

Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 487-496, Warsaw 2013

CRITICAL FLOW VELOCITY IN A PIPE WITH ELECTROMAGNETIC

ACTUATORS

Tomasz Szmidt, Piotr Przybyłowicz

Warsaw University of Technology, Faculty of Automotive and Construction Machinery Engineering, Warsaw, Poland

e-mail: tomasz.szmidt@gmail.com

We study the influence of electromagnetic damping on the dynamics of a pipe conveying
fluid. Pipes supported at both ends as well as cantilever ones (both discharging and aspira-
ting the fluid) are considered. We assume physical parameters of the systems which allow
an experimental verification of results. We develop simple methods of calculation of the
internal and external damping coefficients which are based on known models, and do not
require experiments.The governingpartial equationof thepipe is discretisedwithGalerkin’s
procedure, and the stability of the resultant dynamical system is determined with eigenva-
lues of its linearization. The actuators destabilise the pipe supported at both ends, but can
remarkably improve stability of cantilever ones. The effect of magnetic damping strongly
depends on the position at which actuators are attached to the pipe.

Key words: pipe conveying fluid, dynamic stability, flutter, electromagnetic actuators

1. Introduction

Pipes conveying fluid are an example of nonconservative elastic systems with follower force.
The problem of their dynamics has become a new paradigm inmechanics; moreover, theoretical
predictions can be verified experimentally (Päıdoussis, 2008). Because pipes conveying fluid are
widely used – in water home installations, cooling systems, pipelines, ocean mining – therefore
the results might have practical applications.

Sufficientlyhighflowvelocity results in thebucklingof apipe supportedatbothends– simply
supported (Feodos’ev, 1951; Housner, 1952) and clamped-clamped one (Päıdoussis and Issid,
1974). The external and internal damping of the system depends on the velocity of deflection
of a pipe, thus it has no influence on divergence. On the other hand, a cantilever pipe is always
prone to flutter instability (Gregory andPäıdoussis, 1966a). Then, depending on themass ratio
of the fluid andpipe, both types of energy dissipation can stabilise the systemaswell as decrease
the critical flow velocity (Gregory and Päıdoussis, 1966b).

The elastic Winkler foundation stabilises a non-damped pipe with a flow (Lottati and Kor-
necki, 1986). Partial foundation can increase the critical flow as well as destabilise the pipe, and
also change the type of instability between divergent and oscillatory one (Impollonia andElisha-
koff, 2000), (Elishakoff and Impollonia, 2001).Moreover, a significant effect on the critical flow is
exerted by an intermediate elastic support (Sugiyama et al., 1985), a simple support (Edelstein
and Chen, 1985) and also by a system of several springs or a torsion spring (Päıdoussis, 1998).

Thus an application of electromagnetic actuators, which generate both elastic and dissipa-
tive nonlinear forces, can remarkably affect the stability. The application of such elements for
stabilisation of rotating shafts was proposed byKurnik (1994). It was shown that the magnetic
force passively generated by actuators can bring an increase of the critical rotational velocity.
The efficiency of magnetic stabilisation was also confirmed in systems of columns subjected to
follower load (Kurnik et al., 2009; Szmidt and Przybyłowicz, 2012).


488 T. Szmidt, P. Przybyłowicz

The influence of magnetic forces on nonlinear dynamics of similar structure – a cantilever
beam – was investigated both theoretically and empirically (Moon, 1980; Holmes and Moon,
1983). In 1988 Tang and Dowell studied chaotic motion of a cantilevered pipe subjected to
external magnetic force, approximated by a cubic function of pipe deflection. Here we confine
ourselves toanalysing just the stability of the system.Ontheotherhandweapply electromagnets
(not magnets), and the magnetic force exerted on the pipe is the effect of the actuators state,
whichwemodel indetail.We incorporate realisticmagnetisation curveof steel cores andtake into
account both the effect of magnetic induction and hysteresis. Moreover, we consider aspirating
pipes, which dynamics was not understood in the eighties.

Galerkin’s multimodal discretisation based on the beam eigenfunctions is applied to the
partial differential equation governing the lateral motion of the pipe. The resultant ordinary
equations are coupled with ordinary equations of electro-magnetodynamics of actuators. Then,
the stability is determined by numerical calculations of eigenvalues of the linear system.

2. Analysed system

Consider a fluid-conveying pipewith electromagnetic actuators attached to it (Fig. 1). The pipe
is placed in the direction of the gravitational field g and conveys the stream of water with
velocity V . The support is cantilever, simple at both ends or clamped-clamped. The water is
pumped downwards or aspirated.

Fig. 1. Pipe with a flow and electromagnetic actuators

Assume themotion of the pipe in the plane of symmetry x−w, because for studied systems
a two-dimensional model is sufficient for predicting instability (Modarres-Sadeghi et al., 2008).
The dimensions of the pipe do not change due to the internal fluid pressure or the flow friction.
Lateral deflections are small and the pipe is slender, thus the linear Bernoulli-Euler model is
acceptable. The pipe ismade of a viscoelastic Kelvin-Voigtmaterial. The vibrations frequency is
sufficiently low, so themagnetic induction and its rate of change are approximately constant on
the cross-section of the electromagnet core. Therefore, Bertotti’s model of magnetic hysteresis
is applicable (Przybyłowicz and Szmidt, 2009). We consider a fully-developed turbulent flow,
which can be approximated by the so called plug flow – an infinite elastic rodmoving inside the
pipe.

Incases of thepipe supportedatbothendsandthecantileverpipeconveyingfluidtowards the
free end, the dynamics is governed by the following linear partial equation (Päıdoussis and Issid,
1974) coupled with nonlinear equations of electro-magnetodynamics of actuators (Przybyłowicz
and Szmidt, 2009)


Critical flow velocity in a pipe with electromagnetic actuators 489

(
1+β⋆

∂

∂t

)
EI

∂4w

∂x4
+[MV 2− (m+M)g(L−x)]

∂2w

∂x2
+2MV

∂2w

∂x∂t

+(m+M)g
∂w

∂x
+γ⋆

∂w

∂t
+(m+M +Ma)

∂2w

∂t2
=

A

µ0
(B22 −B

2
1)δe

(AN2

R
+
1

8
σla2
)dB1,2

dt
+
2(z ±w(xe, t))

µ0
B1,2+ lφ

−1(B1,2)=
NU

R

(2.1)

In theequations above, Land EI denote the lengthandflexural rigidity of thepipe, respectively,
m, M, Ma – mass of the pipe, water flowing inside and fluid which surrounds the pipe and is
accelerated by itsmotion, per unit length, β⋆, γ⋆ – coefficients of internal and external damping,
l – length of each of the magnetic circuits, a, A = πa2 – cross-sectional radius and area of the
electromagnet core, N –number ofwire coilswhich arewoundon the cores, R – resistance of the
electric circuit (copper), xe – distance from the electromagnets to the upper end of the pipe, z –
gap in themagnetic circuits (i.e. between the cores attached to the column and electromagnets)
in the middle equilibrium of the column, U – voltage applied to the actuators, σ – electric
conductivity of the steel cores, B = φ(H)= arctan(H/400)T – primarymagnetisation curve of
steel, µ0 – magnetic permeability of vacuum.

The “+” sign at the term expressing the Coriolis effect corresponds to the situation when
the fluid moves downwards. In the case of the aspirating pipe, we change the sign to “−”, so
the flow velocity value V is always positive.

Reversing the flowyields a destabilising effect of theCoriolis force. Furthermore, the reaction
at the free end of a pipe becomes complicated (Päıdoussis et al., 2005). In case of a pipe dischar-
ging the stream, the fluiddoes not change its velocity at the outlet and follows its lateralmotion,
so does not generate any reaction there. In case of an aspirating pipe the fluid is accelerated at
the inlet, thus it brings tensional reaction, tangential to the free end.According to theBernoulli
equation, it decreases by a half the effect of the inertial force of the fluid inside. However, the
lateral reaction of the fluid exerted on the pipe inlet is an open problem. We use the model
assuming that the space-averaged direction of inflow is tangential to the deflected pipe and the
flow field below the entrance does not follow the lateral motion of the pipe. Thismodel overesti-
mated the critical flow velocity observed in the experiment conducted byKuiper andMetrikine
(2008). The other model the could be applicable differs in that the flow field moves with the
inlet, so it does not generate transverse reaction there (this model led to underestimation of the
critical flow in above mentioned experiment).

The dynamics of lateral vibrations of the aspirating pipe is given by

(
1+β⋆

∂

∂t

)
EI

∂4w

∂x4
+
[1
2
MV 2− (M +m)g(L−x)

]∂2w
∂x2
−2MV

∂2w

∂x∂t

+(M +m)g
∂w

∂x
+MV δL

∂w

∂t
+γ⋆

∂w

∂t
+(M +Ma+m)

∂2w

∂t2
=

A

µ0
(B22 −B

2
1)δe

(2.2)

The parameters of analysed systems are presented in Table 1 (by d and D we denote the
internal and external diameter of the pipe). For the plastic pipe, they are the same as in the
experimental work by Kuiper and Metrikine. The material from which the pipe is made and
dimensions of the pipe determine permissible pressure of the supplied water. This pressure and
the pipe geometry determine flowvelocity. The parameterswere selected so that the critical flow
velocity is physically reachable (in the case of an aspirating pipewe took the effect of cavitation
into account).

We assume the approximate solution to the equation of the pipe in a form

w̃(x,t)=
n∑

i=1

W̃i(x)Ti(t) (2.3)


490 T. Szmidt, P. Przybyłowicz

Table 1.Parameters of studied systems (pipes and actuators)

Parameter Steel pipe PVC pipe Unit

L 2 4.75 m
d 0.01 0.07 m
D 0.0111 0.075 m

a 0.002 0.003 m
l 0.3 0.6 m
N 1000 1667 [–]
R 2.38 2.83 Ω

where W̃i, i =1, . . . ,n are the eigenfunctions of a beamwith an appropriate support, normalised
with respect to integrals of their squares. Applying Galerkin’s procedure, we obtain a set of
2n+2first-order differential equations for functions Ti, Ṫi and B1,2. The stability of themiddle
equilibrium is determined by numerical calculations of eigenvalues of the linearized system.

Thevalues of critical flowvelocity presented in Section 4were calculated for n =2,4,6,8,10.
We found that increasing the number of eigenfunctions from 8 to 10 makes almost no visible
change in the shape of stability area, thus 10-modal approximation is accurate enough.

3. Internal and external damping coefficients

We calculate the internal viscous damping coefficient β⋆ by using the coefficient of dissipation
of the material, which is defined as the ratio of energy dissipated as heat due to the internal
friction, to the total energy of deformation (Dyląg et al., 2000).

Consider a cantilever pipe of given geometry, without actuators and fluid inside, when the
external damping and gravitation are absent. Then, its vibrations are given by the equation

β⋆EI
∂5w

∂x4∂t
+EI

∂4w

∂x4
+m

∂2w

∂t2
=0 (3.1)

Assuming the solution in the form of a product

w(x,t) = W(x)T(t) 0¬ x ¬ L t > 0 (3.2)

we obtain two independent differential equations. The solution to W(x) is the sequence of
eigenfunctions Wi characterising the shape of pipe deflection, with corresponding eigenvalues ki
(i = 1, . . . ,∞). We are interested in the first mode, because the values of the coefficients of
dissipation are given for the bending trial. The vibrations of this mode are governed by another
equation obtained through separation of variables

mT̈1(t)+βEIk
4
1Ṫ1(t)+EIk

4
1T(t)= 0 (3.3)

This is the equation of harmonic oscillator, for which one can calculate logarithmic decrement
of damping. On the other hand, this decrement accounts for the half of the material coefficient
of dissipation ψ. From these two relations, we get

β⋆ =
1

k21

√
m

EI

ψ
√
4π2+ψ2/4

(3.4)

The value of the coefficient of dissipation for steel is between 0.01-0.03, so we assume ψ =0.02.
Unfortunately,wehavenot foundthevalueof ψ forPVC, thusweassumethemeanof coefficients


Critical flow velocity in a pipe with electromagnetic actuators 491

Table 2. Calculated coefficients of the internal and external damping β⋆, γ⋆ and the added
mass Ma for the studied pipes and surroundings

Steel pipe PVC pipe
air water air water

γ⋆ [kg/m/s] 0.000966 0.170864 0.002753 0.477095

Ma [kg/m] 0.000180 0.111576 0.006244 4.644865

β⋆ [s] 0.000192 0.002646

for steel and hard rubber, i.e. ψ = 0.1. The values of β⋆ calculated in this way for examined
pipes are given in Table 2.
For themodel of external dampingwe use the one described byKirstein et al. (1998), which

had been known earlier. The coefficient γ⋆ and themass of fluid accelerated by vibrations of the
pipe Ma are calculated with the formulas obtained from the analytical solution to the Navier-
Stokes equation for the surrounding flow outside the pipe. In this model, one assumes that the
pipe performs small harmonic vibrations of frequency ω and is placed far away from the walls
of the tank with fluid.We use the following formulas

γ⋆ =−
1

4
πρD2ωℑ

(
1+
4K1(α)

αK0(α)

)
Ma =

1

4
πρD2ℜ

(
1+
4K1(α)

αK0(α)

)
(3.5)

where

α =
√
iRe Re=

ω

ν

D2

4
(3.6)

Re denotes the kinetic Reynolds number for external flow and K0 and K1 are modified Bessel
functions of the second kind, respectively of the zero and first order.
The problem is what value of frequency ω ought to be used. The goal is the examination of

stability, thus it should be the frequency of upcoming flutter vibrations. The external damping
affects this frequency, so an iterative procedure is required. However, the fluid-conveying pipe
can lose stability in several modes at the same time, whatmakes such a procedure cumbersome
and not necessarily convergent (Kuiper et al., 2007). The authors of that paper proposed a new
model of hydrodynamicdrag, independent of the frequency of vibrations.However, such amodel
is applicable only to flows with high values of the Reynolds number.
In this research, we apply a different approach, which is based on the numerically calculated

frequency of free vibrations of the cantilever pipe filled by still water. The already estimated
internal damping is incorporated, and the effect of actuators is neglected. The analytically
calculated frequency of free vibrations of the first mode with the absence of external damping
constitutes the starting point. In the next steps the procedure calculates the frequency in the
middle of the pipe (basing on the first 10 cycles of free vibrations), and the parameters γ⋆

and Ma. The computations stop when a relative change of γ
⋆ becomes smaller than 1%, what

usually takes place after a few iterations. The obtained values are presented in Table 2.

4. Stability

We studied the influence of magnetic actuators on the stability of: the steel pipe supported at
both ends, steel cantilever pipe conveying water downwards and aspirating pipe made of PVC.
Points marked in figures denote the critical flow velocity for a given voltage, i.e. the border of
stability area. In every figure the right border of the area was found by numerical calculation
of V for which the real part of the decisive eigenvalue changed its sign, for a given value of
voltage U (changed with an appropriate step). When computing the upper border, the flow
velocity V was a parameter and U – a variable.


492 T. Szmidt, P. Przybyłowicz

4.1. Two-side support

We consider the system of a steel pipe with electromagnetic actuators (Table 1). The pipe
supported at both ends loses stability by buckling, and the clamped-clamped support raises
the critical flow twice comparing to the simple support (Päıdoussis and Issid, 1974). Dissipative
forces –which dependon the velocity – do not affect the static bifurcation, so the dynamic effect
ofmagnetic damping (Lenz rule) does not appear. Still, each actuator pulls the pipe, thus exerts
the effect which destabilises the middle equilibrium. As a result, an increase of voltage worsens
the stability of the pipe supported at both ends.

4.2. Cantilever pipe discharging water

In this subsection, we present results for a steel pipe (Table 1) clamped at its upper end. It
has been proved that the cantilever pipe pumping the fluid downwards loses stability by flutter
(Gregory and Päıdoussis, 1966a,b), which let dissipative magnetic forces to appear.

Figures 2 and 3 show the influence of magnetic actuators on the stability area for the pipe
placed in air and in water, respectively. Crossing the right-hand border of the area results in
flutter, and the upper one yields divergence (buckling). One can see that the effect of magnetic
damping is ambiguous. If theactuators are attached at thebottomendof thepipe, theydefinitely
decrease the critical flow. Presumably the pipe is so slender that the static destabilising effect
dominates over the damping one. It is difficult to point out the optimal position of actuators.
From the practical point of view, one can see that the attachment of actuators in the upper part
of the pipe – which is easier in technical implementation – leads to more effective stabilisation.

Fig. 2. Stability of the discharging pipe in air: (a) z =0.00013m, (b) z =0.001m

Fig. 3. Stability of the discharging pipe in water: (a) z =0.00013m, (b) z =0.001m


Critical flow velocity in a pipe with electromagnetic actuators 493

At the zero voltage on actuators, the critical flow velocity for the system immersed in water
is lower than in air (for the given mass ratio of the fluid and the pipe, the external damping
exerts a destabilising effect), and only in that environment the stabilising effect of magnetic
damping can be observed. If the pipe is put in air, the actuators cannot increase the critical flow
velocity.

Decreasing the gap inmagnetic circuits improves the efficiency ofmagnetic damping, because
the system can be stabilised atmuch lower values of U. This is whywe study stiff pipesmade of
steel or PVC rather than rubber pipes, which lose stability at significantly lower flow velocities,
but are susceptible to large-amplitude flutter vibrations.

4.3. Aspirating cantilever pipe

Finally, we present results for the electromagnetically damped pipemade of PVC (Table 1),
through which the water is conveyed upwards. The stability of such a pipe was recently inve-
stigated by Kuiper and Metrikine (2008), both theoretically and experimentally. The observed
critical flow velocity for the pipe partially – at 1.95m depth – submerged in water amounted
to 2.1-2.4m/s, and destabilisation was due to flutter. The model with conventional boundary
conditions (lack of transverse reaction at the free end) predicted the critical velocity of 0.73m/s,
whereas an incorporation of the transverse reaction (as assumed in our work) resulted in overe-
stimation – 6.8m/s. Both models correctly predicted oscillatory instability.

Fig. 4. Stability of the aspirating pipe in air: (a) z =0.001m, (b) z =0.004m

Fig. 5. Stability of the aspirating pipe in water: (a) z =0.001m, (b) z =0.004m

Figures 4 and 5 present stability areas for the pipe put, respectively, in air and in water, for
two values of the gap in magnetic circuits: z = 0.001m and z = 0.004m. At the zero voltage,
the critical flow velocity amounts to ∼2m/s (in air) and 4.1m/s (water). Thus the critical flow


494 T. Szmidt, P. Przybyłowicz

for the pipe partially submerged – as in the experiment by Kuiper and Metrikine – would be
around 3-3.5m/s, which is lower than the value calculated in their paper. The reason lies in
the difference between the assumed models of hydrodynamic drag. Kuiper and Metrikine used
a linear-quadratic model developed earlier by themselves and Battjes (2007), with coefficients
independent of the frequency of vibrations and estimated from auxiliary experiments. In our
research, we use a more widely known linear model, whose parameters have to be estimated
iteratively, as they depend on the frequency.
Once the voltage is applied, the positive effect of electromagnetic damping reveals, and is

muchmore evident than in apipe throughwhich thewater is pumpeddownwards.Aspreviously,
stabilisation is also stronger for the systemwith a lower critical flowvelocity – but here it occurs
in air, not in water. Smaller gap in magnetic circuits allows not only for significant reduction
of the voltage applied to actuators, but also results in higher critical flow. Actuators are most
effective when placed in the upper part of the pipe. However, unlike the discharging case, they
also exert stabilising action when attached at the free end.

5. Summary

We studied the influence of electromagnetic actuators on the dynamics of a fluid-conveying
pipe. Simply supported, clamped-clamped and cantilever (discharging and aspirating) pipes
were investigated. Assumed physical parameters of the systems will enable future experimental
verification of the results.Weproposed simplemethodsof calculation of the internal and external
damping coefficients, which do not require experiments, and are based on knownmodels.
The governing equation was discretised with 10-modal Galerkin’s procedure based on the

beam eigenfunctions. Such number of eigenmodes has proved to be sufficiently convergent. Then
the stability of the resultant dynamical systemwas checked with eigenvalues of its linearization.
The actuators destabilise the pipe supported at both ends, but can improve the stability

of a cantilever one. The efficiency of magnetic damping depends on the external damping of
environment in which the system is immersed. Moreover, it depends on the position at which
the actuators are attached to the pipe.Theoptimal position of the actuators is an openproblem,
but important from the practical point of view position in the upper part of the pipe brings
stabilisation. Decreasing the gap in magnetic circuits improves the efficiency of the method.
Destabilisation of all studied pipes occurs at very high flow velocities. Such extreme flows

are rarely encountered in every-day life and are difficult (yet possible) to achieve.

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496 T. Szmidt, P. Przybyłowicz

Krytyczna prędkość przepływu w przewodzie stabilizowanym aktuatorami

elektromagnetycznymi

Streszczenie

W pracy przeprowadzono badania wpływu tłumienia elektromagnetycznego na właściwości dyna-
miczne przewodu z przepływającympłynem.Rozważonoprzewodypodparte obustronnie swobodnie oraz
zamocowane wspornikowo (dla obydwu przypadków ruchu płynu, tj. rury wyrzucającej czynnik na ze-
wnątrz oraz rury ssącej). Założono takie parametry fizyczne układu, które pozwalają na weryfikację eks-
perymentalną uzyskanychwyników. Zaproponowanoprostemetody pozwalające na oszacowaniewartości
tłumienia zewnętrznego i wewnętrznego oparte na znanych modelach i nie wymagających przeprowa-
dzania doświadczeń. Równanie różniczkowe cząstkowe ruchu przewodu z przepływem zdyskretyzowano
metodąGalerkina, a powyznaczeniuwartościwłasnychukładu zlinearyzowanego, określono jego statecz-
ność.Wwyniku przeprowadzonychanaliz zaobserwowano, że aktuatory elektromagnetycznedestabilizują
przewód obustronnie podparty, ale znacząco poprawiają stateczność rury wspornikowej, przy czym efekt
tłumienia magnetycznego silnie zależy od położenia aktuatorówwzględemmiejsca zamocowania takiego
przewodu.

Manuscript received April 13, 2012; accepted for print January 21, 2013