 112

19
 
 42
253
294
365
784
19

42
41
112

1 0
0 1
Ccc  
1 0

0 1
 3692
 3136

 111π

ˆ   392
K
cc
2
 45
 448

 3π
 8
11 0 11
0 
0
10
η
K cc  
11 0 11

 0 10 0
1
0
1
0
41
112
19

42
5
16
1

6
19 


 0
42


41 
  363π
 784
112 
 ; Ccc  
1 
 3


6 
7


5
39

 
16 
 112

 4054
 12544

0
 273
  1568
1
; K cc  
4
0
 81

 1792
1

3
 27
 224
111
392
3692
3136
3π
8
452
448
452
448
3π
8
9 2
64
3
8
27 3
1568
4054
12544
27 3
224
814
1792

363π
784
0
39
112
3
7
814
1792
273
224
454
256
9 3
32
3π 
8 
 0 1
452 

3π  1 0
448 
 ; K cc 
7  0 1
3 

8 
 1 0

2
9 
64 
3
7
39

112
0

3
16
39 
112 

3

7
;
3 
16 

0 

27 3 
224 
814 
1792 
;
9 3 
32 

454 
256 
0 0
0 0 
;
0 0

0 0


9 2



0

0 
21
20
21

0
49


10  ς 
9 2

0
 20 
21 0  20  ;
; K cc  
0
49



 21
0
 21 0 
10 


0
 20
0  20 

38


92


0
 31 0 
 31 49 30


92


ζ
K cc  
0
 30 
31 0  30  ;
49


 31
0
 31 0 


0
 30
0  30 

For the intermediate masses and springs:
j
ˆ   M aa
M
j
j T
M ac
 K1aai
M acj 
;
K

 1i  1i T
M ccj 
K ac
2i
 K aa
K1aci 
;
K

 2i  2i T
K1cci 
K ac
2i

 K 3aai
K ac
;
K

 3i T
3i
2i 
K cc

K ac
K 3aci 
;
K 3cci 
j
ˆ j  [m j ]
M
aa
pq  N 1 N 1 , m pq  cos  pπ j  cos  qπ j  ;
j
ˆ j  [m
ˆ pq
M
] N 14 , mˆ pqj  cos  pπ j  f q  j  ;
ac
1i
ˆ j  [m j ] , mˆ stj  f s  j  f t  j  ; K1i  k 1i 
M
aa
cc
st 44
 pq  ( N 1)( N 1) , k pq  1i cos  pπi  cos  qπi  ;
2i
2i
2i

K aa
 k pq
 ( N 1)( N 1) , k pq   2i cos  pπi  cos  qπi  ;
3i
3i

K 3aai   k pq
 ( N 1) N 1 , k pq   3i cos  pπi  cos  qπi  ;
1i
 1i cos  pπi  f s i  ;
K1aci  k ps1i 
, k ps
( N 1)4
2i
K ac
 k ps2i 
, k ps2i   2i cos  pπi  f s i  ;
( N 1)4
K 3aci  k ps3i 
, k ps3i  3i cos  pπi  f s i  ;
( N 1)4
K1cci   kst1i  , kst1i  1i f s i  ft i  ; K cc2i   kst2i  , kst2i   2i f s i  ft i  ;
44
44
K 3cci   kst3i  , kst3i   3i f s i  ft i  ;
44
where p, q  0,1, 2 N ; s, t  1, 2,3, 4 ; i  1, 2ns ; j  1, 2nm and  j is the dimensionless
location of the j-th lumped mass and  i is the dimensionless location of the i-th elastic spring.
For the matrices of the nonlinear terms, to simplify the expressions of the matrix elements, a
function vector s  is introduced as follows:
39
s   1,cos  π  ,cos  2π  ,,cos  nπ  , f1 ( ), f2 ( ), f3 ( ), f4 ( )
 s  1, 2 N  5
(A.1)
The matrices of the non-linear terms can then be expressed as:
3


N p   kijp  , kijp    i j p  i j p  d ,  p  1, 2 N  3; i, j  1, 2 N  5
2

0
1
1


N q   kijq  , kijq   i jq  3i jq d ,  q, j  1, 2N  5; i  1, 2N  3
0
 pq   k  , k
pq
ij
1
1


   i j pq  i j pq  i j pq  d ,  i, j, p, q  1, 2N  5
2
2
2

0
1
pq
ij
40
Appendix B:
For the ill-conditioned eigenvalue problem of Eq. (27), considering the asymmetry of matrix
B and referring to [48], a modified algorithm is given as follows:
(1) The symmetrical matrix A (its dimension is M , which is equal to 2 N  10 for the vertical
and transverse vibration, but is equal to 2N  6 for the longitudinal vibration) may become a
diagonal matrix D with the orthogonal transformation Q 0 .
Q T0 AQ 0  D , Q T0 Q 0  
(B.1)
where D is a diagonal matrix with descending elements:
d11  d22 
 dii 
 d MM  0 , 1  i  M
(B.2)
The consequence of the ill-condition of matrix A (its condition number would be greater than
1020 when N  100 ) is that the values of some d ii will be very small compared with other
diagonal elements. Therefore D can be partitioned as:
0  }n1
D
D   11

 0 D22  }n2
(B.3)
where the n2  n2 diagonal matrix D 22 has been chosen such that:
dii   d11 , n1  i  M
where  is a small positive parameter which can be set as 10
(0)
obvious that D(0)
22   D11 . Operator
(B.4)
8
or even smaller when N  50 . It is
denotes the l2 -norm.
(2) For the asymmetrical matrix B , a new matrix B(0) with the orthogonal transformation Q 0
also can be partitioned into the form:
B
(0)
(0)
B11
 Q BQ0   (0)
B21
T
0
(0)
 }n1
B12

}n2
B(0)
22 
And applying the congruent transformation associated with:
41
(B.5)
 D 1/2
Q1   11
 0
0  }n1

  }n2
(B.6)
  0  }n1
A(1)  Q1T DQ1  

0 D22  }n2
B
(1)
(1)
B11
 Q B Q1   (1)
B21
T
1
(0)
(1)
 }n1
B12

}n2
B(0)
22 
(B.7)
(B.8)
It can be observed that:
(1)
B (0)
D22   , and rank( B (1)
21 )= n2 .
22   B 21 ,
(B.9)
(3) Using the orthogonal-triangular decomposition of B (1)
21 by Householder transform [49], then
a n1  n2 matrix R would be obtained:
R }n
T
T
R  Q11
 B(1)21    011  }n2  n ,
  1 2
T
Q11
Q11  
(B.10)
where R11 is a full-rank (  n2 ) triangular matrix.
Applying the transformation associated with:
B (2)
Q 0 }n
Q2   11  1
 0   }n2
(B.11)
  0  }n1
A(2)  QT2 A(1)Q2  

0 D22  }n2
(B.12)
(2)
B11

 QT2 B1Q 2  B (2)
21
 R11

(2)
B12
B (2)
22
0
(2)
 }n2
B13

B (2)
23  }n1  n2

B (0)
22  }n2
(B.13)
(2)
where matrix B13
should be a non-singular matrix for the models in this study, and B (0)
22   R11 .
(4) Finally the computation of eigenvalues of Eq. (27) is transformed into computation of the
(2)
(2)
new equation A x  B x  0 , with D22  0 and B (0)
22  0 , which leads to:
42
B(2)  B(2)  B(2) 1 B(2)  u      u
23
13
12 
n
 22

(B.14)
where u is a part of an eigenvector of Eq. (B.14). Solving the eigenvalue of Eq. (B.14) results in
only M  2n2 complex eigenvalues which may be called the  -stable approximate eigenvalues of
Eq. (27), and the eigenvectors of Eq. (27) take the following form:

 
 2
v  Q0  Q1  Q2  v ; where: v  0, u,  B13
43
1
 2

 B12  u
T
(B.15)