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JCAMECH Vol. 51, No. 2, December 2020, pp 311-322
DOI: 10.22059/jcamech.2020.296012.472
Analysis of the nonlinear axial vibrations of a cantilevered pipe conveying pulsating two-phase flow
Adeshina S. Adegoke a, Omowumi Adewumi a, Akin Fashanu b, Ayowole Oyediran a
a Department of Mechanical Engineering, University of Lagos, Nigeria b Department of Systems Engineering, University of Lagos, Nigeria
1. Introduction
The vibration of pipes due to the dynamic interaction between
the fluid and the pipe is known to be a result of either instability or resonance. The earlier which is because of the decrease in the
effective pipe stiffness with the flow speed Ibrahim [1], and when
the flow velocity attains a critical value, the stiffness vanishes, and
the instability occurs. However, the latter occurs when the pipe
conveys a pulsatile flow resulting in parametric resonance.
Ginsberg [2] is about the earliest publication on the dynamic instability of pipes conveying pulsatile flow for a pinned-pinned
pipe. Chen [3] investigated the effect of small displacements of a
pipe conveying a pressurized flow with pulsating velocity.
Equations of motion were derived for general end conditions and
the Eigenfunction expansion method was used to obtain solutions
for the case of simple supports. It was discovered that in the presence of pulsatile flow, the pipe has regions of dynamic
instability whose boundaries increase with the increased
magnitude of fluctuations. Paidoussis and Issid [4] investigated the
dynamics and stability of flexible pipes-conveying fluid where the
flow velocity is either constant or with a small harmonic
component superposed. For the harmonically varying velocity, stability maps were presented for parametric instabilities using the
Eigenfunction expansion method for pinned or clamped ends
pipes, and also for cantilevered pipes. It was found that as the flow
βββ
Corresponding author. Tel.: [email protected]
velocity increases for both clamped and pinned end pipes,
instability regions increase, while a more complex behavior was obtained for the cantilevered pipes. For all cases, dissipation
reduces or eliminates zones of parametric instability. Paidoussis
and Sundararajan [5] worked on a pipe clamped at both ends and
revealed that the parametric and combination resonance is
exhibited by the pipe when is conveys single-phase flow at a
velocity that is harmonically perturbed. However, Neyfeh and Mook [6] highlighted that nonlinearities are responsible for
various unusual phenomena in the presence of internal and/or
external resonance. Sequel to these early studies on the linear
dynamics of the system, many studies were also published on the
nonlinear dynamics of the subject, notable among these, are the
works of Semler and Paidoussis [7] on the nonlinear analysis of parametric resonance of a planar fluid-conveying cantilevered
pipe. Namachchivaya and Tien [8] on the nonlinear behaviour of
supported pipes conveying pulsating fluid examined in the vicinity
of subharmonic and combination resonance using the method of
averaging. Pranda and Kar [9] studied the nonlinear dynamics of a
hinged-hinged pipe conveying pulsating flow with combination, principal parametric and internal resonance, using the method of
multiple scales. Mohammadi and Rastagoo [10] investigated the
primary and secondary resonance phenomenon in an FG/lipid
nanoplate considering porosity distribution based on the nonlinear
elastic medium. Asemi, Mohammadi and Farajpour [11]
ART ICLE INFO ABST RACT
Article history:
Received: 9 January 2020
Accepted: 24 January 2020
The parametric resonance of the axial vibrations of a cantilever pipe conveying harmonically perturbed two-phase flow is investigated using the method of multiple scale perturbation. The nonlinear coupled and uncoupled planar dynamics of the pipe are examined for a scenario when the axial vibration is parametrically excited by the pulsating frequencies of the two phases conveyed by the pipe. Away from the internal resonance condition, the stability regions are determined analytically. The stability boundaries are found to reduce as the void fraction is increasing. With the amplitude of the harmonic velocity fluctuations of the phases taken as the control parameters, the presence of internal resonance condition results in the occurrence of both axial and transverse resonance peaks due to the transfer of energy between the planar directions. However, an increase in the void fraction is observed to reduce the amplitude of oscillations due to the increase in mass content in the pipe and which further dampens the motions of the pipe.
Keywords:
Axial Vibration
Parametric resonance
Void fraction
Two-phase flow
Perturbation method
Adegoke et al.
312
considered nonlocality and geometric nonlinearity due to
nanosized effect and mid-plane stretching in the study of the
nonlinear stability of orthotropic single-layered graphene sheet. Mohammadi and Rastagoo [12] studied the primary,
superharmonic and subharmonic resonances as a result of the
presence of nonlinearities in the modeling of the vibrations of a
viscoelastic composite nanoplate with three directionally
imperfect porous FG core using the Bubno-Galerkin method and
the multiple scale method. Danesh, Farajpour and Mohammadi [13] investigated the axial vibrations of a tapered nanorod
considering elasticity theory and adopted the differential
quadrature method in solving the governing equations for various
boundary conditions. As demonstrated by various publications, the
effect of nonlinearities is known to be highly crucial in the
understanding of the dynamics and stability of pipes and porous media conveying fluid.
Leaving aside the much more established analysis of the
dynamics of pipes conveying single-phase flows, the question
remains as to how a pulsatile two-phase through a pipe will
influence the dynamic behaviour of the pipe. As seen in the review
of literature, most of the existing publications focused on the transverse vibrations, but the axial oscillations of the pipe can be
of interest also when considering pulsatile flow due to possible
amplification of oscillation amplitude as a result of resonance
phenomenon. This present study investigates the nonlinear axial
vibrations of a cantilever pipe conveying pulsating two-phase flow
with the pulsating frequencies of the two phases parametrically exciting the axial vibrations of the pipe. An approximate analytical
approach will be used to resolve the governing equations by
imposing the method of multiple scales perturbation technique
directly to the systems equations (direct-perturbation method).
2. Problem formulation and modeling
2.1. Assumptions
Considering a cantilever pipe of length (L), with a cross-
sectional area (A), mass per unit length (m) and flexural rigidity
(EI), conveying multiphase flow; flowing parallel to the pipeβs
centre line. The flow is assumed to have a velocity profile can be
represented as a plug flow, the diameter of the pipe is small compared to its length so that the pipe behaves like a Euler-
Bernoulli beam, the motion is planar, deflections of the pipe are
large, but the strains are small, rotatory inertia and shear
deformation are neglected and pipe centerline is assumed to be
extensible.
Figure 1. Systemβs Schematics.
2.2. Equation of motion of an extensible cantilever pipe conveying Two-phase flow
The coupled nonlinear equations of motion of a cantilevered
pipe conveying multiphase flow including the nonlinearity due to
midline stretching is giving by Adegoke and Oyediran [14]:
(π + β ππππ=1 )οΏ½οΏ½ + β MπποΏ½οΏ½
ππ=1 +β 2πππποΏ½οΏ½
β²ππ=1 +
β ππππ=1 ππ
2π’β²β² + β ππποΏ½οΏ½π’β²π
π=1 β πΈπ΄π’β²β² β πΈπΌ(π£β²β²β²β²π£β² + π£β²β²π£β²β²β²) +(π0 βπ β πΈπ΄(πΌβπ) β πΈπ΄)π£
β²π£β²β² β (π0 βπ β πΈπ΄(πΌβπ))β²+
(π+ β ππππ=1 )π = 0, (1)
(π + β ππππ=1 )οΏ½οΏ½ + β 2πππποΏ½οΏ½
β²ππ=1 +β ππ
ππ=1 ππ
2π£β²β² ββ πππππ=1 ππ
2π£β²β² + β ππποΏ½οΏ½π£β²π
π=1 +πΈπΌπ£β²β²β²β² β (π0 β π βπΈπ΄(πΌβπ))π£β²β² β πΈπΌ(3π’β²β²β²π£β²β² + 4π£β²β²β²π’β²β² + 2π’β²π£β²β²β²β² + π£β²π’β²β²β²β² +2π£β²
2π£β²β²β²β² + 8π£β²π£β²β²π£β²β²β² + 2π£β²β²
3) + (π0 βπ β πΈπ΄(πΌβπ)β
πΈπ΄) (π’β²π£β²β² + π£β²π’β²β² +3
2π£β²2π£β²β²) = 0 (2)
With the associated boundary condition
π’(0) = π’β²(πΏ) = 0 (3)
π£(0) = π£β²(0) πππ π£β²β²(πΏ) = π£β²β²β²(πΏ) = 0 (4) Where x is the longitudinal axis, v is the transverse deflection, u is
the axial deflection, n is the number of phases, m is the mass of the pipe, Mj is the masses of the internal fluid phases, EA is axial
stiffness, EI is Bending stiffness, π0 is the tension term, P is the
pressure term, πΌ is the thermal expansivity term, βπ relates to the
temperature difference and βaβ relates to the Poisson ratio (r) as a=1-2r.
2.3. Dimensionless Equation of motion for two-phase Flow
The equation of motion may be reduced to that of two-phase
flow by considering n to be 2 and rendered dimensionless by
introducing the following non-dimensional quantities;
οΏ½οΏ½ =π’
πΏ , οΏ½οΏ½ =
π£
πΏ , π‘ = [
πΈπΌ
π1 +π2 +π]
12β π‘
πΏ2 ,
οΏ½οΏ½π = [π1 +π2πΈπΌ
]
12β
π1πΏ , οΏ½οΏ½π = [π1 +π2πΈπΌ
]
12β
π2πΏ
πΎ = π1 +π2 +π
πΈπΌπΏ3π, π½1 =
π1π1 +π2 +π
,
πΉ1 =π1
π1 +π2, π½2 =
π2π1 +π2 +π
, πΉ2 =π2
π1 +π2,
ππππ πππ:π±0 =πππΏ
2
πΈπΌ , πΉπππ₯ππππππ‘π¦: π±1 =
πΈπ΄πΏ2
πΈπΌ,
ππππ π’ππ: π±2 =ππΏ2
πΈπΌ
οΏ½οΏ½ + πΌπ βπΉ1βπ½1 + πΌπ βπΉ2βπ½2 + 2οΏ½οΏ½πβπΉ1βπ½1 οΏ½οΏ½β² +
2οΏ½οΏ½πβπΉ2βπ½2οΏ½οΏ½β² + πΉ1πΌπ
2οΏ½οΏ½β²β² + πΉ2πΌπ
2οΏ½οΏ½β²β² + πΌπ βπΉ1βπ½1οΏ½οΏ½
β² +
πΌπ βπΉ2βπ½2οΏ½οΏ½β² βπ±1οΏ½οΏ½
β²β² β (οΏ½οΏ½β²β²β²β²οΏ½οΏ½β² + οΏ½οΏ½β²β²οΏ½οΏ½β²β²β²) + (π±0 βπ±2 β
π±1(πΌβπ)β π±1)οΏ½οΏ½β²οΏ½οΏ½β²β² β (π±0 β π±2 βπ±1(πΌβπ))
β²+ πΎ = 0 (5)
οΏ½οΏ½ + 2οΏ½οΏ½πβπΉ1βπ½1οΏ½οΏ½β² + 2οΏ½οΏ½πβπΉ2βπ½2 οΏ½οΏ½
β² +πΉ1πΌπ 2οΏ½οΏ½β²β² +
πΉ2πΌπ 2οΏ½οΏ½β²β² β ππΉ1πΌπ
2οΏ½οΏ½β²β² β ππΉ2πΌπ
2οΏ½οΏ½β²β² + πΌπ βπΉ1βπ½1οΏ½οΏ½
β² +
πΌπ βπΉ2βπ½2οΏ½οΏ½β² β (π±0 β π±2 βπ±1(πΌβπ))οΏ½οΏ½
β²β² + οΏ½οΏ½β²β²β²β² β (3οΏ½οΏ½β²β²β²οΏ½οΏ½β²β² +
4οΏ½οΏ½β²β²β²οΏ½οΏ½β²β² + 2οΏ½οΏ½β²οΏ½οΏ½β²β²β²β² + οΏ½οΏ½β²οΏ½οΏ½β²β²β²β² + 2οΏ½οΏ½β²2οΏ½οΏ½β²β²β²β² + 8οΏ½οΏ½β²οΏ½οΏ½β²β²οΏ½οΏ½β²β²β² + 2οΏ½οΏ½β²β²3)+
(π±0 βπ±2 βπ±1(πΌβπ)β π±1)(οΏ½οΏ½β²οΏ½οΏ½β²β² + οΏ½οΏ½β²οΏ½οΏ½β²β² +
3
2οΏ½οΏ½β²2οΏ½οΏ½β²β²) = 0 (6)
In these equations, οΏ½οΏ½ and οΏ½οΏ½ respectively, are the dimensionless displacements in the longitudinal and transverse direction,
οΏ½οΏ½π and οΏ½οΏ½π are the flow velocities of the constituent phases,
π½1 and π½2 are the mass ratios for each phase which are the same as in single-phase flows as derived by Ghayesh, PaΓ―doussis and
Mixture velocity (VT)
X
Y
Journal of Computational Applied Mechanics, Vol. 51, No. 2, June 2020
313
Amabili [15], πΉ1 and πΉ2 are new mass ratios that are unique to
two-phase flow, relating the fluid masses independent of the mass of the pipe.
Assuming that the velocities are harmonically fluctuating about
their constant mean velocities, the velocities of the phases can be expressed as:
πΌπ = π1 (1+ ΞΌ1 sin(Ξ©1T0)) (7)
πΌπ = π2 (1 + ΞΌ2 sin(Ξ©2T0)) (8) Using these notations,
πΆ11 = βπΉ1βπ½1 , πΆ12 = βπΉ2βπ½2 , πΆ21 = 2βπΉ1βπ½1 , πΆ22 =
2βπΉ2βπ½2 , πΆ31 = πΉ1 , πΆ32 = πΉ2, πΆ5 = π±1, πΆ6 = (π±0 β π±2 βπ±1(πΌβπ)β π±1), πΆ7 = π±0 β π±2 βπ±1(πΌβπ) The equations are reduced as:
οΏ½οΏ½ + πΌπ πΆ11 +πΌπ πΆ12 + οΏ½οΏ½ππΆ21οΏ½οΏ½β² + οΏ½οΏ½ππΆ22οΏ½οΏ½
β² + πΆ31πΌπ 2οΏ½οΏ½β²β² +
πΆ32πΌπ 2οΏ½οΏ½β²β² +πΌπ πΆ11οΏ½οΏ½
β² +πΌπ πΆ12οΏ½οΏ½β² β πΆ5οΏ½οΏ½β²β² β (οΏ½οΏ½β²β²β²β²οΏ½οΏ½β² +
οΏ½οΏ½β²β²οΏ½οΏ½β²β²β²) + πΆ6οΏ½οΏ½β²οΏ½οΏ½β²β² β C7β² + πΎ = 0, (9)
οΏ½οΏ½ + οΏ½οΏ½ππΆ21οΏ½οΏ½β² + οΏ½οΏ½ππΆ22οΏ½οΏ½
β² + πΆ31πΌπ 2οΏ½οΏ½β²β² + πΆ32πΌπ
2οΏ½οΏ½β²β² β
ππΆ31πΌπ 2οΏ½οΏ½β²β² β ππΆ32πΌπ
2οΏ½οΏ½β²β² + πΌπ πΆ11οΏ½οΏ½
β² + πΌπ πΆ12οΏ½οΏ½β² β πΆ8οΏ½οΏ½β²β² +
οΏ½οΏ½β²β²β²β² β (3οΏ½οΏ½β²β²β²οΏ½οΏ½β²β² + 4οΏ½οΏ½β²β²β²οΏ½οΏ½β²β² + 2οΏ½οΏ½β²οΏ½οΏ½β²β²β²β² + οΏ½οΏ½β²οΏ½οΏ½β²β²β²β² + 2οΏ½οΏ½β²2οΏ½οΏ½β²β²β²β² +
8οΏ½οΏ½β²οΏ½οΏ½β²β²οΏ½οΏ½β²β²β² + 2οΏ½οΏ½β²β²3)+ πΆ6 (οΏ½οΏ½β²οΏ½οΏ½β²β² + οΏ½οΏ½β²οΏ½οΏ½β²β² +3
2οΏ½οΏ½β²2οΏ½οΏ½β²β²) = 0. (10)
2.4. The empirical gas-liquid two-phase flow model
The componentβs velocities in terms of the superficial velocities
are expressed as:
ππ = πππ£π, ππ = ππ(1β π£π) (11)
Where ππ and ππ are the superficial flow velocities. Adopting the
Chisholm empirical relations as presented in [16],
Void fraction:
π£π = [1 +β1 β x (1 βππ
ππ) (
1βx
x)(
ππ
ππ)]
β1
=
Volume of gas
Volume of gas+Volume of Liquid (12)
Slip Ratio:
S =Vg
Vl= [1 β x (1 β
Οl
Οg)]1/2
(13)
Where: (x) is the vapor quality and (Οl and Οg) are the densities of
the liquid and gas phases respectively.
The mixture velocity can be expressed as:
VT = Ugvf + Ul(1 β vf) (14)
Individual Velocities:
Vl =VT
S+1, Vg =
SVT
S+1 (15)
For various void fractions (0.1, 0.3, and 0.5) and a series of mixture
velocities, the corresponding slip ratio and individual velocities are estimated and used for numerical calculations.
3. Method of Solution
Multiple-time scale perturbation technique is used to seek an
approximate solution; this approach is applied directly to the
partial differential equations (9) and (10).
u + ππ C11 + ππ C12 + οΏ½οΏ½πC21uβ² + οΏ½οΏ½πC22u
β² + C31ππ2uβ²β² +
C32ππ2uβ²β² + ππ C11u
β² +ππ C12uβ² β C5uβ²β² + Ξ΅(β(vβ²β²β²β²vβ² +
vβ²β²vβ²β²β²) + C6vβ²vβ²β² β C7β² + Ξ³) = 0, (16)
v + οΏ½οΏ½πC21vβ² + οΏ½οΏ½πC22v
β² + C31ππ2vβ²β² + C32ππ
2vβ²β² β
aC31ππ2vβ²β² β aC32ππ
2vβ²β² +ππ C11v
β² +ππ C12vβ² β C7vβ²β² +
vβ²β²β²β²Ξ΅(β(3uβ²β²β²vβ²β² + 4vβ²β²β²uβ²β² + 2uβ²vβ²β²β²β² + vβ²uβ²β²β²β² + 2vβ²2vβ²β²β²β² +
8vβ²vβ²β²vβ²β²β² + 2vβ²β²3) + C6(uβ²vβ²β² + vβ²uβ²β² +3
2vβ²2vβ²β²)) = 0 (17)
Also, perturbing the harmonically fluctuation of the velocity about
the constant mean velocity;
ππ = U1 (1 + Ρμ1 sin(Ξ©1T0)) (18)
ππ = U2 (1 + Ρμ2 sin(Ξ©2T0)) (19)
We seek approximate solutions in the form:
u = u0(T0, T1) + Ξ΅u1(T0, T1) + Ξ΅2u2(T0, T1) + O(Ξ΅) (20)
v = v0(T0, T1) + Ξ΅v1(T0, T1) + Ξ΅2v2(T0, T1) + O(Ξ΅) (21)
Two-time scales are needed π0 = π‘ and π1 = ππ‘. Where Ξ΅ is a
small dimensionless measure of the amplitude of u and v, used as a book-keeping parameter.
The time derivatives are:
π
ππ‘= π·0 + Ξ΅π·1 + Ξ΅
2π·2 + π(π) (22) π2
ππ‘2= π·0
2 + 2Ξ΅π·0π·1 + Ξ΅2(π·1
2 + 2π·0π·2) + π(π) (23)
πβπππ π·π =π
πππ
Substituting Equation (20), Equation (21), Equation (22) and
Equation (23) into Equation (16) and Equation (17) and equating
the coefficients of (Ξ΅) to zero and one respectively:
U-Equation:
π(Ξ΅0). π·02οΏ½οΏ½0 + πΆ21π·0οΏ½οΏ½0
β²π1 + πΆ22π·0οΏ½οΏ½0β²β²οΏ½οΏ½2 +
πΆ31οΏ½οΏ½0β²β²π1
2+ πΆ32οΏ½οΏ½0
β²β²π22β πΆ5οΏ½οΏ½0
β²β² = 0 (24)
π(Ξ΅1). π·02οΏ½οΏ½1 + πΆ21π·0οΏ½οΏ½1
β²π1 + πΆ22π·0οΏ½οΏ½1β²π2 + 2π·0π·1οΏ½οΏ½0 +
πΆ31οΏ½οΏ½1β²β²οΏ½οΏ½1
2+ πΆ32οΏ½οΏ½1
β²β²π22+ C21π·0οΏ½οΏ½1
β²οΏ½οΏ½1 + πΆ22π·0οΏ½οΏ½1β²π2 β
πΆ5οΏ½οΏ½1β²β² β οΏ½οΏ½0
β²β²β²β²οΏ½οΏ½0β² β C7β² + πΎ β οΏ½οΏ½0
β²β²οΏ½οΏ½0β²β²β² + πΆ6οΏ½οΏ½0
β²οΏ½οΏ½0β²β² +
C21π·1οΏ½οΏ½0β²π1 + πΆ22π·1οΏ½οΏ½0
β²οΏ½οΏ½2 + πΆ11πΊ1π1 πππ (πΊ1π0)π1 +
πΆ12πΊ2π2 πππ (πΊ2π0)π2 + 2πΆ31π1 π ππ(πΊ1π0)οΏ½οΏ½12οΏ½οΏ½0
β²β² +
2πΆ32π2 π ππ(πΊ2π0)π22οΏ½οΏ½0
β²β² + πΆ21π1 π ππ(πΊ1π0)π·0π1 οΏ½οΏ½0β² +
πΆ22π2 π ππ(πΊ2π0)π·0οΏ½οΏ½2 οΏ½οΏ½0β² + πΆ41πΊ1π1 πππ (πΊ1π0)π1οΏ½οΏ½0
β² +πΆ42πΊ2π2 πππ (πΊ2π0)π2οΏ½οΏ½0
β² = 0 (25)
Adegoke et al.
314
V-Equation:
π(Ξ΅0). π·02οΏ½οΏ½0 β πΆ7οΏ½οΏ½0β²
β² + οΏ½οΏ½0β²β²β²β² + πΆ21π·0οΏ½οΏ½0
β²π1 +
πΆ22π·0οΏ½οΏ½0β²οΏ½οΏ½2 + πΆ31οΏ½οΏ½0
β²β²π12+ πΆ32οΏ½οΏ½0
β²β²π22β ππΆ31οΏ½οΏ½0
β²β²π12β
ππΆ32 = 0 (26)
π(Ξ΅1). π·02οΏ½οΏ½1 β πΆ7οΏ½οΏ½1
β²β² + οΏ½οΏ½1β²β²β²β² β οΏ½οΏ½0
β²β²β²β²οΏ½οΏ½0β² β 2οΏ½οΏ½0
β²οΏ½οΏ½0β²β²β²β² β
4οΏ½οΏ½0β²β²οΏ½οΏ½0
β²β²β² β 3οΏ½οΏ½0β²β²οΏ½οΏ½0
β²β²β² β 2οΏ½οΏ½03β²β² β 2οΏ½οΏ½0
β²β²β²β²οΏ½οΏ½02β² + 2π·0π·1οΏ½οΏ½0 +
πΆ31οΏ½οΏ½1β²β²π1
2+ πΆ32οΏ½οΏ½1
β²β²π22β 8οΏ½οΏ½0
β²οΏ½οΏ½0β²β²οΏ½οΏ½0
β²β²β² + C6u0β²v0
β²β² +
C6u0β²β²v0
β² +3
2C6v0
2β²v0β²β² + C21D0v0
β²U1 + C22D0v0β²U2 +
C21D1v0β²U1 + C22D1v0
β²U2 β aC31v1β²β²U1
2β aC32v1
β²β²U22+
2C31ΞΌ1 sin(Ξ©1T0)U12v0β²β² + 2C32ΞΌ2 sin(Ξ©2T0)U2
2v0β²β² +
C21ΞΌ1 sin(Ξ©1T0)D0U1 v0β² + C22ΞΌ2 sin(Ξ©2T0)D0U2 v0
β² β
2aC31ΞΌ1 sin(Ξ©1T0)U12v0β²β² β 2aC32ΞΌ2 sin(Ξ©2T0)U2
2v0β²β² +
C41Ξ©1ΞΌ1 cos(Ξ©1T0)U1v0β² + C42Ξ©2ΞΌ2 cos(Ξ©2T0)U2v0
β² =0 (27) The homogeneous solution of the leading order equations
Equation (24) and Equation (27) can be expressed as:
u(x, T0, T1)0 = Ο(x)n exp(iΟnT0) + CC (28) v(x, T0, T1)0 = Ξ·(x)n exp(iΞ»nT0) + CC (29)
Where (CC) is the complex conjugate, Ο(x)n and Ξ·(x)n are the
complex modal functions for the axial and transverse vibrations
for each mode (n) and, Οn and Ξ»n are the eigenvalues for the axial
and transverse vibrations for each mode (n).
3.1. Axial natural frequencies and mode shape
The analytical expression for the axial frequencies is obtained
as:
ππ =2ππβπ.ππ(
π
π)
(πβπ)πΏ, π = 1,2,3,β¦ (30)
Where:
π = πΆ21οΏ½οΏ½12
+πΆ22οΏ½οΏ½22
+βπΆ212οΏ½οΏ½1
2+2πΆ21πΆ22οΏ½οΏ½1οΏ½οΏ½2+πΆ22
2οΏ½οΏ½22β4πΆ31οΏ½οΏ½1
2β4πΆ32οΏ½οΏ½2
2+4πΆ5
2
πΆ5βπΆ31π12βπΆ32οΏ½οΏ½2
2 ,
π =πΆ21οΏ½οΏ½12
+πΆ22οΏ½οΏ½22
ββπΆ212οΏ½οΏ½1
2+2πΆ21πΆ22οΏ½οΏ½1οΏ½οΏ½2+πΆ22
2οΏ½οΏ½22β4πΆ31οΏ½οΏ½1
2β4πΆ32οΏ½οΏ½2
2+4πΆ5
2
πΆ5βπΆ31π12βπΆ32οΏ½οΏ½2
2
With the modal shape expressed as:
π(π₯)π = πΊπ(ππ₯π(ππ1π₯) + ππ₯π(ππ2π₯)) (31)
The constant πΊπ can be obtained using the orthogonality
relationship.
3.2. Transverse natural frequencies and mode shape
Conversely to the axial vibrations, direct analytical estimation is
not possible for the natural frequencies of the transverse vibrations. However, the natural frequencies can be estimated by
solving the quartic equation (32) and the condition of obtaining a non-trivial solution of the boundary condition matrix (33)
simultaneously with a nonlinear numerical routine:
π§4ππ + (πΆ7β πΆ31π12β πΆ32π2
2+ ππΆ31π1
2+
ππΆ32π22)π§2ππ β (πΆ21οΏ½οΏ½1 + πΆ22π2)π§ππππ β π
2π = 0 π =
1,2,3,4 πππ π = 1,2,3,4,5β¦ (32)
Boundary condition matrix:
β
1 1 1 1π§1π π§2π π§3π π§4π
(π§1π)2 . exp(π. π§1π) (π§2π)
2. exp(π. π§2π) (π§3π)2 . exp(π. π§3π) (π§4π)
2. exp(π. π§4π)
(π§1π)3 . exp(π. π§1π) (π§2π)
3. exp(π. π§2π) (π§3π)3 . exp(π. π§3π) (π§4π)
3. exp(π. π§4π)β
β πΊ
[
1π»2ππ»3ππ»4π
]π»1π
= (
0000
)
For a non-trivial solution, the determinant of (G) must varnish, that is:
π·πΈπ(πΊ) = 0 (33)
Where (ππ), are the natural frequencies and (ππ) are the eigenvalues. The mode function of the transverse vibration corresponding to the nth eigenvalue is expressed as:
Ζ(π₯)π = π»1π . [ππ₯ .π§1π .π β (π΄ + π΅ + πΆ + π·) β πΈ] (34)
π΄ = ππ₯ .π§4π .π. [π π§1π .π.(π§1π)
3.π§2πβ π π§1π .π.(π§1π)
3. π§3πβ π π§1π .π . π§4π.(π§1π)
2 .π§2π
(π§2πβ π§4π).(π§3πβ π§4π) .[π π§2π .π.(π§2π)
2β π π§3π .π .(π§3π)2]
π΅ =
ππ₯ .π§4π .π. [π π§1π .π.π§4π.(π§1π)
2.π§3πβ π π§2π .π.π§1π.(π§2π)
3+ π π§2π .π . π§4π. π§1π.(π§2π)2
(π§2πβ π§4π).(π§3πβ π§4π) .[π π§2π .π.(π§2π)
2β π π§3π .π .(π§3π)2]
πΆ = ππ₯ .π§4π .π. [π π§3 .π.π§1π.(π§3π)
3β π π§3 .π.π§4π.π§1π.(π§3π)2+ π π§2π .π .(π§2π)
3.π§3π
(π§2πβ π§4π).(π§3πβ π§4π) .[π π§2π .π.(π§2π)
2β π π§3π .π .(π§3π)2]
π· =
ππ₯ .π§4π .π. [βπ π§2π .π.π§4π .(π§2π)
2.π§3πβ π π§3 .π.π§2π.(π§3π)
3+ π π§3 .π .π§4π.π§2π.(π§3π)2
(π§2πβ π§4π).(π§3πβ π§4π) .[π π§2π .π.(π§2π)
2β π π§3π .π .(π§3π)2]
πΈ = ππ₯ .π§2π .π.(π§1πβ π§4π).[π
π§1 .π. (π§1π)2β π π§3 .π.(π§3π)
2]
(π§2πβ π§4π). [π π§2 .π.(π§2π)
2β π π§3 .π .(π§3π)2]
+
ππ₯ .π§3 .π.(π§1πβ π§4π).[π
π§1π .π. (π§1π)2β ππ§2π.(π§2π)
2]
(π§3πβ π§4π). [π π§2 .π.(π§2π)
2β π π§3 .π .(π§3π)2]
The constant H1 can be obtained using the orthogonality
relationship.
3.3. Axial principal parametric resonance
Substituting equation (28) and equation (29) into the equations (25) and (27) gives;
π·02οΏ½οΏ½1 β πΆ5οΏ½οΏ½1
β²β² + πΆ21π·0οΏ½οΏ½1β²π1 + πΆ22π·0οΏ½οΏ½1
β²π2 + πΆ31οΏ½οΏ½1β²β²π1
2+
πΆ32οΏ½οΏ½1β²β²οΏ½οΏ½2
2= β(πΆ21
ππ(π1)
ππ1
ππ(π₯)
ππ₯π1 + πΆ22
ππ(π1)
ππ1
ππ(π₯)
ππ₯π2 +
2πππ(π1)
ππ1π)ππ₯π(πππ0) + π(π1)
2 (ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
πΆ6ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2)ππ₯π(2πππ0) + [πΆ32π2
π2π(π₯)
ππ₯2ππ₯π(ππΊ2π0)π2
2π β
1
2(πΆ21π1
ππ(π₯)
ππ₯ππ₯π(βππΊ1π0)οΏ½οΏ½1π) +
1
2(πΆ21π1
ππ(π₯)
ππ₯ππ₯π(ππΊ1π0)π1π) β
1
2(πΆ22π2
ππ(π₯)
ππ₯ππ₯π(βππΊ2π0)π2π) +
1
2(πΆ22π2
ππ(π₯)
ππ₯ππ₯π(ππΊ2π0)οΏ½οΏ½2π) β
1
2(πΆ41πΊ1π1
ππ(π₯)
ππ₯ππ₯π(βππΊ1π0)π1)β
1
2(πΆ41πΊ1π1
ππ(π₯)
ππ₯ππ₯π(ππΊ1π0)οΏ½οΏ½1)β
Journal of Computational Applied Mechanics, Vol. 51, No. 2, June 2020
315
1
2(πΆ42πΊ2π2
ππ(π₯)
ππ₯ππ₯π(βππΊ2π0)π2)β
1
2(πΆ42πΊ2π2
ππ(π₯)
ππ₯ππ₯π(ππΊ2π0)π2) β
πΆ32π2π2π(π₯)
ππ₯2ππ₯π(βππΊ2π0)π2
2π β
πΆ31π1π2π(π₯)
ππ₯2ππ₯π(βππΊ1π0)π1
2π +
πΆ31π1π2π(π₯)
ππ₯2ππ₯π(ππΊ1π0)π1
2π] π(π1)ππ₯π(πππ0) +
[πΆ32π2π2π(π₯)
ππ₯2ππ₯π(ππΊ2π0)π2
2π +
1
2(πΆ21π1
ππ(π₯)
ππ₯ππ₯π(ππΊ1π0)οΏ½οΏ½1π)+
1
2(πΆ22π2
ππ(π₯)
ππ₯ππ₯π(π πΊ2π0)π2π) β
1
2(πΆ41πΊ1π1
ππ(π₯)
ππ₯ππ₯π(ππΊ1π0)π1)β
1
2(πΆ42πΊ2π2
ππ(π₯)
ππ₯ππ₯π(ππΊ2π0)π2)+
πΆ31π1π2π(π₯)
ππ₯2ππ₯π(ππΊ1π0)π1
2π] π(π1) ππ₯π(βπππ0) + πππ +
πΆπΆ = 0 (35)
π·02οΏ½οΏ½1 β πΆ7οΏ½οΏ½1β²
β² + οΏ½οΏ½1β²β²β²β² + πΆ21π·0οΏ½οΏ½1
β²π1 + πΆ22π·0οΏ½οΏ½1β²π2 +
πΆ31οΏ½οΏ½1β²β²π1
2+ πΆ32οΏ½οΏ½1
β²β²π22β ππΆ31οΏ½οΏ½1
β²β²π12β ππΆ32οΏ½οΏ½1
β²β²π22=
(βππ(π1)
ππ1(πΆ21
ππ(π₯)
ππ₯π1 + πΆ22
ππ(π₯)
ππ₯π2 + 2π(π₯)ππ) +
6π(π1)2π(π1) (ππ(π₯)
ππ₯)2 ππ(π₯)
ππ₯+ 2π(π1)2π(π1) (
ππ(π₯)
ππ₯)2 π4π(π₯)
ππ₯4+
4π(π1)2π(π1) ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
3πΆ6.π(π1)2π(π1) ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
3
2πΆ6. π(π1)2π(π1) (
ππ(π₯)
ππ₯)2 π2π(π₯)
ππ₯2)ππ₯π(πππ0) +
(2π(π1)π(π1) πΞ¦(π₯)
ππ₯
π4π(π₯)
ππ₯4+ 4π(π1)π(π1) π2Ξ¦(π₯)
ππ₯2
π3π(π₯)
π3+
3π(π1)π(π1) π2π(π₯)
π2
π3Ξ¦(π₯)
π3)ππ₯π(πππ0)ππ₯π(βπππ0) β
(πΆ6π(π1)π(π1) πΞ¦(π₯)
ππ₯
π2π(π₯)
ππ₯2+
πΆ6π(π1)π(π1) ππ(π₯)
ππ₯
π2Ξ¦(π₯)
ππ₯2) ππ₯π(πππ0)ππ₯π(βπππ0) +
[(1
2(πΆ22π2
ππ(π₯)
ππ₯π2π)β
1
2(πΆ42πΊ2π2
ππ(π₯)
ππ₯π2) +
ππΆ32π2π2π(π₯)
ππ₯2π2
2π β πΆ32π2
π2π(π₯)
ππ₯2π2
2π) ππ₯π(βππΊ2π0) +
(1
2(πΆ21π1
ππ(π₯)
ππ₯π1π) β
1
2(πΆ41πΊ1π1
ππ(π₯)
ππ₯π1)+
ππΆ31π1π2π(π₯)
ππ₯2π1
2π β πΆ31π1
π2π(π₯)
ππ₯2π1
2π) ππ₯π(βππΊ1π0) β
(1
2(πΆ21π1
ππ(π₯)
ππ₯π1π) β
1
2(πΆ41πΊ1π1
ππ(π₯)
ππ₯π1)+
ππΆ31π1π2π(π₯)
ππ₯2π1
2π β πΆ31π1
π2π(π₯)
ππ₯2π1
2π) ππ₯π(ππΊ1π0) β
(1
2(πΆ22π2
ππ(π₯)
ππ₯π2π)β
1
2(πΆ42πΊ2π2
ππ(π₯)
ππ₯π2)+
ππΆ32π2π2π(π₯)
ππ₯2π2
2π β
πΆ32π2π2π(π₯)
ππ₯2π2
2π) ππ₯π(ππΊ2π0)]π(π1)ππ₯π(πππ0) +
[(1
2(πΆ21π1
ππ(π₯)
ππ₯π1π)β
1
2(πΆ41πΊ1π1
ππ(π₯)
ππ₯π1) β
ππΆ31π1π2π(π₯)
ππ₯2π1
2π + πΆ31π1
π2π(π₯)
ππ₯2π1
2π) ππ₯π(ππΊ1π0) +
(1
2(πΆ22π2
ππ(π₯)
ππ₯π2π)β
1
2(πΆ42πΊ2π2
ππ(π₯)
ππ₯π2)β
ππΆ32π2π2π(π₯)
ππ₯2π2
2π +
πΆ32π2π2π(π₯)
ππ₯2π2
2π) ππ₯π(ππΊ2π0)]π(π1) ππ₯π(βπππ0) + πππ +
πΆπΆ (36) Here NST denotes non-secular terms. The next task is to determine
the requirements for X(T1) and Y(T1), that permits the solutions
of u1 and v1 to be independent of secular terms. However,
examining equations (35) and (36), it can be observed that various scenarios exist. With a focus on the principal parametric resonance
cases when the pulsation frequencies of the phases Ξ©1 and Ξ©2 are
close to 2Ο but far from 2Ξ». Also, consider the possibility of
having internal resonance (Ο = 2Ξ») and away from internal
resonance (Ο β 2Ξ») relationship between the axial and the
transverse natural frequencies.
The proximity of the nearness can be expressed as:
Ξ©1 = 2π + ππ2 πππ Ξ©2 = 2π + ππ2 (37)
Therefore, Ξ©1 β‘ Ξ©2
Where Ο2 is the detuning parameter.
Substituting the equations of nearness to resonance (37) into
equations (35) and (36), replacing Ξ΅T0 with T1 and collating the
secular terms, the principal parametric resonance conditions are identified and assessed as follows:
3.4. When π is far from ππ
If Ο is far from 2Ξ» then none of the coupled nonlinear terms will generate secular terms, therefore resulting in the uncoupled
response.
The two equations will have bounded solutions only if the
solvability condition holds. The solvability condition demands that
the coefficient of exp(iΟT0) and exp(iΞ»T0) vanishes [17-19], that is, X (T1) and Y (T1) should satisfy the following relation:
β(πΆ21ππ(π1)
ππ1
ππ(π₯)
ππ₯π1 + πΆ22
ππ(π1)
ππ1
ππ(π₯)
ππ₯π2 +
2π(π₯)ππππ(π1)
ππ1) + [πΆ32π2
π2π(π₯)
ππ₯2ππ₯π(ππ2π1)π2
2π +
1
2(πΆ21π1
ππ(π₯)
ππ₯ππ₯π(ππ2π1)π1π) +
1
2(πΆ22π2
ππ(π₯)
ππ₯ππ₯π(ππ2π1)οΏ½οΏ½2π) β
(πΆ41π1ππ(π₯)
ππ₯ππ₯π(ππ2π1)οΏ½οΏ½1π) β
(πΆ42π2ππ(π₯)
ππ₯ππ₯π(ππ2π2)π2π) β
1
2(πΆ41ππ2π1
ππ(π₯)
ππ₯ππ₯π(ππ2π1)π1)β
1
2(πΆ42ππ2π2
ππ(π₯)
ππ₯ππ₯π(ππ2π1)οΏ½οΏ½2)+
πΆ31π1π2π(π₯)
ππ₯2ππ₯π(ππ2π1)π1
2π]π(π1) =0 (38)
(βππ(π1)
ππ1(πΆ21
ππ(π₯)
ππ₯π1 + πΆ22
ππ(π₯)
ππ₯π2 + 2π(π₯)ππ) +
6π(π1)2π(π1) (ππ(π₯)
ππ₯)2 ππ(π₯)
ππ₯+ 2π(π1)2π(π1) (
ππ(π₯)
ππ₯)2 π4π(π₯)
ππ₯4+
4π(π1)2π(π1) ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
3πΆ6.π(π1)2π(π1) ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
Adegoke et al.
316
3
2πΆ6. π(π1)2π(π1) (
ππ(π₯)
ππ₯)2 π2π(π₯)
ππ₯2) = 0 (39)
With the inner product defined for complex functions on [0, 1] as:
β¨π, πβ© = β« ποΏ½οΏ½1
0ππ₯ (40)
Equations (52) and (53) can be cast as:
ππ(π1)
ππ1+ππ(π1) ππ₯π(ππ2π1) = 0 (41)
ππ(π1)
ππ1+ ππ(π1)2π(π1) = 0 (42)
Where:
π =β« βπ(π₯) 10 [πΆ32π2
π2π(π₯)
ππ₯2π22π+1
2(πΆ21π1
ππ(π₯)
ππ₯οΏ½οΏ½1π) ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
+
β« βπ(π₯) 10 [
1
2(πΆ22π2
ππ(π₯)
ππ₯οΏ½οΏ½2π)β(πΆ41π1
ππ(π₯)
ππ₯οΏ½οΏ½1π) ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
+
β« βπ(π₯) 10 [β(πΆ42π2
ππ(π₯)
ππ₯οΏ½οΏ½2π)β
1
2(πΆ41ππ2π1
ππ(π₯)
ππ₯οΏ½οΏ½1) ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
+
β« βπ(π₯) 10 [β
1
2(πΆ42ππ2π2
ππ(π₯)
ππ₯οΏ½οΏ½2)+πΆ31π1
π2π(π₯)
ππ₯2π12π ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
π =β« [6(
ππ(π₯)
ππ₯)2ππ(π₯)
ππ₯+2(
ππ(π₯)
ππ₯)2π4π(π₯)
ππ₯4]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππβ« π(π₯)π(π₯) 10 ππ₯)
+
β« [4ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+8
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
+
β« [8ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3β3πΆ6
ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
+
β« [8ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3β3
2πΆ6(
ππ(π₯)
ππ₯)2π2π(π₯)
ππ₯2]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
Where Q and S are complex numbers such that:
π = ππ + πππΌπππ π = ππ + πππΌ (43)
To estimate the stability region for the axial vibration due to the
principal parametric resonance, X (T1) is expressed in polar form as:
π(π1) = π΅(π1)π(π1π2π
2) πππ π(π1) = π΅(π1) π(
βπ1π2π
2) (44)
Substituting equation (44) into equation (41)
ππ΅(π1)
ππ1+ππ΅(π1) +
π2π(π1) π
2= 0 (45)
With complex amplitudes expressed as;
π = ππ + πππΌ (46)
π΅(π1) = πππΎπ1 πππ π΅(π1) = οΏ½οΏ½ππΎπ1 (47)
Substituting (46) into (47), (47) into (45) and separate to real and
imaginary components;
(πΎ + ππ ππΌ β
π2
2
ππΌ +π2
2πΎ β ππ
)(ππ
ππΌ) = (
00) (48)
To find a non-trivial solution, the determinant of the matrix must
varnish. Therefore;
πΎ = Β±β4ππ
2+4ππΌ
2βπ2
2
2 (49)
Stable solutions require that Ξ³ = 0. Therefore, the stability boundaries can be expressed as:
π2 = β2βππ 2+ππΌ
2 (50)
Therefore; the stability regions can be expressed as:
Ξ©1, Ξ©2 = 2π β 2πβππ 2 +ππΌ2 (51)
3.5. When π is close to ππ
However, to examine the coupled nonlinear dynamics of the
system, which is the scenario when Ο = 2Ξ», another detuning
parameter Ο1 is introduced.
π = 2π + ππ1 (52)
2ππ0 = ππ0β π1π1 πππ (π β π)π0 = ππ0+ π1π1 (53)
The two equations (50) and (51) will have bounded solutions only if the solvability condition holds. The solvability condition
demands that the coefficient of exp(iΟT0) and exp(iΞ»T0) vanishes [17-19], that is, with Ξ΅T0 = T1, X (T1) and Y (T1) should
satisfy the following relation:
β(πΆ21ππ(π1)
ππ1
ππ(π₯)
ππ₯π1 + πΆ22
ππ(π1)
ππ1
ππ(π₯)
ππ₯π2 +
2π(π₯)ππππ(π1)
ππ1) + π(π1)2 (
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
πΆ6ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2)ππ₯π(βπ1π1π) +
[πΆ32π2π2π(π₯)
ππ₯2ππ₯π(ππ2π1)π2
2π +
1
2(πΆ21π1
ππ(π₯)
ππ₯ππ₯π(ππ2π1)π1π) +
1
2(πΆ22π2
ππ(π₯)
ππ₯ππ₯π(ππ2π1)οΏ½οΏ½2π) β
(πΆ41π1ππ(π₯)
ππ₯ππ₯π(ππ2π1)οΏ½οΏ½1π) β
(πΆ42π2ππ(π₯)
ππ₯ππ₯π(ππ2π2)π2π) β
1
2(πΆ41ππ2π1
ππ(π₯)
ππ₯ππ₯π(ππ2π1)π1)β
1
2(πΆ42ππ2π2
ππ(π₯)
ππ₯ππ₯π(ππ2π1)οΏ½οΏ½2)+
πΆ31π1π2π(π₯)
ππ₯2ππ₯π(ππ2π1)π1
2π]π(π1) = 0 (54)
(βππ(π1)
ππ1(πΆ21
ππ(π₯)
ππ₯π1 + πΆ22
ππ(π₯)
ππ₯π2 + 2π(π₯)ππ) +
6π(π1)2π(π1) (ππ(π₯)
ππ₯)2 ππ(π₯)
ππ₯+ 2π(π1)2π(π1) (
ππ(π₯)
ππ₯)2 π4π(π₯)
ππ₯4+
4π(π1)2π(π1) ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3+
Journal of Computational Applied Mechanics, Vol. 51, No. 2, June 2020
317
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
3πΆ6.π(π1)2π(π1) ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2+
8π(π1)2π(π1) ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2
π3π(π₯)
ππ₯3β
3
2πΆ6. π(π1)2π(π1) (
ππ(π₯)
ππ₯)2 π2π(π₯)
ππ₯2)+
(2π(π1)π(π1) πΞ¦(π₯)
ππ₯
π4π(π₯)
ππ₯4+π(π1)π(π1) π4Ξ¦(π₯)
ππ₯4
ππ(π₯)
ππ₯+
4π(π1)π(π1) π2Ξ¦(π₯)
ππ₯2
π3π(π₯)
π3+
3π(π1)π(π1) π2π(π₯)
π2
π3Ξ¦(π₯)
π3)ππ₯π(π1π1π) β
(πΆ6π(π1)π(π1) πΞ¦(π₯)
ππ₯
π2π(π₯)
ππ₯2+
πΆ6π(π1)π(π1) ππ(π₯)
ππ₯
π2Ξ¦(π₯)
ππ₯2) ππ₯π(π1π1π) = 0 (55)
With the inner product defined for complex functions on [0, 1] as:
β¨π, πβ© = β« ποΏ½οΏ½1
0ππ₯ (56)
The equations can be cast as: ππ(π1)
ππ1β π½2π(π1)2ππ₯π(βππ1π1) + π½3π(π1) exp(βππ2π1) = 0
(57) ππ(π1)
ππ1+πΎ2(π(π1)2π(π1) ) + πΎ3(π(π)π(π1) exp(ππ1π1)) =
0 (58)
Where:
π½2 =β« [
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3βπΆ6
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2]π(π₯) ππ₯
10
β« [(πΆ21π1+πΆ22οΏ½οΏ½2)ππ(π₯)
ππ₯+2ππ π(π₯)]π(π₯) ππ₯
10
π½3 =β« βπ(π₯) 10 [πΆ32π2
π2π(π₯)
ππ₯2οΏ½οΏ½22π+1
2(πΆ21π1
ππ(π₯)
ππ₯οΏ½οΏ½1π) ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
+
β« βπ(π₯) 10 [
1
2(πΆ22π2
ππ(π₯)
ππ₯οΏ½οΏ½2π)β(πΆ41π1
ππ(π₯)
ππ₯οΏ½οΏ½1π) ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
+
β« βπ(π₯) 10 [β(πΆ42π2
ππ(π₯)
ππ₯οΏ½οΏ½2π)β
1
2(πΆ41ππ2π1
ππ(π₯)
ππ₯οΏ½οΏ½1) ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
+
β« βπ(π₯) 10 [β
1
2(πΆ42ππ2π2
ππ(π₯)
ππ₯οΏ½οΏ½2)+πΆ31π1
π2π(π₯)
ππ₯2π12π ]ππ₯
(πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯
πΎ2 =β« [6(
ππ(π₯)
ππ₯)2ππ(π₯)
ππ₯+2(
ππ(π₯)
ππ₯)2π4π(π₯)
ππ₯4]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
+
β« [4ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π4π(π₯)
ππ₯4+8
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
+
β« [8ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3β3πΆ6
ππ(π₯)
ππ₯
ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
+
β« [8ππ(π₯)
ππ₯
π2π(π₯)
ππ₯2π3π(π₯)
ππ₯3β3
2πΆ6(
ππ(π₯)
ππ₯)2π2π(π₯)
ππ₯2]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
πΎ3 =β« [2
πΞ¦(π₯)
ππ₯
π4π(π₯)
ππ₯4+4
π2Ξ¦(π₯)
ππ₯2π3π(π₯)
ππ₯3+π4Ξ¦(π₯)
ππ₯4ππ(π₯)
ππ₯]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
+
β« [3π2π(π₯)
ππ₯2π3Ξ¦(π₯)
ππ₯3βπΆ6
πΞ¦(π₯)
ππ₯
π2π(π₯)
ππ₯2βπΆ6
ππ(π₯)
ππ₯
π2Ξ¦(π₯)
ππ₯2]π(π₯) 1
0 ππ₯
β((πΆ21π1+πΆ22οΏ½οΏ½2)β«ππ(π₯)
ππ₯π(π₯) 1
0 ππ₯+2ππ β« π(π₯)π(π₯) 10 ππ₯)
To determine X(T1) and Y(T1), the solution of equations (57) and (58) is expressed in polar form:
π(π1) =1
2πΌπ¦(π1)πππ½π¦(π1) πππ π(π1) =
1
2πΌπ¦(π1)πβππ½π¦(π1) (59)
π(π1) =1
2πΌπ₯(π1)πππ½π₯(π1) πππ π(π1) =
1
2πΌπ₯(π1)πβππ½π₯(π1) (60)
Substituting into the solvability condition and separating real and
imaginary parts. The following set of modulation equation is formed:
0 =ππΌπ₯(π1)
ππ1+ π½3π πΌπ₯(π1)cos(π2) β π½3πΌπΌπ₯(π1)sin(π2)β
π½2π πΌπ¦(π1)2
2cos(π1)β
π½2πΌπΌπ¦(π1)2
2sin(π1) (61)
0 =ππΌπ¦(π1)
ππ1+πΎ2π πΌπ¦(π1)3
4+πΎ3π πΌπ¦(π1)πΌπ₯(π1)
2cos(π1) β
πΎ3πΌπΌπ¦(π1)πΌπ₯(π1)
2sin(π1) (62)
0 = πΌπ₯(π1)ππ½π₯(π1)
ππ1βπ½2πΌπΌπ¦(π1)2
2cos(π1)+
π½2π πΌπ¦(π1)2
2sin(π1) +
π½3πΌπΌπ₯(π1)cos(π2) β π½3π πΌπ₯(π1)sin(π2) (63)
0 = πΌπ¦(π1)ππ½π¦(π1)
ππ1+πΎ2πΌπΌπ¦(π1)3
4+πΎ3πΌπΌπ¦(π1)πΌπ₯(π1)
2cos(π1)+
πΎ3π πΌπ¦(π1)πΌπ₯(π1)
2sin(π1) (64)
Where:
Ο1 = Ξ²x(T1) β 2Ξ²y(T1) + Ο1T1
and Ο2 = Ο2T1β 2Ξ²x(T1)
J2R, J3R, K2R, K3R are the real part of J2, J3, K2, and K3
J2I, J3I, K2I, K3I are the imaginary part of J2, J3, K2, and K3
Seeking for stationary solutions, Ξ±(x)β² = Ξ±(y)β² = Ο1β² = Ο2β² =0 in modulation equations (61) to (64),
0 = 2π½3π πΌπ₯(π1)cos(π2) β 2π½3πΌπΌπ₯(π1)sin(π2)β
π½2π πΌπ¦(π1)2 cos(π1)β π½2πΌπΌπ¦(π1)2 sin(π1) (65)
0 =πΎ2π πΌπ¦(π1)3
4+πΎ3π πΌπ¦(π1)πΌπ₯(π1)
2cos(π1)β
πΎ3πΌπΌπ¦(π1)πΌπ₯(π1)
2sin(π1) (66)
0 = πΌπ₯(π1)π2 β π½2πΌπΌπ¦(π1)2 cos(π1)+
π½2π πΌπ¦(π1)2 sin(π1) +
2π½3πΌπΌπ₯(π1)cos(π2) β2π½3π πΌπ₯(π1)sin(π2) (67)
0 = πΌπ¦(π1)(2π1+π2
4)+
πΎ2πΌπΌπ¦(π1)3
4+ πΎ3πΌπΌπ¦(π1)πΌπ₯(π1)
2cos(π1) +
πΎ3π πΌπ¦(π1)πΌπ₯(π1)
2sin(π1) (68)
The linear solutions can be obtained by setting the coefficient of the nonlinear terms to zero. Therefore,
Adegoke et al.
318
πΌπ₯(π1) = πΌπ¦(π1) = 0 (69) The nonlinear solutions can be obtained by solving for
Ξ±x(T1) and Ξ±y(T1) completely. Equation (62) and (64) can be rewritten as:
β(πΎ2π πΌπ¦(π1)2
2) = π 1 cos(π1+ π1) (70)
β(πΎ2πΌπΌπ¦(π1)2
2+ π1 +
π2
2) = π 1sin(π1+ π1) (71)
Where,
tan(π1) =πΎ3πΌ
πΎ3π , (72)
π 1 = βπΌπ₯(π1)2(πΎ3πΌ2 +πΎ3π 2)) (73)
From,
(sin(π1+ π1))2 + (cos(π1+ π1))2 = 1
(74) We obtained;
πΎ2πΌ2πΌπ¦(π1)4 + 4πΎ2πΌ2π1πΌπ¦(π1)2 + 2πΎ2πΌ2π2πΌπ¦(π1)
2 +πΎ2π 2πΌπ¦(π1)4 + 4π1
2 + 4π1π2 + π22 β 4πΌπ₯(π1)2(πΎ3πΌ2 +
πΎ3π 2) = 0 (75) And:
π1 = tanβ1 [πΎ2πΌπΌπ¦(π1)2+2π1+π2
πΎ2π πΌπ¦(π1)2] β tanβ1 (
πΎ3πΌ
πΎ3π )
(76) Also, equations (61) and (63) can be rewritten as:
(π½2π πΌπ¦(π1)2 cos(π1)+π½2πΌπΌπ¦(π1)2 sin(π1)
2) = π 2cos(π2+ π2) (77)
β(πΌπ₯(π1)2π2
2βπ½2πΌπΌπ¦(π1)2 cos(π1)
2+π½2π πΌπ¦(π1)2 sin(π1)
2) =
π 2 sin(π2+ π2) (78)
Where:
tan(π2) =π½3πΌ
π½3π , (79)
π 2 = βπΌπ₯(π1)2(π½3πΌ2 + π½3π 2) (80)
Transforming the equations (77) and (78) to;
(π½2π πΊ2+π½2πΌ πΊ1
2) = π 2 cos(π2+ π2) (81)
β(πΌπ₯(π1)2π2
2βπ½2πΌ πΊ2
2+π½2π πΊ1
2) = π 2 sin(π2+ π2) (82)
Where:
πΊ1 =2π1 cos(π1)+π2 cos(π1)+πΎ2πΌπΌπ¦(π1)
2 cos(π1)βπΎ2π πΌπ¦(π1)2 sin(π1)
πΎ2π βπΎ2πΌ2πΌπ¦(π1)4+4πΎ2πΌπ1πΌπ¦(π1)
2+2πΎ2πΌπ2πΌπ¦(π1)2+πΎ2π 2πΌπ¦(π1)4+4π1
2+4π1π2+π22
πΎ2π 2πΌπ¦(π1)4
πΊ2 =
2π1 sin(π1)+π2 s in(π1)+πΎ2πΌπΌπ¦(π1)2 cos(π1)+πΎ2π πΌπ¦(π1)2 sin(π1)
πΎ2π βπΎ2πΌ2πΌπ¦(π1)4+4πΎ2πΌπ1πΌπ¦(π1)
2+2πΎ2πΌπ2πΌπ¦(π1)2+πΎ2π 2πΌπ¦(π1)4+4π1
2+4π1π2+π22
πΎ2π 2πΌπ¦(π1)4
From;
(sin(π2+ π2))2 + (cos(π2+ π2))2 = 1 (83) We have:
πΊ12(π½2πΌ2 + π½2π 2) + 2πΊ1π½2π π2πΌπ₯(π1) + πΊ22(π½2πΌ2 + π½2π 2) β
2πΊ2π½2πΌπ2πΌπ₯(π1) + π22πΌπ₯(π1)2 β 4πΌπ₯(π1)2(π½3πΌ2 + π½3π 2) =
0 (84) Resolving equations (75) and (84) for real and positive values of
πΌπ¦(π1)πππ πΌπ₯(π1).
Consequently,
π2 = βtanβ1 (π½3πΌ
π½3π )β
tanβ1 [πΌπ₯(π1)2π2βπ½2πΌπΌπ¦(π1)
2 cos(π1)+π½2π πΌπ¦(π1)2 sin(π1)
π½2π πΌπ¦(π1)2 cos(π1)+π½2πΌπΌπ¦(π1)2 sin(π1)] (85)
Therefore, considering the n-th values of Ξ±x(T1), Ξ±y(T1), Ξ²x(T1)
and Ξ²y(T1) corresponding to the n-th modal functions and the n-th natural frequencies, the n-th solution of the coupled problem is
expressed as:
οΏ½οΏ½(π₯, π‘)π = πΌπ₯(π1)ππ(π₯)π πππ (πππ0+ π½π₯(π1)π) + π(π) (86)
Substituting into the equations (86);
π0 = π‘ , π1 = ππ‘, πΌπ₯(π1)π = πΌπ₯π , πΌπ¦(π1)π = πΌπ¦π , π½π¦(π1)π =π½π₯(π1)πβ π1π+π1ππ1
2, π½π₯(π1)π =
π2ππ1βπ2π
2, Ξ©1 = 2ππ + ππ2π ,
Ξ©2 = 2ππ + ππ2π , Ξ©1 = Ξ©2 = Ξ© πππ π1ππ1 = πππ0β 2πππ0
With the solvability condition fulfilled, the particular solution of
equation (35) for the internal resonance condition is obtained as:
π’1 = C1πΌπ₯(π1) πππ (π½π¦(π1) + π0(π + πΊ)) +
πΆ2πΌπ₯(π1) πππ (π½π₯(π1) + π0(π β πΊ)) + 2C3πππ (π0πΊ) (87)
Where:
πΆ1 =πΆ21π1
ππ(π₯)ππ₯
οΏ½οΏ½1ππ
2βπΆ41Ξ©1π1
ππ(π₯)ππ₯
οΏ½οΏ½1
2βπΆ31
π2π(π₯)
ππ₯2(π1)
2π
β« π(π₯) πΏπ0 (πΆ22Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π+πΆ21Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½1π)ππ₯βΞ©
2+ππ2[πΆ5βπΆ32(π2)
2βπΆ31(π1)2]+
πΆ22π2ππ(π₯)ππ₯
οΏ½οΏ½1ππ
2βπΆ42Ξ©2π2
ππ(π₯)ππ₯
οΏ½οΏ½2
2βπΆ32π2
π2π(π₯)
ππ₯2(π2)
2π
β« π(π₯) πΏπ0 (πΆ22Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π+πΆ21Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π)ππ₯βΞ©
2+ππ2[πΆ5βπΆ32(π2)
2βπΆ31(π1)2]
πΆ2 =πΆ21π1
ππ(π₯)ππ₯
οΏ½οΏ½1ππ
2βπΆ32π2
π2π(π₯)
ππ₯2(π2)
2π+πΆ22π2
ππ(π₯)ππ₯
οΏ½οΏ½2ππ
2
β« π(π₯) πΏπ0 (πΆ22Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π+πΆ21Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π)ππ₯βΞ©
2+ππ2[πΆ5βπΆ32(π2)
2βπΆ31(π1)2]β
πΆ41Ξ©1π1ππ(π₯)ππ₯
οΏ½οΏ½1
2βπΆ42Ξ©2π2
ππ(π₯)ππ₯
οΏ½οΏ½2
2βπΆ31π1
π2π(π₯)
ππ₯2(π1)
2π
β« π(π₯) πΏπ0 (πΆ22Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π+πΆ21Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½1π)ππ₯βΞ©
2+ππ2[πΆ5βπΆ32(π2)
2βπΆ31(οΏ½οΏ½1)2]
πΆ3 =1
2πΆ11Ξ©1π1π1+
1
2πΆ12Ξ©2π2οΏ½οΏ½2
β« π(π₯) πΏπ0 (πΆ22Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½2π+πΆ21Ξ©
ππ(π₯)
ππ₯οΏ½οΏ½1π)ππ₯βΞ©
2+ππ2[πΆ5βπΆ32(οΏ½οΏ½2)
2βπΆ31(π1)2]
The first order approximate solution is expressed as:
οΏ½οΏ½(π₯, π‘) = β πΌπ₯π|π(π₯)π| πππ (π‘Ξ©
2βπ2π
2+ ππ₯π)+
βn=1
Journal of Computational Applied Mechanics, Vol. 51, No. 2, June 2020
319
π(π) (88) Where the phase angles ππ₯π are given by:
tan(ππ₯π) =πΌπ{π(π₯)π}
π π{π(π₯)π},
4. Results and Discussion
This section presents the numerical solutions of the nonlinear dynamics of a cantilever pipe, conveying steady pressurized
air/water two-phase flow.
Table 1: Summary of pipe and flow parameter
Parameter Name
Parameter
Unit
Parameter
Values
External Diameter Do (m) 0.0113772
Internal Diameter Di (m) 0.00925
Length L (m) 0.1467
Pipe density Οpipe (kg/m3) 7800
Gas density ΟGas (kg/m3) 1.225
Water density ΟWater (kg/m3) 1000
Tensile and compressive stiffness EA (N) 7.24E+06
Bending stiffness EI (N) 1.56E+03
4.1. Results for Ο is far from 2Ξ» (Uncoupled axial and transverse vibration)
Numerical examples are presented for the first two modes to examine the effects of the variation in the void fraction of the
conveyed two-phase flow on the parametric stability boundaries based on equation (51).
(a)
(b)
Figure 2: Parametric stability boundaries of mode 1 and mode 2 for varying
void fractions.
In Figures 2a and 2b, stable and unstable boundaries are plotted for the parametric resonance cases for the first and second mode
for three different void fractions. In all the figures, the regions between the boundaries are unstable while other areas are stable.
The stability boundaries are wider for the second mode as compared to the first mode. However, for the first and second
modes, Increase in the void fraction is observed to reduce the
stability boundaries.
4.2. Results for Ο is close to 2Ξ» (Internal resonance case)
The internal resonance case presents a coupling between the axes
which results in the transference of energy between the axes. For varying amplitude of pulsation for the two phases, both phases
slightly detuned by 0.2 from the axial frequency and the axial and transverse frequencies also detuned by 0.2. The amplitude
response curves are plotted for various void fractions.
(a)
0
2
4
6
8
10
12
-6 -4 -2 0 2 4 6
Am
plit
udes
of
puls
atio
n of
pha
se 1
(Β΅
1)
Ο
Amplitude of pulsation of phase 2 (Β΅2=2) for Mode 1
Vf=0.1
Vf=0.3
Vf=0.5
0
2
4
6
8
10
12
-8 -6 -4 -2 0 2 4 6 8
Am
plit
udes
of
puls
atio
n of
pha
se 1
(Β΅
1)
Ο
Amplitude of pulsation of phase 2 (Β΅2=2) for Mode 2
Vf=0.1
Vf=0.3
Vf=0.5
0
48
1216
200
0.2
0.4
0.6
040
80120
160200
Β΅1
ax
Β΅2
Mode 1's axial amplitude response curves for void fraction of 0.1
0.4-0.6
0.2-0.4
0-0.2
Adegoke et al.
320
(b)
(c)
Figure 3: Amplitude response curves of the tipβs axial vibration for mode 1
(a)
(b)
(c)
Figure 4: Amplitude response curves of the tipβs transverse vibration for
mode 1
(a)
(b)
(c)
Figure 5: Amplitude response curves of the tipβs axial vibration for mode 2
04
812
1620
0
0.05
0.1
0.15
0.2
040
80120
160200
Β΅1
ax
Β΅2
Mode 1's axial amplitude response curves for void fraction of 0.3
0.15-0.2
0.1-0.15
0.05-0.1
0-0.05
04
812
1620
0
0.05
0.1
040
80120
160200
Β΅1
ax
Β΅2
Mode 1's axial amplitude response curves for void
fraction of 0.5
0.05-0.1
0-0.05
04
812
1620
0
0.05
0.1
0.15
0.2
040
80120
160200
Β΅1
ay
Β΅2
Mode 1's transverse amplitude response curves for void
fraction of 0.1
0.15-0.2
0.1-0.15
0.05-0.1
0-0.05
04
812
1620
0
0.05
0.1
040
80120
160200
Β΅1
ay
Β΅2
Mode 1's transverse amplitude response curves for void fraction of 0.3
0.05-0.1
0-0.05
0
48
1216
200
0.02
0.04
0.06
0.08
040
80120
160200
Β΅1
ay
Β΅2
Mode 1's transverse amplitude response curves for void
fraction of 0.5
0.06-0.08
0.04-0.06
0.02-0.04
0-0.02
04
812
1620
0
0.001
0.002
0.003
0.004
040
80120
160200
Β΅1
ax
Β΅2
Mode 2's axial amplitude response curves for void fraction of 0.1
0.003-0.004
0.002-0.003
0.001-0.002
0-0.001
04
812
1620
0
0.0005
0.001
0.0015
0.002
040
80120
160200
Β΅1
ax
Β΅2
Mode 2's axial amplitude response curves for void fraction of 0.3
0.0015-0.002
0.001-0.0015
0.0005-0.001
0-0.0005
0
48
1216
200
2
4
6
040
80120
160200
Β΅1
ax
Β΅2
Mode 2's axial amplitude response curves for void fraction of 0.5
4-6
2-4
0-2
Journal of Computational Applied Mechanics, Vol. 51, No. 2, June 2020
321
(a)
(b)
(c)
Figure 6: Amplitude response curves of the tipβs transverse vibration for
mode 2
As a result of the internal coupling between the planar axis, energy
is transferred, oscillation is observed in both the axial and transverse directions as a result of the parametric axial vibrations.
Figures 2, 3, 4, 5 and 6 shows that for both mode 1 and mode 2, an increase in the void fraction is observed to reduce the amplitude of
oscillations due to increasing in mass content in the pipe and which further dampens the motions of the pipe. However, at a high void
fraction of 0.5, Figures 5c and 6c show the occurrence of a resonance peak in the second mode in both axial and transverse
oscillations.
5. Conclusion
In this study, the axial vibrations of a cantilevered pipe have
been investigated. The velocity is assumed to be harmonically
varying about a mean value. The method of multiple scales is
applied to the equation of motion to determine the velocity-
dependent frequencies and the study of the parametric resonance
behavior of the system. Away from the internal resonance
condition, the influence of small fluctuations of flow velocities of
the phases on the stability of the system is examined. The
boundaries separating stable and unstable regions are estimated
and it was observed that an increase in the void fraction reduces
the stability boundaries. With internal resonance, transverse
oscillations will also be generated due to the transfer of energy
from the resonated axial vibrations. However, an increase in void
fraction dampens the motions of the pipe.
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0
48
1216
200
0.01
0.02
0.03
040
80120
160200
Β΅1
ay
Β΅2
Mode 2's transverse amplitude response curves for void fraction of 0.1
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0.01-0.02
0-0.01
0
48
1216
200
0.005
0.01
0.015
040
80120
160200
Β΅1
ay
Β΅2
Mode 2's transverse amplitude response curves for void fraction of 0.3
0.01-0.015
0.005-0.01
0-0.005
0
48
1216
200
0.5
1
040
80120
160200
Β΅1
ay
Β΅2
Mode 2's transverse amplitude response curves for void fraction of 0.5
0.5-1
0-0.5
Adegoke et al.
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