Non-planar vibrations of slightly curved pipes conveying fluid in simple and combination parametric resonances

Journal of Sound and Vibration 413 (2018) 270e290

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Non-planar vibrations of slightly curved pipes conveying fluid in simple and combination parametric resonances
ski a, *, Jan Łuczko b Andrzej Czerwin a

Laboratory of Techno-Climatic Research, Faculty of Mechanical Engineering, Cracow University of Technology, Jana Pawła II 37, 31-864, Krakow, Poland b Institute of Applied Mechanics, Faculty of Mechanical Engineering, Cracow University of Technology, Jana Pawła II 37, 31-864 Krakow, Poland

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 June 2017 Received in revised form 25 September 2017 Accepted 15 October 2017

The paper summarises the experimental investigations and numerical simulations of nonplanar parametric vibrations of a statically deformed pipe. Underpinning the theoretical analysis is a 3D dynamic model of curved pipe. The pipe motion is governed by four nonlinear partial differential equations with periodically varying coefficients. The Galerkin method was applied, the shape function being that governing the beam's natural vibrations. Experiments were conducted in the range of simple and combination parametric resonances, evidencing the possibility of in-plane and out-of-plane vibrations as well as fully non-planar vibrations in the combination resonance range. It is demonstrated that sub-harmonic and quasi-periodic vibrations are likely to be excited. The method suggested allows the spatial modes to be determined basing on results registered at selected points in the pipe. Results are summarised in the form of time histories, phase trajectory plots and spectral diagrams. Dedicated video materials give us a better insight into the investigated phenomena. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Pipe conveying fluid Parametric resonance Flow-induced vibration Non-linear dynamics 3D-motions Hydraulic systems

1. Introduction Pipes conveying fluids are widely used in various sectors of industry. Under certain conditions, however, the interactions between the pipe and fluid flowing with varying velocity lead to dangerous dynamic behaviours, and sometimes the entire hydraulic installation gets damaged. On the other hand the involved phenomena have become an interesting research field and a subject of extensive studies. Development of models of such systems alongside the improved calculation powers permits a more accurate description of even very complicated systems. In comparison to this, a much smaller amount of experimental works is felt. On one hand experimental tests enable the model validation, on the other-they offer us a better insight into the processes which have not been thoroughly researched before. The works by Paidoussis [1,2] and Ibrahim [3] provide an overview of studies investigating the dynamic behaviour of pipes conveying fluids. The problem has been investigated from various perspectives, numerous researchers investigated straight and curved pipes, varying pipe supports (one- or two-ended supports, rigid or elastic), the pipe behaviour during the flow with and without velocity pulsation. Underpinning the computational models have been the string models as well as diverse

* Corresponding author.
ski). E-mail address: [email protected] (A. Czerwin https://doi.org/10.1016/j.jsv.2017.10.026 0022-460X/© 2017 Elsevier Ltd. All rights reserved.


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beam and shell models. The models used are mostly 2D models, though 3D models better capturing the investigating phenomena are now often considered. This study is a continuation of the work [4]. In that article a 3D model of a pipe was proposed, used to investigate the vibrations of statically deformed elastic pipes. The introductory section of that article contains references to other works whose research efforts focused on modelling and investigating the behaviours of curved pipes. Relatively little published literature is available on parametric vibrations of 3D models, especially for pipes fixed at both ends. Most studies investigate the behaviour of cantilevered pipes. Nonlinear models of pipes with a free end were considered by Bajaj and Sethna [5]. Copeland and Moon [6] studied pipes with an additional mass, the works by Steindl and Troger [7,8] focused on flows with motion constraints in the form of various arrays of springs. In the three-part work [9e11], equations were derived that govern 3D motions of pipes under steady flow conditions and the pipe dynamics was investigated for several variants of elastic pipe supports and for a pipe with an end mass attached. This model was verified experimentally by Ghayesh et al. [12,13]. In studies on 3D pipes vibration, the authors often analyze the shape of vibration modes and the motion trajectories of selected points of pipe. Analyses of vibrations of pipes fixed at both ends mostly rely on 2D models. For flows with velocity pulsation, such models well capture the resonance phenomena, allowing those ranges of system parameters to be identified in which parametric vibrations are excited. In the case of combination resonance, however, such simplified models prove inadequate for a detailed description of the occurring phenomena. In the case of non-planar models of pipes fixed at both ends, the main focus is on stability and eigenvalue problems. Chen [14] analysed a linear model of a curved pipe, demonstrating that the stability loss in for the pipe model with a nondeformable axis occurs when the critical velocity is exceeded, similar to straight pipe. Hill and Davis [15] assumed that the pipe axis is deformable and showed that a curved pipe is not be subjected to in-plane and out-of-plane buckling. Misra et al. [16,17] compared various models of curved pipes, concluding that the most adequate model should be that taking into account deformability of the pipe axis. Jung and Chung [18] employed the Galerkin method to determine the natural frequencies of in-plane and out-of-plane vibrations of a nonlinear model of a semi-circular pipe conveying fluid. Taking into account geometric nonlinearity and various types of physical nonlinearities, they demonstrated that the nonlinear model does not buckle even at high flow velocities. Few studies investigate the parametric resonance for 3D models of curved pipes clamped at both ends. For example, Jung et al. [19] explored a nonlinear model of semi-circular pipe with pulsating flows. Recalling the Floequet method, they determined the simple resonance ranges for the first and second modes and for their combination resonance. Nakamura et al. [20] and Yamashita et al. [21] undertook theoretical and experimental investigations of a circular pipe with pulsating flow to highlight the interactions between in-plane and out-of-plane vibrations, stating excitation of out-of-plane parametric resonance. Ni et al. [22] developed a nonlinear model of a curved pipe, determined natural frequencies of its in-plane and outof-plane vibrations and analysed the effects of flow velocity, mass and internal damping on the ranges of parametric resonance. They compare predicted and experimental data, demonstrating the excitation of the out-of-plane parametric resonance. Lu et al. [23] investigated the principal parametric resonance phenomena for a model of a curved pipe with geometrical non-linearities, demonstrating the coupling of in-plane and out-of-plane vibrations. This study summarises results of theoretical and experimental research of slightly curved, elastic pipe. Numerical simulations use a 3D nonlinear model of the system, proposed in Ref. [4], describing the transverse vibrations in two perpendicular directions as well as axial and torsional vibrations. The Gallerkin method is recalled and the approximate solution takes into account eight modes. The main focus is on 3D modes in the range of simple and combination parametric resonance. To visualise the pipe shape during resonance vibrations, the experimental data have been post-processed accordingly.

2. Governing equation In the paper [4] differential equations of motions are derived which govern the dynamic behaviour of a statically deformed pipe conveying pulsating fluid. In Appendix A, the most important assumptions and short explanations of the studied equations are given. The fluid is assumed to be incompressible, the flow velocity remains unchanged along the pipe length and is taken to be constant over the pipe's cross section. Static deformations of an elastic pipe, mostly due to gravity forces, are taken into account. The model of the slightly curved pipe is shown in Fig. 1. The state of equilibrium is defined by the system of the first-order differential equations:

N0 10 ¼ k0 N20  mf l0 Uf20  mp g sin c0




(1)

N0 20 ¼ k0 N10 þ S0  p0 Af  mf Uf20  mg cos c0

(2)

c 0 0 ¼ k0

(3)

k0 0 ¼ N20 =EI3

(4)


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Fig. 1. Model of the system.

u0 10 ¼ cos c0  1=ð1 þ ε0 Þ

(5)

0

u 20 ¼ sin c0
 p0 0 ¼ rf  l0 Uf20 þ g sin c0

(6) (7)

where l0 ¼ l=2di , S0 ¼ T0 þ 2npAf . Symbols and designations given in Eqs. (1)e(7) are compiled in Table 1. The derivatives with respect to the natural coordinate s, measured along the deformed pipe's axis, are indexed with a prime. Boundary conditions for a pipe clamped at both ends are given as:

c0 ð0Þ ¼ c0 ðL0 Þ ¼ 0

(8)

u10 ð0Þ ¼ u10 ðL0 Þ ¼ 0

(9)

u20 ð0Þ ¼ u20 ðL0 Þ ¼ 0

(10)

p0 ðLÞ ¼ p1

(11)

Solving the boundary problem (Eqs. (1)e(11)) by the selected numerical method yields the shape of a deformed pipe centreline (functions c0 ðsÞ, k0 ðsÞ, u10 ðsÞ, u20 ðsÞ), static forces: N10 ðsÞ, N20 ðsÞ and static pressure p0 ðsÞ. Table 1 Parameters of the pipe model. Symbol

Description

a A, Ap di, de E, G g I1, I2¼I3 k0 L, L0 mf, mp, m N10, N20, M30 p, p0, p u1, u2, u3 u10, u20 Uf(t), Uf0 s, x S T0 ε0

dimensionless flow pulsation amplitude internal sectional area and cross-sectional area of a pipe internal and external diameter of the pipe Young modulus, Kirchhoff modulus gravity acceleration due to gravity cross section moment of inertia curvature of the pipe axis in statically deformed state length of non-deformed pipe and length of the statically deformed pipe elementary mass of fluid, mass of a pipe, mass of the hose with fluid axial force, transverse force and bending moment in the state of equilibrium dynamic and static pressure inside the pipe, average pressure p dynamic displacements in tangent, normal and binormal direction static displacements in tangent and normal direction flow velocity, average flow velocity natural coordinate along the pipe axis in deformed and non-deformed state non-linear component of the axial force tension force static axial strain dimensionless coefficient of flow resistance internal damping coefficient Poisson ratio oil and pipe density tangential angle to statically deformed pipe axis dimensionless pulsation frequency

l m n rf, rp c0 up


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3D vibrations of the pipe around its equilibrium position are expressed by displacements u1(s,t), u2(s,t), u3(s,t) respectively in the tangent, normal and binormal direction to the statically deformed centreline(see Fig. 1) and the angle of rotation f1(s,t) around the pipe axis. Equations:

h   i
00    mu€1 þ mf 2 u_ 01  k0 u_ 2 Uf þ u01  k0 u2 U_ f þ u1  2k0 u0 2  k0 0 u2  k20 u1 Uf2 ¼
00 
000 o n 00 00 ¼ ð1 þ mDÞ EAp u1  k0 u0 2  k0 0 u2 þ k0 EI3 u2 þ k0 u1 þ 2k00 u01 þ k0 u1
00     0 þ ðN10 þ S þ w4 Þk0 u02 þ k0 u1  N20 u2 þ k0 u01 þ k00 u1 þ w1  N20 h   i
00    mu€2 þ mf 2 u_ 02 þ k0 u_ 1 Uf þ u02 þ k0 u1 U_ f þ u2 þ 2k0 u01 þ k00 u1  k20 u2 Uf2 ¼
 n  0 o 000 00 000 0 00 0 ¼ ð1 þ mDÞ  EI3 uIV 2 þ k0 u1 þ 3k0 u1 þ 3k0 u1 þ k0 u1 þ k0 EAp u1  k0 u2 
00  0   þ N10 þ w2  N20 k0 u02 þ k0 u1 þ ðN10 þ S þ w4 Þ u2 þ k0 u01 þ k00 u1 þ k0 ðS þ w3 Þ h i 00 mu€3 þ mf 2u_ 03 Uf þ u03 U_ f þ u3 Uf2 ¼ 
o n
00 00 00 0 0 0 0 2 00 0 0 ¼ ð1 þ mDÞ EI2  uIV 3 þ k0 f1 þ 2k0 f1 þ k0 f1 þ GI1 k0 f1 þ k0 f1 þ k0 u3 þ 2k0 k0 u3  0    0 00 00 0 0 0 þ w2  M30 k00  M30 k0 u03  M30 f1 þ N20  M30 f1 þðN10 þ S þ w4  M30 k0 Þu3 þ N10 f1 þ N20 
i
 h
00 00 00 00 rp I1 f€ 1 ¼ ð1 þ mDÞ GI1 f1 þ k0 u3 þ k00 u03 þ EI2 k0 u3  k20 f1  M30 u3  k0 f1

(12)

(13)

(14)

(15)

take into account the effects of static forces and strains and the influence of friction in accordance with the Darcy-Weisbach hypothesis and of internal damping of the Kelvin-Voigt type. Derivatives with respect to time are indicated with a dot whilst D ¼ v=vt is the differential operator. Eqs. (12)e(15) involve a nonlinear term:

S ¼ ð1 þ mDÞ

EAp 2L

ZL h

i 2  2  0 2 u1  k0 u2 þ u02 þ k0 u1 þ u0 3 dx

(16)

0

which can be interpreted as a nonlinear component of the axial force associated with the geometrically nonlinear pipe model. Eqs. (12)e(15) contain functions, defined as follows:


 w1 ¼ mf l0 Uf2  Uf20

(17)

h
i w2 ¼ p00 Af þ mf U_ f þ l0 Uf2  Uf20

(18)

h
i
 w3 ¼ mf U_ f þ l0 Uf2  Uf20 ðs  L0 Þ  mf Uf2  Uf20

(19)

h
i w4 ¼ S0  p0 Af þ mf U_ f þ l0 Uf2  Uf20 ðs  L0 Þ

(20)

which are related mostly to flow velocity and static pressure. An assumption is made in experimental and simulation procedures that flow velocity Uf is a harmonic function of time expressed as:

  Uf ¼ Uf 0 1 þ a sin up t

(21)

Differential equations (Eqs. (1)e(7), (12)e(15)) depend on the coordinate s, measured along the deformed pipe's centreline. For convenience, in the numerical procedure the independent variable x determining the non-deformed centreline point positions is applied. To transformation these equations, the following relationship is used:

ds ¼ 1 þ ε0 dx

(22)

where:

ε0 ¼ N10 =EAp

(23)

Solving Eq. (22) yields the length L0 ¼ sðLÞ of the deformed pipe. Using Eq. (22), the derivatives with respect to the variable s can be represented as derivatives with respect to x. However, when the variables are replaced, Eqs. (1)e(15) become more complicated. Substitution of variables can be easily implemented via numerical algorithm.


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3. Method of analysis The static and dynamic equations can be solved via analytical-numerical methods proposed in the work [4]. Eqs. (1)e(7) with the boundary conditions (Eqs. (8)e(11)) have relevance to describe a nonlinear boundary problem and can be solved numerically without further simplifying assumptions. The system of nonlinear partial differential equations (Eqs. (12)e(15)) are solved by the Galerkin method [13,18,24] whereby the coordinates of the state vector z ¼ ½z1 ; z2 ; z3 ; z4 T ¼ ½u1 ; u2 ; u3 ; 41 T are approximated as follows:

zj ðx; tÞ ¼

N X

4jn ðxÞzjn ðtÞ

(24)

n¼1

where N stands for the number of shape functions. The approximating functions are normalised beam's eigenfunctions. For a pipe clamped at both ends, the functions chosen to approximate transverse vibrations (for j ¼ 2,3) are given as:



1 Un ðLÞ Vn ðxÞ f2n ðxÞ ¼ f3n ðxÞ ¼ pffiffiffi Un ðxÞ  Vn ðLÞ L

(25)

where:

Un ðxÞ ¼ coshðln x=LÞ  cosðln x=LÞ

(26)

Vn ðxÞ ¼ sinhðln x=LÞ  sinðln x=LÞ

(27)

where ln are the roots of the characteristic equation cosh ln cos ln ¼ 1. When handling axial and torsional vibrations (for j ¼ 1,4), the following functions are considered:

f1n ðxÞ ¼ f4n ðxÞ ¼

i pffiffiffiffiffiffiffiffi 2=Lsinðnpx=LÞ

(28)

Solving Eqs. (12)e(15) by the Galerkin method yields a system of ordinary differential equations, which can be written in a concise form (i ¼ 1,2,3,4, n ¼ 1, …,N): 4 X N h X


 i ij ij ij ij Anm z€kn þ Bnm z_jm þ Cnm þ Dnm S zjm þ din S ¼ fni ðtÞ

(29)

j¼1 m¼1

whilst the function:

3 3 X 3 X N X N h   EAp X ij z z þ m z_in zjm þ zin z_jm 5 S¼ P 2L i¼1 j¼1 m¼1 m¼1 nm in jm

(30)

represents the effects of the nonlinear axial force component. Values of coefficients in Eqs. (29) and (30) associated with the system parameters and static solutions are expressed by respective integrals defined in the appendix to the work [4]. For Dijnm ¼ din ¼ 0, Eq. (29) is linear and describes the free vibrations. Analysis of a linearised system permits the natural frequencies to be determined so as to test the stability of system [24]. The system of nonlinear differential equations (Eq. (29)) is integrated via the Runge-Kutty-Verner method of the 5th and 6th order. To achieve a better agreement between the simulation and experimental data, the approximate solution (Eq. (24)) involves eight shape functions ((Eq. (25)) to approximate the transverse vibrations in each direction as well as eight functions (Eq. (28)) approximating axial and torsional vibrations. The numerical procedure was supported by the dedicated programs written in Fortran F77. 4. Experimental testing 4.1. Experimental set-up The experimental set-up is shown in Fig. 2. The pipe is fixed in the horizontal direction with its both ends clamped. Actual values of its parameters are based on experimental results summarised in the work [25]. Major parameters of the hydraulic pipe under test are: length L ¼ 1.91 m, outside diameter de ¼ 0.021 m; inside diameter di ¼ 0.0127 m, Young modulus E ¼ 40 MPa, Poisson ratio n ¼ 0.59, pipe density rp ¼ 1870 kg/m3, the initial tension force T0 ¼ 60 N. The fluid used in the hydraulic system is oil with density rf ¼ 900 kg/m3. The flow of oil with the predetermined average flow rate is generated by a pump in the hydraulic supply unit. Flow pulsation is generated by a proportional direction control valve controlled via signals


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Fig. 2. Experimental set-up.

from a D/A (CV) converter. The average flow velocity implemented during the experiments is Uf0 ¼ 8 m/s, pulsation amplitude a ¼ 0.25, pressure p0 ¼ 2 MPa, oil temperature about 50  C. Flow pulsations are implemented as harmonic signals, their frequencies selected from the range of various types of parametric resonance. The pulsating flow being generated, the phase of vibration excitation was examined, followed by observations of steady-state vibrations. Vibration parameters were measured with six MEMS accelerometers. Acceleration is measured in two directions normal to the pipe's axis: in vertical direction (in-plane) and in horizontal direction (out-of-plane). Acceleration transducers are mounted in six points of the pipe (Fig. 2) placed symmetrically and uniformly spaced (at points xm ¼ mL/7, m ¼ 1,2 … 6). For such configuration of measurement points, the occurrence of the mode node in the investigated frequency range is very unlikely. Twelve signals from the A/D converter are registered and saved by the Data Acquisition software (DAQ), alongside the feedback information from the directional control valve, pressure levels upstream and downstream the pipe (p1 and p2) and the flow rate Qf. To determine velocity and acceleration of vibrations, the acceleration signals are subjected to a single or double numerical integration. 4.2. Approximation of vibration modes In the case of numerical solutions, the modes of transverse vibrations at an arbitrary moment of time can be determined recalling Eq. (24). The issue becomes more complicated when analysing experimental data, because it is problematic how to determine the vibration modes basing on displacements registered at predetermined measurement points. A different methodology could be employed, such as the processing of images registered at specified time moments. However, such methods are time-consuming and not sufficiently accurate, particularly when analysing 3D motions (in the event of combination resonances). In this study the approach is adopted whereby vibration modes are approximated by Eq. (24) with appropriate shape functions, such as those given by Eq. (25). The values of in-plane and out-of-plane transverse displacements obtained from experiments: u2m ¼ z2 ðxm ; tÞ, u3m ¼ z3 ðxm ; tÞ registered at points xm, m ¼ 1,2, …,M (in experiments M ¼ 6) ought to satisfy Eq. (24). Accordingly, these equations can be rewritten as: M X

fjn ðxm Þzjn ðtÞ ¼ ujm

(31)

n¼1

Solving the system of 2 M algebraic linear equations (Eq. (31)) for subsequent time moments yields the function zjn ðtÞ, (j ¼ 2,3, n ¼ 1, …,M), thus revealing the shape of the pipe's axis during oscillations.


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Fig. 3. Comparison of approximated modes and pipe images during simple resonances.

For comparison, Fig. 3 shows the vibration modes in three subsequent simple resonances (in-plane motions) derived from signals registered at M ¼ 6 measurement points and the corresponding image of the investigated pipe section. 5. Results of the simulations and the experiments 5.1. Parametric resonance phenomenon Both numerical simulations and experimental data implicate the likelihood of a parametric resonance effect, a threat to the hydraulic system yet attracting a great deal of researchers' attention. For established values of the pipe parameters, of particular importance is an influence of the flow parameters, such as average flow rate, amplitude and frequency of pulsations. In work [4] estimated the simple and combination parametric resonance regions whilst investigating the dynamic behaviour of the system in a wide range of those parameters. This study investigates the influence of pulsation frequency on the parametric resonance effect. To get a better insight into the resonance phenomena, the mode components are analysed by the spectral analysis methods. As regards the numerical simulation data, the problem is relatively simple. Numerical integration of differential equations (Eq. (29)) yields the functions zjn(t), j ¼ 1, …4, n ¼ 1, …,N, that describe the axial, transverse and torsional vibration of an elastic pipe, by Eq. (24). The main focus is on analysis of those solutions to highlight certain physical phenomena taking place in the resonance range and to give us a better insight into in-plane (j ¼ 2) and out-of-plane (j ¼ 3) transverse vibrations. To identify the component modes, the spectral analysis of numerical solutions zjn(t) is performed, via the FFT. When handling experimental data, the problem becomes more complex. The adopted methodology relies on timediscretised signals registered at M measurement points (e.g. M ¼ 6). Signal components can be obtained by the band-pass filtering method resorting to filters of frequency equal to components frequency and small band width. Afterwards each component is to be approximated by the procedure outlined in section 4.2. This approach requires that the frequencies of relevant components should be first estimated. An inverse procedure can be adopted and the problem handled in a similar manner as in numerical solutions. Basing on the signals from the transducers, the full mode can be first approximated via solving Eq. (31), followed by spectral analysis of thus determined functions zjn ðtÞ, j ¼ 2,3 that determine the actual shape of inplane and out-of-plane modes at discrete time moments. In the steady-state vibrations, the spectra of investigated signals have several major components (in some cases the number of components K ¼ 2), hence the functions zjn ðtÞ, j ¼ 2,3, n ¼ 1,2, …,N can be rewritten as:

zjn ðtÞ ¼

K X

zkjn ðtÞ

(32)

k¼1

where:


zkjn ðtÞ ¼ akjn cos uk t  akjn Eq. (24) can be expressed as:

(33)


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zj ðx; tÞ ¼

K X

zkj ðx; tÞ

277

(34)

k¼1

where the functions:

zkj ðx; tÞ ¼

N X

fjn ðxÞzkjn ðtÞ

(35)

n¼1

describe the changes of the mode corresponding to frequency uk. This mode is dependent on all shape functions (Eq. (25)) recalled to approximate the pipe's transverse motions. Let us introduce the indexes Ak expressing the amplitudes of component vibrations with frequency uk ¼ 2pfk, defined as:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX   2   2 Ak ¼ t ak2n þ ak3n

(36)

n¼1

Fig. 4a plots the frequency components fk in the function of pulsation frequency fp (in the interval 2.5 Hz - 44 Hz, with the step 0.05 Hz). Fig. 4b shows the indexes Ak, providing information about amplitudes of vibration with the frequency fk. These diagrams are plotted for Uf0 ¼ 8 m/s and a ¼ 0.25. The types of lines in Fig. 4a (spectrogram) are associated with particular spectrum components. The yellow line (fp) designates the component with the frequency of pulsation, the green line (fs) represents the component with frequency 0.5fp (i.e. the fundamental frequency in simple resonances); the blue line (fc1) and dashed red line (fc2) are associated with principal components in combination resonances (fc2-higher mode, fc1 - lower mode). A light green line (fp þ fs), cyan line (fp þ fc1) and pink dashed line (fp þ fc2) represent the components with frequencies equal to the sum of pulsation frequency and the main component of the resonance frequency. Curved plots with the same designations (Fig. 3b) show vibration amplitudes. The analysis of a diagram shown in Fig. 4b reveals several resonance ranges dominated by one (plot fs) or two modes (fc1 and fc2), particularly in a higher frequency range. These are simple and combination parametric resonance ranges, respectively. A primary simple parametric resonance is most likely to occur for the excitation frequency fp, nearing the value 2fn, where fn is natural frequency of the system. In further sections this type of resonance will be denoted by rnen. For pulsation frequency satisfying the condition fp z fn þ fm, the primary combination resonance occurs, designated as rnem. The simple parametric resonance ranges for subsequent modes (r1-1, r2-2, r3-3, r4-4, r5-5) are separated by the combination parametric resonance ranges for neighbouring modes (r1-2, r2-3, r3-4, r4-5). Apart from the main component with frequency nearing fn, simple resonance ranges rnen, reveal the presence of a component fp and a component with frequency equal to

Fig. 4. Vibration components in the function of pulsation frequency (Uf0 ¼ 8 m/s, a ¼ 0.25): (a) frequency of vibration components; (b) amplitude of vibration components.

278


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fn þ fp. In the combination resonance range rn-(nþ1), the spectrum features two main components (fn, fnþ1) associated with respective free vibration modes, a component with the pulsation frequency and components with summated frequencies. In the resonance ranges of the type rn-(nþ1), the arithmetic mean of frequency components is equal to the halved excitation frequency. Main components, both in simple and combination resonance ranges, are associated with respective natural frequencies of the system, though in most cases they are not equal to them. Those frequencies tend to increase almost linearly with increase in pulsation frequency fp in the range of values slightly below and above the natural frequency. Amplitudes of the components increase with increasing excitation frequency. The component with the pulsation frequency occurs in the entire frequency range. The plot of its amplitude in the function of frequency reveals clear-cut resonance ranges, near the natural frequency of the pipe. With increasing frequencies, these resonances become less pronounced, which is attributable to internal damping in the system.

5.2. Out-of-plane simple resonance r1e1 Vibrations in simple resonances are mostly planar and their modes resemble the respective free vibration modes. For the first mode resonance region r1-1, both theoretical and experimental data implicate the occurrence of independent in-plane and out-plane resonances. Predicted frequencies of in-plane and out-of-plane natural vibrations are similar in value. Those out obtained from the analysis of the linear model are: fin 1 ¼ 3.79 Hz and f1 ¼ 3.68 Hz. For pulsation frequency fp¼7.4 Hz, nearly twice as high as fout in the investigated system, out-of-plane vibrations are 1 excited. Fig. 5 summarises the selected characteristics of the pipe vibration in this resonance. Fig. 5a plots the experimental data whilst numerical results are summarised in Fig. 5b. In upper section there is a 3D view of the pipe at the selected time instant (thick line) and within a given time interval (thin lines). There are motion trajectories of control points on the pipe, corresponding to the sensors' positions. Signals measured at six points are used to reconstruct the pipe motions, in accordance with the methodology outlined in section 4.1. Below in the 3D view are the pipe projections onto the vertical plane (in-plane) and the horizontal plane (out-of-plane), with indicated projections of motion trajectories. In the lower section there are time histories and phase trajectories at the point x ¼ 2L/7.
maps which permit the evaluation of the Plotted points representing the time instants kTp ¼ k/fp (k ¼ 1,2 …) form the Poincare vibration character.


maps in out-of-plane resonance r1-1 (Uf0 ¼ 8 m/ Fig. 5. Vibration mode shapes, motion trajectories at selected points, time histories, phase portraits and Poincare s, a ¼ 0.25): (a) results of the experiment (fp¼7.25 Hz); (b) results of the simulation (fp¼7.4 Hz).


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In out-of-plane resonance vibrations in the horizontal plane tend to predominate and their mode is similar to the first free vibration mode. A gyroscopic effect can be observed in the shape of a travelling wave. As the value of the second natural frequency f2 ¼ 7.46 Hz is nearly twice as high as fout 1 , the impacts of the second mode resonance develop, giving rise to pipe motions in the vertical plane. The final effect is manifested by motions observed in a 3D view, exhibiting spatial features and slightly curved motion trajectories. Time histories and phase trajectories registered out-of-plane (brown curves) implicate that in the steady-state vibrations are periodic. Two stroboscopic points are indicative of the second-order sub-harmonic vibrations (their period being twice as high as that of the applied excitation). In-plane vibrations (blue curves) are also periodic and their frequency is equal to pulsation frequency (one stroboscopic point). Visualisations of these processes can be seen in the video 1, based on the experimental data, showing the excitation phase of vibrations and steady-state vibrations in resonance. Supplementary video related to this article can be found at https://doi.org/10.1016/j.jsv.2017.10.026.

5.3. In-plane simple resonance r1e1 In the range of slightly higher frequencies, the in-plane resonance r1e1 occurs in the vertical plane. Similar to Figs. 5 and 6 summarises the results of parametric vibration analyses for pulsation frequency fp ¼ 7.6 Hz, nearing the doubled frequency of free vibrations in-plane (fin 1 ¼ 3.79 Hz). In this case experimental results (Fig. 6a) and simulation data (Fig. 6b) slightly differ in qualitative terms. In both cases vibrations are generated in the vertical plane and have shape of the first mode. Experimental results (Fig. 6a) reveal a minor deflection of the trajectory, indicative of a small contribution of vibrations in the horizontal plane, their mode similar to the second mode. Fig. 7 shows the change of the vibration mode obtained from simulations. Dark and bright shades are used to denote the pipe positions at subsequent time instants, revealing the gyroscopic effect associated with the fluid flow. The motions of points on the pipe along the pipe differ in phase and direction. Registered point displacements are characteristic of those of a travelling wave. Pipe vibration in their excitation phase and its steady-state vibrations are visualised in Video 1. 5.4. Combination resonance r2-3 More complex and interesting vibration modes are registered in combination resonance. These are fully non-planar vibrations and motion trajectories of the pipe's points form characteristic figures. Fig. 8 illustrates the pipe behaviour during the combination resonance r2-3. Similar to Fig. 6, it shows vibration modes and motion trajectories at selected points (3D view and respective projections). Fig. 8a summarises experimental data whilst

Fig. 6. Vibration mode shapes, motion trajectories at selected points in the in-plane resonance r1-1 (Uf0 ¼ 8 m/s, a ¼ 0.25): (a) results of the experiment (fp ¼ 7.75 Hz); (b) results of the simulation (fp ¼ 7.6 Hz).

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Fig. 7. Changing vibration mode in the in-plane resonance r1e1, simulation results (Uf0 ¼ 8 m/s, a ¼ 0.25, fp ¼ 7.6 Hz).

Fig. 8. Vibration mode shapes and motion trajectories at selected points in combination resonance r2-3 (Uf0 ¼ 8 m/s, a ¼ 0.25): (a) results of the experiment (fp¼19.0 Hz); (b) results of the simulation (fp¼19.5 Hz).

numerical simulation results are given in Fig. 8b. The motion of the pipe's centreline was reconstructed via the solution to Eq. (31). Analysis of results confirms the non-planar motion of the pipe. Motion trajectories of the pipe points have complicated shapes, whilst all trajectories feature five vertices (n ¼ 5). The shapes of motion trajectories registered along the pipe axis are changeable and the actual shape of the pipe at the given time instant is not a planar but a spatial curve (all points of the trajectory are not in one plane). The mode shape changes over time, assuming intricate 3D forms. Comparison of experimental observations and simulation data shows they are in good agreement, thus evidencing the adequacy of the adopted dynamical model of the pipe. The enclosed animated movie (video 2) shows the full cycle of pipe vibrations (5 periods of excitation). Supplementary video related to this article can be found at https://doi.org/10.1016/j.jsv.2017.10.026. Fig. 9 illustrates the changes of the in-plane and out-of plane modes over time, within 5 periods of the excitation (one full cycle). Subsequent figures (counting from below) show the shape of the pipe axis projections at selected, uniformly spaced time instants (kTp/4, k ¼ 1,2 …), revealing a smooth transition from the second to the third mode, which gives rise to the effect resembling the wave travelling in two alternating directions. Similar effects are registered in the two planes (in-plane and out-of plane), and there is a phase shift between the two directions. These effects can be observed in the enclosed video 2. The spectrogram shown in Fig. 4 reveals two dominating frequency components associated with natural frequencies of the two subsequent modes, and several components whose contribution is rather minor. Fig. 10a shows the spectrum of a displacement signal registered in the experiments, Fig. 10b and c summarise the time histories and phase trajectories at the point 5L/7 for in-plane and out-of-plane motions. Frequencies of main components are: f2 ¼ 7.69 Hz and f3 ¼ 11.35 Hz and the sum of f2 and f3 is close to the pulsation frequency fp¼19 Hz. Apart from frequency components associated with the second or third mode, the spectrum reveals a


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Fig. 9. Vibration mode changing over time in the resonance r2-3, results of the experiment (fp ¼ 19 Hz, Uf0 ¼ 8 m/s, a ¼ 0.25): (a) in-plane modes; (b) out-of-plane modes.

Fig. 10. Analysis of vibrations at point 5L/7 in the resonance r2-3, results of the experiment (fp ¼ 19 Hz, Uf0 ¼ 8 m/s, a ¼ 0.25): (a) displacement spectrum; (b) time
maps. histories; (c) phase portraits and Poincare

component with the pulsation frequency as well as frequency components f2þfp and f3þfp. Amplitudes of additional components are by at least one order of magnitude smaller than those of the main components and their influence on the actual mode shape is negligible. Furthermore, phase trajectories in Fig. 10c are indicative of a quasi-periodic vibration profile (a closed curve over a long time interval). Quasi-periodic vibrations are attributed to the ratio of main frequency components being an irrational number. Explanation of the intricate shapes of motion trajectories in combination resonance requires the analysis of components mode. Fig. 11 summarises experimental data, showing 3D modes and their in-plane and out-of-plane projections and motion trajectories of the two main components mode associated with frequencies f2 ¼ 7.69 Hz and f3 ¼ 11.35 Hz. These modes correspond to the second mode of free vibration (with a single node) and the third mode (two nodes). These modes, whist retaining their shape, rotate around the pipe axis and that is why the component trajectories of the control points are roughly circular. Frequency of mode rotation is equal to that of the given component whist the directions of their rotation are opposite. At the given time instant the component mode is not an ideal planar curve, but exhibits spatial curvature which is evidenced by the distribution of points on the trajectories (Fig. 11b). Despite the symmetry of structure and pipe attachments, the shape of each trajectory (Fig. 8) does not reveal any symmetry along the pipe length. That is the consequence of non-symmetry of mode components with respect to the pipe centre (Fig. 11). Nodes and anti-nodes of mode components are slightly shifted with respect to the flow direction and the amplitude of the second mode with two anti-nodes is changed, which is the result


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Fig. 11. Main components of the mode in the combination resonance r2-3, results of the experiment (Uf0 ¼ 8 m/s, a ¼ 0.25, fp¼19.0 Hz): (a) modes and motion trajectories in 3D view; (b) in-plane and out-of-plane modes.

of gyroscopic effects associated with the fluid flow. As the contribution of other spectral modes is minor, the resultant mode being the sum of two mode components is similar to the mode shown in Fig. 8. Video 3 shows the changes of modes and motion trajectories over time. Supplementary video related to this article can be found at https://doi.org/10.1016/j.jsv.2017.10.026. The dominating effect of two components on the vibration mode and motion trajectories is characteristic of all combination resonances considered in this study. It is worthwhile to consider the shape of trajectories from the standpoint of geometry, treating the resultant trajectory as the sum of two components. In a general case, the actual shape of the trajectory at the point on the pipe x ¼ xm is governed by Eqs. (32)e(35) which lead to the following relationships:

zj ðxm ; tÞ ¼

K X N X


fjn ðxm Þakjn cos uk t  akjn

(37)

k¼1 n¼1

that govern the transverse vibrations in-plane (j ¼ 2) and out-of-plane (j ¼ 3). Phases akjn can be arbitrary in a general case, yet     in the combination resonance ranges the phase difference satisfies the approximate condition ak3n  ak2n zp=2. Recalling this condition, Eq. (37) for K ¼ 2 can be rewritten as:


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in in in uin ¼ Ain 1 cos u1 t  a1 þ A2 cos u2 t  a2

(38)

    out out uout ¼ Aout  Aout 1 sin u1 t  a1 2 sin u2 t  a2

(39)

where: uin ¼ z2 ðxm ; tÞ, uout ¼ z3 ðxm ; tÞ. From the standpoint of geometry, Eqs. (39) and (39) are parametric equations of out in out in in out out trajectories. For Ain 1 ¼ A1 and A2 ¼ A2 and a1 ¼ a2 ¼ a1 ¼ a2 the trajectory components are circular. The shape of the resultant trajectory at the given point of the pipe is associated mainly with the ratio of mode components' amplitudes at this in out out point, i.e. Ain 2 =A1 or A2 =A1 . When the ratio f2/f1 of frequencies is a rational number (e.g. f2/f1 ¼ n/m, Fig. 12), a curve is obtained with the number of in out out vertices equal to l ¼ n þ m. This curve does not rotate over time. Fig. 12 shows the curves obtained for Ain 2 =A1 ¼ A2 =A1 ¼ 1, and for selected ratios of frequency components. When the ratio f2/f1 is an irrational number, the curve rotates round its centre point and the rate of this rotation is dependent on the actual detune d ¼ f2/f1en/m. When the detune value is positive, the rotation is counter-clockwise, otherwise the curve rotates clockwise. Fig. 13 shows the shapes of trajectories for various detune values. The actual shape of the resultant trajectory is dependent, to a large extent, on the ratio of amplitude components. Fig. 14 in out out shows several curves obtained for various ratios of component amplitudes g ¼ Ain 2 =A1 ¼ A2 =A1 : In the cases considered in the study the component trajectories are circles and the resultant trajectory is a symmetrical figure. Deformation of component trajectories results in non-symmetry of the resultant trajectory. Such deformations can be out in out the consequence of the difference between amplitudes Ain 1  A1 or A2  A2 , or the effect of phase shifts. in out in Fig. 15 illustrates several cases of non-symmetrical trajectories. The difference in amplitudes Aout 1  A1 or A2  A2 causes the circle to be deformed, turning it into an ellipse, its semi-axes vertical and horizontal (Fig. 15b and c). Phase shifts also out cause the circles to be transformed into an ellipse, its semi-axes inclined and oblique (Fig. 15d and e for Ain 1 ¼ A1 ¼ 1.0, out Ain 2 ¼ A2 ¼ 0.8). Let us return to the combination resonance r2-3. The ratio of main frequency components f3/f2 ¼ 1.473 is an irrational number, approaching 3/2. Therefore the number of vertices in the trajectory should be equal to n ¼ 5, which is confirmed by observations of trajectories obtained experimentally and in numerical procedures (Figs. 8 and 16). Detune d ¼ f2/f1en/m ¼ 0.027 is negative, so the entire trajectory will rotate clockwise. Trajectories obtained from measurements (Fig. 16) are not perfect circles, hence their minor asymmetry. Analysis of the pipe's motion registered within the time interval equal to 25 periods of pulsations (5 cycles) implicates the rotation of the trajectory (Fig. 17). This effect can be also observed in video 3. The study investigating how vibrations are excited in a parametric resonance r2-3 (Fig. 18) reveals that the rate of amplitude increase for in-plane and out-of-plane vibrations are different. In-plane vibrations develop at a faster rate and for that reason the shape of motion trajectories will change at the stage of vibration excitation. At first they extend in the vertical direction. When the state-state vibrations are reached, motion trajectories assume their final shape. This behaviour is illustrated in Fig. 18 where vibrations at the point L/7 are examined. The process of vibration excitation at all control points can be viewed on video 2.

5.5. Higher-order combination resonances The elastic pipe under test features a relatively low stiffness, that is why the progression of subsequent natural frequencies is almost an arithmetic series (as in the string model), particularly in the low frequency range. For such frequency distribution (i.e. for fn/fnþ1 ¼ n/nþ1) in respective combination resonances of the type rn-(nþ1), the trajectories of points in the pipe ought to feature l ¼ 2nþ1vertices, so the number of vertices in resonance r2-3 ought to be l ¼ 5 (Fig. 8) and whilst in resonance r4-5, the number will be l ¼ 9. However, simulation and experimental data (Fig. 19) indicate that in resonance r4-5 the number of vertices becomes l ¼ 11. Higher frequencies of the beam model's vibration no longer follow the pattern of arithmetic series and so the ratio of the fifth frequency f5 ¼ 19.9 Hz Hz to the fourth f4 ¼ 16.48 is equal to f5/f4 ¼ 1.208 and is value is better approximated by the fraction 6/5 ¼ 1.2 than by 5/4 ¼ 1.25. For that reason the trajectory will feature 11 vertices.

in out out Fig. 12. Trajectories for Ain 2 =A1 ¼ A2 =A1 ¼ 1 and for following ratios of frequency components: (a) f2/f1 ¼ 2:1 (l ¼ 3); (b) f2/f1 ¼ 3:2 (l ¼ 5); (c) f2/f1 ¼ 4:3 (l ¼ 7); (d) f2/f1 ¼ 5:4 (l ¼ 9); (e) f2/f1 ¼ 6:5 (l ¼ 11).

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in out out Fig. 13. Trajectories for Ain 2 =A1 ¼ A2 =A1 ¼ 1, f2/f1 ¼ 3:2 and for various detune values:(a) d ¼ 0.045; (b) d ¼ 0.015; (c) d ¼ 0; (d) d ¼ 0.015; (e) d ¼ 0.045.

Fig. 14. Trajectories for f2/f1 ¼ 3:2 and various ratios of component amplitudes: (a) g ¼ 0.2, (b) g ¼ 0.5, (c) g ¼ 0.8, (d) g ¼ 1.0, (e) g ¼ 1.2, (f) g ¼ 2, (g) g ¼ 5.

Fig. 15. Resultant trajectories and their components for f2/f1 ¼ 3:2: (a) A1in ¼ A1out ¼ 1.0; A2in ¼ A2out ¼ 0.8, (b) A1in ¼ 1.0; A1out ¼ 1.2; A2in ¼ A2out ¼ 0.8, (c) A1in ¼ 1.0; in out in in out out in in out out Aout 1 ¼ 1.2; A2 ¼ 1.0; A2 ¼ 0.8, (d) a1 ¼ 0.2; a2 ¼ a1 ¼ a2 ¼ 0, (e) a2 ¼ 0.2; a1 ¼ a1 ¼ a2 ¼ 0.

in out out Fig. 16. Trajectories and their components in the combination resonance r2-3 (f3/f2 ¼ 1.473) at points: (a) x ¼ L/7; Ain 1 ¼ 11.1, A2 ¼ 6.44, A1 ¼ 9.71, A2 ¼ 5.57 in out out in in out out in in out out [mm]; ain 1 ¼ a2 ¼ 0, a1 ¼ 0.075, a2 ¼ 0.025; (b) x ¼ 2L/7; A1 ¼ 15.1, A2 ¼ 3.62, A1 ¼ 13, A2 ¼ 3.19 [mm]; a1 ¼ a2 ¼ 0, a1 ¼ 0.065, a2 ¼ in out out in out in out 0.05; (c) x ¼ 4L/7; Ain 1 ¼ 7.6, A2 ¼ 5.81, A1 ¼ 6.7, A2 ¼ 5.1 [mm]; a1 ¼ a2 ¼ 0, a2 ¼ 0.02, a1 ¼ 0.08.

Fig. 19 summarises the experimental data in combination resonance r4-5. On the left-hand side (Fig. 19a) the pipe's 3D motions are analysed, the right-hand side (Fig. 19b) provides the components modes. The pipe motion in this case is nonplanar, which is characteristic for all combination resonances. Trajectories of points on the pipe rotate counter-clockwise (low value of negative detune d ¼ f5/f4e6/5 ¼ 0.007). A vibrating pipe assumes intricate non-planar shapes, with a well pronounced ‘travelling wave’ effect. The main components are associated with the fourth and fifth natural mode (Fig. 19b). The fourth mode features three distinct nodes shifted in the direction of the fluid flow and the nearer to the pipe beginning, the larger the shift. The fifth form exhibits four nodes and its shape in right-hand side of the pipe is clearly deformed. The nodes are shifted in the direction of the fluid flow. The node shift and mode deformations may be attributable to gyroscopic effects associated with fluid flow. Video 4 shows the changes of resultant modes and motion trajectories of control points. Supplementary video related to this article can be found at https://doi.org/10.1016/j.jsv.2017.10.026.


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Fig. 17. Rotation of the motion trajectory in parametric combination resonance r2-3, experimental data (Uf0 ¼ 8 m/s, a ¼ 0.25, fp¼19.0 Hz).

Fig. 18. Vibration excitation in resonance r2-3, experimental data (Uf0 ¼ 8 m/s, a ¼ 0.25, fp¼19.0 Hz): time histories and trajectories of motion at the point L/7.

Fig. 19. Vibration modes and motion trajectories at selected points in combination resonance r4-5, experimental observations (fp¼36.4 Hz Uf0 ¼ 8 m/s, a ¼ 0.25): (a) pipe vibration modes, (b) main components modes.

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Fig. 20. Analysis of vibration at point 5L/7 in the resonance r4-5, experimental data (fp ¼ 36.4 Hz, Uf0 ¼ 8 m/s, a ¼ 0.25): (a) displacement spectrum, (b) time
maps. histories, (c) phase portraits and Poincare

Spectral analysis (Fig. 20) reveals the distribution of the components characteristic in combination resonances. The sum of two main components f4 ¼ 16.48 Hz and f5 ¼ 19.9 Hz is approximately equal to the pulsation frequency (fp ¼ 36.4 Hz). As the ratio of component frequencies is an irrational number f5/f4 ¼ 1.207, vibrations are quasi-periodic, which is evidenced by
map shown in Fig. 20c. phase portrait and Poincare

6. Conclusions This study summarises the theoretical and experimental investigations of the parametric resonance of an elastic pipe caused by pulsating fluid flow. The analysis uses a 3D model of pipe capturing its curvature (static deformation) due to gravity forces. Incorporating the pipe's curvature and geometrical nonlinearities results in coupling of in-plane and out-plane vibrations, which in certain conditions give rise to non-planar vibrations. Experimental observations agree well with numerical simulation results, confirming the adequacy of the adopted model. Various types of parametric resonances are observed in certain ranges of pulsation frequency. Simple resonance (involving one mode) are generated when the pulsation frequency is nearly twice as high as the system's natural frequency whilst the combination resonances (involving two modes) occur when the pulsation frequency is equal to the sum of the two natural frequencies. In simple resonances the actual modes assume the shape of the respective free vibration modes and are usually periodic (sub-harmonic). A thorough analysis of the mode shape changing over time reveals the occurrence of the ‘travelling wave’ effect, caused by gyroscopic effects (Coriolis forces) associated with fluid flow in a hydraulic system. Pipe curvature gives rise to independent resonance of the first mode in-plane and out-plane, being the consequence of a minor difference between the respective natural frequencies. The main achievement of this study lies in identification and analysis of non-planar vibrations in the combination resonance ranges. Numerical simulations confirmed by experimental observations revealed complex 3D form of the pipe's vibrations. The studies of such phenomena in pipes clamped at both ends have not been reported in literature on the subjects known to the authors. In combination resonances, the resultant mode is the superposition of two component modes associated with the system's free vibration modes. The component modes are not planar (as in simple resonances), but tend to rotate on 3D planes whilst the trajectories of a given point on a pipe are nearly circular. Component modes rotate with different angular velocities and in opposite directions, and the resultant mode has a complex 3D form, which varies cyclically in time. When the ratio of angular velocities of component modes is not a rational number, the resultant mode will additionally turn around the pipe's centreline. Trajectories of motion of particular points form characteristic figures, their shapes depending on the type of combination resonance (r1-2, r2-3, r3-4, r4-5, …) and the ratio of component modes' amplitudes. As this ratio along the pipe's length is varied, at each point of the pipe they will be different. On account of non-symmetry of certain component modes with respect to the centre point in the pipe (attributable to gyroscopic effects) as well as the node shifting along the pipe's axis, the resultant mode will not be symmetrical over the entire pipe length. In combination resonances vibrations are quasi-periodic, which is the result of superposition of vibrations with frequencies that are irrational.


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The analysis of motion trajectories and the vibration profile confirmed the good agreement between experimental observations and numerical simulation data, which proves the adequacy of the adopted model and methodology. Another major accomplishment is the proposed method of reconstructing the vibration mode on the basis of measurements at selected points of the pipe. Approximation of modes recalling the appropriate shape functions permits visualisation of complicated, time-variant shapes, giving us a better insight into the investigated phenomena. Results obtained by the proposed method, supported by simulation data and pipe photos at selected time instants confirm its adequacy. Thus the method can become a cost-effective alternative to complicated and expensive 3D image analyses. Appendix A In the work [4] equations of statics and dynamics of the pipe conveying fluid were derived, using rotation matrices, which determine the orientations of the respective coordinate systems. After introducing some simplifications, equations (Eqs. (12)e(15)) describing system vibrations were obtained. The assumptions and the most important parts of equation derivation are explained below. Scalar form of equations is used. The fluid element with of mass of a unit mf and the pipe element with a mass mp is shown in Figure A.1. Apart from gravity force mfgds and pressure force pAf the fluid element is subjected to normal nids and tangential tids components of internal forces in the fluid-pipe system. The pipe element is subjected to axial force Ni1 and transverse forces Ni2 and Ni3 (vector Ni ¼ [Ni1, Ni2, Ni3]) as well as twisting moment Mi1 and bending moments Mi2 and Mi3 (vector Mi ¼ [Mi1, Mi2, Mi3]) as well as gravity force mpgds, tension force S0 and internal forces.

Fig. A.1. Indication of forces for pipe element with fluid.

Differential equations of vibrations can be obtained using the principles of linear and angular momentum for the pipe element with fluid. In this approach, the internal forces ti and ni between the fluid and the pipe do not affect the obtained form of the equations. Taking into account only linear and quadratic terms, the simplified form of differential equations is as follows:


mp ap1 þ mf af 1 ¼ N 0 i1  p0 Af  k0 Ni2  N0 i2 f3 þ N 0 i3 f2  Ni1 þ S0  pAf k0 f3  Ni2 f0 3 þ mg sin c0

(A.1)

 

mp ap2 þ mf af 2 ¼ N 0 i2 þ Ni1 þ S0  pAf ðk0 þ f0 3 Þ þ N 0 i1  p0 Af f3  N 0 i3 f1  Ni2 k0 f3 þ mg cos c0

(A.2)

 

mp ap3 þ mf af 3 ¼ N 0 i3  N 0 i1  p0 Af f2 þ N 0 i2 f1  Ni1 þ S0  pAf f0 2 þ Ni2 f0 1

(A.3)

rp I1 f€ 1 ¼ M0 i1  k0 Mi2 þ Mi3 ðf0 2 þ k0 f1 Þ

(A.4)

where apk ¼ u€k and:


00 af 1 ¼ u€1 þ 2ðu_ 0 1  k0 u_ 2 ÞUf þ ðu0 1  k0 u2 þ 1ÞU_ f þ u 1  2k0 u0 2  k0 0 u2  k20 u1 Uf2

(A.5)


00 af 2 ¼ u€2 þ 2ðu_ 0 2 þ k0 u_ 1 ÞUf þ ðu0 2 þ k0 u1 ÞU_ f þ u 2 þ 2k0 u0 1 þ k0 0 u1  k20 u2 Uf2 þ k0 Uf2

(A.6)

00 af 3 ¼ u€3 þ 2u_ 0 3 Uf þ u0 3 U_ f þ u 3 Uf2

(A.7)

To determine the simplified relationship between fluid pressure p and flow rate Uf, as proposed by Païdoussis [1], the influence of vibrations on the pressure value was omitted. Based on the principle of momentum for fluid element, taking account of flow resistance according to the Darcy-Weisbach hypothesis (ti ¼ mfl0U2f), the equation follows:


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p0 Af ¼ mf U_ f  mf l0 Uf2 þ mf g sin c0

(A.8)

which can be interpreted as a projection corresponding vector equation on the tangent axis to the pipe. From equation (Eq. (A.8)), for the constant flow (U_ ¼ 0, U ¼ U ), the equation follows: f

f

f0

p0 0 Af ¼ mf l0 Uf20 þ mf g sin c0

(A.9)

that is also equation (Eq. (7)), which determines the change of static pressure. The equations (Eq. (A.8) and (A.9)) also result in dependencies:


 ðp0  p0 0 ÞAf ¼ mf U_ f  mf l0 Uf2  Uf20

(A.10)

and after the integration in the interval (s, L0), the following relationship:

h
i pðs; tÞAf ¼ p0 ðsÞAf  mf U_ f þ l0 Uf2  Uf20 ðs  L0 Þ

(A.11)

In the case of elastic pipes, the gravitational forces can cause a considerable deformations of the pipe in the vertical plane. Equations (Eqs. (A.1, A.2)) were used to determine the shape of the statically deformed pipe, assuming that generalized internal forces are the sum of static and dynamic forces (Nik ¼ Ni0þNk and Mik ¼ Mi0þMk) however, due to the plane state of the deformation, different from zero are only: N10, N20 and M30. For steady flow without vibration equations Eq. (A.1)) and (Eq. (A.2) take the form:

0 ¼ N0 10  p0 0 Af  k0 N20 þ mg sin c0

(A.12)


mf k0 Uf20 ¼ N0 20 þ N10 þ S0  p0 Af k0 þ mg cos c0

(A.13)

from where after taking account of Eq. (A.9), equations (Eq. (1), Eq. (2)) result. Curvature k0 is given by:

k0 ¼ c 0 0

(A.14)

and from the equations: N20 ¼ eM30’ and M30 ¼ EI3k0 can be determined the static component of transverse force

N20 ¼ EI3 k0 0

(A.15)

After minor transformations of formulas Eq. (A.14) and Eq. (A.15), equations (Eqs. (3) and (4)) are derived. Equations Eq. (5) and Eq. (6) can be obtained from geometric relationships:

u0 10 þ

1 ¼ cos c0 1 þ ε0

u0 20 ¼ sin c0

(A.16) (A.17)

between the coordinates of the vector u0 and the angle c0, which defines tangent to the axis of the statically deformed pipe. After calculation N10 from the equations (Eqs. (1)e(7)), the static deformation ε0 ¼ 1eds/dx can be calculated using the formula:

N10 ¼ EAp ε0

(A.18)

Taking into account the Voigt-Kelvin hypothesis of internal damping, internal moments caused by vibrations are determined by the equations:

M1 ¼ ð1 þ mDÞGI1 ð40 1  k0 42 Þ

(A.19)

M2 ¼ ð1 þ mDÞEI2 ð40 2 þ k0 41 Þ

(A.20)

M3 ¼ M30 þ ð1 þ mDÞEI3 40 3

(A.21)

From a principle of angular momentum, by omitting the rotatory inertia of the pipe element and preserving only linear members, the following relationships between transverse forces and bending moments occur:

N2 ¼ M 0 3

(A.22)


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N3 ¼ M0 2 þ k0 M1  M30 ð40 1  k0 42 Þ

289

(A.23)

In determining the axial force N1 takes into account the influence of geometric nonlinearity, assuming the formula:

1 1 1 ε ¼ ðu0 1  k0 u2 Þ þ ðu0 1  k0 u2 Þ2 þ ðu0 2 þ k0 u1 Þ2 þ u02 2 2 2 3

(A.24)

defining the axial deformation. As suggested by Holmes [4], the nonlinear deformation component (Eq. (A.24)) was replaced by the averaged value along the length of the pipe. From the general formula N1¼(1þmD)EApε we get the relation:

  N1 ¼ ð1 þ mDÞ EAp ðu0 1  k0 u2 Þ þ S

(A.25)

where the nonlinear element S is defined by the formula (Eq. (16)). In order to obtain the final form of equations of vibrations (Eqs. (12)e(15)), the equations (Eqs. (A.1)e(A.4)) are used again. After substituting: Nik ¼ Ni0þNk, Mik ¼ Mi0þMk and using the equations (Eqs. (A.5-A.25)) we obtain:

mp ap1 þ mf af 1 ¼ N 0 1  ðp0  p0 0 ÞAf  k0 N2  N0 20 43  N100 k0 43  N20 40 3 mp ap2 þ mf af 2

i  h
¼ N 0 2 þ N1 þ S  ðp  p0 ÞAf þ mf Uf20 k0 þ N100 40 3 þ N 0 10  p0 Af 43  N20 k0 43

(A.26) (A.27)


mp ap3 þ mf af 3 ¼ N 0 3  N0 10  p0 Af 42 þ N0 20 41  N100 40 2 þ N20 40 1

(A.28)

rp I1 4€ 1 ¼ M0 1  k0 M2 þ M30 ð40 2 þ k0 41 Þ

(A.29)

where:


N100 ¼ N10 þ S0 þ S  pAf

(A.30)

where for the Euler-Bernoulli beam model there are relationships:

43 ¼ u0 2 þ k0 u1

(A.31)

42 ¼ u0 3

(A.32)

The system of equations (Eqs. (A.26eA.29)) include linear components, depending on the generalized coordinates and the nonlinear component associated with the axial force S. After the laborious transformations it can be converted to form (Eqs. (12)e(15)). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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